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<p>The space-time interval equation is this:</p>
<p>$$\Delta s^2=\Delta x^2+\Delta y^2+\Delta z^2-(c\Delta t)^2$$</p>
<p>Where, $\Delta x, \Delta y, \Delta z$ and $\Delta t$ represent the distances along various coordinates according to an observer, and $\Delta s$ is the space-time interval. All observers agree on the space-time interval, it is constant. My question is <strong>why is it squared?</strong> If we had in equation like this:</p>
<p>$$\Delta s'=\Delta x^2+\Delta y^2+\Delta z^2-(c\Delta t)^2$$</p>
<p>$\Delta s'$ would be constant as well. It would also never be imaginary. It would have units of $[length]^2$ instead of $[length]$ though.</p>
<p>Is there a theoretical or practical reason that we define the space-time interval based on squaring, or is it just to make it look similar to Pythagoras' theorem/give it simpler units or something else entirely?</p>
| 3,815 |
<p>I am doing some <a href="http://physics.stackexchange.com/questions/28563/hours-of-light-per-day-based-on-latitude-longitude-formula">calculations</a> to see how many hours of light does a specific location (identified by latitude and longitude) has in a specific day of the year.
Contrary to my expectations I get a slight variance for different years. See the data calculated for this point in Svalbard (77.553604,23.670272) in the same day for different years. The values are calculated using the same formula and the same coordinates.</p>
<pre><code>2005-02-22 5.84500
2006-02-22 5.76472
2007-02-22 5.68389
2008-02-22 5.60222
2009-02-22 5.85472
2010-02-22 5.77472
2011-02-22 5.69389
2012-02-22 5.61250
</code></pre>
<p>Is this normal and why so?</p>
| 3,816 |
<p>$s$ is <em>sharp</em>, $p$ for <em>principal</em>, $d$ for <em>diffuse</em>, $f$ for <em>fundamental</em>.</p>
<p>Where do all those term come from? I do not see any link with the corresponding shapes.</p>
| 3,817 |
<p>I've recently been learning about Fourier optics, specifically, that a thin lens can produce the Fourier transform of an object on a screen located in the focal plane.</p>
<p>With this in mind, does the lens in a human eye produce a Fourier transform on the retina?</p>
<p>Any help appreciated</p>
| 3,818 |
<p>I'm studying fluid mechanics and I have the following doubt: on the book, the author first deduces the differential form of the equation of balance of momentum. First he argues that if the fluid is ideal and is contained in a region $D\subset \mathbb{R}^3$ and if the pressure is $p : D\times \mathbb{R}\to \mathbb{R}$ then the stress force per unit volume is $-\nabla p$ and the body force per unit volume is $\rho \mathbf{b}$ (with $\mathbf{b}$ being the body force per unit mass). In that case the law becomes</p>
<p>$$\rho \dfrac{D\mathbf{u}}{Dt} = -\nabla p + \rho\mathbf{b}$$</p>
<p>Then with some manipulations he derives the integral form of the law. Basically he considers a <strong>fixed</strong> region $W$ and consider the total momentum contained in $W$ as</p>
<p>$$\int_W \rho \mathbf{u} \ dV$$</p>
<p>In that case the integral form can be obtained differentiating that. So, we gain the following law:</p>
<p>$$\dfrac{d}{dt}\int_W \rho \mathbf{u} \ dV = - \int_{\partial W} (p\mathbf{n}+\rho \mathbf{u}(\mathbf{u}\cdot \mathbf{n}))\ dV + \int_W \rho \mathbf{b} \ dV$$</p>
<p>This law is over a fixed region. So we choose a region on the fluid, compute the momentum inside it and see how it varies as the fluid flows through it.</p>
<p>Now, to assume as little differentiability as possible, the author proceeds to obtain one integral form of the law directly from basic principles. If $\varphi$ is the fluid flow map, $\varphi_t = \varphi (\cdot, t)$, then he states the following:</p>
<p>$$\dfrac{d}{dt} \int_{\varphi_t(W)} \rho \mathbf{u} \ dV = \mathbf{S}_{\partial \varphi_t(W)} + \int_{\varphi_t(W)} \rho \mathbf{b} \ dV$$</p>
<p>Where $\mathbf{S}_{\partial \varphi_t(W)}$ is the total force at time $t$ exterted on the fluid contained in $\varphi_t(W)$ by means of stress on its boundary $\partial \varphi_t(W)$. </p>
<p>This is quite clear: we pick a region, look at the mometum and the rate of change of momentum should be equal to the total force applied to it. Now this region is <strong>time-dependent</strong>. That is, instead of looking at a fixed region and at the fluid flowing through it, the author is following a chunk of fluid that moves with time.</p>
<p>My question is: why, when deriving the integral form of the balance of momentum law from the differential form, we consider a fixed region and why, when stating it from basic principles, we consider a region varying with time? Is there some connection between this and the differences between the Lagrangian and Eulerian points of view?</p>
| 3,819 |
<p>I went to a restaurant yesterday and while I was eating dessert, I saw a <strong>standard stainless steel spoon (note the alliteration!</strong>) balancing on a plate. It was balancing on its handle, and I started wondering, </p>
<p>Why is that particular point the center of gravity for the spoon?
Thanks.</p>
| 3,820 |
<p>Do all the planets in our solar system have the same angular speed? Physics teacher says yes, my research is not crystal clear. I want to make sure I have the right information for future reference.</p>
| 3,821 |
<p>In regular quantum mechanics of particles, I have the Schrodinger evolution picture for a <strong>general state</strong></p>
<p>$$ i\hbar \frac{d}{dt} \left|\psi(t)\right> = \hat H \left|\psi(t)\right> $$</p>
<p>then we take the inner product with respect to $ \left< x\right| $ to obtain the equation in the position representation.</p>
<p>$$ i\hbar \frac{d}{dt} \psi(x,t) = -\frac{\hbar^2}{2m}\nabla^2\psi(x,t) - V(x)\psi(x,t) $$</p>
<p>where $$ \psi(x,t) = \left< x |\psi(t)\right> $$ and $$ \left< x |\hat H|\psi(t)\right> = -\frac{\hbar^2}{2m}\nabla^2\psi(x,t) - V(x)\psi(x,t) $$</p>
<p>In the case of quantum fields, the position and momentum and demoted from operator status to simply paramaters and the field operator (here for KG real field) is given by </p>
<p>$$ \hat \psi(\vec x) = \int \frac{d^3p}{(2\pi)^3} \frac{1}{\sqrt{2\omega_p}} \Bigl[\hat a(\vec p) e^{i\vec k. \vec x} + \hat a^\dagger(\vec p) e^{-i\vec k. \vec x} \Bigr] $$
in the Schrodinger picture. The states are generated from vacuum by the operator</p>
<p>$$ \hat \psi(x) \left|0\right> = \int \frac{d^3p}{(2\pi)^3}\frac{1}{2E_p}e^{-i\vec p.\vec x} \left|\vec p\right>$$
Now what exactly are these states, are they eigenstates of some operator (seems like the field operator) or quite arbitrary ? In what representation is the field operator given here ?</p>
<p>EDIT :</p>
<p>I think I understood the first part of the question, these states are the eigen-states of Hamiltonian of the field. But the question about the choice of basis of the operator still remains.</p>
| 3,822 |
<p>Why is there a size limitation on human/animal growth? Assuming the technology exists for man to grow to 200 feet high, it's pretty much a given that the stress on the skeletal structure and joints wouldn't be possible to support the mass or move...but WHY is this? if our current skeletal structures and joints can support our weight as is, wouldn't a much larger skeletal structure do the same assuming it's growing in proportion with the rest of the body? And why wouldn't a giant person be able to move like normal sized humans do? (I'm honestly thinking Ant Man, or even the non-biological sense of mechs/gundams/jaegers)...I'm just having a hard time grasping why if it were possible to grow to gigantic sizes or create giant robots, why it then wouldn't be possible for them to move. </p>
| 3,823 |
<p>I - and others - observe that when a musical performance ends, the echo of the last chord appears to rise in pitch by up to a quarter tone while the echo decays. This effect appears to be independent of the type of performance - orchestral or choral. I am tempted to assume that this is related to what is known in electro-acoustic circles as Space Echo, but I cannot see why this should occur.</p>
| 3,824 |
<p>Would the excess charge on a conductor move to surface until the electric field inside become zero if the Coulomb law was for example $\frac{1}{r^3}$? If yes, would the distribution $\sigma(x,y)$ be different from when it is $\frac{1}{r^2}$?</p>
| 3,825 |
<p>Let's say I have a law like this,
$$D=\frac{c}{r}$$
where $c$ is a constant, $r$ a distance in meter. my measures of $r$ are [$0.02m$, $0.01m$], then $<r>=0.015m$ and $\delta r = \pm 0.005m$. So now if I want to calculate $D+\delta D$ should I use $+\delta r$ or $- \delta r$ in my equation?</p>
<p>because if I use $+\delta r$ I get a smaller value than if I use $-\delta r$ since $r$ divide $c$</p>
<p>edit: in my real problem I have a lot of data, all is fine when I use the minus delta. I just want to be sure...</p>
| 3,826 |
<p>I recently read about the <a href="http://en.wikipedia.org/wiki/Kessler_syndrome" rel="nofollow">Kessler Syndrome</a> and am thinking about writing a story set in a world suffering from it. In the interest of realism, I am curious about the secondary effects which would be produced by this kind of scenario.</p>
<ul>
<li>About what distance would the debris cloud reside? How deep would it be?</li>
<li>Would the debris cloud destroy satellites? Some satellites? All satellites? Could you build a satellite which would survive it?</li>
<li>What would be the effects on the ground? Would there be frequent meteor showers? Would the meteor showers cause damage? Would the sun or stars be obscured?</li>
<li>How long would the cloud last? How could it be manually eliminated?</li>
</ul>
<p>I understand that there are a huge number of variables here, and that many of the answers depend upon how the syndrome got its start. I'm just curious about the best-case and worst-case scenarios. For the best-case scenario, imagine that the <a href="http://en.wikipedia.org/wiki/International_space_station" rel="nofollow">International Space Station</a> explodes. For worst-case, imagine that <a href="http://en.wikipedia.org/wiki/433_Eros" rel="nofollow">433 Eros</a> is in geosynchronous orbit and explodes.</p>
<p>I also understand that this is a subjective question. I am trying to make it a <a href="http://blog.stackoverflow.com/2010/09/good-subjective-bad-subjective/" rel="nofollow">good subjective question</a>.</p>
| 3,827 |
<p>The <a href="http://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation">Schrödinger equation</a> describes the quantum mechanics of a single massive non-relativistic particle. The <a href="http://en.wikipedia.org/wiki/Dirac_equation">Dirac equation</a> governs a single massive relativistic spin-½ particle. The photon is a massless, relativistic spin-1 particle.</p>
<p>What is the equivalent equation giving the quantum mechanics of a single photon?</p>
| 3,828 |
<p>In interferometry (specifically, in the domain of <a href="http://en.wikipedia.org/wiki/Fabry_Perot" rel="nofollow">Fabry-Perot cavities</a>), the function $$f(\phi) = \frac{1}{1 + F \sin^2 \phi}$$ , which describes the shape of the resonant structure of the cavity, is often called the "Airy function" (for instance, in <a href="http://scienceworld.wolfram.com/physics/AiryFunction.html" rel="nofollow">Wolfram Mathworld</a>). However, it is obviously quite different from the <a href="http://en.wikipedia.org/wiki/Airy_function" rel="nofollow">special functions Ai(x)</a> that usually go by that name.</p>
<p>This function resembles probability density function of the <a href="http://en.wikipedia.org/wiki/Wrapped_Cauchy_distribution" rel="nofollow">wrapped Cauchy distribution</a>.</p>
<p><strong>How did it get the name "Airy function"?</strong></p>
<p>I've heard that Fabry and Perot gave it this name in one of their original papers (maybe <a href="http://hal.archives-ouvertes.fr/docs/00/23/96/30/PDF/ajp-jphystap_1892_1_313_0.pdf" rel="nofollow">this one</a>? PDF, in French, which I can't read), in honor of (the same) George Biddell Airy who had earlier considered similar interferometers. It would be great if someone could help ferret out the first reference to that function by this name.</p>
| 3,829 |
<p>Within the context of Einstein space-times, we know that the contraction of the Weyl tensor across a set of indices always vanishes, like so :</p>
<p>$$C{^{\alpha }}_{\mu \alpha \nu }=0$$</p>
<p>From a purely mathematical standpoint this should be straightforward ( but perhaps tedious ) enough to prove from the definition of the conformal tensor in terms of the Riemann tensor and its contractions. However, I am wondering what the physical and/or geometric meaning and significance - if any - of this vanishing contraction really is ? I am a very visual person and learner, so an intuitive geometric understanding of this would be very helpful to me.</p>
| 3,830 |
<p>Both energy and mass gives has gravity. If an object receives energy, it will appear heavier and space will curve slightly more around that object.</p>
<p>That energy could be potential energy, or static energy.</p>
<p>If an object A is accelerated toward a black hole B, it gains velocity and thus energy. It will probably not observe that increase in energy locally, but a distant observer C should see an increase in gravity from that object as it gains kinetic energy. Since the black hole B also is accelerated in the opposite direction, observer C should ALSO observe an increase in gravity from the black hole.</p>
<p>From this I assert that:</p>
<ol>
<li><p>A pair of distant objects should amplify each others gravity when observed by a third party.</p></li>
<li><p>Object A will see an increase in gravity from B as it comes closer to B and vice versa, meaning that Newtons formula should be off by a tiny margin.</p></li>
<li><p>Observer C would assume objects A and B are heavier than they actually are when observed from A or B. Observer C would the use "wrong" numbers to calculate other nearby objects trajectories etc.</p></li>
</ol>
<p>Would observer C also actually SEE different trajectories in assertion 3? It seems illogical that we'd see an asteroid collide into the surface of a planet, while it didn't collide from a different frame of reference due to them not observing the same increase in kinetic energy...</p>
<p>Clarification:</p>
<p>It is my understanding the gravity around an object increases as it is given more energy. The amount of kinetic energy an object has, is relative to the observer.</p>
<p>Example: Two exoplanets are moving in opposite directions. Between them they will have a lot of kinetic energy. Planet 1 then passes near a star, and its trajectory and velocity changes so that it is parallel to the trajectory and velocity of planet 2.</p>
<p>Between planets 1 and 2, there is no longer any kinetic energy.</p>
<p>Then an asteroid comes close to planet 1. From earth, we see planet 1 having a certain mass and a certain energy level - and both of those factors affect the trajectory of the asteroid. The new trajectory we observe for the asteroid means that the asteroid will collide with our moon.</p>
<p>Observed from planet 2, planet 1 does not have the same energy level. It still has the same mass. So planet 2 then sees the asteroid getting a different trajectory, and sees it actually hitting earth instead of the moon.</p>
<p>The question is: what would observers on planet 2 and earth actually see? Which photons would reach them?</p>
<p>Could it in some way be so that observers on planet 2 actually see a devastation on earth? Sounds like a paradox to me.</p>
| 112 |
<p>Does anyone know where I can find a pedagogical explanation of large-N factorization in SU(N) gauge theories or nonlinear O(N) sigma models (in the latter case the trace corresponds to a dot product).</p>
<p>The reference I am using is Polyakov's gauge fields and strings. However, I find the explanation of large-N factorization there quite opaque.</p>
<p>Thanks.</p>
| 3,831 |
<p>In the second edition of <em>Classical dynamics of particles and systems</em> by Jerry B. Marion, it is said that the van der Pol equation
$$\ddot{x}-\mu\left({x_0}^2-x^2\right)\dot{x}+{\omega_0}^2x=0$$
where $\mu$ is a positive parameter of small value, has a limit cycle with amplitude $|x_0|$ (I'm paraphrasing this from the first edition in spanish) and he even illustrates it as</p>
<p><img src="http://i.stack.imgur.com/AU9dy.png" alt="Marion van der Pol illustration"></p>
<p>In newer editions this has been removed, but they add nothing new whatsoever. From the vdP equation alone it seems that this is right, but I noticed that the limit cycle does not have this amplitude in any $\mu$ case, e.g. for $\mu=0.1$ and $x_0=1$, the plot in phases space looks like (in Mathematica)</p>
<p><img src="http://i.stack.imgur.com/4s2dl.png" alt="van der Pol with $\mu=0.1$, $x_0=1$"></p>
<p>I tried different values and the behaviour is the same, e.g. for $\mu=5$ and $x_0=9$, the plot for position in time is</p>
<p><img src="http://i.stack.imgur.com/6DDgq.png" alt="van der Pol with $\mu=5$, $x_0=9$"></p>
<p>In both cases the amplitude seems to be $2|x_0|$. I think I intuitively understand what's going on by seeing the field lines of the oscillator and how it changes behaviour (damping) when it crosses $x=x_0$, e.g. for $\mu=1$ and $x_0=2$,</p>
<p><img src="http://i.stack.imgur.com/LFXcL.png" alt="van der Pol with $\mu=1$, $x_0=2$"></p>
<p><strong>But how can I find the <em>real</em> amplitude of the limit cycle and why is it not simply $|x_0|$ as the van der Pol equation suggest?</strong></p>
| 3,832 |
<p>Assuming we have a plane with a propeller driven by an engine like <a href="http://selair.selkirk.bc.ca/training/aerodynamics/graphics/piston-turn-propeller.gif" rel="nofollow">this one</a> .
How would the torque reaction be generated if the piston is aligned with the plane center of mass? What I mean is, where are the reaction forces located, in which part of the engine?</p>
| 3,833 |
<p>This question seems a lot like one of those "Phylosoraptor" memes all over the Internet, and it might be very silly, but I've been thinking about this for a while.</p>
<p><strong>Is it possible that space is not actually expanding but rather, the speed of light is decreasing throughout the entire Universe?</strong> ...as if light were traveling through a medium that would be changing its properties over time.</p>
<p>I guess there are several phenomenons that can be explained just as well by assuming that either space is expanding, either light is slowing down, but I do not know enough physics (and phenomenons that could contradict such a claim) in order to rule out this possibility.</p>
| 113 |
<p>Let </p>
<p>$$\mathbf{H} = H_x \mathbf{u}_x + H_y \mathbf{u}_y + H_z \mathbf{u}_z$$</p>
<p>be a vector field whose components are defined with respect to the unit vectors $\mathbf{u}_x$, $\mathbf{u}_y$ and $\mathbf{u}_z$, so in the $(x,y,z)$ system of coordinates.</p>
<p><strong>Question 1</strong>: If we computed its curl in a <em>new</em> system of coordinates, that is $(x' = x, y' = y, z' = -z)$, how would we do it?</p>
<p>$$\nabla \times \mathbf{H} = \mathrm{det}
\begin{vmatrix}
\mathbf{u}_{x'} & \mathbf{u}_{y'} & \mathbf{u}_{z'}\\
H_x? & H_y? & H_z?\\
\displaystyle \frac{\partial}{\partial x'} & \frac{\partial}{\partial y'} & \frac{\partial}{\partial z'}
\end{vmatrix}
$$</p>
<p>In other words, which quantities should we put in the second line? $H_x$, $H_y$ and $H_z$ are along $\mathbf{u}_x$, $\mathbf{u}_y$ and $\mathbf{u}_z$ and <em>not</em> $\mathbf{u}_{x'}$, $\mathbf{u}_{y'}$ and $\mathbf{u}_{z'}$.</p>
<p><strong>Question 2</strong>: And if we would like to keep $H_x$, $H_y$ and $H_z$ in the second line, which would be their meaning in this computation?</p>
<p>These questions are strictly relative (but not equal) to my <a href="http://physics.stackexchange.com/questions/113373/obtain-the-same-maxwells-equation-after-a-change-of-coordinates">previous</a> one.</p>
| 3,834 |
<p>I have came to understand that humid air will help prevent electrostatic forces that can propel dust and cause it to cling to surfaces.</p>
<p>My first question: is this above statement true?</p>
<p>If the answer to the first question is true, then please continue: I have attempted to find designs for the various types of humidifiers but have not been successful. After examining a simple facial steamer and using some thought experiments, I have a crude design to propose, but am open to any other proposed designs reasonable for DIY construction. I understand there are many method to vaporize water.</p>
<p>My current design is to have a stainless steel, rectangular tank with all sides enclosed and two holes drilled in the top. One hole will be used for inbound air while the other hole will be connected to a fan which will pull humid air out. On the bottom of the tank will be attached a flexible silicone heater that will heat the water to X temperature and enhance evaporation.</p>
<p>My second question: is this design effective?</p>
<p>My third question: given the simplicity of the proposed design, are any other designs better for DIY?</p>
| 3,835 |
<p>In a question: </p>
<blockquote>
<p>To construct a solenoid, you wrap insulated wire uniformly around a plastic tube 12cm in diameter and 50cm in length. You would like a 2.2 A current to produce a 2.6 kG magnetic field inside your solenoid.</p>
<p>What is the total length of wire you will need to meet these specifications?</p>
</blockquote>
<p>In the provided answer: </p>
<p>$$NC = nL C = (\frac{B}{\mu_0 I}) L (\pi d)$$</p>
<p>Where: </p>
<ul>
<li>$N$ is total number of turns</li>
<li>$n$ is number of loops</li>
<li>$C$ is circumference</li>
</ul>
<p>But I dont understand why is there an $L$</p>
<hr>
<p>I did it by: </p>
<p>$$B = \mu_0 n I$$</p>
<p>Then length of wire needed is just $nC$ whats wrong? </p>
| 3,836 |
<p>my question is about how photons travel from a light source and hit an object.</p>
<p>When you look at an object being hit by light the whole surface becomes brighter.</p>
<p>What i'm trying to understand is why the entire surface lights up.</p>
<p>When the photons leave the light source they scatter out to hit objects, this makes me logicically imaging that a surface will light up based on where those light packets hit.</p>
<p>And not all parts of a surface will light up because the surface area of the light source is too small to match up with each corresponding part of a surface. </p>
<p>Here is a quick text example of what im saying.</p>
<p>LightSource = *</p>
<p>Photons = / or | or \</p>
<p>Surface = _</p>
<p>Expected bright surfaces = %</p>
<pre><code> *
/ | \
_________
% % %
</code></pre>
<p>So because the light source's surface area was only big enough able to put out 3 streams of light i expect there to be areas not being hit by light.</p>
<p>So what i want someone to explain to me is why in real life, the entire surface lights up.</p>
<p>Sorry for such a stupid question, i usually keep these kinds of questions to myself but i decided to reach out.</p>
| 3,837 |
<p>The heuristic argument for Hawking Radiation is, that a virtual pair-production happens just at the event horizon. One particle goes into the black hole, while the other can be observed as radiation.</p>
<p>I never quite understood this explanation. Say, a virtual pair is created. From the point of view of the "radiation particle" the other particle will need an infinite amount of time in order to reach the event horizon. But the virtual pair production violates energy conservation, so the particles can only exist for a finite amount of time (Heisenberg uncertainty principle). Then, they must annihilate each other again, without emitting any radiation.</p>
<p>I often heard the "explanation" that one particle "tunnels" through the event horizon. But isn't this again a flawed argument? Tunnelling is an effect known from flat spacetime and it makes particles cross potentials that they could not cross classically. How could this help to "jump over" an infinite time interval? For me it seems like superimposing ideas from flat spacetime on curved spacetime without any justification.</p>
<p>While I understand, that the original derivation from Hawking follows a different argument, one should, in principle, be able to attach his results to a physical process. I worked through his derivation and while I must admit that I didn't understand every step, it seems to me that a lot of assumptions have to be made along the way in order to produce the results. Also the whole derivation is based on Quantum Field Theory in curved spacetime, which stands on shaky grounds, since nobody can say, where it is valid and where not - we don't have a complete theory of Quantum Gravity and we simply can't say in which situations QFT in curved spacetime is a good approximation.</p>
<p>As you can see, I am really confused about this issue. Every sort of help is appreciated!</p>
| 3,838 |
<p>I have two 4,000V, 2.5mA, DC power supplies and am attempting to use them in such a way to cause a 6x4x1-inch ABS plastic object to repel dust from the ambient air and prevent this dust from settling on the object.</p>
<p>I understand that two points that have opposite electrical charges will attract each other and that two points that have like electrical charges will repel each other. Using one or both of my power supplies and the plastic object described above, I am looking to somehow charge any potential dust particles that are in a small room and apply the opposite charge to the plastic object.</p>
<p>First Question:
Assuming all potential dust particles in the small room can be charged and the plastic object can be given an opposite charge, it should repel the dust particles and prevent them from settling on it. Is this correct?</p>
<p>Second Question:
How can I apply a charge to all potential dust particles in the small room?</p>
<p>Third Question:
Is it possible to apply a charge to a plastic object of this size? Does the ABS plastic present too high resistance for an object this size?</p>
<p>Fourth Question:
If the answer to the third question is yes, what ways can be used to enhance the conductivity of the plastic part. Perhaps a process such as dipping the plastic part in a liquid suspension of copper powder. Any other ideas?</p>
| 3,839 |
<p>In AdS/CFT a charged Black hole is probably someway equivalent to introducing a chemical potential (<a href="http://physics.stackexchange.com/questions/7470/chemical-potential">Chemical potential</a>) at the boundary theory. Is there a quick way to see how it is or how does this correspondence work? </p>
<p>I am sure it's well argued, but I haven't found a satisfactory explanation in any paper. Specially I will like to understand by connecting it with the thermodynamic case. In addition, it will be very helpful, if you can suggest me any introductory references or something with a nice explanation. Thanks!</p>
| 3,840 |
<p>The hall conductivity $\sigma_{xy}$ seems to reflect to some extent the response of a system in direction $\hat{y}$ to certain perturbation (electric field for example) restricted in $\hat{x}$ direction. </p>
<p>My question is, does a nonzero $\sigma_{xy}$ imply anything about the the physics of edge response, i.e. if given a half infinite system with an edge at $x=0$, what would be the effect on y direction? Would there be a current in y direction along the edge?</p>
<p>Thanks!</p>
| 3,841 |
<p>A small ball of radius $r$ performes small oscillations within a hollow cylinder of radius $R$. What would be the angular frequency of the oscillations given that the rolling is without slipping?
The angle between the radius connecting the center of the hollow cylinder to the ground (my $y$ axis) and the line connecting that center to the point of contact between the ball and the cylinder is $\phi$. The positive $x$ axis is to the right. The position of the ball may be written thus:
$$y = R - (R-r)\cos\phi \\ x = (R-r)\sin\phi$$
The ball has translational as well as rotational kinetic energy, hence, the total kinetic energy, $T$, should be:
$$T=\frac12m(\dot{x}^2 + \dot{y}^2) + \frac12I\dot{\phi}^2$$
where $I$ is the moment of inertia of the small ball. Hence, $T$ is equal to $$T=\frac12m[(R-r)^2\dot{\phi}^2] + \frac12I\dot{\phi}^2$$
The potential energy, $V$, should be $mgy$. </p>
<p>Now, the Lagrangian, $L$, should be $T-V$, hence
$$L=\frac12m[(R-r)^2\dot{\phi}^2] + \frac12I\dot{\phi}^2 - \frac12mg(R-r)\phi^2$$
(under small oscillations approximation). Now, from the Lagrangian, using the Euler-Lagrange formalism, the angular frequency could be easily determined. Is that correct? I believe it isn't yet am not really sure why. I'd be grateful for some comments on this solution.</p>
| 3,842 |
<p>This question is metallurgical engineering, but I had a similar doubt regarding density of liquids and what causing it.</p>
<p>Forged parts refines defects, dislocations will be moved strengthening the metal. But will the density of forged metal change?</p>
<p>My earlier question was, what causes liquids to have different densities?</p>
| 3,843 |
<p>What unexplored areas (known unknowns) are there in atomic layer deposition (ALD)? What unexplored applications of ALD are there? It seems like people use it a lot for coatings of either insulators or sometimes transparent conducting contacts. What else could you do with it?</p>
| 3,844 |
<p>When I was in some physics -lesson, probably something to do with Quantum Physics -- the teacher said that certain Maxwell equations would change if the Higg's boson is found. It is also possible that I have mixed something, he may have meant magnetic monotones. Anyway does there exist any change to Maxwell equations after the Higg's boson?</p>
| 3,845 |
<p>Could anybody suggest a reference for the present stellar models? <em>In particular</em>, I would appreciate references containing the core temperatures and pressures of neutron stars...</p>
| 3,846 |
<p>Victor Stenger argues that the apparent randomness in quantum mechanics is a result of the randomness in the macroscopic detectors (similar to the randomness in the laws of thermodynamics) and is not something that is inherent to quantum mechanics (this is my interpretation of Stenger). This would imply that QM is ultimately deterministic. See <a href="http://www.colorado.edu/philosophy/vstenger/Timeless/08-timeless.pdf">http://www.colorado.edu/philosophy/vstenger/Timeless/08-timeless.pdf</a></p>
<p>If this is true, then would Shor's factorization algorithm still work for large integers? More generally, how would scalable quantum computing be affected?</p>
| 3,847 |
<p>I am reading Zee's <em>Quantum field theory in a nutshell.</em> On time reversal he has</p>
<blockquote>
<p>Consider the transformation $t\rightarrow t'= -t$. We want to find $\Psi'(t')$ such that $i(\partial/\partial t′)\Psi′( t′) = H\Psi'(t′)$. Write $\Psi′(t′) = T\Psi(t)$, where $T$ is some operator to be determined (up to some arbitrary phase factor $\eta$). Plugging in, we have $i[ \partial/ \partial( − t)] T\Psi( t) = HT\Psi( t)$. Multiply by $T^{-1}$, and we obtain $T^{ − 1}( − i) T( \partial/ \partial t)\Psi( t) = T^{−1}HT( t)\Psi(t)$.</p>
</blockquote>
<p>My question is: Has he assumed that $T$ and $\partial/\partial t$ commute and if so why is it valid to do that? </p>
| 3,848 |
<p>Say a canon is where the circle is, and it shoots two different canonballs at different angles, but at the same speed, which angle would make the cannonball hit the ground first?</p>
<p><img src="http://i.stack.imgur.com/6LZic.png" alt="enter image description here"></p>
<p>Intuitively I'd think they'd hit at the same time, however, I remember a formula describing the second coordinate $y$: $y = - 1/2 \cdot g \cdot t^2 + v_{0y} \cdot t$</p>
<p>Where $t$ is the time, and $v_{0y}$ is the starting velocity. Thus a high $v_{0y}$, would lead to a greater total time $t$, thus it would hit A first. This seems to be missing something though.</p>
| 3,849 |
<p>Sorry if this is a silly question (engineer here), but I was wondering if the math in particle physics assumes that unitarity applies even between measurements. In other words, I take it that the evolution of quantum states is governed by an operator that ensures the probability of all possible events adds up to 1 <em>at all times</em>. What I'm wondering -- is there an operator for which this does not apply <em>at all times</em> but still gives measurement results with probability between 0 and 1?</p>
| 3,850 |
<p>Two clocks are located at either end of a two light-hour long pole and motionless relative to the pole. Each clock transmits its time and notes that the other clock shows a reading two hours behind its own. That is, the clocks can be considered synchronised with each other.
There is a flashbulb at the midpoint of both clocks. It goes off, and when each clock sees it (one hour later), it starts accelerating toward the other and each at the same rate (applying the same amount of thrust for the same local time). They do this for a short time until reaching a steady speed of 0.4c, relative to the flashbulb.
Now according to the most recent generation of relativity experts, each clock should observe the other running more slowly. This needs to be the case because time dilation is based on the square of velocity so direction of travel is unimportant.<br>
As they approach each other, the observed time difference will reduce because it takes less time for the transmitted signal to arrive. But the observed clock rate (after adjusting for doppler shift) will be slower at all times. By logical extension then, when they finally pass alongside or stop adjacent to each other, each clock should observe the timestamp of the other clock to be less than its own.
Now obviously that outcome can’t be acceptable. Therefore we must conclude that any time dilation observed during their passage is nothing other than an illusion, and certainly not ‘real’ according to any experimental measurement, since only the final side-by-side comparison counts. It goes without saying then that if the clocks were instead moving apart from each other then any observed time dilation must also be an illusion.</p>
<p>Am I correct?</p>
| 3,851 |
<p>Consider a uniform disk rolling without slipping with a certain constant angular velocity.Firstly it is moving in sufficiently rough surface.What will happen if it crosses the rough surface and just enters the smooth frictionless surface in its way?Will it be in the state of pure rotation or attains translatory motion or remains in pure rolling state.Please explain.</p>
| 3,852 |
<p>As I understand, the CIExy graphic maps "greenness", or rather middle-wavelengthness, to the Y axis and "redness", or rather long-wavelengtsness, to the X axis. The trapping used to reduce the 3d gamut of the human eye to 2d is that the gamut is specified for a certain constant luminosity, and long-wavelength + middle-wavelength + short-wavelength equals the said luminosity, thus the third coordinate can be extrapolated from the first two.</p>
<p>An example of this plot can be seen <a href="http://en.wikipedia.org/wiki/File%3aCIExy1931_srgb_gamut.png" rel="nofollow">here, on wikipedia</a>.</p>
<p>The curved edge of the human vision gamut gives the single wavelength spectrum. If this is the case, how come the RGB gamut (triangular) does not intersect the curved edge of the human vision gamut? The light coming off the 3 LED categories in the display should be of a nigh-single wavelength (as LEDs have a single, very sharp emission peak). </p>
| 3,853 |
<p>What is the difference between <a href="http://en.wikipedia.org/wiki/Stress_%28mechanics%29" rel="nofollow">stress</a> and <a href="http://en.wikipedia.org/wiki/Pressure" rel="nofollow">pressure</a>? Are there any intuitive examples that explain the difference between the two? How about an example of when pressure and stress are not equal?</p>
| 3,854 |
<p>That's all there is to it. Well... I was daydreaming about making gloves that had the ability to shock people when you punched them. I thought how it would be better if they didn't need a battery and I thought I wonder if you could put two repelling magnets on the knuckle that when forced to come together (by a punch for example) would generate the electricity that shocked your opponent. Problem is I don't know if that's even possible and Google didn't seem to be able to answer my question. Also, not really capable of making these gloves but I don't like daydreaming about fantastical situations that don't make sense.</p>
| 3,855 |
<p>The <a href="http://hyperphysics.phy-astr.gsu.edu/hbase/uncer.html" rel="nofollow">Uncertainty Principle</a> is a relationship between measurements of pairs of attributes, position and momentum, as well as energy and time. Perfect precision of one attribute's measurement leads to a full lack of knowledge about the other attribute (most likely, because there is no solid quantity for the other attribute). This appears to be a fundamental attribute of the world at large. <a href="http://en.wikipedia.org/wiki/De_Broglie%E2%80%93Bohm_theory" rel="nofollow">Bohmian mechanics</a> posits hidden variables, and makes the claim that particles (ordinary objects) have attributes at all times.</p>
<p><a href="http://plato.stanford.edu/entries/russell/" rel="nofollow">Bertrand Russell</a> developed an argument against <a href="http://plato.stanford.edu/entries/paradox-zeno/" rel="nofollow">Zeno's paradoxes</a> which he called the "at-at theory of motion."</p>
<blockquote>
<p>Bertrand Russell offered what is known as the "at-at theory of motion". It agrees that there can be no motion "during" a durationless instant, and contends that all that is required for motion is that the arrow be at one point at one time, at another point another time, and at appropriate points between those two points for intervening times. In this view motion is a function of position with respect to time.</p>
</blockquote>
<p>In order to measure anything's position, you would need to have a single precise instant of measurement. The same would apply to energy, which comes in discrete units. To measure momentum (mass times velocity) with the at-at theory, you need to take at least two measurements. To take a time measurement, such as comparing elapsed time as measured by two different timepieces, you would also necessarily need at least two measurements.</p>
<p>The Uncertainty Principle gives a precise mathematical formulation of an apparent fact of nature, which is unlikely to be something that a <a href="http://plato.stanford.edu/entries/thought-experiment/" rel="nofollow">thought experiment</a> could derive. However, the nature of the relationships appear to follow as natural consequences of Russell's response to Zeno. (And if space is quantized, it may be a fact of nature.)</p>
<p>Does the Uncertainty Principle actually follow from Bohmian mechanics and the at-at theory of motion? If not, what am I missing?</p>
| 3,856 |
<p>Special Relativity (<a href="http://en.wikipedia.org/wiki/Special_relativity" rel="nofollow">SR</a>) paradoxes are old-hat. But as I read explanations, they tend to resolve issues of simultaneity by applying the appropriate math... but that seems to me to be proving the theory by positing the theory. Just because the math is valid doesn't mean an argument is sound. All I have to do is believe warping space and time against curves and yes... it all works out.</p>
<p>For instance... A space ship is traveling toward an asteroid and away from a stationary body at 60 percent the speed of light while at the same time an asteroid is traveling directly toward the space ship and toward the stationary body at 60 percent the speed of light.</p>
<p>So the typical question is, "when does the space ship meet its demise?... distance / (1.2*c)?" <em>My question is distinctly different.</em> Why is it we are confident that the conclusion matches reality... i.e. how do we know the space ship doesn't just blow up before it thought it should based on only the perception and measurements available to it which seem to be governed by SR?</p>
| 3,857 |
<p>Why did scientists study black body radiations from something as complicated as a hollow container rather than the radiation from something simple like a thin solid cylinder?</p>
| 3,858 |
<p>Please give your explanation in layman terms, without throwing any complex equations at me.</p>
<p>We have a 5000 liter tank dug in ground and a water pump connected to this tank that draws water from it and fills a overhead tank. We also have a tap at the opening of the pump. (A T joint is connected to the output of the pump, one goes to overhead tank and the other has a tap attached to it). </p>
<p>Now, people draw water from this tap without turning on the pump (We try a lot to prevent this, yet people stubbornly do it). Nowadays when we turn on the pump, unless we pour water into the tap (About a liter of water) after turning on the pump we are not getting water output from the pump. A plumber suggested to us that it is because of Air lock and said to turn the tap upside down, like its a fountain :| Because of some physical constraints we cannot do this and it will be freaking expensive to do it.</p>
<p>I am somehow not convinced that it is a solution to the problem. Could someone tell me what is happening here and how to solve this problem ?</p>
| 3,859 |
<p><a href="http://en.wikipedia.org/wiki/Planck_time" rel="nofollow">Planck time</a> is really a weird topic, if we try to find out that is it time or time interval.
It is the time taken by light to travel a Planck LENGTH so it must be time interval. But we also know that Planck length is MOST FUNDAMENTAL and it can't be differentiated anymore. So what's the reality?</p>
| 114 |
<p>I've been reading about <a href="http://en.wikipedia.org/wiki/Scalar-tensor_theory" rel="nofollow">scalar-tensor</a> theories of gravity, such as Brans-Dicke theory, and I started thinking about the scalar field. Now, I know that the Higgs field is a scalar field, and of course has a quanta, the Higgs boson. According to Wikipedia, the quanta of a scalar field will always be a spin-0 boson. What would the general properties of a boson arising from the scalar field in <a href="http://en.wikipedia.org/wiki/Scalar-tensor_theory" rel="nofollow">scalar-tensor theories?</a> How would it interact with other particles (if at all)?</p>
| 3,860 |
<p>I am given the following problem:</p>
<blockquote>
<p>If an airplane propeller rotates at 2000 rev/min while the airplane flies at a speed of 480 km/h relative to the ground, what is the linear speed of a point on the tip of the propeller, at radius 1.5m, as seen by (a) the pilot and (b) an observer on the ground? The plane’s velocity is parallel to the propeller’s axis of rotation. </p>
</blockquote>
<p>I was able to solve part a pretty easily by just using the formulas that relate linear speed with angular speed, but I didn’t get part b correct. I thought that the answers would be the same from both perspectives because the speed of the plane does not contribute to the rotation of the propeller, but I’m assuming that is the wrong way to think about it (because that produces the wrong answer). Can someone explain why this is the wrong approach? </p>
<p>The solution to this problem involved noting that "The plane’s velocity $v_p$ and the velocity of the tip $v_t$ (found in the plane’s frame of
reference), in any of the tip’s positions, must be perpendicular to each other." How is this relevant to the problem? </p>
| 3,861 |
<p><strong>Questions:</strong></p>
<p>Three $x$ m long rods form an equilateral triangle. Two of the rods are charged to $+q$ C and the third to $-q$ C. What is the electric field strength at the center of an equilateral triangle?</p>
<p><strong>Attempt:</strong></p>
<p>I know how to find the electric field strength due to one rod, I just divide the rod into pieces, each of length $dx$ and charge $dq$ then integrate from 0 to x.</p>
<p>After I find the electric field at the center due to each rod, how do I find the resultant electric field?</p>
| 3,862 |
<blockquote>
<p>During an interval of time, a tennis ball is moved so that the angle between the velocity and the acceleration of the ball is kept at a constant 120º. Which statement is true about the tennis ball during this interval of time?
Choose one answer.</p>
<p>a. Its speed decreases and it is changing its direction of travel.<br>
b. Its speed remains constant, but it is changing its direction of travel.<br>
c. Its speed decreases and it is not changing its direction of travel.<br>
d. Its speed increases and it is changing its direction of travel.<br>
e. Its speed remains constant and it is not changing its direction of travel</p>
</blockquote>
<p>In my mind the ball is travelling with a negative velocity, south, all in the y component. At the time it is being accelerated at 120 degrees, thus it is slowing down in the negative Y direction. "Its speed decreases and it is not changing direction of travel." That is, it will eventually change its direction of travel, but just because it is accelerating in the opposite direction of the current vector does not mean that it has changed direction; yet. The answer is given as "a." But couldn't "a." or "c." be true? </p>
| 3,863 |
<p>Is it valid to approximate the function $$ Z(t)=\sum_{n}e^{-tE_{n}} ,\ t\ge 0$$ </p>
<p>by the integral over phase space: $$ \frac{ 1}{2\sqrt \pi}\int_{0}^{\infty}dxe^{-tV(x)}?$$</p>
<p>For example, in order to evaluate the zeta function over eigenvalues</p>
<p>$ \Gamma (s) \zeta _{H} (s,q) = \int_{0}^{\infty}dtZ(t)\exp(-qt)Z(t)$</p>
<p>with 'q' a positive number, and $ \zeta _{H} (s,q)= \sum_{n}(q+E_{n})^{-s} $</p>
<p>Is this approximation faithful and reliable?</p>
| 3,864 |
<blockquote>
<p>An object is thrown horizontally with a velocity of 30 m/s from the top of a tower. It undergoes a constant downward acceleration of 10 m/s2. The magnitude of its instantaneous velocity after 4.0 sec, in meters per second, is:</p>
</blockquote>
<p>To approach this question I first thought to myself that the velocity in the y-component after 4s is going to equal 10+2(10)+3(10)+4(10); 100m/s. The x velocity will remain constant. Thus the velocity at t=4 would be the resultant vector of 100m/s in the y and 30m/s in the x, which equals 104.4m/s. I am wondering where I am going wrong with my reasoning here? </p>
| 3,865 |
<p>Basically I have a set of vectors of unit length, $\{\nu_i\}$, describing the movement of phonons (all orthogonal to each other), $\{\omega_i\}$. Lets say I only have two atoms, $m_1$ and $m_2$. In this case with only two atoms I have $N=3\times2=6$ modes. In the following it really doesn't matter how many modes I have.</p>
<p>The displacement of a single atomic mode would then be $\nu_i\times \sqrt{\frac{\hbar}{m_1\omega}}=r$, from the harmonic oscillators characteristic length, $l_i=\sqrt{\frac{\hbar}{m_i\omega}}$, see <a href="http://en.wikiversity.org/wiki/Quantum_harmonic_oscillator" rel="nofollow">Quantum Harmonic oscillator</a>, if it were a single mode.</p>
<p>Thus I have a matrix of orthogonal mode vectors, $V=\{\nu_i\times l_i\}$, which can be used to create a displacement in cartesian coordinates, $R=\{r_i\}$ by knowing the vector of phonon displacements, $U=\{u_i\}$. This means that solving the linear equation:
\begin{equation}
V\cdot U=R
\end{equation}
I can find the displaced coordinates by knowing the phonon displacement $U$. </p>
<p>For same mass $m_1=m_2$, I have suspected that I can use the reduced mass for both atoms, thus yielding $l_i=\sqrt{\frac{\hbar}{m_i\omega_i/\sqrt2}}$.<br>
But what about $m_1\neq m_2$? I suspected I could use the reduced mass again:
\begin{equation}
\mu=\frac{m_1m_2}{m_1+m_2},
\end{equation}
however, how to apply it for two different mass objects? </p>
<p>If you have a general application $N$ particles I would be much obliged.</p>
| 3,866 |
<p>Following up on <a href="http://physics.stackexchange.com/questions/65838/lorentz-transformation-of-single-particle-states">this question</a>: Weinberg says </p>
<blockquote>
<p>In general, it may be possible by using suitable linear combinations of the $\psi_{p,\sigma}$ to choose the $\sigma$ labels in such a way that $C_{\sigma'\sigma}(\Lambda, p)$ is block-diagonal; in other words, so that the $\psi_{p,\sigma}$ with $\sigma$ within any one block by themselves furnish a representation of the <b>inhomogenous</b> Lorentz group.</p>
</blockquote>
<p>But why inhomogeneous Lorentz group if, in the first place, we performed a homogeneous Lorentz transformation on the states, via $U(\Lambda)$? I also want to be clear what is meant by the states "furnishing" a representation.</p>
<p>Regarding the above confusion, the same scenario again shows up during the discussion on the little group. Here's a little background: $k$ is a "standard" 4-momentum, so that we can express any arbitrary 4-momentum $p$ as $p^{\mu} = L^{\mu}_{\nu}(p) k^{\nu}$, where $L$ is a Lorentz transformation dependent on $p$. We consider the subgroup of Lorentz transformations $W$ that leave $k$ invariant (little group), and find that:</p>
<p>$U(W)\psi_{k \sigma} = \sum_{\sigma'} D_{\sigma' \sigma}(W)\psi_{k \sigma'}$. Then he says:</p>
<blockquote>
<p>The <b>coefficients $D(W)$</b> furnish a representation of the little group; i.e., for any elements $W$ and $W'$ , we get $D_{\sigma' \sigma}(W'W) = \sum_{\sigma''}D_{\sigma' \sigma''}(W)D_{\sigma''\sigma}(W')$.</p>
</blockquote>
<p>So is it that even in the first part about the Lorentz group, $C$ matrices furnish the representation and not $\psi$?</p>
<p>Also, for the very simplified case if $C_{\sigma'\sigma}(\Lambda, p)$ is completely diagonal, would I be correct in saying the following in such a case, for any $\sigma$?</p>
<p>$$U(\Lambda)\psi_{p,\sigma} = k_{\sigma}(\Lambda, p)\psi_{\Lambda p, \sigma}$$</p>
<p>Only in this case it is clear to me that $U(\Lambda)$ forms a representation of Lorentz group, since $\psi_{p,\sigma}$ are mapped to $\psi_{\Lambda p, \sigma}$.</p>
| 3,867 |
<p>As in <a href="http://physics.stackexchange.com/questions/65834/why-are-non-momentum-dofs-of-single-particle-states-discretely-labeled">this question</a>, let $\psi_{p,\sigma}$ be a single-particle 4-momentum eigenstate, with $\sigma$ being a discrete label of other degrees of freedom.</p>
<p>Weinberg discusses the effect of a homogenous Lorentz transformation $U(\Lambda, 0)$ or $U(\Lambda)$ on these states, and concludes that $U(\Lambda)\psi_{p,\sigma}$ is a linear combination of $\psi_{\Lambda p,\sigma'}$.</p>
<p>$$U(\Lambda)\psi_{p,\sigma} = \sum_{\sigma'} C_{\sigma'\sigma}(\Lambda, p)\psi_{\Lambda p,\sigma'}$$</p>
<p>Again, is there any physical information that we can extract from this? (I realize that $\psi_{\Lambda p,\sigma'}$ represent physical states after Lorentz transformation).</p>
| 3,868 |
<p>In evaluating the vacuum structure of quantum field theories you need to find the minima of the effective potential including perturbative and nonperturbative corrections where possible.</p>
<p>In supersymmetric theories, you often see the claim that the Kähler potential is the suitable quantity of interest (as the superpotential does not receive quantum corrections).
For simplicity, let's consider just the case of a single chiral superfield:
$\Phi(x,\theta)=\phi(x)+\theta^\alpha\psi_\alpha(x) + \theta^2 f(x)$
and its complex conjugate. The low-energy action functional that includes the Kähler and superpotential is
$$
S[\bar\Phi,\Phi] = \int\!\!\!\mathrm{d}^8z\;K(\bar\Phi,\Phi)
+ \int\!\!\!\mathrm{d}^6z\;W(\Phi) + \int\!\!\!\mathrm{d}^6\bar{z}\;\bar{W}(\bar\Phi)
$$
Keeping only the scalar fields and no spacetime derivatives, the components are
$$\begin{align}
S[\bar\Phi,\Phi]\big|_{\text{eff.pot.}} = &\int\!\!\!\mathrm{d}^4x\Big(\bar{f}f\,\frac{\partial^2K(\bar\phi,\phi)}{\partial\phi\partial{\bar\phi}} + f\,W'(\phi) + \bar{f}\, W(\phi)\Big) \\
\xrightarrow{f\to f(\phi)}
-\!&\int\!\!\!\mathrm{d}^4x\Big(\frac{\partial^2K(\bar\phi,\phi)}{\partial\phi\partial{\bar\phi}}\Big)^{-1}|W'(\phi)|^2
=: -\!\int\!\!\!\mathrm{d}^4x \ V(\bar\phi,\phi)
\end{align}$$
where in the second line we solve the (simple) equations of motion for the auxiliary field.
The vacua are then the minuma of the effective potential $V(\bar\phi,\phi)$.</p>
<p><strong>However</strong>, if you read the old (<a href="http://inspirehep.net/record/5138">up</a> <a href="http://inspirehep.net/record/179081">to</a> <a href="http://inspirehep.net/record/177500">mid</a> <a href="http://inspirehep.net/record/182277">80</a><a href="http://inspirehep.net/record/189554">s</a>) literature on supersymmetry they calculate the effective potential using all of the scalars in the theory, i.e. the Coleman-Weinberg type effective potential using the background/external fields $\Phi(x,\theta)=\phi(x) + \theta^2 f(x)$. This leads to an effective potential
$U(\bar\phi,\phi,\bar{f},f)$ which is more than quadratic in the auxiliary fields, so clearly not equivalent to calculating just the Kähler potential. The equivalent superfield object is the <em>Kähler potential + auxiliary fields' potential</em>, as defined in "<a href="http://inspirehep.net/record/376182">Supersymmetric effective potential: Superfield approach</a>" (or <a href="http://inspirehep.net/record/485478">here</a>). It can be written as
$$
S[\bar\Phi,\Phi] = \int\!\!\!\mathrm{d}^8z\;\big(K(\bar\Phi,\Phi) + F(\bar\Phi,\Phi,D^2\Phi,\bar{D}^2\bar{\Phi})\big)
+ \int\!\!\!\mathrm{d}^6z\;W(\Phi) + \int\!\!\!\mathrm{d}^6\bar{z}\;\bar{W}(\bar\Phi)
$$
where
$F(\bar\Phi,\Phi,D^2\Phi,\bar{D}^2\bar{\Phi})$ is at least cubic in $D^2\Phi,\bar{D}^2\bar{\Phi}$.
The projection to low-energy scalar components of the above gives the effective potential $U(\bar\phi,\phi,\bar{f},f)$ that is in general non-polynomial in the auxiliary fields and so clearly harder to calculate and work with than the quadratic result given above.</p>
<hr>
<p><strong>So my question is</strong>: when did this shift to calculating only the Kähler potential happen and is there a good reason you can ignore the corrections of higher order in the auxiliary fields?</p>
| 3,869 |
<p>In a critical theory with dynamical critical exponent $z \neq 1 $, which amongst frequency, $\omega$, and dispersion, $E(\vec{k})$, may be referred to as ''energy''? I'm confused about this since in general $\omega$ and $E(\vec{k})$ can have different scaling dimensions. Some clarification would be very appreciated.</p>
| 3,870 |
<p>My question is in reference to the action in equation 4.130 of Becker, Becker and Schwartz.
It reads as, </p>
<p>$S_{matter}= \frac{1}{2\pi}\int (2\partial X^\mu \bar{\partial}X_\mu + \frac{1}{2}\psi^\mu \bar{\partial} \psi_\mu + \frac{1}{2}\tilde{\psi}^\mu \partial \tilde{\psi}_\mu)d^2z$</p>
<ul>
<li><p>Its not clear to me as to why this should be the same as the Gervais-Sakita (GS) action as it seems to be claimed to be. Firstly what is the definition of $\tilde{\psi}$? (..no where before in that book do I see that to have been defined..) Their comment just below the action is that this is related to the $\psi_+$ and $\psi_-$ defined earlier but then it doesn't reduce to the GS action. </p></li>
<li><p>What is the definition of the "bosonic energy momentum tensor" ($T_B(z)$) and the "fermionic energy momentum tensor" ($T_F(z)$)? I don't see that defined earlier in that book either. </p></li>
</ul>
<p>I am not able to derive from the above action the following claimed expressions for the tensors as in equation 4.131 and 4.133,</p>
<p>$T_B(z) = -2\partial X^\mu(z)\partial X_\mu (z) - \frac{1}{2}\psi^\mu(z)\partial \psi _\mu (z) = \sum _{n=-\infty} ^{\infty} \frac{L_n}{z^{n+2}}$</p>
<p>and </p>
<p>$T_F(z) = 2i\psi^{\mu} (z) \partial X_{\mu} (z) = \sum _{r=-\infty}^{\infty} \frac{G_r}{z^{r+\frac{3}{2}}}$ </p>
<ul>
<li><p>It would be helpful if someone can motivate the particular definition of $L_n$ and $G_r$ as above and especially as to why this $T_B(z)$ and $T_F(z)$ are said to be holomorphic when apparently in the summation expression it seems that arbitrarily large negative powers of $z$ will occur - though I guess unitarity would constraint that. </p></li>
<li><p>Why is this action called "gauge-fixed"? In what sense is it so? </p></li>
</ul>
| 3,871 |
<p>Following the treatment of Weinberg, chapter 2, we consider $\psi_{p,\sigma}$ as single-particle eigenstates of the 4-momentum. Weinberg says that $\sigma$ labels all other degrees of freedom and we take this label to be discrete for one-particle states. So what exactly is the physical implication of discrete and continuous labeling of other degrees of freedom? And why is discrete labeling physically pertinent to single-particle states?</p>
| 3,872 |
<p>I know that I can just read off the phase diagram for water (for the surface atmospheric pressure on each object). But could there possibly be some nuances that someone might miss just from viewing the diagram?</p>
| 3,873 |
<p>Will the <a href="http://www.jwst.nasa.gov/" rel="nofollow">James Webb Space Telescope</a> (JWST) be able to capture, or ever be used for, anything similar to the <a href="http://en.wikipedia.org/wiki/Hubble_Ultra-Deep_Field" rel="nofollow">Hubble Ultra Deep Field</a> (HUDF)?</p>
| 3,874 |
<p>Currently there is a <a href="http://astronomy.stackexchange.com/questions/914/what-is-an-approximation-of-the-average-number-of-supernovae-every-century-in-the">unique chance</a> for amateur astronomers to observe a very near <a href="http://en.wikipedia.org/wiki/Type_Ia_supernova">type Ia supernova</a>, named <a href="http://en.wikipedia.org/wiki/SN_2011fe">PTF 11kly</a>.</p>
<p>As <a href="http://astronomy.stackexchange.com/questions/958/what-objects-states-of-objects-with-absolute-magnitude-do-we-know-of">standard candles</a> are very important to measure distances in the universe, can you give a brief overview of how the luminosity of "PTF 11kly" will probably develop according to current star models?</p>
<p>Where do our current models lack accuracy? Is this single supernova so special, that it can fill some gaps in our knowledge?</p>
| 3,875 |
<p>What knowledge of computer science should I have, to be able to pursue research in quantum computing. I am a Physics undergrad and would take three core courses in QM, before the completion of my degree. SO I guess necessary QM would be done. What about computer science?</p>
| 3,876 |
<p>I've studied differential geometry just enough to be confident with differential forms. Now I want to see application of this formalism in thermodynamics.</p>
<p>I'm looking for a small reference, to learn familiar concepts of (equilibrium ?) thermodynamics formulated through differential forms.</p>
<p>Once again, it shouldn't be a complete book, a chapter at max, or an article.</p>
<p><strong>UPD</strong> Although I've accepted David's answer, have a look at the Nick's one and my comment on it.</p>
| 3,877 |
<p>I am rather confused because it would seem that mathematical conclusions I have drawn here goes against my physical intuition, though both aren't too reliable to begin with.</p>
<p>We have a potential step described by $$V(x)=\begin{cases}0& x\le0\\V_0 & x>0\end{cases}$$</p>
<p>and a wavefunction $\psi(x)$ that satisfies the equation $${\hbar^2\over 2m}{\partial^2 \over \partial x^2}\psi(x)+V(x)\psi(x)=E\psi(x)$$</p>
<p>I wish to find the probability of reflection. By continuity constraints at $x=0$ I have arrived at the reflection amplitude being $$R={k-q\over k+q}$$ where $k=\sqrt{2mE\over \hbar^2}$ and $q=\sqrt{2m(E-V_0)\over \hbar^2}$ then we let $V_0\to -\infty$ giving $q\to \infty$, so $R\to -1$</p>
<p>$\implies |R|^2\to 1$</p>
<p>But I would have guessed that $|R|^2$ should vanish at the limit so that the incident wave is totally transmitted!</p>
<p>Could someone please explain?</p>
| 3,878 |
<p>A billard ball is struck with a cue. The line of action of the applied impulse is horizontal and passes through the center of the ball. The initial velocity $v_0$ of the ball, its radius $R$, its mass $M$ and coefficient of friction $\mu_k$ between the ball and the table are all known. How far will the ball move before it ceases to slip on the table?</p>
| 3,879 |
<p>I'm CS major trying to learn QFT on my own . I'm trying to make an efficient study plan .The problem is that I've never read any textbook from cover to cover and solved all the problems .What of the following is the most productive approach:</p>
<p>A-To start reading textbooks from cover to cover like reading most of an electrodynamics book and solving all the problems before tackling QFT</p>
<p>B-To just start reading QM and if I encounter something that requires Magnetostatics for example I go to an electrodynamics book to understand it even though I've never dealt with Magnetostatics before? And after I learn harmonic oscillators ,I open a book on QFT and read the chapter on classical klein-gordon field ? and then possibly, open a book on solid state to understand vibrations in a solid etc. and If I don't know complex analysis ,I wait until I encounter a problem that require complex analysis (Like the propagators) to learn it?</p>
| 115 |
<p>I have a question which is inspired by considering the light field coming off an incandescent lightbulb. As a blackbody radiation field, the light is in thermal equilibrium at temperature $T$, which implies that each normal mode has a <em>mean</em> energy given by Planck's law, and a random phase. Thus, if I were to look, microscopically, at the electric field, I would see a fairly complicated random function that can only really be considered constant at timescales $\tau\ll \hbar/k_B T$ (at which the corresponding modes have next to no amplitude and therefore do not affect the electric field's time dependence). </p>
<p>This illustrates a general aspect of thermal and thermodynamic equilibrium: they are only relevant concepts when the systems involved are looked at on timescales far longer than their relevant dynamics.</p>
<p>My question, then, is this: are examples of slowly-varying systems (where by "slowly" I mean on the timescales of seconds, or preferably longer) that can be considered to be in thermal equilibrium on timescales longer than that known?</p>
| 3,880 |
<p>In an oscillations exercise there is a spring attached to another spring, attached to a block.</p>
<p>Long story short: I have to find the global $k$.
In the solutions it says:
"Because the springs are massless, they act similar to a rope under tension, and the same force $F$ is exerted by each spring."</p>
<p>I don't really understand how the springs being massless, is an argument for saying that they exert the same force (being the global force) upon the block.</p>
<p>Anyone care to elaborate?</p>
| 3,881 |
<p><a href="http://www.keelynet.com/tesla/00685958.pdf" rel="nofollow">http://www.keelynet.com/tesla/00685958.pdf</a></p>
<p>An elevated, insulated copper plate, a patented Tesla Condenser, and a voltage inverter, we've got free electricity right?</p>
<p>admittedly small, but as a proportion to potential difference in the height of the mast and possibly the depth of the ground pole.</p>
<p>Anyone really know what's going on here and why it's not implemented?</p>
| 3,882 |
<p>On page 135 in Srednicki he defines the functional integral </p>
<p>$$Z(J) = \int\mathcal{D}\phi\,\exp\Big[\mathrm{i}\big(S+\int\mathrm{d^4}y \,J_a\phi_a\big)\Big], \tag{A}$$ </p>
<p>where $S$ and $J_a$ are the action and sources respectively (sum over $a$). What I don't seem to get is that when he cosiders a small variation $\delta Z$ he get the variation of the action inside an integral as follows (I get it without the integral):</p>
<p>$$0=\delta Z(J) = \mathrm{i}Z(J) \times \Bigg[\int \mathrm{d^4}x\Big(\,\frac{\delta S}{\delta \phi_a(x)}+J_a(x)\Big)\delta \phi_a(x)\Bigg]\tag{B}$$</p>
<p>My attempt: </p>
<p>$$0=\delta Z(J) = \frac{\delta Z}{\delta\phi_b(x)}\delta\phi_b(x)\\[10mm]=\int \mathcal{D}\phi\,\delta\phi_b(x)\Bigg[\frac{\delta}{\delta\phi_b(x)}\mathrm{e}^{\mathrm{i}(S+\int\mathrm{d^4}y\,J_a(y)\phi_a(y))}\Bigg].\tag{C}$$</p>
<p>The box becomes: </p>
<p>$$\Bigg[\frac{\delta}{\delta\phi_b(x)}\mathrm{e}^{\mathrm{i}(S+\int\mathrm{d^4}y\,J_a(y)\phi_a(y))}\Bigg] = \frac{\delta}{\delta\phi_a(y)}\mathrm{e}^{\mathrm{i}(S+\int\mathrm{d^4}y\,J_a(y)\phi_a(y))}\frac{\delta\phi_a(y)}{\delta\phi_b(x)}\\[10mm]=\delta_{ab}\delta^4(x-y)\mathrm{e}^{\mathrm{i}(S+\int\mathrm{d^4}y\,J_a(y)\phi_a(y))}\times\mathrm{i}\underbrace{\frac{\delta}{\delta\phi_a(y)}\Big(S+\int\mathrm{d^4}y\,J_a(y)\phi_a(y)\Big)}_{\Lambda}. \tag{D}$$</p>
<p>Lambda becomes (?)
$$\Lambda = \frac{\delta S}{\delta \phi_a(y)}+\int\mathrm{d^4}y J_a(y). \tag{E}$$</p>
<p>What I'm I doing wrong here? </p>
| 3,883 |
<p>I just need a confirmation on a problem that I am dealing. In my problem I have a statioary kaon which spontaenously splits into two pions which are headed in different directions.</p>
<p>Is it possible to write the energy conservation like this: </p>
<p>\begin{align}
E_1 &= E_2\\
\sqrt{{E_{01}}^2 + (p_1c)^2} &= 2\sqrt{{E_{02}}^2 + (p_2c)^2}\longleftarrow \substack{\text{since $p_1 =0$}}\\
{E_{01}} &= 2\sqrt{{E_{02}}^2 + (p_2c)^2}\\
\end{align}</p>
<p>I am especialy interested in a factor of 2 before the square root.</p>
<hr>
<p><strong>EDIT:</strong></p>
<p>What about if Kaon would split into 2 diferent particles A and B? </p>
<p>\begin{align}
E_1 &= E_2\\
\sqrt{{E_{01}}^2 + (p_1c)^2} &= \sqrt{{E_{0A}}^2 + (p_Ac)^2} + \sqrt{{E_{0B}}^2 + (p_Bc)^2}
\end{align}</p>
<p>where $p_A = p_B$?</p>
| 3,884 |
<p>I am not quite understanding what needs to be done but I am assuming that I must combine the two equations and get rid of the variable $T$ in order to make this equation of $f$ below:</p>
<p>(Note:These formulas are in the regards to inclined planes)</p>
<p><strong>Equation of $f$:</strong>
$$f=g(m-M\sin(\theta))-(m+M)a$$</p>
<p><strong>Equation #1:</strong>
$$T-Mg\sin(\theta)-f=Ma$$</p>
<p><strong>Equation #2:</strong> $$mg-T=ma$$</p>
<p>Here is the work I have done so far:</p>
<p>$$T-Mg\sin(\theta)-f-Ma=mg-T-ma$$</p>
<p>(I add and subtract the variables to one side where I get)</p>
<p>$$f=2T-g(m-M\sin(\theta))-(m+M)a$$</p>
<p>The only part that is confusing me is, how do I get rid of the variable $T$ in this equation?</p>
<p>(NOTE: I am not looking for the answer but only how to deal with this type of problem so that I may know what to do in the future if this problem comes around. Also please edit this post for the sake of the quality of the content of this stack exchange site)</p>
<p>I thank everyone in advance for the contribution to answering this question</p>
| 3,885 |
<p>Let us consider a routine mechanics problem.A block of $ 10kg $ rests on another block of $ 2kg $ on a frictionless table.Let $ \mu = 0.2 $ be the coefficient of friction(let's neglect the difference between coefficients of static and kinetic friction here)between the surfaces of the two blocks.
<img src="http://i.stack.imgur.com/Iwfb3.png" alt="image"></p>
<p>A horizontal force of $ 10N $ is applied on the lower block.Now,the maximum force of friction( $F_\mu $)that can be applied on either block by the other surface is equal to $$ F_\mu\,_\max = \mu R $$ where $R$ is the normal contact force between the two blocks.As the blocks don't accelerate vertically, $$ R= 10g \approx 100N $$ Therefore, $$ F_\mu\,_\max = 0.2 \times 100 = 20N $$ The newton's 2nd law for the lower block in the horizontal direction would be $$ 10 - Friction = 2kg \times a_2 $$ and that for the upper block would be $$ Friction = 10kg \times a_1 $$ Clearly,the maximum force of friction is more than the force applied($10N$).So,frictional force will ensure that none of the blocks have any acceleration.</p>
<p>But,if you look at both the blocks as a single system,the only net external force is $F=10N$ and hence the acceleration of the centre of mass of the system is $ a= \frac{10}{10+2} = \frac{5}{6} m/s^2 .$</p>
<p>Since both of the the above results contradict each other,(unless the blocks accelerate in the opposite directions with same magnitudes of acceleration,which is not the case),I assume that its fairly wrong to say that the blocks have zero individual accelerations(with respect to ground).I seriously suspect that the acceleration of either block in my equation is of the frame of reference of the other block,rather than ground;but I fail to see how.I think my question is mostly a confusion in writing newton's 2nd law for different reference frames when friction is involved.</p>
| 3,886 |
<p>I worked 20years in a very high magnetic field at an Aluminum foundry. I'm now in my 50's and I don't know or am I aware of any negative effects from this. Is there anything I should be on the look out for?
I am also trying to understand Tesla's understanding of Earth's magnetic field an tapping into that power.
Can we tap into the thermodynamics of our natural magnetic field without destroying the protective properties of it?</p>
| 116 |
<p>I don't understand how to solve this:</p>
<p>A $\pi^+$ decays into a muon and neutrino. Find the pion's energy if</p>
<ul>
<li>max $E_\nu$ / min $E_\nu$ = 100/1;</li>
<li>$m_\nu = 0$</li>
<li>$m_\pi*c^2 = 140\text{ peta-eV}$</li>
</ul>
| 3,887 |
<p>Can anyone please provide an intuitive explanation of why phase shift of 180 degrees occurs in the Electric Field of a EM wave,when reflected from an optically denser medium?</p>
<p>I tried searching for it but everywhere the result is just used.The reason behind it is never specified.</p>
| 358 |
<p>I read that there is an effort to define a kilogram in terms that can exactly be reproduced in a lab. Why has it taken so long to get this done? It seems this would be fairly important.</p>
<h1>Edit</h1>
<p>Today I got around to finding the references.<br>
You can see the international prototype of the kilogram IPK is the artifact whose mass defines at present the SI unit of mass <a href="http://www.bipm.org/en/scientific/mass/prototype.html" rel="nofollow">here</a>. You can also go <a href="http://www.bipm.org/en/si/" rel="nofollow">here</a> where you will see a link <em>on the possible future revision to the SI</em>. It sounds like this may happen in the next few years. I am not surprised that it takes years of R&D and millions of dollars to do this. Well that's peanuts compared to what was spent on LHC. I am not saying LHC was not worth the cost. It just seems we could have a modern definition of a kilogram by now if a few governments wanted to make it a priority. Does anyone here know if this effort has been given the investment that it deserves?</p>
| 184 |
<p>In one dimension -</p>
<p>How can one prove that the Hammiltonian and the parity operator commute in the case where the potential is symmetric (an even function)?</p>
<p>i.e. that [H, P] = 0 for V(x)=V(-x)</p>
| 3,888 |
<p>Say you had a very radioactive element in a confined area: could that element (hypothetically speaking) go through beta decay, then, once it has too many protons could it immediately go through electron capture to become that same radioactive element OR become another unstable element that will again go through beta decay?</p>
| 3,889 |
<p>I don't quite understand what a self-sustained discharge is. I figure it means that the processes involved are self supporting and generate themselves, so that I don't have to put energy into the system and generate an outside voltage at the electrodes. Can an experimental device ever by fully self-sustained? If yes, wouldn't that mean I can't control when it's running or now?</p>
| 3,890 |
<p>In my Computer science class I was given a problem where I have to simulate a bouncing ball using "real physics". I have been trying to find a equation that will simulate the height of the bounce given a gravity and an arbitrary mass. And I will need to calculate the next bounce and it's height. A lot of the equations I've found require a time. But I don't really care about a time, all I want is to get the next height after the previous bounce until it finally hits height of 0 or close to it. I haven't taken a physics class since high school( 5 years ago) and that was basic physics.</p>
| 3,891 |
<p>I am trying to use <a href="http://archive.eso.org/cms/tools-documentation/jsky/" rel="nofollow">JSkyCat</a> to mark a set of coordinates on a FITS image, and have found the dialog for choosing a catalog file to load. However, I am having trouble finding any documentation on what format it expects (and my guesses have been unsuccessful).</p>
| 3,892 |
<p>I know that we now have telescopes which can capture the images of the stars and galaxies millions of light-years away from us.</p>
<p>Does the telescope capture the past image of the star, i.e. the light which it emitted centuries ago? </p>
<p>What guarantee is there that the star is still alive? </p>
<p>What basis do organizations like NASA plan missions for evading such stars?</p>
| 3,893 |
<p>On Earth, North is determined by the magnetic poles of our planet. Is there such a thing as "North" in outerspace? To put it another way, is there any other way for astronauts to navigate besides starcharts? For instance, if an astronauts spaceship were to be placed somewhere (outside of our solar system) in the milkyway galaxy, would there be a way for them to orient themselves?</p>
| 3,894 |
<p>On an elliptical treadmill a regular person can easily burn 1000 calories in one hour (treadmill reports calories burnt). This translates into:
$$(1\times 10^3\mathrm{cal/hr}\times 4.2\times10^3\mathrm{J/cal})/3.6\times 10^3\mathrm{s/hr} \approx 1.2 \; \mathrm{kW} \approx 1.5 \; \mathrm{hp}$$
On the other hand, <a href="http://en.wikipedia.org/wiki/Human-powered_equipment" rel="nofollow">Wikipedia says</a> "A trained cyclist can produce about 400 watts of mechanical power for an hour or more..."
Is the problem that the treadmill gives wrong numbers? Or it is true that running using legs and arms - on elliptical machine or cross-country skiing which seems to be similar - a human can produce a lot more mechanical power than cycling? I thought the maximum power is set by the cardiovascular system, so it would be the same, running or cycling.</p>
| 3,895 |
<p>Why is it that neutrons evaporate from nuclei more easily than protons do? </p>
<p>Intuitively, since protons are electrostatically repelled (in addition to whatever nuclear forces they have in common with neutrons), one would expect protons to be ejected more readily than neutrons. (Maybe this is even what does happen for small nuclei, but apparently not large nuclei.)</p>
<p>It seems to be said in common parlance that the Coulomb force/barrier acts to contain the protons. Which is counter-intuitive. </p>
<p>On the other hand, at least some textbooks acknowledge the coulomb force trying to push protons away from others in the nucleus, and so they infer that the nuclear force (e.g., residual strong force) must act more strongly on protons than what it does on neutrons. (At least in big nuclei, "beta stable nuclei", nuclei with an excess of neutrons..) So how does this nuclear force distinguish between protons and neutrons?</p>
| 3,896 |
<p>The equation of state for a perfect fluid is that $p=\omega \rho c^{2}$, where $p$ is the pressure, $\rho$ is the density, $c$ is the vacuum speed of light, and $\omega$ is called the equation of state parameter. $\omega$ may be constant or varying in time. I'm looking for broad answers (and references if possible) to the following:</p>
<ul>
<li>Why is it important that the equation of state parameter of dark energy is measured?</li>
<li>What will it tell us? </li>
<li>What are the implications?</li>
</ul>
| 3,897 |
<p>Why can't the expansion of the universe be thought as the Big Bang itself still in progress? Why do we need to introduce dark energy? The Big Bang was powerful, and that explosion itself could still be continuing right?</p>
| 3,898 |
<p>In the SM with gauge group U(1)xSU(2)xSU(3), those factors are associated to the gauge bosons associated with a local symmetry and the Higgs field provides masses to the elementary fermions AND the W,Z bosons (the photon is inert) through SSB. My question is, the Higgs boson has (likely) a mass of about 127 GeV, what kind of symmetries prevent the Higgs to become very heavy through quantum radiative corrections? SUSY can make the work, but is there ANY clever alternative symmetry that protects the Higgs particle to become a superheavy particle via quantum corrections to the mass? I believe that the (minimal) Higgs boson has NO local symmetry associated to it, as far as I know...</p>
| 3,899 |
<p>I'm reading about the acousto-optic effect and on the Acousto-Optical Tunable Filters on particular and wanted to understand the physics under its working. I found this paper</p>
<p><a href="http://dx.doi.org/10.1016/S0079-6727(03)00083-1" rel="nofollow">http://dx.doi.org/10.1016/S0079-6727(03)00083-1</a></p>
<p>where they talk about the interactions of polarized light within a crystal. They say</p>
<blockquote>
<p>"For instance, if the incident radiation is an ordinary ray (o-ray), it will be converted into an extraordinary ray (e-ray) upon interaction with the acoustic wave."</p>
</blockquote>
<p>and a few paragraphs later</p>
<blockquote>
<p>"Considering the entering o-ray [...], photons interact with phonons [...] and are converted to e-rays by a 90° polarization rotation."</p>
</blockquote>
<p>They do not explain the origins of this polarization rotation (and I imagine that it's beyond my knowledge) but is this rotation of 90° always true and why of 90° particularly?</p>
| 3,900 |
<p>I have heard my friends referring to quantity ma as the "force of acceleration" whereas my teacher told us it can't be referred as so. Is it correct to refer that quantity a force? If not, how can we better describe this quantity?</p>
| 3,901 |
<p>I am trying to solve a situation, where I'd like to know how much energy would be needed to push a mass of water out of a container. Here's an image to help understand:</p>
<p><img src="http://i.stack.imgur.com/3tyr0.png" alt="enter image description here"></p>
<p>The water tank has a height of 2 meters, length of 6 meters, and width of 4 meters. If a piston, much like a coffee press but watertight, pushes the water with only the hole on top of the container, as shown in the drawing, as an exit point, how much energy would be needed for that device to go all the way to the other side of the container? Assuming the hole is 1 square meter, if that can help. </p>
<p>I understand there are a lot of forces to take into consideration, such as friction, hydrostatic pressure, and more, but I'm trying to figure out if it would require a lot of energy or would it remain minimal? What is the force that will require the most energy? Hydrostatic pressure?</p>
<p>If there is a formula that would enable me to figure this out, that is all I'm asking, I can try and do the math myself, but I can't figure out where to start!</p>
<p>Thanks a lot for the help!</p>
| 3,902 |
<p>Let a two dimensional system be in the state $\phi=|0\rangle\langle0|$, for any basis $M$ spanned by the orthogonal vectors $|\psi_0\rangle,|\psi_1\rangle$, we can measure $\phi$ in basis $M$ and obtain "0" and "1" with probabilities $p_0=\mathrm{tr}(|\psi_0\rangle\langle\psi_0|\phi)$, $p_1=1-p_0$. </p>
<p>My question is, if I want to select $M$ randomly, is there some commonly understood way of choosing a random basis? My second question is, what would be the distribution of $p_0,p_1$ resulting with this random selection of the basis?</p>
| 3,903 |
<p>Particles can be represented as <a href="http://en.wikipedia.org/wiki/Wave_packet" rel="nofollow">wave packet</a>. So how do particles get <a href="http://en.wikipedia.org/wiki/Compton_scattering" rel="nofollow">scattered</a>?
Waves superimpose on one another, they don't bounce off of on one another.</p>
| 3,904 |
<p>This is a long shot, but while the 100YSS conference is going on at Houston, i haven't been able to get a grip on myself and think in other, more mundane and short-term rewards as normal people do. The main thing that has been taking up my mind is scalable ways to stabilize and store positronium in a way that the mass of storage device is negligible to the antimatter fuel</p>
<p>It is known that excited Ps atoms with high n (rydberg or rydberg-like) have <a href="http://arxiv.org/abs/physics/9712028" rel="nofollow">estimated lifetimes of the order of years</a>. Since positronium seems to be quite reactive (di and tri-positronium molecules have already been observed) I've been wondering if it could be possible to grow large crystals from positrons and electrons in a simple cubic lattice, where <strong>Na</strong> and <strong>Cl</strong> in the lattice being replaced by $e^{+}$ and $e^{-}$.</p>
<blockquote>
<p><strong>Question:</strong> is possible that positronium can have a crystalline bulk phase with enough ion distances to avoid annihilation?</p>
</blockquote>
<p>The reason why i'm interested in positronium and not antihydrogen, which looks relatively easier to manipulate, is that proton-antiproton annihilation is extremely messy, with more than half of the mass-energy decaying into end products like neutrinos, which won't contribute anything to kinetic energy. In contrast, positronium decays in relatively soft gamma rays of 511 KeV (and 380 KeV for orthopositronium) that in theory we could reflect with extremely low grazing angles with efficiencies up to 90%</p>
<p><strong>As an addendum to the larger question of positronium storage methods,</strong> i can't avoid mentioning that there have been other ways proposed over the years to keep the antimatter from touching the walls. If we want to store positron and electron separated, we have to deal with the Brillouin density limit that affects all existing and future Penning traps ($10^{12} e/cm^{3}$ for a field of 1 Tesla), so a simple calculation shows that a cylindrical container with weight ratio of 1 between container and antimatter masses, with 100 Tesla and 10 cm of thickness <em>will have to have a radius of $10^{12}$ meters, roughly the radius of Neptune orbit!</em> so in order to explore more viable alternatives, we are forced to consider only neutral antimatter (unless we find materials able to sustain over large regions magnetic fields 5-6 orders of magnitude higher than the current state of the art)</p>
<p><a href="http://www.waoline.com/science/astronautique/Links/High_Density_Storage_Antimatter-Dr_Steven-D_Howe-Dr_Gerald_A_Smith-Synergistic_Technologies_Inc%281%29.pdf" rel="nofollow">This report</a> gives a nice overview of some of the alternatives that have been explored, the most interesting is <a href="http://en.wikipedia.org/wiki/Paraelectricity" rel="nofollow">paraelectricity</a> (which is an effect where induced electric fields create virtual mirror charges on the wall that repels charges and possibly dipoles). <a href="http://en.wikipedia.org/wiki/Quantum_reflection" rel="nofollow">Quantum reflection</a> between antimatter BEC and wall is also mentioned, but i have reservations toward this one, since even if positronium BEC will form theoretically at higher temperatures (roughly 1-10 Kelvin), it is still complex to keep a large mass of BEC isolated enough from the environment (think of accelerations on a hypothetical ship) to make this a viable alternative. Basically if you sneeze, you and your ship will become a mininova.</p>
<p>There is <a href="http://books.google.com/patents/US6813330.pdf" rel="nofollow">one patent online</a> proposing using <a href="http://en.wikipedia.org/wiki/Photonic_crystal" rel="nofollow">photonic bandgap crystals</a> to disallow the decay band of rydberg positronium, but it seems to me that requires cavities that are of the order of one wavelength of the decay radiation, which put us again in a scenario with low ratios of container-fuel masses</p>
<p>PS: Glad i've finally written this down, hopefully i'll be able to take my mind off a bit from this subject</p>
| 3,905 |
<p>I find the idea of the wave function being 'just' a collection of numbers (probabilities) quite alluring, and elegant in explaining away the whole 'collapse' business (see Luboš' answer to <a href="http://physics.stackexchange.com/questions/10068/on-the-nature-of-the-collapse-of-the-wave-function">this question</a>).</p>
<p>I realize though, you can only stretch an analogy so far. Yes, learning location of a wanted criminal does not in fact 'collapse' his wave function, but is there any analogy which explains wave function interfering with itself (as per double split experiment)?</p>
| 3,906 |
<p>In "The Elegant Universe," two individuals, George and Gracie, are noted to be moving relative to one another in space and wearing clocks. They sync their clocks upon passing each other. Each observes the other's clock to be ticking more slowly. Gracie communicates with George via cell. </p>
<blockquote>
<p>Gracie communicates her time to George as he recedes into the distance. Initially, Gracie thinks that George will hear her reported time before his clock reaches the same time. But then, she takes into account travel time and realizes the travel time will more than compensate, and George will actually hear her time after his clock has passed that time. </p>
</blockquote>
<p>Then, the author notes that </p>
<blockquote>
<p>Gracie realizes that even if George takes the travel time into account, he will conclude from Gracie's communication that her clock is running slower than his. </p>
</blockquote>
<p>I'm quite confused by the last statement (specifically the phrase "<b>even</b> if George takes the travel time into account"). I thought travel time is actually the reason why George will hear Gracie's time after that time has elapsed on his clock. The word "even" seems to suggest that travel time of the communication is not the reason why this will occur. What is meant by "even?" </p>
<p>EDIT: I think that I understand now while trying to rewrite this question. I misinterpreted the setup in the book as communication travel leading to the receiver (George) getting Gracie's time after that time has elapsed on his clock. </p>
<p>But the time difference is greater than just travel time. The additional delay is due to Gracie (from George's perspective) transmitting the time after that time has already elapsed. After all, Gracie's clock is moving more slowly (due to time dilation) relative to his clock and that accounts for the extra delay (beyond the travel time). Is that correct?</p>
| 117 |
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