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<p>I'm trying to understand the concepts of time dilation and spatial compression. I've been using the classic example of firing a photon of light inside a ship (spaceship, boxcar, etc.) moving at a constant velocity to the second observer. But when I do the Lorentz, I get different values depending upon the direction of travel of the photon relative to the ship.</p>
<p>My understanding is that time dilation for the two observers should be a constant, as should spatial compression, since their relative velocities are constant (at least for the duration of the experiment).</p>
<p>When I fire the photon in the direction of travel of the ship, the measurements all work out as expected. Time on the ship is passing slower. Then when I fire the photon backward in the direction of travel of the ship, all the values that should be constant have different values. So I assumed I'd done the transforms wrong. I redid them and still got different values. hen I simplified the equations by picking values that make the transforms very basic. Ship length = 1 light second for the ship observer. Ship speed = 0.1 light year for the second observer. Even then I get variances in the values that I expect to be constants. </p>
<p>My problem might be in measuring the distance the photon traveled for each observer.
In firing forward, the second observer sees the photon travel the compressed length of the ship plus the distance the ship travels. The ship observer see it travel the length of the uncompressed ship. When firing backwards, the second observer sees the photon travel the compressed length of the ship minus the distance the ship travels. Since the ship observer sees the photon travel the same distance and amount of time regardless of direction, the second observer must also see it travel the same time in both directions. But the distance is shortened for the secondary observer.</p>
<p>So either the time dilation or the spatial compression would have to change for the equations to work. But my original premise was those are constants because the relative velocity is assigned to be constant for the experiment! Am I still doing the transforms wrong, or do I have a bad assumption in which values become constants by forcing relative velocity to be constant?</p>
<p>I've reviewed dozens of the time paradox explanations. But those arise from the two observers returning to one frame of reference. I've let my ship sail on beyond the end of the experiment indefinitely, purposely to avoid those problems from interfering in learning how to do the transforms correctly. Now I'm not sure I didn't step into a different paradox. Maybe this should be solved in GR instead?</p> | 4,279 | [
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<p>We know Newton's three laws:</p>
<ol>
<li><p>A object at rest will remain at rest, and an object in motion will remain in motion unless a external force acts upon it.</p></li>
<li><p>If an unbalanced force acts on a object, the object will accelerate in the direction of the net force.</p></li>
<li><p>If an object $A$ exerts a force on object $B$, then object $B$ will exert a equal force to object $A$ in the opposite direction. $F_{a\text{ on }b} = -F_{b\text{ on }a}$</p></li>
</ol>
<p>What I think is that as you throw the snow, it begins moving at a constant velocity. However, when the shovel stops moving, the snow will remain in motion, causing it to accelerate/fly toward the snow bank, according to Newton's first law. But is the snow <strong>accelerating</strong> when it leaves the shovel?
How does this apply to Newton's second law? Thanks!</p> | 4,280 | [
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<p>I'm still in high school, and while I can't complain about the quality of my teachers (all of them have done at least a bachelor, some a masters) I usually am cautious to believe what they say straight away. Since I'm interested quite a bit in physics, I know more about it than other subjects and I spot things I disagree with more often, and this is the most recent thing:</p>
<p>While discussing photons, my teacher made a couple of statements which might be true but sound foreign to me:</p>
<ul>
<li><p>He said that under certain conditions, photons have mass. I didn't think this was true at all. I think he said this to avoid confusion regarding $E=mc^2$, however, in my opinion it only adds to the confusion since objects with mass can't travel with the speed of light, and light does have a tendency to travel with the speed of light.. I myself understand how photons can have a momentum while having no mass because I lurk this site, but my classmates don't.</p></li>
<li><p>He said photons don't actually exist, but are handy to envision. This dazzled my mind. Even more so since he followed this statement by explaining the photo-electric effect, which to me seems like a proof of the existence of photons as the quantum of light. He might have done this to avoid confusion regarding the wave-particle duality.</p></li>
</ul>
<p>This all seems very odd to me and I hope some of you can clarify. </p> | 4,281 | [
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<p>I know I have posted <a href="http://physics.stackexchange.com/q/103787/2451">this</a> question <a href="http://physics.stackexchange.com/q/101179/2451">before</a> some time ago. But no one could help so I decided to put my problem in another background.</p>
<p>The Schrödinger equation of a free scalar field is given by</p>
<p>$i\partial_{t}\Psi[\Phi,t]~=~\underset{A}{\underbrace{\frac{1}{2}\int d^{3}x\left(-\frac{\delta^{2}}{\delta\Phi^{2}(\vec{x})}+|\nabla\Phi|^{2}+m^{2}\Phi^{2}\right)}}\Psi[\Phi,t]$.</p>
<p>This is the Schrödinger representation of QFT.</p>
<p>Now I want to know, whether the operator $A$ on the r.h.s is essentially <a href="http://en.wikipedia.org/wiki/Self-adjoint_operator">self-adjoint</a>?</p>
<p>Any idea or advise?</p>
<p>My problem is how to handle the functional derivative here.</p> | 4,282 | [
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<p>I don't quite understand this quote from Stephen J. Gould's <em>Ever since Darwin</em>, where he talks about the compensating physical characteristics of organisms for their size. </p>
<blockquote>
<p>Other essential features of organisms change even more rapidly
with increasing size than the ratio of surface to volume. Kinetic
energy, in some situations, increases as length raised to the fifth
power. If a child half your height falls down, its head will hit
with not half, but only 1/32 the energy of yours in a similar fall. In return,
we are protected from the physical force of its tantrums, for
the child can strike with, not half, but only 1/32 of the energy
we can muster.</p>
</blockquote>
<p>Mass increases with the third power of length, and even if we allow for a shorter fall (even though that is not clear from the wording) for the child (who is half as tall), I can only come up with the fourth power. </p> | 4,283 | [
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<p>Consider the <a href="http://en.wikipedia.org/wiki/Metric_space#Definition" rel="nofollow">metric space</a> $(M, d \,)$ where set $M$ contains sufficiently many (at least five) distinct elements,<br>
and consider the assignment $c_f$ of coordinates to (the elements of) set $M$,</p>
<p>$c_f \, : \, M \leftrightarrow {\mathbb{R}}^3; \, c_f[ P ] := \{ x_P, y_P, z_P \}$</p>
<p>such that distinct coordinates values are assigned to distinct elements of set $M$, and<br>
such that for the function</p>
<p>$f \, : \, ({\mathbb{R}}^3 \times {\mathbb{R}}^3) \rightarrow {\mathbb{R}};$<br>
$f[ \{ x_P, y_P, z_P \}, \{ x_Q, y_Q, z_Q \} ] := $
${\sqrt{ (x_Q - x_P)^2 + (y_Q - y_P)^2 + (z_Q - z_P)^2 }} \equiv {\sqrt{ \sum_{ k \in \{ x \, y \, z \} } (k_Q - k_P)^2 }}$</p>
<p>and for any three distinct elements $A$, $B$, and $J$ $\in M$ holds</p>
<p>$f[ c_f[ A ], c_f[ J ] ] \, d[ B, J ] = f[ c_f[ B ], c_f[ J ] ] \, d[ A, J ]$.</p>
<p>Is the metric space $(M, d \,)$ therefore flat?</p>
<p>(i.e. in the sense of vanishing <a href="http://en.wikipedia.org/wiki/Cayley-Menger_determinants" rel="nofollow">Cayley-Menger determinants</a> of distance ratios between any five elements of set $M$.)</p> | 4,284 | [
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] |
<p>The Lagrangian for electromagnetic field has the following expression:
$$
L = -\frac{1}{c^{2}}A_{\alpha}j^{\alpha} - \frac{1}{8 \pi c}(\partial_{\alpha} A_{\beta})(\partial^{\alpha}A^{\beta})
$$</p>
<p>(I used Lorentz calibration $\partial_{\alpha} A^{\alpha} = 0 $).</p>
<p>If I add the summand $\frac{\mu^{2}}{8 \pi c}A_{\alpha}A^{\alpha}$, I'll get an equations for field (which characterized by some 4-vector $A^{\alpha}$ (not electromagnetic (!!!))) of strong interaction and (for static case) the expression for Yukawa potential. So what is the physical meaning of summand written above? </p>
<p>This term is somehow characteristic of the mass of the interaction carriers, but I don't understand the physical meaning of $A_{\alpha}A^{\alpha}$.</p> | 4,285 | [
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<p>I have not been able to find a resource to tell me the standard notation for a normalized scalar value. Normalized vectors (i.e. unit vectors) are typically denoted by placing a hat over the variable, something like: </p>
<p>$${\bf \hat{e} = \dfrac{e}{||e||} }$$</p>
<p>However, does the same apply to normalizing (and nondimensionalizing) a scalar? Would it be correct to write: </p>
<p>$$ \hat{L} = \dfrac{L}{L_0} $$</p>
<p>This is assuming that $L$ and $L_0$ are just scalar lengths. If I am defining my own notation, is it <em>verboten</em> to call this something like $\bar{L}$ (with a bar)?</p>
<p>If it makes any difference, I am a mechanical engineer and this will be going in my thesis.</p> | 4,286 | [
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<p>This question is regarding the quantum circuit in the picture below.</p>
<p>Suppose we have the set up below, where U performs the operation $U:\mid x \rangle \mid y \rangle \rightarrow \mid x \rangle\mid y \oplus f(x) \rangle$.</p>
<p>We define the function $f(x)$ on the 3-bit string $\mid x \rangle= \mid x_1 x_2 x_3 \rangle$ with $f(x)=a\cdot x=a_1 x_1 \oplus a_2 x_2 \oplus a_3 x_3$, where $\oplus$ is addition modulo 2 and $a$ is the fixed three bit string $\mid a \rangle=\mid a_1 a_2 a_3 \rangle$.</p>
<p>The objective is to show how this circuit finds $a$.</p>
<p>So far I have that the 3 zero qubits pass through 3 Hadamard gates, putting them in the bell state $\beta_{00}=\frac{1}{\sqrt2}(\mid 0 \rangle +\mid 1 \rangle)$. So at point 1 on the circuit the system has state:</p>
<p>$$\frac14(\mid 0 \rangle +\mid 1 \rangle)(\mid 0 \rangle +\mid 1 \rangle)(\mid 0 \rangle +\mid 1 \rangle)(\mid 0 \rangle -\mid 1 \rangle)=\frac14 \sum_{x\in\{0,1\}^3} \mid x \rangle(\mid 0 \rangle -\mid 1 \rangle$$</p>
<p>Where we have found the RHS by expanding out and summing over all the possible 3 but qubits.</p>
<p>So next we consider what happens when the system passes through the $U$ gate, we find:</p>
<p>$$\frac14 \sum_{x\in\{0,1\}^3} \mid x \rangle(\mid f(x) \rangle -\mid 1 \oplus f(x) \rangle )$$</p>
<p>Here we notice that is $f(x)$ is $0$ nothing changes, but if it is $1$ it acts as a factor of $-1$, therefore the system is:</p>
<p>$$\frac14 \sum_{x\in\{0,1\}^3} (-1)^{f(x)}\mid x \rangle(\mid 0 \rangle -\mid 1 \rangle )$$</p>
<p>Now $f(x)$ outputs sums of the elements of $a$ depending on the value of $x$, for example for $\mid x \rangle = \mid 110 \rangle$, $f(x)=f(110)=a_1 \oplus a_2$. Would anyone be able to help me understand how this helps us find the three values for $a$ at our measured gates? The main issue i'm finding is finding a useful of writing an expression for the system after it has passed through the last three Hadamard gates, one which we can use to find $a$, any help would be greatly appreciated!</p>
<p><img src="http://i.stack.imgur.com/nSGNe.jpg" alt="Figure 1"></p> | 4,287 | [
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<p>In quantum mechanics can the mass and the linear momentum of a particle be measured precisely or do they commute ?</p> | 4,288 | [
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<p>$V=IR$</p>
<p>Right?</p>
<p>$100 (\rm{V}) = 0 (\rm{A}) \times 100 (\rm{\Omega})$</p>
<p>Lets say something has $100 \rm V$ potential But since this object is surrounded by air and current is not flowing therefore there has to be $100 \rm{\Omega}$. Right? But then the equation does not work as 100 does not equal 0? </p>
<p>You get infinite resistance</p> | 4,289 | [
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<p>Does the gravitation of Earth have a limit? when a body projected vertically with $v=11km/s$ (escape velocity) from Earth's surface does this means that it does not return back to Earth? </p> | 4,290 | [
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<p>As far as I know, the electromagnetic force only interacts on particles with electrical charge, but I was told that the electromagnetic force was involved in the following reaction:</p>
<p>$$\Sigma^0\rightarrow\Lambda^0+\gamma.$$</p>
<p>How can this be, when the electric charge of both the $\Sigma^0$ particle and the $\Lambda^0$ particle is $0$? Or is it another fundamental interaction that is involved in the reaction? Thank you.</p> | 4,291 | [
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<p>In my physics textbook there is an example of using capacitor switches in computer keyboard:</p>
<blockquote>
<p>Pressing the key pushes two capacitor plates closer together, increasing their capacitance. A larger capacitor can hold more charge, so a momentary current carries charge from the battery (or power supply) to the capacitor. This current is sensed, and the keystroke is then recorded.</p>
</blockquote>
<p>That makes perfect sense, and is kind of neat. What I am curious about, is what happens to that extra charge afterwards. Is there some sort of discharge mechanism? I suppose, that would be also necessary to differentiate between single keystrokes and continuous depression (register stroke current, then register the discharge). What would happen to the capacitor if there was no such discharge mechanism, but its capacitance was suddenly reduced?</p>
<p>If capacitance is reduced, and the charge stays the same, then, according to $Q = C \Delta V_C$, the difference of potentials on plates of capacitor should increase and exceed that of a power supply thus reversing the current. Is that what is happening, and the keystrokes are recorded by sensing not only the existence of the current, but also its direction?</p> | 4,292 | [
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<p>Guys I couldn't catch a point of multiverse theory..</p>
<p>Theory: If space-time goes on forever, then it must start repeating at some point, because there are a finite number of ways particles can be arranged in space and time..</p>
<p>Question: It seems like as if multiverse theory stands on a optical argument. We call there is a next universe because we are not seeing that since there is a speed limit of light and our optical universe is 14 billions light yrs in diameter(or whatever it be..since we can not assume our space time a sphere or 2dimensional plane).Doesn't it sound weird? If there is a space between two universe then what is the value of calling it a system of two universe, Isn't it just a single universe?</p>
<p>Question:Is it a good idea to call there is a next universe because it is exactly alike?</p>
<p>Question: Another question is How does the multiverse theory provides solution to the grandfather paradox? Isn't it like we fold the two dimensional spacetime graph where universe exactly alike to our universe exits??If my above argument is correct then how is it valid?? Note that I am not questioning multiverse theory this time..So please Don't give answer to this question proving my above arguments wrong..Let this be independent question..</p>
<p>I know I am wrong But please Make me clear.</p> | 4,293 | [
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<p>It is given that acceleration is constant, so can we infer that average speed and velocity are the same?? Moreover, circular motion is out of the question, as the function of x(t) where x=displacement, suggests, that for any t>=0, displacement can not be zero...</p>
<p>This is the conceptual problem I am facing in a question:
My teacher was reading out the question, and it was asked only to find the avg velocity from the acceleration. She, on her own, added a part to it, asking us to also find avg speed, and then, while discussing solutions, said that a graph must be made in order to solve this...so do you think that it is absolutely necessary? Moreover, if my premise is flawed, then how can the graph even help??</p>
<p>Thanks in advance!</p> | 4,294 | [
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<p><a href="http://phys.org/news/2013-04-gravity-lingua-franca-relativity-quantum.html" rel="nofollow">Lingua Franca links relativity and quantum theories with spectral geometry</a></p>
<p>Could someone give me a short synopsis of this article in laymens terms? What implications does this have in the physics community? Is this work ground breaking or just the start of something that might be?</p>
<p>From what I can understand this physicist related two types of maths that we use to model the world around us. Which has been hard to do, because the types of maths are incompatible with each other. </p>
<p>But what is this spectral geometry? How does it relate the two types of maths in relativity and quantum theory? What might those maths be? What about the two has made it hard to unite them?</p> | 4,295 | [
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<p>Two very interesting new papers on arXiv last night by Lev Vaidman and friends lead me to ask about the differences between Cramer's transactional interpretation of quantum mechanics (TIQM) and the two state vector formalism (TSVF) advanced by Aharonov, Vaidman and others. At a first look, they both seem very similar to me.<br>
The two papers:<br>
<a href="http://arxiv.org/abs/1304.7474" rel="nofollow">http://arxiv.org/abs/1304.7474</a> “The past of a quantum particle”, and<br>
<a href="http://arxiv.org/abs/1304.7469" rel="nofollow">http://arxiv.org/abs/1304.7469</a> “Asking a particle where it has been”.<br>
From these two papers, you can find further references for TSVF, including its origin.<br>
For TIQM, you can start with Wikipedia, and also trace back to the origin.
<a href="http://en.wikipedia.org/wiki/Transactional_interpretation" rel="nofollow">http://en.wikipedia.org/wiki/Transactional_interpretation</a> </p> | 4,296 | [
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<p>In a <a href="http://arxiv.org/abs/0909.4593" rel="nofollow">paper</a> by V. Gurarie et al. , the theorem of inclusion is used to prove that there is no direct phase transition between Mott insulator and spuerfluid in presence of disorder. In Fig. 2 of that paper, why if $\Delta > \Delta_c $, there exist domains in phase B which locally look like phase A? Similarly, why for $\Delta < \Delta_c $, there exist domains in phase A which locally look like phase B?</p> | 4,297 | [
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<p>Do atoms have any uniquely identifying characteristic besides their history?</p>
<p>For example, if we had detailed information about a specific carbon atom from one of Planck's fingerprints, and could time-travel to the cosmic event in which the atom formed, would it contain information with which we could positively identify that they two are the same?</p> | 4,298 | [
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<p>I know there is evidence that it is not predetermined and I tried reading articles on it but most of them either don't explain the intuition behind the experiment or they speak in a foreign language (That language being science). If you could explain the intuition behind the experiment and also give an analogy that would be great.</p> | 4,299 | [
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<p>Since we know that all accelerated charges radiate energy and we also know that all matter is made up of protons and electrons which are all the while doing accelerated motion.So from this can i conclude that every piece of matter radiates energy because of jiggling motion of atoms?</p> | 132 | [
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<p>Okay, I know that in quantum mechanics the quantum observable is obtained from the classical observable by the prescription </p>
<p>$$ X \rightarrow x,\quad P \rightarrow -i\hbar\frac{\partial}{\partial x} $$ </p>
<p>in the position basis. Now my question is, what if $x$ or $p$ appears in the denominator in a classical expression? How to promote this to a quantum expression? What would be the meaning of division by an operator?</p>
<p>Edit: Thank you for your responses. My expression likely contains a mixture of x and p. For eg., it could contain terms like $$\frac{p}{x^2}$$ or $$\frac{xp}{(x^2 + a^2)^{3/2}}$$. How to resolve products of non-commuting operators like x,p in a satisfactory way?</p> | 4,300 | [
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<p>How many degrees of freedom does an asymmetric top have if it is rotating about a fixed point?What are the generalised coordinates used then?</p> | 4,301 | [
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<p>Heisenberg famously derived his uncertainty principle by considering the disturbance that a measurement would have on a small enough system.</p>
<p>Of course in the mathematical formalism of Quantum Mechanics the relationship is derived from more basic principles.</p>
<p>How does String Theory account for it? Heuristically, and mathematically?</p> | 4,302 | [
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<p>I have read some popsci articles and documentaries about the early universe and they often explain how various features of the universe came about and at what time. For example hydrogen atoms came about after hundreds of thousands of years.</p>
<p>Now, the official SI time unit, the second, is based on the caesium atom's properties. How can we talk about time lengths in seconds or years if there are no atoms yet in the universe? (Or going back further where there aren't even electrons.)</p>
<p>Clearly, there must be a way to do it. Then the next question is: why don't we use those field properties or whatever that existed before atoms to define the SI second? If atoms are not needed to get "the beat of time" in the early universe because something else can provide the tick then we could also use those things today instead of relying on the caesium atom for the definition of the second.</p>
<p>I hope it's clear what I mean. Thanks!</p> | 4,303 | [
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<p>I was reading "Nature of space and time" by Penrose and Hawking, pg.13, </p>
<blockquote>
<p>If $\rho=\rho_0$ at $\nu=\nu_0$, then the RNP equation</p>
<p>$\frac{d\rho}{d\nu} = \rho^2 + \sigma^{ij}\sigma_{ij} + \frac{1}{n} R_{\mu\nu} l^\mu l^\nu$
implies that the convergence $\rho$ will become infinite at a point $q$ within an affine parameter distance$\frac{1}{\rho_0}$ if the null geodesic can be extended that far.</p>
<p><em>if $\rho=\rho_0$ at $\nu=\nu_0$ then $\rho$ is greater than or equal to $\frac{1}{\rho^{-1} + \nu_0-\nu}$. Thus there is a conjugate point before $\nu=\nu_0 + \rho^{-1}$.</em></p>
</blockquote>
<p>I dont understand many terms here. Firstly, what is affine parameter distance? And I am at loss as to how does one get the 2nd relation between $\rho$ and $\frac{1}{\rho^{-1} + \nu_0-\nu}$. How can you derive it? Frankly, I dont understand ANYTHING about how does this equation come, though I suspect it just the Frobenius theorem. Because that is how you get conjugate points in spacetime.</p>
<p>Please give me DETAILED asnwers, as I have mentioned before, I am not too comfortable with it. I dont understand anything in blockquotes other than the RNP equation.</p>
<p>Thanks in advance!!!</p> | 4,304 | [
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<p>Let's say I have a hollow conductive rod, 10mm diameter (O.D.), and I place a magnet of known strength 50mm up the shaft. What is the microTesla (mT) or Gauss (G) of the magnetic field (or flux density, or whatever it would be) <strong>of the shaft</strong> 50mm away from the magnet attached to it?</p>
<p>Here's what I'm attempting to do:</p>
<p>I want to attach a magnet to my manual transmission's shifter lever, and then use hall effect sensors to detect it's position and display the selected gear on a display. I have the Arduino code written and the display wired. I'm having a problem understanding what sensitivity Hall Effect Sensors to get, what strength and size magnet(s) to get, and how to go about figuring how they relate to each other.</p>
<p>Essentially, I don't want the magnet too close to the 6 Hall Sensors, nor do I want a huge, expensive magnet. I have 216 neodymium buckeyballs I can encircle the shaft with, if I used them would it matter which polarity I had them arranged in? If I used a single magnet, would I place the poles alongside or perpendicular to the rod? How would I go about ball-parking the mT of the magnetic field 50mm (or whatever distance) down the rod, away from the magnet to get an idea of the sensitivity (rated in mV/mT for the hall effect sensors)?</p>
<p>How does a magnetic field propagate through a conductor? How would I position the magnet(s) to maximize the mT down the rod of my shifter? How do I place the Hall Effect Sensors (which read mT perpendicular to their face) to maximize the readings?</p>
<p>Ideally, I'm hoping to place the magnet(s) far enough away that what I'm reading is the transmitted 'magnetism' of the rod itself, is this an improper way to look at it (or even possible)? If it's not, is it the case that I have to bring the actual magnets near the hall effect sensors, and I can't 'transfer' it to the rod?</p> | 4,305 | [
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<p>In the Schrödinger's cat experiment 'there is a tiny bit of radioactive substance, so small that perhaps in the course of the hour, one of the atoms decays'. The rest of the experiment magnifies this into a macroscopic superposition, but I want to know more about the claim that the radioactive decay produces a superposition.</p>
<p>Firstly, has this been experimentally tested? Something along the lines of accelerating radioactive ions so there is a chance that they will decay while in flight (and so change trajectory), and then combining the decayed and undecayed parts to look for interference.</p>
<p>Secondly, the tiny bit of radioactive substance will still contain large numbers of atoms. Won't this cause problems? If the atoms were in a Bose-Einstein condensate, then I would expect that there could be a superposition of 'one (unspecified) atom decayed' and 'no atoms decayed', but they're not, so a specific atom will decay. Won't that mess things up?</p> | 4,306 | [
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<p>This is for anyone with experience in optics/imaging/photography as well as anyone who likes to puzzle over tricky physics problems. </p>
<p>As the title suggests, this is about combining two (for all practical purposes) identical light beams in an optical system to one beam of twice the intensity. Mind you, I'm not talking about monochromatic laser beams, although the underlying problem would be the same.
As an example, imagine a fancy imaging system that you've constructed and with which you look at objects, which are fairly dim. Therefore, you'd like to enhance the image quality by collecting as much light as possible coming from that source by using not one, but multiple copies of your fancy device. You then project the beams from those device onto, let's say, a single CCD chip and thereby end up with a higher signal-to-noise ratio. You only have one camera available, so just buying a few extra cams and superposing the images on your PC is NOT an option. </p>
<p>Now, the crux of this problem is: How does one combine multiple identical beams into one, while keeping the intensity loss (that one can certainly expect) to a minimum? </p>
<p>In general, there seems to be two basic approaches to tackle this problem: </p>
<ol>
<li><p>Don't bother with beam combining, instead, project the beams from different angles onto the CCD and somehow deal with the varying distortion/defocussing of the resulting images caused by the different angles of incident. </p></li>
<li><p>Try to combine the beams into one. You then won't have to deal with the troubles arising from different angles as in the first strategy. </p></li>
</ol>
<p>Intuitively, I prefer option 2, but after pondering on it for a week, I found the problem of combining identical beams surprisingly non-trivial. </p>
<p>Maybe anyone of you guys here has had to deal with a similar problem or maybe you just happen to have a really nice idea how to solve it. Let me know what you think, I will also try to explain some of the (flawed) ideas I had a bit later!</p> | 4,307 | [
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<p>I read about the quantum quench problem in condensed matter physics.
But what does really mean? Has anybody a good explanation about the origin of quantum quench problem?</p> | 4,308 | [
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<p>Does <a href="http://en.wikipedia.org/wiki/Newton%27s_laws_of_motion" rel="nofollow">Newton's laws</a> of <a href="http://en.wikipedia.org/wiki/Newton%27s_law_of_universal_gravitation" rel="nofollow">gravitation</a> and <a href="http://en.wikipedia.org/wiki/Kepler%27s_laws_of_planetary_motion" rel="nofollow">Kepler's laws</a> give exactly the same orbit for two bodies?</p>
<p>Could someone please explain the derivation if so?</p> | 4,309 | [
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<p>I'm studying Jackiw's "Fractional Charge and Zero Modes for Planar Systems in a Magnetic Field" DOI: <a href="http://journals.aps.org/prd/abstract/10.1103/PhysRevD.29.2375" rel="nofollow">10.1103/PhysRevD.33.2500</a></p>
<p>but I have difficulties at some points. One of the problems is</p>
<p>$$\langle j^0\rangle=\pm\frac{e}{4\pi}B$$</p>
<p>where $j$ is four-current, $B$ is magnetic field. How can I derive this result (and why we used only $j^0$ component)?.</p> | 4,310 | [
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<p>Is <a href="http://en.wikipedia.org/wiki/Temperature">temperature</a> of a single molecule defined?</p>
<p>This question just cropped up in my mind as I have often heard of laws being violated when it comes to the scale of a single molecule. Does this happen in case of temperature too?</p> | 133 | [
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<p>I have been asking around at my school and at the high school and at EWU but no one can answer this question: can a magnet or a magnetic field push gravity?</p> | 4,311 | [
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<p>If we get the dispersion relation from the Fourier transform of the lattice vectors then how do we get electrons information? Specifically, for the $k=0$ point of the graph, does this mean the electron has zero momentum(I am pretty sure that electrons don't have zero momentum in that case due to Heisenberg's uncertainty principle)? </p>
<p>I was thinking (since crystal momentum and electron momentum are different) that the $k$ is how the overall electrons wave packets move in regards to crystal momentum, or that crystal momentum dictates the electrons momentum.</p>
<p>As you can probably tell I am still very shaky on this, so anything is really appreciated!</p> | 4,312 | [
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<p>Why Thermodynamic cycles are used to obtain work for example carnot engine rather than single isothermal process which is part of cycle and can be used to obtain more work.And why isothermal process do not violate second law even when there is no friction or any irreversibility.I know they violate second law in actual world due to friction .But in theory reversible processes are possible which do not violate second law,so why it doesn't.</p> | 4,313 | [
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<p>I don't know much about black holes physics and so I find the Schwarzschild equations with a few contradictions. In particular I am trying to understand this little puzzle. The Schwarzschild Newtonian gravitational field equation is expressed as follows (see <a href="http://en.wikipedia.org/wiki/Schwarzschild_radius" rel="nofollow">http://en.wikipedia.org/wiki/Schwarzschild_radius</a>):</p>
<p>$\frac { r^2 }{r_s} \frac {g}{c^2} = \frac {1}{2}$ </p>
<p>So once a particle is close to the event horizon such that $r\to r_s$ the equation becomes:</p>
<p>$ r \frac {g}{c^2} = \frac {1}{2}$ </p>
<p>But then at that point light cannot escape so I would think that $g\to c/t$ where $t=1 sec$. So the equation would now approximate to:</p>
<p>$r = \frac {c/t}{2}$</p>
<p>But this implies that $r$ is generalized for all black holes regardless the mass. If $r$ is indeed $r\equiv r_s$ then that contradicts the other Schwarzschild formula where $r$ depends on $m$:</p>
<p>$r_s = \frac {2 G M} {c^2}$</p>
<p>Also based on what we know about $r_s$ in massive black holes, a radius of 150,000 km is quite small. </p>
<p>What's wrong with this picture?</p> | 4,314 | [
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<p>I have obtained a series of data that is about density distribution of liquid in three dimensions, density$=f(x,y,z)$. How can I know the detail information about the way that density changes with $x$ and $y$ and $z$? And how should I do to get the positions of minimum and maximum of liquid density? </p> | 4,315 | [
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<p>I run into wikipedia articles about ergosphere of rotating black holes. What if some massive body passes nearby some black hole: is something like ergosphere produced, or is the event horizon distorted by the gravity of the passing body?</p> | 4,316 | [
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<p>When trying to calculate the lift force generated by a simple rectangular blade, I've found the following equation: $$F = \omega^2 L^2 l\rho\sin^2\phi$$ in which $\omega$ is the angular velocity, $L$ is the length of the helix, $l$ is the width of the helix (both in meters), $\rho$ is the air density at normal conditions, and $\phi$ is the angular deviation of the helix related to the rotating axis. So a 4-helix propeller would lead to $F=4\omega^2 L^2 l\rho\sin^2\phi$ and so on. </p>
<p>However, when I substitute the variables with real values, it seems to me very unreal. For example, if the 4-blades propeller works at $13000 RPM$, $\rho=1.293 kg/m^3$, $\phi=1^\circ$, $l=10^{-2}m$, $L=5\times 10^{-2}m$, leads to $$F=4\times(120\pi\times13000)^2\times(5\times10^{-2})^2\times(10^{-2})\times1.293\times(\sin1^\circ)^2$$$$\therefore F=9.46\times10^5N$$</p>
<p>So in this case, a simple $1cm\times 5cm$ propeller would carry $946 tons$ is surely wrong. So, where is the issue in the formulae?</p>
<p><strong>Update</strong></p>
<p>There was a mistake of my calculation, mr. @Mark Eichenlaub made me see this. The right calculation is $$F=4\times(2\pi\times13000/60)^2\times(5\times10^{-2})^2\times(10^{-2})\times1.293\times(\sin1^\circ)^2$$$$\therefore F=0.072995N$$ which is quite reasonable. So, for lifting a $0.5 kg$ load ($\implies P=9.81\times 0.5=4.9N$) we should get longer blades or change the $\phi$ inclination. Thanks.</p> | 4,317 | [
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<p>What's the difference between something being labeled a "nano-particle" or it being called a "molecule"?</p> | 4,318 | [
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<p>While studying alternating currents I could read and observe through an oscilloscope that there can be phase difference between emf and current. But, is a phase difference of 180 degrees possible in a series LCR circuit? </p> | 4,319 | [
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<p>We may choose a non-rotating earth as our reference frame and ask ourselves: how about the planetary and stellar motions. A star at a distance of 10 million light years would turn around the earth in 24h with a velocity of 10^18 m/s. </p>
<p>A friend once told me that actually articles have been published delving into this problem, e.g. to prove that fictitious forces emerge from the choice of such a bizar reference frame that ensure that the earth is still (somewhat) flattened at it's poles.</p>
<p>Questions:</p>
<p>1) Does anybody know of such a publication?</p>
<p>2) I know that even such speeds of 10^18 m/s are not in contradiction with relativity because a limiting velocity only exists for exchange of information, which apparantly does not occur.<br>
Still: could anybody explain why such bizar velocities are allowed?</p> | 4,320 | [
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<p>The electrical field for a charge density $\varrho(r)$ is
$${\bf E}({\bf r})=\frac{1}{4\pi\varepsilon_0}\int \varrho({\bf r}')\frac{{\bf r}-{\bf r}'}{\hspace{.1cm }|{\bf r}-{\bf r}'|^3} \mathrm d^3r'.$$</p>
<p>Given a single particle of charge $q_C$ fixed in the centre of the coordinate system, a test charge $q_T$ feels a field which goes as $\propto\frac{1}{r^2}$. This field diverges as one gets closer (${\bf r}\to{\bf 0}$). On the other hand, if we have a big wall of charges with density $\varrho_W$, then its electrical field is constant and doesn't diverge as I get closer ($z\to0$) - for example imagine a capacitor place in the xy-plane.</p>
<p>Say I have a charge $q_T$ at position ${\bf x}(0):=(0,0,d)$ above charged plate in the xy-plane. It's eighter infinitely big of or radius $R$. Now I start moving the particle along the given trajectory ${\bf x}(t):=(0,0,d-v\ t)$. Here $d$ and $v$ are fixed constants. What is the value of ${\bf E}({\bf x}(t))$?</p>
<p>What is the electrical field in the case where I have a capacitor plane containing many charges, one of which is perfectly fixed at ${\bf 0}$?</p> | 4,321 | [
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<p>What are examples of endeavors, in the history of mankind, to understand physical phenomena with models which were proved to be incorrect later, reformed significantly, or are still under development?</p> | 4,322 | [
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<p>Matter-- I guess I know what it is ;) somehow, at least intuitively. So, I can feel it in terms of the weight when picking something up. It may be explained by gravity which is itself is defined by definition of the matter!</p>
<p>What is <a href="http://en.wikipedia.org/wiki/Antimatter">anti-matter</a>?</p>
<p>Can you explain it to me? </p>
<ul>
<li>Conceptually simplified</li>
<li>Real world evidence</li>
</ul> | 4,323 | [
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<blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="http://physics.stackexchange.com/questions/9165/which-mechanics-book-is-the-best-for-beginner-in-math-major">Which Mechanics book is the best for beginner in math major?</a> </p>
</blockquote>
<p>I am looking for suitable ways to learn mechanics in mathematician's perspective.
I went through:</p>
<ul>
<li>multivariable calculus from Spivak,
<ul>
<li>real analysis from Pugh, </li>
<li>differential equations from Hirsh/Smale/Devaney (mostly focusing on linear system, existence & uniqueness, nonlinear dynamical system, bifurcation, and brief touch on chaos) (so no application covered) </li>
<li>differential geometry from Pressley (but I hate pressley, so I am going to review through doCarmo)</li>
</ul></li>
<li>topology from Willard (but not all of them)</li>
</ul>
<p>The problem is I did not take freshman physics coures (because of annoying labs;;)</p>
<p>My goal is to be able to read Abraham/Marsden's Foundations of Mechanics or something
of that level.</p>
<p>I was thinking of reading differential equations book's applications section first and... idk.</p>
<p>What books do you think is suitable for me to start learning classical mechanics?</p>
<p>P.S. Some people mentioned Arnold's Mathematical Methods of Classical Mechanics,
but do you think it is self-contained in terms of physical intuition required?</p> | 98 | [
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<p>How do the Planets and Sun get their initial rotation?
Why do Venus and Mercury rotate so slowly compared to other planets and why does Venus rotate in a different direction to Mercury, Earth and Mars?</p> | 4,324 | [
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<p>This might be a very simple question. I just want someone to point me the right direction to understand things like this:
$$
\langle x|x'\rangle=\delta(x-x') \\
\psi(x)=\langle x|\psi\rangle \\
\tilde{\psi}(p) = \langle p|\psi\rangle \\
\langle x|p\rangle=\frac{1}{\sqrt{2\pi \hbar}}\exp(ipx/\hbar)
$$
I am using Griffiths textbook, but it's too confusing. Where (on the internet) can I find a easy approach to understand these representations? </p> | 4,325 | [
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<p>This is kind of related to this,
<a href="http://physics.stackexchange.com/questions/53768/defining-a-cft-using-beta-functions">Defining a CFT using beta-functions</a></p>
<p>So what would be the right definition of a CFT even classically? </p>
<ul>
<li>Is it true that classically one will call a theory scale invariant only if the action is invariant under scale transformations? (and not by the Lagrangian density) </li>
</ul>
<p>For example under the scale transformations $x' = \lambda x$, in $3+1$ the scalar field goes as $\phi'(x') = \lambda \phi(x)$ and in $1+1$ it goes as $\phi'(x') = \phi (x)$. This means that in $3+1$ the action of the massless scalar field is not scale invariant but in $1+1$ it is but the Lagrangian density goes the otherway. But from the point of view of beta-functions isn't it more consistent to call the $1+1$ theory as a CFT but not the one in $3+1$? </p>
<p>Isn't a massless scalar field theory in $3+1$ guaranteed to produce mass by RG flow whereas the $1+1$ theory will not? </p> | 4,326 | [
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<p>In the Lagrangian for a charged particle in an electromagnetic field </p>
<p>$$L = \frac{1}{2}mu^2 - q(\phi - \frac{\vec{A}}{c}\cdot \vec{u})$$</p>
<p>the energy of the particle is contained in the kinetic term, the rest being interaction terms of the particle with the electromagnetic field. If it's travelling at some velocity, then it will generate its own magnetic field and therefore possess a magnetic self energy, so which term in $L$ contains this?</p> | 4,327 | [
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<p>I have always been puzzled by how do you arrive at Lagrangians? That is, how do you know that the functional you need to get Newton's equations is </p>
<p>$L$ = $T-V(x)$</p>
<p>Do you derive the Lagrangian first somehow or do you just guess the one which would satisfy the action to give equation's of motion?(because then you would need to know the equations of motion first)</p>
<p>Also, is the Lagrangian always equivalent to something resembling $T-V$, or there are other ways to determine it?
More generally,
<em>is the action known before you know the equations of motion, or vice versa</em>? What I really wish to know is, was the Einstein Hilbert action discovered before field equations were? If so, <em>how</em>?</p>
<p>(<em>My major doubt is how do you come across the Lagrangian if you don't know the equations of motion, and you can't guess $L$ then</em>) </p> | 4,328 | [
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<p>Imagine you have two homogeneous spheres with the same diameter of $d=0.1 m$. They have the same mass $m = 1 kg$. The distance between the centers of mass is $r= 1 m$. Their electrical charge shall be disregarded. At $t=0$ the spheres do not have any relative motion to each other. Due to gravitation they will accelerate and start moving towards each other. After some time they will touch each other. </p>
<p><em>How to calculate analytically the time it takes the two spheres to meet each other.</em> I'm not interested in a numerical solution. </p>
<p>I have already tried several ways but I don't get to a solution.</p>
<hr>
<p>Imagine that the 2 spheres have different masses and diameters.
$m_{1}=2 kg$, $m_{2}=5 kg$, $d_{1}=0.03 m$, and $d_{2}=0.3m$.
<em>How to calculate analytically when and where the 2 spheres are going to meet?</em></p>
<hr>
<p><em>How do you calculate the second problem taking the theory of relativity into account?</em> I know that it will not change the result that much but I am interested in the mathematical solution.</p> | 134 | [
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<p>Is it ok to say $145\,{\rm k\,MPa}$ for $145\, {\rm GPa}$. We are so used to comparing stresses in ${\rm MPa}$ that I want to keep things relative to this unit. So would it be a no-no to do so. </p> | 4,329 | [
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<p>All of the quantum algorithms I've seen so far require a turing-complete quantum computer, at least as far as I can tell. Are there any quantum algorithms that require only a <a href="https://en.wikipedia.org/wiki/Quantum_finite_state_machine" rel="nofollow">quantum finite automaton</a>? If so, how does their asymptotic complexity compare to the classical versions of those algorithms?</p> | 4,330 | [
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<p>I'm a Mechanical Engineering student and I'm working on my senior project, so I need help. My project is about designing a solar dish having a diameter of 1.5 meters and a focal length of 60cm. so at the focal point, a circular coil (copper pipe) will be folded, in order to have a superheated steam as an output. What I'm struggling to know is: How to calculate the Power at the focal point?</p> | 175 | [
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<p>I believe that answer to my question is rather trivial but I can't seem to get my head around it. In context of ADM formulation of gravity (or any other differential geometry context, I guess) the covariant derivative of a normal vector to the hypersurface/foliation/slice, in any direction within the hypersurface gives extrinsic curvature. In components K_ij would be e.g. covariant derivative of normal vector in direction "i" projected to direction "j" where both basis vectors "i" and "j" lie in the hypersurface.
My question is, how come that vector which is covariant derivative of the normal vector along one of these "i"'s always has orthogonal component equal to zero?</p>
<p><strong>I.e. how come the covariant derivative of normal vector in direction "i" projected to the normal vector itself always equals zero?</strong> </p>
<p>It isn't true for any other vector, as a matter of fact, using compatibility of a metric from definition of extrinsic curvature K_ij=(cov(n,i)|j) we can easily see that (n|cov(j,i))=K_ij so that covariant derivative of any other vector "j" along "i" does have a component along normal "n" and it is exactly the extrinsic curvature.</p>
<p>Consequently, I'd assume that the above property of not having an orthogonal component of covariant derivative is kind of a defining property of a normal vector. I just wonder where does this follow from.</p>
<p>Thank you!</p>
<p>Cheers!</p> | 4,331 | [
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<p>This question is about the generalization of Coulomb's law to continuous bodies of charge. The basic statement of Coulomb's Law involves two discrete charges $q_1$ an $q_2$:</p>
<p>$$\vec{F}_i = \frac{1}{4 \pi \epsilon_0} \frac{q_1 q_2}{r_{12}} \hat{r}_i $$</p>
<p>Here $i$ represents the charge on which the force is exerted, and $\hat{r}_i$ represents the unit displacement vector between the other charge and the charge $i$.</p>
<p>Many treatment of electrostatics extend this law to the case that one charge is not discrete, but rather a continuous body. The force on the discrete charge $Q$ is then:</p>
<p>$$\vec{F} = \frac{Q}{4 \pi \epsilon_0} \int \frac{dq}{r^2} \hat{r} $$</p>
<p>Here $dq$ is the infinitesimal charge element of the continuous body, while $r$ and $\hat{r}$ represent the distance and displacement vectors between $dq$ and $Q$.</p>
<p>Continuing this way, we could probably propose an expression for force between two continuous bodies of charge, like so:</p>
<p>$$\vec{F} = \frac{1}{4 \pi \epsilon_0} \int \int \frac{dq_1 dq_2}{r^2} \hat{r}$$</p>
<p>However, I have not really seen this expression in the literature/treatments of electrostatics. Does anyone know why this is the case? Is the expression not useful, or are there no applications demanding the above expression? </p> | 4,332 | [
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<p>The <a href="http://en.wikipedia.org/wiki/Higgs_boson" rel="nofollow">Higgs boson</a> is a hypothetical elementary particle predicted by the Standard Model (SM) of particle physics. It belongs to a class of particles known as bosons</p>
<p>What Will Happens If physicist Find the Higgs Particle that thought to be fundamental is not fundamental?</p> | 4,333 | [
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<p>"Stability" is generally taken to be the justification for requiring that the spectrum of the Hamiltonian should be bounded below. The spectrum of the Hamiltonian is <em>not</em> bounded below for thermal sectors, however, but thermal states are nonetheless taken to be stable because they satisfy thermodynamic constraints. In classical Physics, we would say that the thermal state has lowest <em>free energy</em>, which is a thermodynamic concept distinct from Hamiltonian operators that generate time-like translations.</p>
<p>The entropy component of free energy, meanwhile, is a <em>nonlinear</em> functional of the quantum state (presuming that the definition of entropy in quantum field theory would be at least this much like von Neumann's definition in terms of density operators), so we can reasonably expect the sum of the energy and entropy components to have a minimum in the state of greatest symmetry, as we see for thermal states. [It seems particularly notable in this context that the entropy is not an observable in the usual quantum mechanical sense of a linear functional of the quantum state.]</p>
<p>The presence of irreducible randomness in quantum mechanics presumably puts quantum field theory as much in the conceptual space of thermodynamics as in the conceptual space of classical mechanics, despite the quasi-functorial relationship of "quantization", so <em>perhaps</em> we should expect there to be some relevance of thermodynamic concepts.</p>
<p>Given this background (assuming, indeed, that no part of it is <em>too</em> tendentious), <strong>why should we think that requiring the Hamiltonian to have a spectrum that is bounded below should have anything to do with stability in the case of an interacting field?</strong> The fact that we can construct a vacuum sector for free fields in which the spectrum of the Hamiltonian is bounded below does not seem enough justification for interacting fields that introduce nontrivial biases towards statistically more complex states.</p>
<p>This question is partly motivated by <a href="https://johncarlosbaez.wordpress.com/2011/12/22/quantropy/" rel="nofollow">John Baez' discussion of "quantropy" on Azimuth</a>. I am also interested in the idea that if we release ourselves from the requirement that the Hamiltonian of interacting fields must have a spectrum that is bounded below, then we will have to look for analogues of the KMS condition for thermal states that restore some kind of analytic structure for interacting fields.</p>
<p>I asked a related Question <a href="http://physics.stackexchange.com/questions/5812/there-seems-to-be-no-definition-of-stability-in-axiomatic-qft-is-there-and">here</a>, almost a year ago. I don't see an answer to the present Question in the citations given in Tim van Beek's Answer there.</p> | 4,334 | [
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<p>How can triangulation be used to calculate the approximate distance to very distant celestial bodies like stars, globular clusters, etc.? And can it be used to measure the distance to a Black Hole?
(Can someone please help me with tags?)</p> | 4,335 | [
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<p>When using the fluctuating exchange approximation (FLEX) as a dynamical mean field theory (DMFT) solver, <a href="http://rmp.aps.org/abstract/RMP/v78/i3/p865_1" rel="nofollow">Kotliar, et al.</a> (p. 898) suggest that it is only reliable for when the interaction strength, $U$, is less than half the bandwidth. How would one verify this? Also, is there a general technique for establishing this type of limit?</p>
<p>To clarify, DMFT is an approximation to the Anderson impurity model, and FLEX is a perturbative expansion in the interaction strength about the band, low interaction strength limit.</p> | 4,336 | [
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<p>Inspired by physics.SE: <a href="http://physics.stackexchange.com/questions/15571/does-the-dimensionality-of-phase-space-go-up-as-the-universe-expands/15613">http://physics.stackexchange.com/questions/15571/does-the-dimensionality-of-phase-space-go-up-as-the-universe-expands/15613</a></p>
<p>It made me wonder about symplectic structures in GR, specifically, is there something like a Louiville form? In my dilettante understanding, the existence of the ADM formulation essentially answers that for generic cases, but it is unclear to me how boundaries change this. Specifically, I know that if one has an interior boundary, then generally the evolution is not hamiltonian; on the other hand, if the interior boundary is an isolated horizon, then the it is hamiltonian iff the first law of blackhole thermodynamics is obeyed (see <a href="http://arxiv.org/abs/gr-qc/0407042">http://arxiv.org/abs/gr-qc/0407042</a>). </p>
<p>The sharper form of the question is thus what happens cosmologically? </p>
<p>(And as usual for a research level (?) question: what are the Google-able search terms to find out more about this?) </p> | 4,337 | [
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<p>Excuse me, I have calculated $a^g$ a lot of times, using the relation between $:\;:$ and ${}^{{}_\circ}_{{}^\circ} \; {}^{{}_\circ}_{{}^\circ}$. But I can't get the same result with the book.
It is not too hard to get
$$
:b(z)c(z'):-{}^{{}_\circ}_{{}^\circ} b(z)c(z'){}^{{}_\circ}_{{}^\circ} =
\frac{\left(\frac{z'}{z}\right)^{\lambda-1}-1}{z-z'}.
$$
Which can be found in Problem 2.13 on $P_{76}$.
Take the limit $z\rightarrow z'$, we find
$$
:b(z)c(z):-{}^{{}_\circ}_{{}^\circ} b(z)c(z){}^{{}_\circ}_{{}^\circ} = \lim_{z\rightarrow z'}\frac{\left(\frac{z'}{z}\right)^{\lambda-1}-1}{z-z'}=\lim_{z\rightarrow z'}\frac{\left(1-\frac{z-z'}{z}\right)^{\lambda-1}-1}{z-z'}=\lim_{z\rightarrow z'}\frac{\left(1-(\lambda-1)\frac{z-z'}{z}\right)-1}{z-z'}
=\frac{1-\lambda}{z}.
$$
$$
\partial (:b(z)c(z):)-\partial ({}^{{}_\circ}_{{}^\circ} b(z)c(z){}^{{}_\circ}_{{}^\circ}) = -\frac{1-\lambda}{z^2}
$$
and
$$
:\partial b(z)c(z):-{}^{{}_\circ}_{{}^\circ} \partial b(z)c(z){}^{{}_\circ}_{{}^\circ} = \lim_{z\rightarrow z'}\partial\frac{\left(\frac{z'}{z}\right)^{\lambda-1}-1}{z-z'}
=\lim_{z\rightarrow z'}\left(-\frac{\left(\frac{z'}{z}\right)^{\lambda-1}-1}{(z-z')^2}+\frac{(1-\lambda)\left(\frac{z'}{z}\right)^{\lambda-1}}{z(z-z')}\right)
=\frac{(\lambda-1)^2}{z^2}.
$$
The energy-momentum tensor is $T(z)=:(\partial b)c:-\lambda\partial(:bc:)$.</p>
<p>Using the above results, we can express $T(z)$ in <em>creation-annihilation normal ordering</em> ${}^{{}_\circ}_{{}^\circ}\;{}^{{}_\circ}_{{}^\circ}$
$$
T(z)={}^{{}_\circ}_{{}^\circ} \partial b(z)c(z){}^{{}_\circ}_{{}^\circ}+\frac{(\lambda-1)^2}{z^2}-\lambda\left(\partial ({}^{{}_\circ}_{{}^\circ} b(z)c(z){}^{{}_\circ}_{{}^\circ})-\frac{1-\lambda}{z^2}\right)
={}^{{}_\circ}_{{}^\circ} \partial b(z)c(z){}^{{}_\circ}_{{}^\circ}-\lambda\partial ({}^{{}_\circ}_{{}^\circ} b(z)c(z){}^{{}_\circ}_{{}^\circ})+\frac{1-\lambda}{z^2}
$$
$$
\Rightarrow a^g=1-\lambda
$$
But in Polchinski's book,$a^g=\frac{1}{2}\lambda(1-\lambda)$. What wrong with my derivation?</p> | 4,338 | [
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<p>Normally supersymmetric quantum field theories have Lagrangians which are supersymmetric only on-shell, i.e. with the field equations imposed. In many cases this can be solved by introducing auxilary fields (field which don't carry dynamical degrees of freedom, i.e. which on-shell become a function of the other fields). However, there are cases where no such formulation is known, e.g. N=4 super-Yang-Mills in 4D.</p>
<p>Since the path integral is an integral over all field configurations, most of them off-shell, naively there is no reason for it to preserve the on-shell symmetry. Nevertheless the symmetry is preserved in the quantum theory.</p>
<p>Of course it is possible to avoid the problem by resorting to a "Hamiltonian" approach. That is, the space of on-shell field configurations is the phase space of the theory and it is (at least formally) possible to quantize it. However, one would like to have an understanding of the symmetries survival in a path integral approach. So:</p>
<blockquote>
<p>How can we understand the presence of on-shell symmetry after quantization from a path integral point of view?</p>
</blockquote> | 4,339 | [
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<p>Fluid flows become turbulent beyond a certain velocity. The velocity is almost always with respect to a fixed boundary. However, an observer in a frame of reference travelling with the fluid will also experience turbulence when the velocity of fluid with respect to the boundary exceeds a certain value. Does this indicate that fluid flow is not invariant under a Galilean transformation?</p> | 4,340 | [
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<p>The following question is probably very elementary:
<strong>whether molecules of <a href="http://en.wikipedia.org/wiki/Ideal_gas" rel="nofollow">ideal gases</a> may have optic properties?</strong>
As far as I understand, when one discusses optic properties, one assumes that molecules of the material have some inner structure, in particular different energy levels.</p>
<p><strong>The question is whether existence of such an inner structure may contradict the assumption that the gas is ideal.</strong></p> | 4,341 | [
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<p>In a horizontal surface, a block (cube) is sliding due to a sudden push. When the block slides, there is frictional force which is acting on the block. </p>
<p>Frictional force will have a torque around the center of mass, so why does the block not rotate/roll around (a horizontal axis through) the center of mass instead of sliding?</p> | 4,342 | [
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<p>Is there any way to find the energy of a particle through its decay probability?</p> | 4,343 | [
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<p>The singlet state of two qubits is anticorrelated in every basis. For example, in the Pauli bases, it can be expressed,</p>
<p>$\frac{1}{\sqrt{2}} ( | 01 \rangle - | 10 \rangle) = \frac{1}{\sqrt{2}} ( | +- \rangle - | -+ \rangle) = \frac{1}{\sqrt{2}} ( | \circlearrowright \circlearrowleft \rangle - | \circlearrowleft\circlearrowright \rangle)$.</p>
<p>However, there is no state that is correlated in every basis. The best we can get is correlation in two Pauli bases, and anti-correlation in the third. For example,</p>
<p>$\frac{1}{\sqrt{2}} ( | 00 \rangle + | 11 \rangle) = \frac{1}{\sqrt{2}} ( | ++ \rangle + | -- \rangle) = \frac{1}{\sqrt{2}} ( | \circlearrowright \circlearrowleft \rangle + | \circlearrowleft\circlearrowright \rangle)$.</p>
<p>Why is this? Would it violate some principal, like allowing for superluminal communication?</p> | 4,344 | [
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<p>I am doing some experiments on musical strings (guitar, piano, etc.). After performing a Fourier Transform on the sound recorded from those string vibrations, I find that the fundamental frequency is not absolutely the component with the largest amplitude (or energy).</p>
<p>I learnt from introductory physics courses that fundamental frequency is the "major" component of a vibrating string. So why does it sometimes does not possess the highest amplitude?</p> | 4,345 | [
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<p>Gauge bosons are represented by $A_{\mu}$, where $\mu = 0,1,2,3$. So in general there are 4 degrees of freedom. But in reality, a photon (gauge boson) has two degrees of freedom (two polarization states). </p>
<p>So, when someone asks about on-shell and off-shell degrees of freedom, I thought they are 2 and 4. But I read that the off-shell d.of. are 3. </p>
<p>And my question is how to see this?</p> | 4,346 | [
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<p>Imagine we have two different differentiable functions $f(t)$ and $g(t)$ where $t$ generally represents the time, if there exists the following limit as
$$ \lim\limits_{t\rightarrow \infty } \frac{\| \dot{f}(t) \|}{\|\dot{g}(t)\|}=c$$
Then, is there any appropriate physical explanation for this limit ?
If ignore the limit, the $\frac{\| \dot{f}(t) \|}{\|\dot{g}(t)\|}$ is sort of instantaneous reletive absolute rate of change of $f(t)$ over $g(t)$, but what is that when limit involves in ?</p> | 4,347 | [
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<p>If energy required to accelerate a particle to the speed of light is infinite then where do they get it from?</p>
<p>But first if photon's are massless, then why do they collide to some other thing and get reflected as in our daily life or get deflected after colliding to an atom in the sun and just move inside the sun for a long time before escaping it? </p> | 4,348 | [
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<p>Dopfer Momentum-EPR experiment (1998) seems to provide a interesting tweak in the EPR experiment.</p>
<p>To read more details on this experiment, see:</p>
<p><a href="http://www.hep.yorku.ca/menary/courses/phys2040/misc/foundations.pdf" rel="nofollow">http://www.hep.yorku.ca/menary/courses/phys2040/misc/foundations.pdf</a> (page S290)
<a href="http://www.physics.ohio-state.edu/~lisa/CramerSymposium/talks/Cramer.pdf" rel="nofollow">http://www.physics.ohio-state.edu/~lisa/CramerSymposium/talks/Cramer.pdf</a> (page 11)</p>
<p>Also, a discussion of the experiment, the coincidence counter, and how the conditions of the no-signaling theorems might not apply to this setup:</p>
<p><a href="http://casimirinstitute.net/coherence/Jensen.pdf" rel="nofollow">http://casimirinstitute.net/coherence/Jensen.pdf</a></p>
<p>In summary, the experiment sends two entangled photons A and B toward two separate arms. Arm A has a lens and a Heisenberg detector, which can be placed either at the focal plane or the image plane. Arm B is sent on a two-slit filter. The observed results of this experiment are as follows:</p>
<p>1) if the Heisenberg detector at arm A is placed at the focal plane, the output of the two-slit filter at arm B is an interference pattern</p>
<p>2) if the Heisenberg detector at arm A is placed at the image plane (twice the focal plane) the output of the two-slit filter at arm B is an incoherent sum of intensities from each slit</p>
<p>As suggested in the 3rd paper link I posted, if you use time bins, there really doesn't seem to be <em>any need at all</em> to rely on the coincidence counter, as you can study the interference pattern on each time bin in isolation from photons received in other time bins</p>
<p>Am I confused? How is the coincidence count being used at all? notice that the interference pattern is spatial, not temporal!</p> | 4,349 | [
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<p>Long ago I learned that a <a href="http://en.wikipedia.org/wiki/Plasma_%28physics%29" rel="nofollow">plasma</a> was a distinct state of matter after solid, liquid and gas, and also that it was achieved by imparting heat to a the matter. But most references describe a plasma as an ionized gas. So I'm having trouble understanding, what then, does it mean to be a distinct phase of matter? Is ionization, as opposed to heat, all that's required to make a gas a plasma? If so, what makes a plasma more distinguished than, say, an ionized liquid?</p> | 4,350 | [
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<p>I just had a confusion. Does the mass of the body actually increase when it is moving with a certain velocity? Or does it only look like the mass has increase to another observer. How can the actual mass of the body increase. Please correct me if I am wrong but I feel that it only seems to the observer that the mass has increased but it does not increase in real.</p>
<p>The increase of actual mass would imply that that its velocity will decrease to comply with conservation of momentum. But how can the velocity decrease if no external force is applied?</p> | 54 | [
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<p>There are videos and articles on the internet which demonstrate that water flows down a flush clockwise in Northern Hemisphere and anti-clockwise in Southern Hemisphere.</p>
<p>Here are a couple of links which claim to demonstrate this fact :
<a href="http://www.youtube.com/watch?v=Pb69HENUZs8" rel="nofollow">http://www.youtube.com/watch?v=Pb69HENUZs8</a>
<a href="http://www.youtube.com/watch?v=z9uN9rcgJ1s" rel="nofollow">http://www.youtube.com/watch?v=z9uN9rcgJ1s</a></p>
<p>There are others which call this a total myth, and one of them is a very popular website.
Link : <a href="http://science.howstuffworks.com/science-vs-myth/everyday-myths/rotation-earth-toilet-baseball2.htm" rel="nofollow">http://science.howstuffworks.com/science-vs-myth/everyday-myths/rotation-earth-toilet-baseball2.htm</a></p>
<p>Interestingly, I got introduced to this by a Children's Fact Book. I am not quite convinced that it is a myth. </p>
<p><strong>Can you suggest a way to proceed towards an analytical treatment?</strong></p> | 135 | [
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<p>I was near ($\approx40m$) an overhead power line and I heard a sound coming from the cables of the power line; I think the sound was made by the vibrations of the power cables due to the wind but I am not sure. The wind was very light.</p>
<p>The sound was not the "buzz" asked about <a href="http://physics.stackexchange.com/questions/12629/why-do-power-lines-buzz?rq=1">here</a>.</p>
<p><img src="http://i.stack.imgur.com/azt0a.jpg" alt="enter image description here"></p>
<p>My question is: assuming the sound was generated by the wind, is it possible to estimate the speed of wind from the sound properties (i.e. its <a href="http://en.wikipedia.org/wiki/Spectrogram" rel="nofollow">spectrogram</a>) and the mechanical properties of the cable?</p>
<p>If yes, how accurate will be the estimate? </p>
<p>If yes, can you provide some back-of-the-envelope calculation?</p> | 4,351 | [
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<p>Assume I have two disks, $p_1$ and $p_2$, of radius $r$, with their own velocities (preferably in $(x,y)$ form, but $(m, \theta)$ works too) and masses (unit-less, but same unit) collide in two dimensions, how can I compute their resulting velocities?</p>
<p>I was looking around on the internet and it seems like every calculation assumes that one of them is at rest, but both of mine will be moving.</p>
<p>Wikipedia has <a href="http://en.wikipedia.org/wiki/Elastic_collision#Two-_and_three-dimensional" rel="nofollow">this bit</a>, but it assumes that I know how to calculate the angle of deflection of the system, $\theta$, but that's even more confusing.</p>
<p>I'm pretty lost. What do I do?</p> | 136 | [
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<p>Why can't we accelerate to the speed of light? It's just a speed and nothing else. Universe also crossed the speed of light at the time of big bang. Is this is just a interpretation or there is any fact behind it? I am asking it as I don't know much about this theory.</p> | 137 | [
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<p>How can I size an electromagnet? For example if I would to attract a mass of $x$ kg what are the calculations that I must do to size the ferromagnetic core and of course the solenoid? And thus, how can I compute the force of the magnetic field generated?</p>
<ul>
<li>Current: AC 230V @ 50Hz.</li>
<li>Material: Iron.</li>
<li>Distance: 30 cm.</li>
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<p><strong>I have a few questions related to the emission of electromagnetic radiation by black bodies. </strong>
<hr/>
Consider the following image:</p>
<p><img src="http://s22.postimg.org/77bujyudt/image.png" alt="fig.1"></p>
<p>On the above image I have drawn the rays of light that are emmited by black bodies assuming that they have only 8 points of emmision (They are marked with red dots).<br />
<em>Which of these two images shows the real situation? The single point emits the radiation in all directions or only in one direction? </em></p>
<hr />
<p>If my intuition is not wrong and the image on the right is correct consider the next image where I assume that there is only one point of emmission on the object:</p>
<p><img src="http://s23.postimg.org/m87phx7dn/image.png" alt="fig.1"></p>
<ul>
<li><em>Is the intensity of radiation the same in every direction?</em></li>
<li><em>Does the radiation in each direction have the same intensity-wavelength distribution (presented in the image below)? </em></li>
</ul>
<p><img src="http://s24.postimg.org/b53ar97ad/image.png" alt="fig.3"></p> | 4,353 | [
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<p>A problem has been presented that goes like this:</p>
<p>Particles normally exist as several mathematical possibilities rather than one actual object. It is said that in the absence of observation, particles exist in a superposition of possibilities rather than one actual thing. But when we look they are not in such an indefinite state.</p>
<p>The problem states that the way to solve finding out why there is something definite when we look is decoherence. But it is said that this doesn't really solve the problem because whatever particles were used to collapse another, what was used to collapse that particle? And so on and on.</p>
<p>And the reason for this is because the wave function of a particle cannot be unentangled from that of whatever is used to measure it. When one photon is measured by another they entangle. If one particle measures another, it inherits part of its wave function, and that particle which is supposed to be measuring, cannot be fully explained without what it is measuring. </p>
<p>So you need another measuring device to collapse that initial measuring particle to a definite state. But then you need something else to collapse that measuring apparatus as well, and so on and so on. This creates a chain of material objects in a superposition of measuring, which is known as a Von Neumann chain.</p>
<p>Since quantum laws are what describe all material objects, some other particles or measuring apparatus is always needed to collapse the next one in line. You keep going back until you get to something nonlocal. Outside the entire material system, which escapes this chain by not being bound by the same physical laws, and is able to cause final collapse of everything in the chain, which is argued to be a conscious observer. Something beyond the material with the ability to collapse the entire physical system.</p>
<p>Is this true? If not, why not.</p> | 4,354 | [
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<p>Let's assume we have a perfect single-photon source: a device emitting exactly one photon at a time, with defined energy and direction. Let's shoot a photon: we know exactly the position of the photon (starting point and time, velocity) and it's momentum (energy and velocity). Would such a device violate uncertainty principle? Where is the trap?</p>
<hr>
<p>Just to clarify things, my question essentially is: <em>a particle (e.g., a photon) prepared in an eigenstate of momentum can be found everywhere (at least along the direction of momentum)?</em></p> | 4,355 | [
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<p>Suppose an object A is traveling at a velocity of 100 m/s, and another object B is traveling at 105 m/s. With both the objects traveling through the same direction, taking A as a reference frame, the velocity of B would be 5 m/s (Is this actually right?). But, when they're traveling in opposite directions, how would one measure the velocity of B (with A as reference frame)..? Does it actually take a negative sign? - Sorry, if I have a misunderstanding...</p> | 4,356 | [
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<p>Morrison writes in "Morrison, Michael A. : Understanding quantum physics : a user's manual" </p>
<blockquote>
<p>$ |\Psi(x,t)|^2 \xrightarrow[x\rightarrow\pm \infty ]{} 0$ at all
times <em>t</em> [bound state] </p>
<p>$ |\Psi(x,t)|^2 \xrightarrow[x\rightarrow\pm \infty ]{} 0$ at any
particular time <em>t</em> [unbound state]</p>
</blockquote>
<p>So I can imagine that "all" means the entirety of all times, but do I not get "all" when summing over all particular states?</p>
<p>I also understand that in a bound state, the wave is never at the infinity position, but the wave of an unbound state may exist there.</p> | 4,357 | [
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<p>Please see the following photos. (I cannot post them...)</p>
<p><a href="http://i1163.photobucket.com/albums/q554/startanewww/CIMG4545.jpg" rel="nofollow">http://i1163.photobucket.com/albums/q554/startanewww/CIMG4545.jpg</a></p>
<p><a href="http://i1163.photobucket.com/albums/q554/startanewww/CIMG4546.jpg" rel="nofollow">http://i1163.photobucket.com/albums/q554/startanewww/CIMG4546.jpg</a></p>
<p>From the first photo, the book mentioned <em>Don't cover the paper with glass because glass absorbs enough ultraviolet light to slow the damage process.</em> (line 8)</p>
<p>My questions:</p>
<ul>
<li><p>Why can glass absorb ultraviolet light?</p></li>
<li><p>Will papers with dark colours fade fastest? Why is that the case? (see the second last paragraph)</p></li>
</ul> | 4,358 | [
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<p>I'm trying to show that $H_D = -i\boldsymbol{\alpha}.\nabla+\beta m$ is hermitian. </p>
<p>Its given that
$$
\gamma^{0\dagger}=\gamma^0
$$
$$
\boldsymbol\gamma^\dagger=-\boldsymbol\gamma
$$</p>
<p>What i've done is:
$$
H_D^\dagger = +i\boldsymbol\alpha^\dagger.\nabla+\beta^\dagger m
$$
$$
=+i(\gamma^0\boldsymbol\gamma)^\dagger.\nabla+\beta m
$$
$$
=+i(\boldsymbol\gamma^\dagger\gamma^{0\dagger}).\nabla+\beta m
$$
$$=-i(\boldsymbol\gamma\gamma^0).\nabla+\beta m
$$</p>
<p>Well clearly i'm doing somthing wrong because $\boldsymbol\gamma\gamma^0\neq\gamma^0\boldsymbol\gamma$</p> | 4,359 | [
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<p>Please see the following photo. (I cannot post it)
<a href="http://i1163.photobucket.com/albums/q554/startanewww/CIMG4548.jpg" rel="nofollow">http://i1163.photobucket.com/albums/q554/startanewww/CIMG4548.jpg</a></p>
<p>Why is atomic bomb in a "fish-like" shape? (I don't know how to describe it) Is it specially designed for some purposes?</p>
<p>Thanks a lot.</p> | 4,360 | [
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] |
<p>This is with regards to adiabatic magnetisation. </p> | 4,361 | [
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<p>Having read Art Hobsons paper on Quantum field theory, he states " the field collapses into a field of atomic size" This seems to be stating that each field quanta is a different quantum field? Like 2 electrons are 2 electron fields, rather than the 2 electrons come from the same field. I thought they all emerged from the same field rather than 2 of them have there own/be there own seperate field. </p>
<p>Also he say field quanta are infinitly extended and has its energy spread over light years. This has confused me also, I havent seen any infintely sized electrons about, and as electrons are field quanta, how can field quanta be countable, yet infinite in size?</p>
<p>please help...</p> | 554 | [
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<p>I've been having this confusing thought for so long now it would be amazing if someone could answer me.</p>
<p>Imagine this asterisk * . As you see, from the center point, lines go outwards, just like a sun will emit rays of light in all direction.</p>
<p>BUT, theoretically, there should be a "finite" amount of photons it sends in space, which means that the farther you are from that sun, let's say 1 million light years, the less likely are your eyes of catching the photons emitted from the source?</p>
<p>So does this make the theory of the photon bad? Since we can see a star millions of light years away, no matter where we stand on the ground of this planet. This means that the star can emit at least one photon every millimeter or less millions of light years away so this would mean that it would need an "impossible" density of photons sent at the source on the sun's surface in every direction of the cosmos. </p>
<p>How's this possible?</p> | 4,362 | [
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<p>I have been thinking about the definition of the notation $\cal N$ and its relation to the number of supercharges in <a href="http://en.wikipedia.org/wiki/Super-Poincar%C3%A9_algebra">SUSY</a>, but still feel a little confused. In dimension 2, we usually denote, for example, $\cal N = (2,2)$ supersymmetry, where we have 2 chiral supercharge and 2 anti-chiral supercharge; but in higher dimensions we just refer to $\cal N = 1$, etc. What is the difference and why we make such different notation?</p>
<p>Also, I would appreciate of one can explain the exact meaning of $\cal N$, for example in 4 dimensions, and how they are related to number of supercharge $Q$ and independent spinors.</p> | 4,363 | [
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<p>In this question, I'm not talking about particle spin.</p>
<p>I guess, when an object rotates, its atoms also rotate. When an atom rotates, its particles must move in space. </p>
<p>I wonder that if the particles have a direction.</p>
<p>Can they rotate or do they just move position around the axis (middle) of a proton so we consider that the proton rotates?</p>
<p>Let's think about single particles like electron instead of composite particles like hadrons.</p>
<p>Can electrons rotate ?</p>
<p>Edit : I think this is a simple and good question but I couldn't get a sufficient answer yet .</p> | 4,364 | [
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<p>I've imagined this little scenario to help me conceptualize things.</p>
<p>Let's say we have a doughnut-shaped object with a hole whose diameter is greater than that of a sphere. Let's say that the sphere is vertically aligned with the center of the doughnut and is horizontally gravitating towards it according to Newton's law of universal gravitation. What will happen?</p>
<p>1) The sphere will pass through the doughnut, travel a certain distance (but how far) and accelerate back towards the doughnut (and oscillate back and forward).</p>
<p>2) The sphere will stop as it reaches the center of the doughnut. The reasoning behind this is that the distance between the two objects will be zero and hence the acceleration which is inversely proportional to the distance will be infinity (but in both directions?). This doesn't sound right.</p>
<p>3) Something else</p> | 4,365 | [
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<p>If the Moon had gravity as good as Earth and a magnetic field could it have supported life? Because if the Moon had gravity, it could have retained water more than is present today on the surface. </p>
<p>If the Earth is in the habitable zone, does the Moon also lie in the habitable zone?</p> | 4,366 | [
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<p>I was searching the internet when I found this article <a href="http://arxiv.org/abs/0804.1764" rel="nofollow">http://arxiv.org/abs/0804.1764</a>, about achieving high voltage and power to ignite thermonuclear reaction by inertial confinement, the plan was to achieve it in a ultra vacuum and something that have to do with "taylor flow " which includes quantum physics which I really dont know anything about.
So I searched more and all the sites ( including wikipedia ) where talking about maximum voltage related to gas surrounding, why can't we surround it with liquid with a much higher dielectric strength?</p> | 4,367 | [
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<p>This isn't a homework question, but it might as well be. The problem I have been pondering is:</p>
<blockquote>
<p>If a disc (or children's roundabout if you like), of radius r, mass m, is spun around it's center with an initial force F, and thereafter there is the friction force (of either the axle or air resistance or both) of f, then how long will it take to come to a stop?</p>
</blockquote>
<p>I have thought about it and have come up with not much. My first way is thus:</p>
<p>$F = ma$, so</p>
<p>$a = F/m$, the initial acceleration ( or should that be $(F-f)/m$ ?).</p>
<p>And then the deceleration is $a = -f/m$.</p>
<p>I'm not sure how to calculate the initial linear velocity, but assuming I have it, $u$, say, then I could say that after time $t$ the velocity is
$v = u -at = u -ft/m$, where f is the friction force.</p>
<p>So then the disc would stop spinning when $ t= um/f$. I am aware that this is wrong (well, it might work if we were dealing with linear motion). Straight away it seems wrong because it doesn't take into account the radius of the disc and also the slow down seems linear, when from observation it seems rotating discs slow down and taper off to a standstill. But that is as far as I got. I have tried to use angular motion equations (well $\omega r = v$) but I am stuck at this point, and of course, finding the initial velocity.</p>
<p>Any help is appreciated.</p> | 4,368 | [
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