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<p>I want to simulate and test set of <strong>2D designs</strong> that basically have <strong>pulley/gear/chain-linked</strong> systems under Gravity (For e.g. to check how a pulley would rotate given particular weights, of course I'm not talking about such small systems, but systems where different pulleys/gears/weights are linked together).</p>
<p><em>[Frictional/mechanical properties can be neglected for my purposes]</em></p>
<p>Can anyone recommend any softwares <strong>(Opensource/paid)</strong> that can be used to easily design pulleys, gears, some chains(for e.g. to link a pulley to a gear) and simulate their behavior when the system is placed in a gravitational field?</p>
<p>I can construct the physical replicas of the designs and test them but it's rather cumbersome and a waste of resources to do so. Your help is much appreciated.</p> | 5,923 | [
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<p>I am having trouble understanding the origin of the bosonic stimulated emission. How can I qualitatively understand why bosons Boson's attract each other into similar quantum states.</p>
<p>The furtherst I was able to get was to simulate the feedback process, in my mind, but I don't have the foggiest of ideas why this rule must be the case. I haven't taken any classes on quantum or statistical mechanics.</p>
<blockquote>
<p>Explicitly, if N bosons occupy a given state, the transition rates
into that state are proportional to (N + 1).</p>
</blockquote> | 5,924 | [
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<p>Equation of continuity says us that if we insert some fluid in a tube, the same amount of fluid will come out from the other end. If we make a small hole in a hose pipe, water will come out with a great speed. The bigger the hole, the slower the speed. This is a direct consequence of the equation of continuity.</p>
<p>But at the case of water tap, when we start to turn on the tap slightly, the velocity of water is slow. As we turn on the tap more, the speed increases. This is contradictory with equation of continuity.</p> | 913 | [
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<p>Is there any fundamental physical reason (thermodynamics/entropy?) behind the fact that there doesn't exist home appliance for fast cool down of food/drinks?</p>
<p>I know that there are some methods (liquid nitrogen, etc.) used in the kitchen, but I mean something much more common, like microwave oven but with opposite function.</p>
<p><strong>Update</strong> Thanks for the responses so far. I understand that the temperature difference is the key. However, I do not understand why it is so much harder/slower to generate large temperature difference on the cool side?</p> | 5,925 | [
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<p>What exactly do people mean when they talk about the scale dependence of the effective potential ($V$)? I explain the motivation for my question (and hence my confusion) below. Please correct me as appropriate.</p>
<hr>
<p>If one defines the effective potential as the non-derivative part ($p^2 \rightarrow 0$) of the 1-PI effective action, then something like the Callan-Symanzik equation will imply that
$$\frac{d}{d \log \mu} V = 0 \implies \left[ \frac{\partial}{\partial \log \mu} + \beta_i \frac{\partial}{\partial \lambda_i} + \gamma \frac{\partial}{\partial \log \phi} \right] V = 0$$</p>
<p>If the value of the effective potential at any field value (and zero momentum) is scale independent, then shouldn't the vacuum be stable at all scales, if we know it's stable at some scale?</p>
<hr>
<p>All that I've said above holds so long as perturbation theory has been used correctly.</p>
<p>I've also seen some people "RG-improve" the effective potential by resumming the leading logs across all loop order. But then, the moment you do a partial resummation across loop orders, is there any reason for the thing you calculate to be scale independent? It's also not clear what the physical interpretation of such a scale-dependent quantity should be -- so why should one take the appearance of another vacuum seriously -- after all, you don't see any such thing when you do your calculations at some "low" scale -- and physical observables (at zero momentum) better not be RG-scale dependent?</p> | 5,926 | [
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<p>I am given an initial x and y position and initial velocity and I was asked to graph the trajectory in 1 second intervals.</p>
<p>This is what I have so far:</p>
<p>If $x_0 = 1, v_{0x} = 70, y_0 = 0, v_{0y} = 80, a_x = 0, a_y = -9.8$, and time will be 0,1,2,3... and so on.</p>
<p>Using these equations on every second you can find the plot will be a bell shaped with the highest point being ~ 325 m at about 600 seconds: </p>
<p>$$ x = x0 + (v_{0x})t + 1/2((a_x)t^2) $$
$$ y = y0 + (v_{0y})t + 1/2((a_y)t^2) $$</p>
<p>Usually in physics, we are taught in perfect condition with no air resistance. But what if there was air resistance?
How would it affect this problem? How can i add it to my calculations and see what the difference is?</p> | 7 | [
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<p>Assume an ideal board, supported by two joists. Where should those two joists be optimally placed?</p>
<p>Instinctively, I'd say at somewhat less than 25% and somewhat more than 75% of the extend of the board.</p>
<p>Presumably this also depends on whether the board can move freely with respect to the joists, or whether the board is attached to the joists.</p>
<p>I'm unclear even where to start analyzing this problem ...</p> | 5,927 | [
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<p>Can anyone suggest a video or book that will allow the layman to grasp physics concepts in a conceptual manner? </p> | 5,928 | [
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<p>How does physics scattering experiments relate to real life? And what does the scientist gain from such experiments? I am having a hard time figuring the answer out. Please help.</p> | 5,929 | [
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<p>I understand NASA is now experimenting with a tweaked Alcubierre drive.</p>
<p>Apperantly some physicists predict crazy heat (hotter than the sun) inside the warp bubble caused by Hawking radiation (recent paper by Carlos Barceló, Stefano Finazzi, and Stefano Liberati).</p>
<p>One of the biggest problems with nuclear fusion is the creation of very high temperatures.</p>
<p>Can we utilize the Hawking radiation in a warp bubble to make nuclear fusion happen?</p> | 5,930 | [
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<p>I have always heard that the Earth, due to its structure, cannot explode. Now, I'm quite fascinated by astronomy physics but I admit I only hold some more than basic knowledge. </p>
<p>In any case, even though I have interiorized this concept, I have no basis to prove it. If you search on the internet, you get both theories (No it won't vs <a href="http://bioresonant.com/news.htm">yes it will</a>), so you understand that even if one is the current accepted theory, I'm not sure I know why it is the case.</p>
<p>Considering this, what concrete reasons can we bring forward in its favour? And why?</p> | 5,931 | [
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<p>Witten's 1981 paper <a href="http://www.sciencedirect.com/science/article/pii/0550321381900213">"Search for a realistic Kaluza-Klein theory"</a> is frequently cited for its observation that, in a compactification of d=11 supergravity on a manifold with SU(3) x SU(2) x U(1) isometry, left-handed and right-handed fermions would transform in the same way under the gauge group, in opposition to experiment. Years later, in the era of M-theory, <a href="http://arxiv.org/abs/hep-th/0109152">Acharya and Witten</a> were able to get chiral fermions by compactifying on a manifold with singularities. But Witten 1981 also mentioned, at the end, that coupling to a torsion field might produce chiral asymmetry. (Subsequent explorations of the idea: <a href="http://prd.aps.org/abstract/PRD/v33/i2/p363_1">for</a> and <a href="http://jmp.aip.org/resource/1/jmapaq/v25/i9/p2696_s1">against</a>.) </p>
<p>Now I'm wondering: <strong>Can torsion in M-theory revive the 1970s version of Kaluza-Klein theory?</strong> A <a href="http://arxiv.org/abs/1105.2299">very recent paper</a> constructed CFT duals for M-theory on AdS<sub>4</sub> x M<sup>111</sup> with torsion flux, M<sup>111</sup> being one of the "M<sup>pqr</sup> manifolds" with SU(3) x SU(2) x U(1) isometry group that was considered in Witten 1981. (The M<sup>pqr</sup> notation is Witten's; in a more recent notation this manifold is designated M<sup>3,2</sup> or sometimes Y<sup>2,3</sup>.) Can M-theory on such a background give rise to chiral fermions? </p> | 5,932 | [
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<p>Do quantum states contain exponentially more information than classical states? It might seem so at first sight, but what about in light of <a href="http://www.scottaaronson.com/talks/advucsb.ppt">this talk</a>?</p> | 5,933 | [
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<p>The <a href="http://en.wikipedia.org/wiki/Anthropic_principle">anthropic principle</a> is based upon Bayesian reasoning applied to the ensemble of universes, or parts thereof, conditioned upon the existence of conscious observers. That still leaves us with the problem of determining the prior probabilities. This is the famous measure problem. How is this measure determined in the current cosmology literature? </p> | 5,934 | [
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<p>I read that the non-commutativity of the quantum operators leads to the <a href="http://en.wikipedia.org/wiki/Uncertainty_principle#Mathematical_derivations">uncertainty principle</a>.</p>
<p>What I don't understand is how both things hang together. Is it that when you measure one thing first and than the other you get a predictably different result than the measuring the other way round?</p>
<p>I know what non-commutativity means (even the minus operator is non-commutative) and I think I understand the uncertainty principle (when you measure one thing the measurement of the other thing is kind of blurred - and vice versa) - but I don't get the connection.</p>
<p>Perhaps you could give a very easy everyday example with non-commuting operators (like subtraction or division) and how this induces uncertainty and/or give an example with commuting operators (addition or multiplication) and show that there would be no uncertainty involved.</p>
<p>Thank you.</p> | 923 | [
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<p>I was reading Wikipedia which stated </p>
<blockquote>
<p>....Voyager 1's current relative velocity is 17.062 km/s, or 61,452 kilometres per hour (38,185 mph).....</p>
</blockquote>
<p>It travels away from sun. So sun's gravity must slow it down. What propels Voyager to that speed?</p> | 5,935 | [
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<p>Schwarzschild singularities are described by the Kantowski-Sachs metric with a contracting S<sup>2</sup>. Of course, T-duality doesn't apply to S<sup>2</sup>. But what about a Kasner-type singularity with two contracting spatial dimensions compactified over a torus T<sup>2</sup>, and an expanding spatial dimension? The T-dual of the torus gives rise to a geometry which is expanding in all spatial directions.</p> | 5,936 | [
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<p>It is known that pure heavy water will quickly get contaminated by light hydrogen atoms if exposed to air due to isotope exchange.</p>
<p>Does the same thing happens when heavy water is stored in plastic container, which also contains light hydrogen atoms?</p> | 5,937 | [
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<p>Two towns are at the same elevation and are connected by two roads of the same length. One road is flat, the other road goes up and down some hills. Will an automobile always get the best mileage driving between the two towns on the flat road versus the hilly one, if operated by a perfect driver with knowledge of the route? Is there any hilly road on which a better mileage can be achieved?</p> | 5,938 | [
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<p>It might seem obvious but i can't imagine how is gravitational pull is different from acceleration due to gravity?</p> | 5,939 | [
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<p>How do the wheels of a train have sufficient grip on a metal track? I mean both of the surfaces are smooth (and not flexible) and it is okay if there is no inclination, but how about on an inclined track?</p> | 5,940 | [
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<p>We all learn in grade school that electrons are negatively-charged particles that inhabit the space around the nucleus of an atom, that protons are positively-charged and are embedded within the nucleus along with neutrons, which have no charge. I have read a little about electron orbitals and some of the quantum mechanics behind why electrons only occupy certain energy levels. However...</p>
<p>How does the electromagnetic force work in maintaining the positions of the electrons? Since positive and negative charges attract each other, why is it that the electrons don't collide with the protons in the nucleus? Are there ever instances where electrons and protons <em>do</em> collide, and, if so, what occurs?</p> | 132 | [
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<p>This probably is dumb question, but it has been giving me issues for days.
I was thinking about a point of mass m in a circular orbit around a planet. The question is: giving a radial impulse, what is the change in the shape of the orbit? Will it change in another circular orbit? Or maybe will undergo a spiral motion ?
I'm sorry for the dumb question and for bad english.</p> | 5,941 | [
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<p>I am a layman. I am aware that the <a href="http://en.wikipedia.org/wiki/Alcubierre_drive" rel="nofollow">Alcubierre Drive</a> has not yet been proven to be possible, but there is something about the concept itself that I am confused about. If there is no movement within the bubble, how is time measured inside the bubble? If there is no velocity, what is time measured against?</p> | 5,942 | [
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<p>What are the advantages/disadvantages of a transmission vs a reflection grating? It seems like a transmission grating would be easiest to use. I'm trying to get a spectrum from Thomson scattered light in a plasma. The broader the spectrum of the scattered light, the hotter the plasma. It's a weak effect, so it's important to keep as much light as possible. It would also be important that the image of light is not distorted (at least not along a chosen axis... obviously one direction will have the light spread).</p> | 5,943 | [
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<p>I am calculating for many hours and I am really confused with this exercise.</p>
<p>Consider a comoving observer sitting at constant spatial coordinates$(r∗,θ∗,φ*)$, around a Schwarzschild black hole of mass $M$. The observer drops a beacon into the black hole (straight down, along a radial trajectory). The beacon emits radiation at a constant wavelength $λ_{\mathrm{em}}$ (in the beacon’s rest frame).</p>
<p>From Schutz's book, I think "comoving" might be wrong, because it is only true when the object flies on geodesic which has to be a circle and that is not guaranteed.</p>
<p>The Schwarzschild metric:</p>
<p>$$ds^2=-(1-2M/r)dt^2+(1-2M/r)^{-1}dr^2+r^2d\Omega$$</p>
<p>I tried to solve this in this way:</p>
<p>In the observers MCRF, he has the four velocity:</p>
<p>$$U_{\operatorname{obs}}=\vec{(1,0,0,0)}$$</p>
<p>Now we calculate the energy of the object by the observer:</p>
<p>$$E/m=\bar{E}=-U_{\mu}^PU^{\mu}_{\operatorname{obs}}=-U_{0}^P$$</p>
<p>with $U^p$ the objects velocity measured by the observer. Then we found </p>
<p>$$U_{0}^P=-g^{00}\bar{E}=\dot{t}$$</p>
<p>The last equating can you show with respect to the proper time.</p>
<p>The other component follows from:</p>
<p>$$U_{\mu}^PU^{\mu}_P=-1$$</p>
<p>From this equation you can compute:</p>
<p>$$(U^r)^2=-(1-2G/R)+\bar{E}$$
something like that…</p>
<p>But now I have a question.</p>
<p>I am in the MCRF of the observer hence a local initial system with a flat metric. But I used the in this point the Schwarzschild metric not the Minkowski one.
I thought you can do this, because the Schwarzschild metric is valid there in every point. And when you use the Minkowski one you get that the velocity gets infinity on $r =2M$ and this is wrong.</p> | 5,944 | [
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<p>How does a magnifying glass work? I know it creates a virtual image of the observed object but how is it possible that humans can see the virtual image?</p> | 209 | [
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<p>Reason behind <a href="http://en.wikipedia.org/wiki/Canonical_quantization" rel="nofollow">canonical quantization</a> in QFT?</p>
<p>In the scalar field theory we simply promote the scalar field, $\phi(x)$ to a set of operators: $\hat{\phi}(x)$. What is the reason behind this?</p> | 5,945 | [
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<p>I would like to have an image (in any kind of space), where I see the path of a "light" source. In my understanding the most common, directed source would be a laser pointer. </p>
<p>a) Is this correct? If not, what would be an other option?</p>
<p>As far as I know, to be able to "see" the path of the light, the material through the light passes should reflect some of it into my "camera", like smoke.</p>
<p>b) Is this correct? Do I always need this "material"? Does the air reflect any of it?</p> | 5,946 | [
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<p>Trick dice have been used in gambling and magic shows. These are dice that, when rolled, will land with a preferred number facing upwards. This makes the result of rolling the dice non-random.</p>
<p>I was able to find a few ways to make trick dice, and I'm sure there are many more methods.</p>
<ol>
<li><a href="http://technabob.com/blog/2010/03/19/how-to-make-trick-dice/" rel="nofollow">Melt the dice in an oven</a>.</li>
<li><a href="http://www.instructables.com/id/Loaded-Dice/" rel="nofollow">Drill and fill a hole with heavier material</a>.</li>
</ol>
<p>While these methods do work, they are not flawless. It's possible for the dice to still produce an unwanted result.</p>
<p>I was wondering with new materials like <a href="http://www.youtube.com/watch?v=FrapkkuQGIk" rel="nofollow">liquid metal</a> and such things. Is it possible to create a truly flawless die, which when thrown on a flat surface will always produce the same result?</p> | 5,947 | [
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] |
<p>So far 20 <a href="http://en.wikipedia.org/wiki/Synthetic_element">synthetic elements</a> have been synthesized. <strong>All are unstable</strong>, decaying with half-lives between years and milliseconds.</p>
<p>Why is that?</p> | 5,948 | [
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] |
<p><a href="http://en.wikipedia.org/wiki/Kramers_theorem" rel="nofollow">Kramers theorem</a> rely on odd total number of electrons. In reality, total number of electrons is about 10^23. Can those electrons be so smart to count the total number precisely and decide to form Kramers doublets or not? </p> | 5,949 | [
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<p>Consider an explosion which is taking place in a vacuum, The exploding object is very small with a huge amount of energy and high density (like a big bang explosion) and the object explodes from the energy internally. I wanted to know how the mass is going to distribute after huge explosion in every direction. Does the mass distribute equally in all direction with same velocity or differently in different direction.</p> | 5,950 | [
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<p>What causes <a href="http://youtu.be/EM69cCGZF8o" rel="nofollow">those radiation readings</a> in my radiation meter (See the linked video)? Metal pieces were magnetized.</p>
<p>Used radiation detector is <a href="http://rads.stackoverflow.com/amzn/click/B00051E906" rel="nofollow">RADEX RD1503</a>. "Radiation" is not generated if the tube has been wrapped inside an aluminium foil. Should induction occur through thin aluminium foil? At least magnet holds through that foil and plastic tube.</p> | 5,951 | [
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<p>I am not a technical guy and I have no scientific knowloedge in <code>physics</code> but I have been reading books, watching videos in order to understand our cosmology and existance. Forgive me if I ask nonsense questions, if I do I'd do it because of my lack of knowledge.</p>
<p>As we all know <code>gravitational waves</code> have been found. I have also been reading <code>Hidden Reality</code> of <code>Brian Greene</code> which I think a little bit confusing (it is funny that his videos are much more understandable than his books). I understand that GW occur on very high levels of energy (although I don't why it requires such energy levels). Based on this, what I want to ask are:</p>
<ul>
<li>Why GW requires such energy levels?</li>
<li>Why attractive gravity doesn't create GW?</li>
<li><code>removed</code></li>
</ul>
<p>I hope I explained me and my questions clear. </p>
<p>Regards</p> | 5,952 | [
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<p>If non-zero cosmological constant interpreted as a repulsive field, what would be the properties of the excitation of such field, i.e. the particle which serves as the field's quantum?</p>
<p>What would be its spin, mass, possible interactions and other properties?</p> | 5,953 | [
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<p>In many textbooks it is said that mass renormalization of the electron mass is only logarithmic $\delta m \sim m\, log(\Lambda/m)$ because it is protected by the chiral symmetry.
I understand that in case of massless fermions to keep them massless in the renormalization procedure it must be like this. Or differently said, the renormalization procedure respects the axial current conservation. </p>
<p>But is there a compulsory reason for the renormalization procedure to respect the axial current conversation ? Does every renormalization procedure respect that ? Apparently Pauli-Villars and dimensional renormalization do it, but what for other procedures ?
I also know that in triangular Feynman diagrams anomalies occur which do break the axial current conversation. So why can't it happen for something simpler like the electron mass respectively self-energy ?</p> | 5,954 | [
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<p><img src="http://i.stack.imgur.com/fuDZG.png" alt="enter image description here"></p>
<p>In the solution, it says we have $dr/dt= -V$ (polar coordinates)
How? I can't see how this can be possible, we know that $r(t)=V/\omega(t)$, and that's it.</p> | 5,955 | [
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<p>The unit for angular acceleration $\alpha$ is:</p>
<p>$$\mathrm{rad/s^2}$$</p>
<p>The unit for torque is $\mathrm{Nm}$:</p>
<p>$$\mathrm{kg\ m^2/s^2}$$</p>
<p>And their relationship with Inertia is:</p>
<p>$$I = \tau/\alpha$$</p>
<p>So shouldn't the unit for for Inertia be:</p>
<p>$$\mathrm{kg\ m^2/rad}$$</p>
<p>yet everywhere I read says it is simply $\mathrm{kg\ m^2}$ instead. How does the $\mathrm{rad}$ unit fall off?</p> | 210 | [
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<p>If we touch the live wire and ground at the same time, we will get a shock. But the current goes from live to ground and not to neutral i.e, circuit is open. Then how can we get a shock?</p>
<p><img src="http://i.stack.imgur.com/fLKbj.png" alt="Image"></p>
<p>This is the same circuit: will the current flow?<br>
<img src="http://i.stack.imgur.com/nryo4.png" alt="Image"></p>
<p>I think current should not flow. So, what is the reason that we still get shocks when the circuit is open?</p> | 5,956 | [
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<p><--burst of thrust------|ME in a ship>----->>--<<--laser radar--<<---->>---||destination</p>
<p>In the above, I imagine myself in a ship, marking my speed to a far off destination with laser radar. In addition, someone at the destination is clocking me as well with a separate laser.</p>
<p>So, moving toward my destination at speed S1, I'm at rest. Then I fire a burst of thrust and experience acceleration, a, for time, t, and arrive at a new speed S2, right?</p>
<p>This is all just simple physics right?</p>
<p>My question is, where does special relativity play into this and how would it be perceived by A) me in the ship and B) the person at the destination clocking the ship?</p>
<p>Would it be reasonable to want to graph velocity and distance to destination over time, given regular bursts of thrust, for both me in the ship and the observer at the destination? If so, any tips/equations to use?</p>
<p>PS: Although this might sound like a homework problem (or maybe I'm giving myself too much credit), it's actually just a lunchtime conversation :)</p> | 5,957 | [
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<p>Suppose that $U(3)$ was the gauge group. We can decompose this as
$U(3)=U(1)\times SU(3)$,
which implies that in addition to the $SU(3)$ that has eight generators corresponding to eight gluons, there would be an additional generator for $U(1)$.</p>
<p>Would that generator behave like a photon? (no self interaction and no color) </p> | 5,958 | [
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<p>This is really a question more about mining ventilation than true "physics." </p>
<p>I occasionally explore a southwest desert mine, but am aware of the hazards - typically passing over half a dozen vintage mines as unsafe (including anything vertical) before finding one stable enough and acceptable for exploration. (I won’t call it “safe”….just acceptable.) My understanding is that, the air in desert mines is usually unproblematic. But should running water - and certainly stagnant water - be present, the presence of harmful $CO_2$ pockets from plant/algae respiration or decomposition should be acknowledged. Carbon monoxide can be an issue as well, something I usually associate with combustion (miners' carbide illumination, wooden torch illumination, primitive engines), but perhaps there are other $CO$ sources in a mine that I'm unaware of. Unlike natural caves which often have multiple inlets, mines with single entrances have no opportunity for flow-through air ventilation. But they nonetheless often have unhindered mountainside openings which are exposed to decades of windstorms. I'm curious, when we are $200$ ft. deep within a hypothetical dry desert mine with one unhindered opening, wouldn't the air deep inside the mine essentially be unchanged from 1930, with the original respiration and combustion products fully present (though a little diluted)? </p>
<p>Secondly, while that may be POSSIBLE, how LIKELY is this situation? </p>
<p>(I assume gas monitors have possibly provided general Yes/No answers to this question over the decades.) </p> | 5,959 | [
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<p>According to a website, $T_3 = T_2 + T_1$. Also, the rope and pulley device are mass less. But I don't really understand this. Let me try and explain how I see the problem.</p>
<p>$M_2$ receives tension in the up direction. This tension is the result of the resultant force of $M_1$ on the rope.</p>
<p>$M_1$ receives tension in the up direction. This tension is the result of the reactant force of tension in $T_2$</p>
<p>The forces on the rope connected to the masses are: $T_{1m}$, $T_{2m}$ and these are both resultant forces from said masses. But since they are technically in different directions, they cancel each other out.</p>
<p>So the net force on the rope is zero.</p>
<p>But why is $T_3 = T_2 + T_1$?</p>
<p>I do not understand this at all. It just doesn't make sense to me. </p>
<p><img src="http://i.stack.imgur.com/lRZc2.gif" alt="enter image description here"></p> | 5,960 | [
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<p>I'm working on a problem in 3d field theory and I'm confused about how to write the superfields. Specifically, I'm not sure if the signs and coefficients of terms are purely a matter of convention or if they need to be chosen more carefully. </p>
<p>For example in <a href="http://arxiv.org/abs/0806.1519" rel="nofollow">0806.1519v3, page 18</a> we have the 3d chiral, anti-chiral and vector superfields respectively: </p>
<p>(1) $\Phi = \phi(x_{L}) + \sqrt{2} \theta \psi(x_{L}) + \theta^{2} F(x_{L})$ </p>
<p>(2) $\Phi^{\dagger} = \bar{\Phi} = \bar{\phi}(x_{R}) - \sqrt{2} \bar{\theta} \bar{\psi}(x_{R}) - \bar{\theta}^{2} \bar{F}(x_{R})$ </p>
<p>(3) $V = 2i\theta \bar{\theta} \sigma (x) + 2 \theta \gamma^{\mu} \bar{\theta} A_{\mu}(x) + \sqrt{2} i \theta^{2} \bar{\theta} \bar{\chi}(x) - \sqrt{2} i \bar{\theta}^{2} \theta \chi(x) + \theta^{2} \bar{\theta}^{2} D(x) $</p>
<p>However one could take the alternative approach of beginning with the 4d superfields of Wess and Bagger in their book "Supersymmetry and Supergravity":</p>
<p>(4) $\Phi = \phi(x_{L}) + \sqrt{2} \theta \psi(x_{L}) + \theta^{2} F(x_{L})$ </p>
<p>(5) $\Phi^{\dagger} = \bar{\Phi} = \bar{\phi}(x_{R}) + \sqrt{2} \bar{\theta} \bar{\psi}(x_{R}) + \bar{\theta}^{2} \bar{F}(x_{R})$ </p>
<p>(6) $V = - \theta \gamma^{\mu} \bar{\theta} A_{\mu}(x) + i \theta^{2} \bar{\theta} \bar{\chi}(x) - i \bar{\theta}^{2} \theta \chi(x) + \dfrac{1}{2} \theta^{2} \bar{\theta}^{2} D(x) $</p>
<p>Its clear that simply performing a dimensional reduction of (4), (5) and (6) will not give us (1), (2) and (3).</p>
<p>In fact one could look at "Introduction to Supersymmetry" by Müller-Kirsten and Wiedemann and see that, even sticking to the 4d case, different signs and coefficients are used compared to (4), (5) and (6).</p>
<p>Now normally I'd simply accept that there's some difference in notation and not consider it a big deal. However, when we write the superfields in actions we have superfield multiplications (e.g: $\Phi^{\dagger} e^{qV} \Phi$). These give important terms when we multiply out the consituents of the superfields. Some such terms are also present in the non-susy case. Having performed a few calculations it has become clear that different conventions for the superfields result in different signs and coefficients for these resulting terms (unsurprisingly). I would have thought such differences to be physically important since they arise in the action. When calculating probability amplitudes, for example, would such signs and coefficients not play a crucial role? In non-supersymmetric field theory the terms tend to have very specific signs and coefficients, I would have thought the superfields would have had to make sure such exact terms are reproduced when superfields are multiplied together. </p> | 5,961 | [
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<p>Everyone knows that the mass of a system is less than the mass of its components, with the equation:</p>
<p>$M = \sum_i m_i - BE(M) $</p>
<p>Now, if we consider a general decay, lets say</p>
<p>$A \rightarrow \sum_i B_i$</p>
<p>then, for the conservation of the first component of the four-vector momentum we obtain a necessary condition for the decay:</p>
<p>$ M(A) \ge \sum_i M(B_i) $</p>
<p>Isn't this last condition in contradiction with the first one?
The only answer I get is a particle that can decay is not an eigenstate of the system. For this reason it shouldn't have a defined binding energy. So, for example, Tritium, that can decay, hasn't a BE? This appears so illogical if we think about nuclei with an half-life of years or more. Moreover I know this energy is experimentally defined and is bigger that 3He's BE.</p> | 5,962 | [
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<p>What is the term used to describe the behaviour of (for example) a water droplet in free fall when it has reached a certain speed and then the force on the droplet causes the larger droplet to disperse into many smaller sized droplets. This also occurs in a Taylor Cone when using electrostatic spray. The charge builds in the liquid until it eventually reaches a critical charge and then the negatively charged droplet releases.</p> | 5,963 | [
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<p>Maybe mine is a silly question, but are there <strong>mathematical similarities</strong> or <strong>common roots</strong> between the <a href="http://en.wikipedia.org/wiki/Christoffel_symbols" rel="nofollow">Christoffel symbols</a>:</p>
<p>$ \nabla - \partial = \Gamma $</p>
<p>and the <a href="http://en.wikipedia.org/wiki/Gamma_matrices" rel="nofollow">Dirac matrices</a> $ ( \gamma^\mu \gamma^\nu + \gamma^\nu \gamma^\mu = 2g^{\mu\nu}I )$ obtained through:</p>
<p>$ (\gamma^\nu \partial_\nu)(\gamma^\mu \partial_\mu) = \partial^\mu \partial_\mu $</p>
<p><strong>EDIT:</strong> What I mean is that the Dirac matrices are obtained by trying to match two different derivatives, so I was wondering if that had some common ground with the Christoffel symbols that are defined as the difference between the connection and the coordinate derivative.</p> | 5,964 | [
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<p>I'm looking at a way to prove that one recovers, under ad hoc assumptions, classical mechanics from quantum theory. Usually, we can find in textbooks that the propagator </p>
<p>$K(x,x_0;t)=\langle x|e^{-i \hat H t/\hbar}|x_0\rangle$ </p>
<p>is given in the classical limit by using the path integral formulation </p>
<p>$K(x,x_0;t)\propto e^{i S[q_c(t)]}$</p>
<p>where $S[q]$ is the classical action and $q_c(t)$ is the classical path from $x_0$ to $x$ in a time t (which minimizes $S$).</p>
<p>But this is not very satisfying, as $K$ is a probability amplitude, which is really not what we would like in the classical limit. The least we would like to obtain is that the probability to be in $x$ at time $t$ is given by</p>
<p>$P_t(x)=\delta(x-q_c(t))$ $\,\,\,\,\,\,\,\,$ (1)</p>
<p>or something equivalent. My naive approach to recover this result would be the following. The propagator is not enough, as the "classicalness" of the motion should be given by the initial state. It seems natural to choose the initial wave-function to be a wave-packet peaked around the initial position (at x=0 from now on) with spread $\Delta x$ and with momentum $p$, for example </p>
<p>$\psi(x)=\dfrac{e^{-\frac{x^2}{2\Delta x^2}-i p x/\hbar}}{\pi^{1/4}\sqrt{\Delta x}}$.</p>
<p>We would say that the dynamics will be classical at least if $\Delta p\propto \hbar/\Delta x \ll p$. There is maybe other constraints, for instance the the dynamics is classical only at times $t$ long enough, but this is still not clear to me.</p>
<p>Finally, the classical probability is given by (or at least proportional to)</p>
<p>$P_t(x)=|\int d x_0 K(x,x_0;t) \psi(x_0)|^2\propto |\int d x_0 e^{i S[q_c(t)]} \psi(x_0)|^2$. $\,\,\,\,\,\,\,\,$ (2)</p>
<p>My question is : is there a way to show/prove that under these assumptions, we can get (1) from (2) for any hamiltonian ?
I have looked at the simplest case of a free particle, and it seems to work (I still have some issues to get the final result, but my feeling is that it works). If it helps, I could post the calculation later. But a general proof would be great. </p> | 5,965 | [
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<p>What is the difference between Quantum Physics, <a href="http://en.wikipedia.org/wiki/Quantum_theory" rel="nofollow">Quantum Theory</a>, <a href="http://en.wikipedia.org/wiki/Quantum_mechanics" rel="nofollow">Quantum Mechanics</a>, and <a href="http://en.wikipedia.org/wiki/Quantum_field_theory" rel="nofollow">Quantum Field Theory</a>? Are they the same subject? I believe that they are not the same subject! Maybe there is not big difference between those subjects but I need to know what is main difference between those subjects and what is main intersection? Also I need to know which one is big subject relative to another? </p> | 5,966 | [
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<p>This is the last part of a derivation of the equation for an ideal gas undergoing reversible adiabatic expansion. I'm trying to prove that $T V^{\gamma-1}$ is constant, but my result is that $T V^{1-\gamma}$ is constant. Where am I going wrong?</p>
<p>I don't know what notation is standard, so please tell me if I should clarify something. I think this is just a mathematical mistake though.</p>
<p>$$C_vd T = pdV = \frac{NT}{V}dV$$
$$\frac{dT}{T} = \frac{N}{C_V}\frac{dV}{V} = \frac{C_p - C_V}{C_V}\frac{dV}{V}$$
$$\frac{dT}{T} = (\gamma-1) \frac{dV}{V}$$
$$d\ln T = (\gamma-1) d \ln V $$</p>
<p>Integrate from $T_0$ to $T_1$ and $V_0$ to $V_1$ (start state to end state of the process):</p>
<p>$$\ln T_1 - \ln T_0 = (\gamma-1)( \ln V_1 - \ln V_0 ) $$
$$\ln \frac{ T_1 }{T_0} = (\gamma-1)\ln \frac{ V_1}{ V_0 } $$
$$\frac{ T_1 }{T_0} = \left(\frac{ V_1}{ V_0 }\right)^{\gamma-1} $$</p>
<p>Now put the start state and the end state on different sides:</p>
<p>$$ \frac{T_1}{V_1^{\gamma-1}} = \frac{T_0}{V_0^{\gamma-1}} $$
$$ TV_1^{1-\gamma} = T_0 V_0^{1-\gamma} $$</p>
<p>Therefore, $T V^{1-\gamma}$ must be constant throughout the process.</p> | 5,967 | [
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<p>I used to have a vague feeling that the residue theorem is a close analogy to 2D electrostatics in which the residues themselves play a role of point charges. However, the equations don't seem to add up. If we start from 2D electrostatics given by $$\frac{\partial E_x}{\partial x} + \frac{\partial E_y}{\partial y} = \frac{\rho}{\epsilon_0},$$ where the charge density $\rho = \sum_i q_i \delta(\vec{r}-\vec{r}_i)$ consists of point charges $q_i$ located at positions $\vec{r}_i$, and integrate over the area bounded by some curve $\mathcal{C}$, we find (using Green's theorem) $$\int_\mathcal{C} (E_x\, dy - E_y\, dx) = \frac{1}{\epsilon_0} \sum_i q_i.$$ Now, I would like to interpret the RHS as a sum of residua $2\pi i\sum_i \text{Res}\, f(z_i)$ of some analytic function $f(z_i)$ so that I would have the correspondence $$q_i = 2\pi i\epsilon_0 \text{Res}\, f(z_i).$$ For this to hold, the LHS would have to satisfy $$\int_\mathcal{C} (E_x\, dy - E_y\, dx) = \int_\mathcal{C} f(z)\, dz,$$ however, it is painfully obvious that the differential form $$E_x\, dy - E_y\, dx = -\frac{1}{2}(E_y+iE_x)dz + \frac{1}{2}(-E_y + iE_x)dz^*$$ can never be brought to the form $f(z)dz$ for an analytic $f(z)$.</p>
<p>So, it would appear that there really isn't any direct analogy between 2D Gauss law and the residue theorem? Or am I missing something?</p> | 5,968 | [
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<p>If temperature makes particles vibrate faster, and movement is limited by the speed of light, then temperature must be limited as well I would assume. Why there is no limits?</p> | 237 | [
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<p>As I remembered, at the 2 poles of a battery, positive or negative electric charges are gathered. So there'll be electric field existing within the battery. This filed is neutralized by the chemical power of the battery so the electric charges will stay at the poles.</p>
<p>Since there are electric charges at both poles, there must also be electric fields outside the battery. What happens when we connect a metal wire between the 2 poles of a battery? I vaguely remembered that the wire has the ability to restrain and reshape the electric field and make it within the wire, maybe like a electric field tube. But is that true?</p> | 78 | [
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<p>This question arises in a somewhat naive form because I am largely unfamiliar with String Theory. I do know that it incorporates higher space dimensions where I shall take the overall dimensionality to be 10 in this question, for concreteness. Now the traditional <a href="http://en.wikipedia.org/wiki/Penrose%E2%80%93Hawking_singularity_theorems" rel="nofollow">Hawking-Penrose Singularity</a> results apply to the the General Relativity manifold of 3+1 dimensions; with the 4D Schwarzchild solution providing an example of a Singularity and Black Hole.</p>
<p>So the question is: do singularities (and maybe associated Event Horizons) necessarily form in all 10 dimensions?</p>
<p>Examining this question for myself I see that <a href="http://arxiv.org/abs/math/0603190" rel="nofollow"> this paper for mathematicians</a> introduces an $N$ dimensional Schwarzchild metric and in theorem 3.15 an $N$ dimensional Hawking-Penrose singularity theorem. However this cannot answer directly to the intentions of the String theory models. For example it is mathematically possible to extend 4D Schwarzchild to 10D differently by adding a 6D Euclidean metric. So one question is whether this modified 10D Schwarzchild even meets the conditions for the $N$ dimensional Hawking-Penrose theorem. Although such a modification is not likely acceptable as a String Theory extension, it shows that we can consider some cases:</p>
<p>a) All 4D singularities / Event Horizons are actually 10 D ones.</p>
<p>b) Some/all 4D singularities / Event Horizons are "surface phenomena" in String theory - the underlying Bulk Volume is singularity free.</p>
<p>EDIT: Expressed a bit more formally this is saying that the String Theory has a singularity free solution $\Phi$ in 10D, but when $\Phi$ is restricted or reduced to 3+1D it is one of the known singular solutions of GR.</p>
<p>c) Some Singularities in String Theory Bulk (the 6D part) can arise without a corresponding 4D singularity (akin to a "deep earth earthquake" in 10D space-time, perhaps)?</p> | 5,969 | [
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<p>It seems like everything in the universe is in motion, at least relative to some other object. That leads me to believe that all objects are in motion. But how do we measure motion when we are talking about galaxies, is there any reference point that's not moving?</p> | 5,970 | [
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<p>I am a graduate student in experimental physics, currently writing up my dissertation and beginning to apply to post-doc positions. While it is very easy to find out about open positions in my particular sub-field (via various mailing lists), it is more difficult to find out about potentially interesting positions in other sub-fields.</p>
<p><strong>What is a good way to identify potentially interesting post-doc position openings outside of one's sub-field?</strong></p> | 5,971 | [
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<p>I'm working with an electronic temperature logger that is being affected by heat generated internally.</p>
<p>How does one come up with a calibration equation to calculate a more accurate reading of ambient temperature based on what the temperature sensor reads, taking into account its own power consumption?</p>
<p>details:</p>
<p>After a few hours and in equilibrium, the sensor reports values that are actually 1 degree Celsius higher than the ambient room temperature (22C) measured by a calibrated device. The sensor is accurate to 0.1 degree C at reporting the temperature of the device itself (which due to heat generated by the electronics has gotten warmer) </p>
<p>The device consumes ~0.1 watts of power, weighs about 200g and has an average specific heat capacity of 1.0 j/g (weighted mix of glass, abs, fr-4, copper). Dimensions are 1"x 3"x 4". </p>
<p>What I've got so far is this heating calculation: 200g * 1c * 1.0j/g / 0.1w / 60s = ~33 minutes to heat up 1 degree. </p>
<p>I'm assuming what we need is to figure out Sensor value - Heat-generated + Heat-dissipated to arrive at actual temperature. Which will require measure the K in newton's law? then what?</p>
<p>I'd really appreciate you help here.</p> | 5,972 | [
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<p>Suppose an infinitely long cylindrical shell of radius $a$ carries a surface charge density $\sigma_0$ and is surrounded by a coaxial cable of inner radius $b$ and outer radius $c$ with uniform charge density $\rho_0$ such that the combined system is neutral.</p>
<p>I am asked to calculate the electrostatic energy in two equivalent ways, using
$$W=\frac{1}{2}\int\rho V\,\mathrm{d}\tau$$
and
$$W=\frac{\epsilon_0}{2}\int E^2\,\mathrm{d}\tau.$$
For some reason, which I have struggled (in vain) to locate, I am not obtaining the same result, and I would appreciate it if someone could help locate precisely what went wrong.</p>
<p>I think this is rather trivial, but it is tedious, and I must have overlooked something somewhere, embarrassingly enough.</p>
<p>First, I calculate $\rho_0$ in terms of $\sigma_0$. Consider some length $L$ along the system. The charge along the inner shell is $2\pi aL\sigma_0$ and the charge along the coaxial cable is $\pi c^2L\rho_0-\pi b^2L\rho_0=\pi L\rho_0(c^2-b^2)$. Setting their sum to be equal to 0,
$$\rho_0=-\frac{2a\sigma_0}{c^2-b^2}$$</p>
<p>Next, I apply Gauss's law to find the electric field of the inner cylindrical shell and of the coaxial cable independently and sum them together to obtain the electric field for the system as a whole.</p>
<p>Consider first the inner cylindrical shell and construct a Gaussian cylinder of length L. Due to symmetry, the electric field must only point in the $\hat{\mathbf{s}}$ direction and is a function of $s$ only. For $s<a$, Gauss's law implies that the electric field is zero. For $s>a$, the charge enclosed is $Q=2\pi aL\sigma_0$, so $|\mathbf{E}|2\pi sL=Q/\epsilon_0$ yields $\mathbf{E}=(a\sigma_0/s\epsilon_0)\hat{\mathbf{s}}$.</p>
<p>Then consider the coaxial cable and a Gaussian cylinder of length L. Inside the cable, Gauss's law tells us the electric field is 0. For $b<s<c$, the enclosed charge is $Q=\rho_0\pi L(s^2-b^2)$, and the surface integral is equal to $|\mathbf{E}|2\pi sL$, so the electric field is $\mathbf{E}=(\rho_0/\epsilon_0)(s^2-b^2)(1/2s)\hat{\mathbf{s}}$, and similarly for $s>c$, it is $\mathbf{E}=(\rho_0/\epsilon_0)(c^2-b^2)(1/2s)\hat{\mathbf{s}}$.</p>
<p>Now, I combine these expressions to arrive at a piecewise function for the electric field.</p>
<p>$$
\mathbf{E}(\mathbf{r})=\left\{
\begin{array}{lr}
0 & s<a\\
\frac{a\sigma_0}{s\epsilon_0}\hat{\mathbf{s}} & a<s<b\\
\frac{a\sigma_0}{\epsilon_0}\frac{1}{s}\left(1+\frac{s^2-b^2}{b^2-c^2}\right)\hat{\mathbf{s}} & b<s<c\\
0 & c<s
\end{array}
\right.
$$</p>
<p>Now, from the electric field, we will calculate the potential, taking $\infty$ as our reference point and working our way inwards, to speak. The potential for $s>c$ is of course 0, because the electric field there is also 0.</p>
<p>Suppose that $b<s<c$. Then the potential is $V=V_1(s)$, where
$$
\begin{align}
V_1(s)&=-\int_c^s\left(\frac{a\sigma_0}{s'\epsilon_0}+\frac{\rho_0(s'^2-b^2)}{2s'\epsilon_0}\right)\,ds'=-\frac{a\sigma_0}{\epsilon_0}\int_c^s\frac{1}{s'}\left(1+\frac{s'^2-b^2}{b^2-c^2}\right)\,ds'=-\frac{a\sigma_0}{\epsilon_0}\left.\left(\frac{s'^2-2c^2\ln s}{2b^2-2c^2}\right)\right|_c^s\\
&=\frac{a\sigma_0}{2\epsilon_0(b^2-c^2)}\left(c^2-s^2+2c^2\ln(s/c)\right)=\frac{a\sigma_0}{\epsilon_0}\frac{1}{2}\left(\frac{c^2-s^2+2c^2\ln(s/c)}{b^2-c^2}\right)
\end{align}
$$</p>
<p>Suppose that $a<s<b$. Then the potential is $V=V_1(b)+V_2(s)$, where
$$V_1(b)=\frac{a\sigma_0}{2\epsilon_0(b^2-c^2)}\left(c^2-b^2+c^2\ln(b/c)\right)=\frac{a\sigma_0}{\epsilon_0}\left(\frac{c^2\ln(b/c)}{b^2-c^2}-\frac{1}{2}\right)$$
$$
\begin{align}
V_2(s)&=-\int_b^s\frac{a\sigma_0}{s'\epsilon_0}\,ds'=\frac{a\sigma_0}{\epsilon_0}\ln(b/s)
\end{align}
$$
Suppose that $s<a$. Then the potential is $V=V_1(b)+V_2(a)$, where
$$V_2(a)=\frac{a\sigma_0}{\epsilon_0}\ln(b/a)$$</p>
<p>To summarize,</p>
<p>$$
V(s)=\left\{
\begin{array}{lr}
\frac{a\sigma_0}{\epsilon_0}\left(\frac{c^2\ln(b/c)}{b^2-c^2}-\frac{1}{2}\right)+\frac{a\sigma_0}{\epsilon_0}\ln(b/a) & s<a\\
\frac{a\sigma_0}{\epsilon_0}\left(\frac{c^2\ln(b/c)}{b^2-c^2}-\frac{1}{2}\right)+\frac{a\sigma_0}{\epsilon_0}\ln(b/s) & a<s<b\\
\frac{a\sigma_0}{\epsilon_0}\frac{1}{2}\left(\frac{c^2-s^2+2c^2\ln(s/c)}{b^2-c^2}\right) & b<s<c\\
0 & c<s
\end{array}
\right.
$$</p>
<p>Next, I will proceed to calculate the electrostatic energy. Since I wish to obtain the energy per unit length, I will omit the integration along the $z$ axis from $z_0$ to $z_0+L$, because the multiplicative factor of $L$ will end up disappearing in the end anyway.</p>
<p>I can calculate the electrostatic energy with $W=\frac{1}{2}\int\rho V\,\mathrm{d}\tau$. We only need to consider the regions where $\rho$ is nonzero, namely the cylindrical surface and $b<s<c$. We have
$$
\begin{align}
W&=\frac{1}{2}\int\rho V\,d\tau\\
&=\pi a\sigma_0\frac{a\sigma_0}{\epsilon_0}\left(\frac{c^2\ln(b/c)}{b^2-c^2}-\frac{1}{2}\right) + \pi a\sigma_0\frac{a\sigma_0}{\epsilon_0}\ln(b/a) \\
&\quad + 2\pi\frac{1}{2}\frac{-2a\sigma_0}{c^2-b^2}\int_b^c\frac{a\sigma_0}{2\epsilon_0(b^2-c^2)}s\left(c^2-s^2+c^2\ln(s/c)\right)\,ds\\
&=\frac{\pi a^2\sigma_0^2}{\epsilon_0}\left[\left(\frac{c^2\ln(b/c)}{b^2-c^2}-\frac{1}{2}\right)+\ln(b/a)+\frac{1}{(b^2-c^2)^2}\int_b^cs\left(c^2-s^2+2c^2\ln(s/c)\right)\,ds\right]\\
&=\frac{\pi a^2\sigma_0^2}{\epsilon_0}\left[\left(\frac{c^2\ln(b/c)}{b^2-c^2}-\frac{1}{2}\right)+\ln(b/a)+\frac{1}{(b^2-c^2)^2}\left.\left(c^2s^2\ln(s/c)-\frac{1}{4}s^4\right)\right|_b^c\right]\\
&=\frac{\pi a^2\sigma_0^2}{\epsilon_0}\left[\left(\frac{c^2\ln(b/c)}{b^2-c^2}-\frac{1}{2}\right)+\ln(b/a)+\frac{1}{(b^2-c^2)^2}\left(-b^2c^2\ln(b/c)-\frac{1}{4}c^4+\frac{1}{4}b^4\right)\right]
\end{align}
$$</p>
<p>Now, I will calculate the electrostatic energy with $W=\frac{\epsilon_0}{2}\int E^2\,\mathrm{d}\tau$. We have
$$
\begin{align}
W&=\frac{\epsilon_0}{2}\int E^2\,d\tau=\pi\epsilon_0\int_0^\infty E^2s\,ds=\pi\epsilon_0\frac{a^2\sigma_0^2}{\epsilon_0^2}\int_a^bs\frac{1}{s^2}\,ds + \pi\epsilon_0\frac{a^2\sigma_0^2}{\epsilon_0^2}\int_b^c\frac{1}{s}\left(1-\frac{s^2-b^2}{c^2-b^2}\right)^2\,ds\\
&=\frac{\pi a^2\sigma_0^2}{\epsilon_0}\left(\ln(b/a)+\frac{1}{(b^2-c^2)^2}\left.\left(c^4\ln s-c^2s^2+s^4/4\right)\right|_b^c\right)\\
&=\frac{\pi a^2\sigma_0^2}{\epsilon_0}\left(\ln(b/a)+\frac{c^4\ln c-c^4\ln b-c^4+b^2c^2+(1/4)c^4-(1/4)b^4}{(b^2-c^2)^2}\right)
\end{align}
$$</p>
<p>Upon examination, it seems they are not equal. Obviously, this is wrong.</p>
<p>Edit: I fixed it! They're equal now!</p> | 5,973 | [
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<p>When you are sitting in a room where there is a source of bad smell, such as somebody smoking or some other source of bad smell, it is often a solution to simply move to another spot where bad smell is not present. Assuming you are not actually the source of the smell, this will work for a while until you notice the smell has somehow migrated to exactly the spot where you are now sitting. Frustrating. </p>
<p>This got me thinking about the fluid mechanics of this problem. Treat bad smell as a gas that is (perhaps continuously) emitted at a certain fixed source. One explanation could be that human breathes and perhaps creates a pressure differential that causes the smell to move around. Is there any truth to this? Please provide a reasoned argument with reference to the relevant thermodynamic and/or fluid quantities in answering the question. Theoretical explanation is desired, but extra kudos if you know of an experiment.</p> | 5,974 | [
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<p>In the book "Reflections on Relativity" by Kevin Brown, there is a chapter called "Relatively Straight", in which he derives the geodesic equations using the Euler equation. <a href="http://mathpages.com/rr/s5-04/5-04.htm" rel="nofollow">Online version</a></p>
<p>Just after the second mention of the Euler equation (about 80% down), there is the following text: "Therefore, we can apply Euler's equation to immediately give the equations of geodesic paths on the surface with the specified metric</p>
<p>$$ \frac{\partial F}{\partial x^\sigma} - \frac{d}{d\lambda}\frac{\partial F}{\partial \dot{x}^\sigma}$$</p>
<p>For an n-dimensional space this represents n equations, one for each of the coordinates $x_1, x_2, ..., x_n$. Letting $w = (\frac{ds}{dl})^2 = F^2 = g_{\alpha\beta}\dot{x}^\alpha\dot{x}^\beta$ this can be written as</p>
<p>$$\frac{\partial w^{1/2}}{\partial x^\sigma} - \frac{d}{d\lambda}\frac{\partial w^{1/2}}{\partial \dot{x}^\sigma} = \frac{d}{d\lambda}\frac{\partial w}{\partial \dot{x}^\sigma} - \frac{\partial w}{\partial x^\sigma} - \frac{1}{2w}\frac{dw}{d\lambda} \frac{\partial w}{\partial \dot{x}^\sigma} = 0 $$</p>
<p>I get the substitution of sqrt(w) for F on the LHS, but can't see how he obtains the middle expression. I have tried using the product/chain rules as is usual with these things, but just cannot see what he is doing here.</p>
<p>I can usually follow Kevin's work, with a bit of effort, but this one seems a little trickier than I am used to. Can anyone help me to understand the trick?</p> | 5,975 | [
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<p>Plasma consists of positive(ions) and negative(electrons) charges. Ions repel each other, so do the electrons, in the mean time ions attract electrons and vice versa, what will be the net result? Does the plasma tend to collapse or expand or stay in constant density? Why is it stable? </p> | 5,976 | [
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<p>When someone comes in from the cold to a heated room, it sometimes feels like there is coldness radiating away from that person. Is there a sense in which we can say that coldness radiates similarly to heat radiation? Does it make sense to have a theory of thermodynamics of coldness in which heat is replaced with coldness (-heat)?</p> | 5,977 | [
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<p>In classical mechanics, there are three types of invariance: invariance, form invariance and gauge invariance.
I am looking for a precise definition of these terms, but all I can find are sentences like "a physical law is form invariant if the form of the law does not change under coordinate transformations" or "a quantity is invariant if the transformed quantity is the same function of spacetime", which I don't understand and which sound more like alchemy than physics to me.</p>
<p>So, let's say we have a physical system with phase space $S = \mathbb{R}^{2n}$, a Hamilton function $H: S×ℝ→ℝ, (x,t)↦H(x,t)$, some observable $G: S×ℝ→ℝ^m, (x,t)↦G(x,t)$ and the solution $(q, p): ℝ→S, t ↦ (q(t), p(t))$ to Hamilton's equations $\frac{dp_k(t)}{dt}=-\frac{∂H(p(t), q(t), t)}{∂q_k}$ and $\frac{dq_k(t)}{dt} = \frac{∂H(q(t), p(t), t}{∂p_k}$ for some initial value.</p>
<p>Now, let's look at a $C^∞$-diffeomorphism $φ: S×ℝ→S, (q, p, t) → (q', p')$, that is a function $φ:S×ℝ→S$, which is arbitrarily often differentiable, invertible, and whose inverse function is also arbitrarily often differentiable (a "smooth" coordinate transformation).</p>
<p>Here are the definitions as far as I understand:</p>
<p>$G$ is called invariant with respect to φ if and only if
$∀ t ∈ ℝ: G(q(t), p(t), t) = G(φ(q(t), p(t), t), t)$.</p>
<p>$H$ is called gauge-invariant with respect to φ if and only if there is a function $(H': S×ℝ→ℝ)$ for which the canonical equations $\frac{p'_k(t)}{dt} = -\frac{∂H'(q'_k(t), p'_k(t), t)}{∂q'_k}$, $\frac{q'_k(t)}{dt} = \frac{∂H'(q'_k(t), p'_k(t), t)}{∂p'_k}$, where $(q'(t), p'(t)) = φ(q(t), p(t), t)$), hold.</p>
<p>Are these definitions correct? If not: <strong>In precise mathematical terms, what does it mean for G to be form invariant, invariant, or for H to be gauge invariant?</strong></p> | 5,978 | [
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<p>Can we scatter the lightning before it hits the ground?Can we scatter the lightning by connecting many wires to a lightning rod?</p> | 5,979 | [
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<p>I am currently working on a (functional) analysis problem refining Pekar's Ansatz (or adiabatic approximation, as it is called in his beautiful 1961 manuscript "Research in Electron Theory of Crystals"). </p>
<p>Anyways, I have two related questions, which the members of this community may find simple. </p>
<p>The Fröhlich Hamiltonian is given as follows in three dimensions</p>
<p>$$H=\mathbf{p^{2}}+\sum_{k}a_{k}^{\dagger}a_{k}-\biggl(\frac{4\pi\alpha}{V}\biggr)^{\frac{1}{2}}\sum_{k}\biggl[\frac{a_{k}}{|\mathbf{k}|}e^{i\mathbf{k\cdot x}}+\frac{a_{k}^{\dagger}}{|\mathbf{k}|}e^{-i\mathbf{k\cdot x}}\biggr]$$</p>
<p>The physical scenario here is an electron moving in a 3-dimensional crystal. Each $k$ signifies a (vibrational) mode of the crystal. </p>
<p>If we restrict ourselves to just a 1-dimensional crystal, why is it that the Hamiltonian can be written as follows: </p>
<p>$$H=\mathbf{p^{2}}+\sum_{k}a_{k}^{\dagger}a_{k}-\biggl(\frac{4\pi\alpha}{V}\biggr)^{\frac{1}{2}}\sum_{k}\biggl[a_{k}e^{i\mathbf{k\cdot x}}+a_{k}^{\dagger}e^{-i\mathbf{k\cdot x}}\biggr]$$</p>
<p>Namely, why do we drop the $|\mathbf{k}|$ factor in the third term? </p>
<p>Furthermore, I see how the creation and annihilation operators work on the (bosonic) Fock space (referring to the crystal here), especially when we write the creation operator in the form $\sum_{k=0}^{\infty}\frac{(a^{\dagger})^{k}}{\sqrt{k!}}\left|0\right\rangle =\left|k\right\rangle$. Namely, the creation operator is jumping from one tensored state in Fock Space to the next. However, I also see the form $a_{k}=\frac{1}{\sqrt{2}}\bigl(k+\frac{d}{dk}\bigr)$. How are the two forms connected? How do you intuitively think of the latter form? For example, I thought of the former form as the creation operator jumping from one state in fock space to the next, but the latter form I am not quite sure. </p> | 5,980 | [
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<p>I'm trying to figure out how much distance does a ball of balsa wood covers until reaching terminal velocity, being released from a bottom of a pool.</p>
<p>To my understanding, I need to first figure out the terminal velocity. For that, I take:</p>
<p>$$F_\text{drag} + mg = F_\text{buoyancy}$$</p>
<p>$$F_\text{drag} = C_d A \frac{\rho_m V_t^2}{2}$$
$$F_\text{buoyancy} = V \rho_m g$$
$$mg = V \rho_s g$$</p>
<p>$\rho_m$ is the liquid density, and $\rho_s$ is the ball. $C_d$ is the drag coefficient. A the surface of the ball, $V$ the volume of the ball.</p>
<p>From this equation I reached:
$$v_\text{terminal}=\sqrt{\frac{2g}{\gamma}\left(1-\frac{\rho_s}{\rho_m}\right)}$$</p>
<p>$\gamma$ being defined as shape coefficient:
$$ \gamma = C_d A / V$$</p>
<p>That looks good, and gives ok results. But when I try to go ahead and find the time / distance I get weird data.
I start from this equation:
$$m\frac{dv}{dt}=F_b-F_d-mg$$
Solving:
$$\frac{dv}{dt}=g\left(\frac{\rho_m}{\rho_s}-1\right) - 0.5\frac{\rho_m}{\rho_s}\gamma v^2$$
with the integral border $v_t$ I found couple of lines before, I get an arctanh of 1, meaning infinity.</p>
<p>I checked the integral, can't find anything wrong. What am I missing here?</p> | 5,981 | [
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<p>Somehow when I google about the nuclear bombs I find a lot of books and resources that seem to explain everything about how those bombs are made. But sometimes I often hear that countries that want to make them usually don't have the technology to do them. But I always questioned myself, as why?! I mean it looks to me that almost all information are out there in books and research papers, so are there remaining secrets or so?!</p>
<p>EDIT:
My question is not related to politics at all!!! What I meant by secrets, I meant physical secrets. I'm talking about scientific secrets that are still not exposed or people are unable to find solutions to.</p> | 5,982 | [
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<p>Newton's second law says $F=ma$.
Now if we put $F=0$ we get $a=0$ which is Newton's first law.
So why do we need Newton's first law ? </p>
<p>Before asking I did some searching and I got this: Newtons first law is necessary to define inertial reference frame on which the second law can be applied.</p>
<p>But why can't we just use Newton's second law to define an inertial frame? So if $F=0$ but $a$ is not equal to 0 (or vice versa), the frame is non-inertial.</p>
<p>One can say (can one?) we cannot apply the second law to define a reference frame because it is only applicable to inertial frames. Thus unless we know in advance that a frame is inertial, we can't apply the second law.</p>
<p>But then why this is not the problem for the first law?</p>
<p>We don't need to know it in advance about the frame of reference to apply the first law. Because we take the first law as definition of an inertial reference frame.</p>
<p>Similarly if we take the second law as the definition of an inertial frame, it should not be necessary to know whether the frame is inertial or not to apply the second law (to check that the frame is inertial).</p> | 5,983 | [
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<p>This is really a one-and-a-half part question.</p>
<p>I know that when paint is mixed with a solvent or used with a primer, it sometimes wrinkles. As I understand, a key physical phenomena here is a non-uniform evaporation of the primer or solvent that gives rise to different evaporation rates at the paint surface. This can create a non-uniform temperature difference. This in turn can create a thermocapillarity (surface tension effects due to non-uniform temperature distribution) or solutocapillarity (surface tension effects due to non-uniform solvent/solute concentration) effects that wrinkle the paint surface.</p>
<p>So is this a solvent problem or a primer problem? Am I understanding the physics right but the nature of primer and solvent wrong?</p> | 5,984 | [
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<p>A large electroscope is made with "leaves" that are $78$-cm-long wires with tiny $24$-g spheres at the ends. When charged, nearly all the charge resides on the spheres.</p>
<p>If the wires each make a $26^{\circ}$ angle with the vertical, what total charge Q must have been applied to the electroscope? Ignore the mass of the wires.</p>
<p>We know that the formulas to use are:</p>
<p>$$F = mg \quad \text{ and } \quad F = \dfrac{kq_1q_2}{r^2}$$</p>
<p>Working out the force equations, we obtain:</p>
<p>$$\Sigma F_x = F_T \cos(\theta) - mg = 0 \rightarrow F_T = \dfrac{mg}{\cos(\theta)}$$
$$\Sigma F_y = F_T \sin(\theta) - F_E = 0 \rightarrow F_E = F_T\sin(\theta) = mg\tan(\theta)$$</p>
<p>So</p>
<p>$$F_E = k\dfrac{(\frac{Q}{2})^2}{d^2} = mg\tan(\theta) \rightarrow Q = 2d\sqrt{\dfrac{mg\tan(\theta)}{k}}$$</p>
<p>By substitution</p>
<p>$$Q = 2(7.8 \times 10^{-1} \mbox{m})\sqrt{\dfrac{(24 \times 10^{-3} \mbox{kg})(9.8 \mbox{m}/\mbox{s}^2)(\tan(26^{\circ}))}{9.0 \times 10^{9} \mbox{N}\cdot\mbox{m}^2/\mbox{C}^2}}$$</p>
<p>I got $5.6 \times 10^{-6}$, but it's incorrect.</p>
<p>Any suggestions or advices here?</p> | 5,985 | [
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<p>what is the key difference between the windings of the electric motor and electric generator?</p> | 5,986 | [
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<p>I've read that maximally entangled qubit states are a good "unit" of bipartite entanglement since it is possible to create any other entangled state from them using local operations and classical communication (LOCC) provided sufficiently many copies are available. What would the protocol be to construct a maximally entangled qutrit ($\vert \psi \rangle_{AB} = \frac{1}{\sqrt{3}}(\vert 00 \rangle + \vert 11 \rangle + \vert 22 \rangle)$) between two space separated parties from a set of $n$ Bell states ($\vert \Phi^+ \rangle^{\otimes n}_{AB} = 2^{-n/2} (\vert 00 \rangle + \vert 11 \rangle)^{\otimes n}$) initially shared by those parties using only LOCC?</p>
<p>If you can, please include in your answer why local operations alone would not be sufficient.</p> | 5,987 | [
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<p>I am working on an algorithm for a real-time simulation. </p>
<p>I would like to calculate to extremely permissive tolerances approximate values for the stress within a 2D geometry. It will not be difficult to triangulate so we can assume a mesh or grid is available. </p>
<p>From what I have read, it seems that in real applications it is essentially impossible to directly measure stress, and it can only be deduced by observing strain or deformation. Is this true? Does this mean that there exists no method, analytical or numerical, that will allow me to produce meaningful numeric values? </p>
<p>To be specific, I am working in only 2 dimensions, with a rigid polygonal shape. I'd like to calculate the distribution of stresses within the geometry when an external force is applied to a point on the boundary.</p>
<p>As an example, consider the geometry of the capital letter L. If a force is applied at the top, in a westerly direction (left), then there would be tensile stress applied at the inside corner of the L, and little bit of compression at the outside corner. There would also be some high values found near the point of application. </p>
<p>I arrived at those conclusions by intuition but is there an algorithm I can use to calculate this numerically or analytically without generating a mesh and deforming it first? I'm okay with having to produce a mesh, but requiring me to simulate deformation before I can produce values for the stress tensor is just not feasible for a real-time solution.</p>
<p>I have come up with an idea for an algorithm that I will try to implement, which will hopefully be sufficient for my purposes. I thought about how the very concept of stress is at odds with a rigid body system. It seemed like an irreconcilable paradox. But I realized that I can have rigid objects, but I can simply let them break at critical moments, and the broken pieces can continue to be rigid. Rigid objects break at points on their geometry which are the weakest. Assuming they are made of the same material and density, areas of small cross-section are weak: Momentum transfer and the clash of internal forces occur through these cross-sections, and when there is not enough area to transmit these forces, failure occurs. </p>
<p>So, I can take my geometry and use some fuzzy sampling and determine "weak areas" by looking for the shortest cross sections (in 2D the split planes are simply line segments). Then, if I can calculate the amount of momentum that is required to pass through the segment in order to keep the object rigid, and compare that with a predetermined value, I think it might just work. </p>
<p>The problem with this is that I'm not producing a stress tensor field. I am relying almost entirely on heuristics, and I'm producing a split plane by sampling from a random set and seeing which ones behave most realistically. It might possibly look good with tweaking but it's so non-physical that I really want to find a more robust method.</p> | 5,988 | [
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<p>In a Physics lesson today our teacher performed a demonstration to show how quinine in tonic water glows blue when under UV light. He showed us the same demonstration using water in a Pyrex beaker, and with just the Pyrex beaker itself in order to compare the effects.</p>
<p>Under UV light the Pyrex beaker glowed orange. Why did this happen?</p>
<p>And more generally, what determines the colour of visible light emitted by the object under UV light?</p>
<p>If you were able to shed some light (:D) on this it would be appreciated.</p> | 5,989 | [
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<p>There are proofs in the literature that QFT including microcausality is <em>sufficient</em> for it not to be possible to send signals by making quantum mechanical measurements associated with regions of space-time that are space-like separated, but is there a proof that microcausality is <em>necessary</em> for no-signaling?</p>
<p>I take microcausality to be trivial commutativity of measurement operators associated with space-like separated regions of space-time. In terms of operator-valued distributions $\hat\phi(x)$ and $\hat\phi(y)$ at space-like separated points $x$ and $y$, $\left[\hat\phi(x),\hat\phi(y)\right]=0$.</p>
<p>An elementary observation is that classical ideal measurements satisfy microcausality just insofar as all ideal measurements are mutually commutative, irrespective of space-like, light-like, or time-like separation. For a classical theory, ideal measurements have no effect either on the measured system or on other ideal measurements, so ideal measurements cannot be used to send messages to other ideal experimenters. Whether a classical theory admits space-like signaling <em>within</em> the measured system, in contrast to signaling by the use of ideal measurements applied by observers who are essentially outside the measured system, is determined by the dynamics. Applying this observation to QFT, I note that microcausality is not sufficient <em>on its own</em> —without any dynamics, which I take to be provided in QFT by the spectrum condition— to prove no-signaling at space-like separation.</p>
<p>As an auxiliary question, what proofs of sufficiency do people most often cite and/or think most well-stated and/or robust?</p> | 717 | [
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<p>How can I diagonalize the following operator?</p>
<p>$$\lambda \hat{\vec{\sigma}}\cdot\vec{r}$$</p>
<p>where $\lambda$ is a real constant, $\hat{\vec{\sigma}}=(\hat{\sigma_{x}},\hat{\sigma_{y}},\hat{\sigma_{z}}) $ is the Pauli operator and $\vec{r}=(x,y,z)$ is the position operator.</p> | 5,990 | [
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] |
<p>If you go to the Atlas experiment <a href="http://atlas.ch/" rel="nofollow">http://atlas.ch/</a> and click the status button, there's an AFS reading at the bottom with a current value</p>
<p>50ns_228b+1small_214_12_180_36bpi_8inj</p>
<p>The 50ns seems to refer to the bunch spacing, the 228b the number of bunches, but what do the other numbers refer to?</p> | 5,991 | [
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<p>Physically, quantum entanglement is ranged from full long-range entanglement (Bose-Einstein condensate), described by a basis of states that look like this:</p>
<p>$$ |\Psi\rangle = |\phi_{i_{0} i_{1} ... i_{N}}\rangle $$</p>
<p>to full-decoherence (a Maxwell-Boltzmann ideal gas) which a basis of states that look like this:</p>
<p>$$ |\Psi\rangle = \prod_{i}{ |\phi_{i}\rangle } $$</p>
<p>And in the middle of this range we have 2-particle entanglement terms, 3-particle entanglement, etc.</p>
<p>So it seems natural to arrange the wavefunction as a series looking like:</p>
<p>$$ |\Psi\rangle = \sum{ \prod_{i}{ |\phi_{i}\rangle } } + \sum{ \lbrace \prod_{i_{0} ,i_{1} > i_{0}}{ |\phi_{i_{0} i_{1}}\rangle } \rbrace } + \sum{ \lbrace \prod_{i_{0} ,i_{1} > i_{0} , i_{2} > i_{1}}{ |\phi_{i_{0} i_{1} i_{2}}\rangle } \rbrace } + \dotsb $$</p>
<p><strong>BEGIN EDIT</strong> I feel that i need to put in a bit more clear footing the mathematics behind this separation. Let's take an arbitrary state vector $|\Psi\rangle$, we might write it like this:</p>
<p>$$ |\Psi\rangle = \prod_{i}{ c^{0}_{i} |\phi_{i}\rangle } + |\Psi_{Remainder}\rangle $$</p>
<p>That is, we write the state vector as a vector that is completely separable and a remainder that is not. <em>This decomposition is unique</em>. Proof: take another decomposition with $\widehat{c^{0}_{i}}$, take the difference between both decompositions and verify that both remainders are completely separable, which is against the definition</p>
<p>The idea is that one should be able to further this decomposition of the remainder state vector, for instance lets take a vector with zero completely separable part (that is, we are on the equivalence class of our remainder above) and attempt to write it like:</p>
<p>$$ |\Psi_{R_0}\rangle = \prod_{i_{0} ,i_{1} > i_{0}}{ c^{1}_{i_0 i_1} |\phi_{i_{0} i_{1}}\rangle } + |\Psi_{R_1}\rangle $$</p>
<p>Analogously, one can prove that the $c^{1}_{i_0 i_1}$ are unique and depend only on the $|\Psi_{R_0}\rangle$ vector. And we don't need to do any symmetrization operation to get this result, so this is an universal decomposition of the state vector (for bosons and fermions)</p>
<p>(note: i'm aware that this leaves out a lot of products with mixed entanglement, i.e: some single-particle states multiplied by two-particle entangled states, but since they don't add anything to this particular argument i choosed to leave them aside)</p>
<p><strong>END EDIT</strong></p>
<p><strong>BEGIN 2ND EDIT</strong></p>
<p>separability of states is a property that is invariant under unitary transformations, so a separable state is not equivalent to a separable one in any basis you choose.</p>
<p>I've investigated a bit more and the problem in general of knowing if a state is separable or not is known as QSP (quantum separability problem). For a definition please look at <a href="http://arxiv.org/abs/cs/0603047" rel="nofollow">this paper</a></p>
<p><strong>END 2ND EDIT</strong></p>
<p>Question:
Where can i read more about this sort of expansion and do you know if there is a computational framework to estimate the relative magnitudes of each term (for instance, i would expect that for Bose-Einstein condensates you need to keep all the terms of the expansion, while for relatively high-temperature solids you would be able to get away with 3 or 4 terms)</p> | 5,992 | [
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<p>How much body temperature can a motor withstand along with the ambient temperature to work without failing?</p> | 5,993 | [
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<p>Can the value of Newton's <a href="http://en.wikipedia.org/wiki/Gravitational_constant" rel="nofollow">Gravitational Constant</a> $G$ be measured within a stably bound atom?</p>
<p>PLEASE NOTE: Since scattering experiments do not involve stably bound systems, their results are not germane to the specific question asked here.</p> | 5,994 | [
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<p>Does the mass of both the parent object, and the child object affect the speed at which the child object orbits the parent object?</p>
<p>I thought it didn't (something like $T^2 \approx R^3$) until I saw an planet on the iphone exoplanet app, that is closer to it's star than a planet in another system, yet takes longer to complete one orbit. Both of the planets were a similar mass, as were the stars.</p> | 375 | [
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<p>There are two experiments that are often used to explain Quantum Mechanics: the two-slit experiment and the EPR paradox. I am curious what would happen if you combined them.</p>
<p>Imagine an experiment where you fire pairs of entangled particles at two simultaneous two-slit setups. If you used detectors, you could find out how the entangled particles' paths correlate. Perhaps you'd be able to deduce, based on the result of one detector, which slit the other particle went through. Now, if you were to run the experiments with a detector on one side and no detector on the other side, would the unobserved particles still form an interference pattern, even though you know which slit they would have went through?</p>
<p>My intuition is that the answer is yes. Despite being entangled, the particles should not have correlated actions, otherwise we would have invented faster-that-light communication. You could create a device that constantly fired entangled particles toward two far-apart worlds, and if one world suddenly started observing the particles on their side, the particles arriving at the other world would instantly cease to create an interference pattern.</p> | 5,995 | [
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<p>I have a hard time understanding why light waves of different wavelengths diffract in a different manner. According to Huygens' principle, every point on the wavefront is a source of a secondary wave. So if we have a white light going through, say, a single slit (light rays parallel to each other and perpendicular to slit's plane), all what's supposed to happen is a plain diffraction, just like of any other wave. That is, the wave will progress spherically, but it will still be a white light. Why instead we get a splitting of different wavelengths? In other words, how does light color affect diffraction geometrically?</p> | 5,996 | [
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<p>Please refer to the question "Resolution of the EPR paradox using relativity of simultaneity" I hope that this question will be clearer.</p>
<p><strong>EPR photon experiments are a paradox for scientists. Can someone tell me why?</strong></p>
<p>Distances and traveling time of photons are always measured on the basis of light speed c (300.000 km/sec.). However, according to time dilation formulas of special relativity (T'=T/γ) these measured values are no <strong>real</strong> values, but only <strong>observed</strong> values, observed by any observer. An application of these formulas (and more precisely: of the factor <em>reciprocal gamma</em>) yields that the <strong>real</strong> speed of light is <strong>not defined</strong> (zero-by-zero division) and the distances and times are shrinking down to zero. As a result it seems clear that all events of photons are taking place simultaneously, and thus photons may be here and there simultaneously - thus no paradox at all!</p>
<p>I learned already that possibly relativity and time dilation do not apply to particles moving at light speed. But what is the sound reason for such an exception if on the other hand this is prohibiting a resolution of the EPR paradox?</p> | 5,997 | [
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<p>I know there are two ways of measuring the angle of a prism with a goniometer:</p>
<blockquote>
<p>let the collimator shine (monochromatic) light on 2 sides of the prism and measure the angle between the 2 reflected lightbeams. </p>
</blockquote>
<p>or </p>
<blockquote>
<p>Let the lightbeam hit one side of the prism, note down the position of the the table of the goniometer where the prism is on (=$\beta_1$) and check the location of the reflected lightbeam, turn the table around counter-clockwise until the reflected lightbeam is in the same place and note down that position (=$\beta_2$).</p>
</blockquote>
<p>Now, my problem is that, while I understand completely how to perform these actions, I don't understand why:<br>
In the first case the measured angle is the angle of the prism times two.<br>
In the second case when subtracting $|\beta_1-\beta_2$| from $180°$ you get the angle of the prism.</p> | 5,998 | [
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<p>I'm studying a problem and I encountered a strange problem:</p>
<blockquote>
<p><em>When a ball bounces how much time does the ball spend while touching the floor?</em></p>
</blockquote>
<p>To be more clear I suppose that when a ball bounces the actual bounce can't start EXACTLY when the ball touches the floor but the ball touches the floor, the Energy from the fall is given to the floor then the floor gives the energy back and the ball bounce; but in what time?</p> | 5,999 | [
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<p>In page 488 of Peskin and Schroeder, it is stated (emphasis mine):</p>
<blockquote>
<p>It is not difficult to check using (15.27) and (15.21) that, even for <em>finite</em> transformations, the covariant derivative has the same transformation law as the field on which it acts.</p>
</blockquote>
<p>I was trying to indeed verify that.</p>
<p>This is what I have tried:</p>
<ol>
<li>$V\left(x\right)\in SU\left(2\right)^{\mathbb{R}^4}$</li>
<li>$\psi\left(x\right)\mapsto V\left(x\right)\psi\left(x\right)$.</li>
<li>$D_\mu\left(x\right)\equiv\partial_\mu-igA_{\mu,\,j}\left(x\right)\frac{\sigma^j}{2}$.</li>
<li>$igA_{\mu,\,j}\left(x\right)\frac{\sigma^j}{2}\mapsto V\left(x\right)igA_{\mu,\,j}\left(x\right)\frac{\sigma^j}{2}\left[V\left(x\right)^\dagger\right]-V\left(x\right)\left\{\partial_\mu\left[V\left(x\right)^\dagger\right]\right\}$</li>
</ol>
<p>Thus:</p>
<p>\begin{align} D_\mu\left(x\right)\psi\left(x\right) \mapsto & \left\{\partial_\mu -V\left(x\right)igA_{\mu,\,j}\left(x\right)\frac{\sigma^j}{2}\left[V\left(x\right)^\dagger\right]+V\left(x\right)\left\{\partial_\mu\left[V\left(x\right)^\dagger\right]\right\}\right\}V\left(x\right)\psi\left(x\right) = \\\ &= \left[\partial_\mu V\left(x\right)\right]\psi\left(x\right)+V\left(x\right)\partial_\mu\psi\left(x\right)-V\left(x\right)igA_{\mu,\,j}\left(x\right)\frac{\sigma^j}{2}\psi\left(x\right)+V\left(x\right)\left\{\partial_\mu\left[V\left(x\right)\dagger\right]\right\}V\left(x\right)\psi\left(x\right) = & \\\ &= V\left(x\right)D_\mu\left(x\right)\psi\left(x\right)+\left\{\left[\partial_\mu V\left(x\right)\right] + V\left(x\right)\left\{\partial_\mu\left[V\left(x\right)\dagger\right]\right\}V\left(x\right)\right\}\psi\left(x\right) \end{align}</p>
<p>So as far as I understand, $\boxed{\left[\partial_\mu V\left(x\right)\right] + V\left(x\right)\left\{\partial_\mu\left[V\left(x\right)\dagger\right]\right\}V\left(x\right)\stackrel{?}{=}0}$ should be zero.</p>
<p>To prove that I have used the fact that $VV\dagger=1$: </p>
<p>\begin{align} \partial_\mu\left[ V\left(x\right)\right] + V\left(x\right)\left\{\partial_\mu\left[V\left(x\right)\dagger\right]\right\}V\left(x\right) &= \\ \partial_\mu\left[ V\left(x\right)1\right] + V\left(x\right)\left\{\partial_\mu\left[V\left(x\right)\dagger\right]\right\}V\left(x\right) &= \\ \partial_\mu\left[ V\left(x\right)V\left(x\right)^\dagger V\left(x\right)\right] + V\left(x\right)\left\{\partial_\mu\left[V\left(x\right)\dagger\right]\right\}V\left(x\right) &= \\ \left[\partial_\mu V\left(x\right)\right]V\left(x\right)^\dagger V\left(x\right) +V\left(x\right)\left[ \partial_\mu V\left(x\right)^\dagger \right]V\left(x\right) +V\left(x\right)V\left(x\right)^\dagger\left[ \partial_\mu V\left(x\right)\right] + V\left(x\right)\left\{\partial_\mu\left[V\left(x\right)\dagger\right]\right\}V\left(x\right) &= \\2\left\{\partial_\mu\left[ V\left(x\right)\right] + V\left(x\right)\left\{\partial_\mu\left[V\left(x\right)\dagger\right]\right\}V\left(x\right)\right\}\end{align}</p>
<p>Is this all correct?</p> | 6,000 | [
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<p>In a badly-adjusted (too cold) fridge, I notice that things like celery & lettuce are more likely to be spoiled by becoming frozen if they're put on a <em>middle</em> shelf, rather than the salad drawer at the <em>bottom</em>.</p>
<p>I realise the salad drawer is "enclosed". But it's far from air-tight, so over time I would expect denser cold air to sink to the bottom and end up filling the salad drawer.</p>
<p>I don't know if it's relevant that I've only seen this phenomenon in fridges with an ice-making compartment at the <em>top</em> of the fridge (as opposed to a "larder fridge" that doesn't have an ice-making compartment). </p>
<p>I know <a href="http://www.goalfinder.com/blog/template_permalink.asp?id=97" rel="nofollow"><strong><em>water</em></strong> expands as it cools below 4°C</a> (the specific part of the temperature scale relevant here), but I'm pretty sure <em>air</em> doesn't do this. So why should the bottom of a fridge end up being the <em>warmest</em>, rather than the <em>coldest</em> part?</p> | 6,001 | [
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<p>I tried to solve a problem using two different ways and I had some trouble, the problem is:</p>
<p>We define a symmetry transform of the expected value of $\vec{P}$ like this:
$$\langle \psi|\vec{P}|\psi \rangle \rightarrow \langle \psi|\vec{P}+m\vec{V_0}|\psi \rangle$$</p>
<p>being $\vec{V_0}$ a constant vector.</p>
<p>If we define the generators $\vec{K}$ like $$\vec{K}=-m \vec{T} $$
prove that we obtain the following relations.</p>
<p>$$ [K_j,P_k]=-im\delta_{jk} $$
$$ [K_j,K_k]=0 $$</p>
<p><strong>And my two solutions are</strong></p>
<p>Using $$U^-1 \vec{P} U = (1-i \vec{\epsilon} \vec{T}) \vec{P} (1+i \vec{\epsilon} \vec{T}) = \vec{P} + m\vec{V_0} = \vec{P} + \vec{\epsilon} $$ i end up with (where $\vec{\epsilon} \equiv m\vec{V_0} $)
$$ [\vec{K},\vec{P}] = -im $$
My problem is if we expand those vectors like this
$$ \vec{K}\vec{P}-\vec{P}\vec{K} = K_iP_i+K_jP_j+K_kP_k-K_iP_i-K_jP_j-K_kP_k = [K_i,P_i]+[K_j,P_j]+[K_k,P_k]=3[K_i,P_i] $$</p>
<p>and this gives $[K_i,P_i]=\frac{-im}{3} $ wich is wrong</p>
<p>But if I start with</p>
<p>$$(1-i \epsilon_i T_i) P_i (1+i \epsilon_i T_i) = P_i + m V_{i0} $$</p>
<p>we obtain the good answer</p>
<p>$$[K_i,P_i]=-im $$</p>
<p>so, my question is, whats wrong with the first method and why it doesn't give me the right answer.</p> | 6,002 | [
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<p>I was reading Planetary Motion (page 117) in Barry Spain's <em>Tensor calculus</em>, and stupidly enough, I didn't understand this. The equations are :</p>
<p>$$\frac{d^2\psi}{d\sigma^2} + \frac{2}{r}\frac{dr}{d\sigma}\frac{d\psi}{d\sigma} = 0,$$</p>
<p>$$\frac{d^2t}{d\sigma^2} + \frac{2m}{c^2r}\left(1-\frac{2m}{c^2r}\right)^{-1}\frac{dr}{d\sigma}\frac{dt}{d\sigma} = 0,$$</p>
<p>And the next statement reads</p>
<p>"we integrate the above to get</p>
<p>$$r^2\frac{d\psi}{d\sigma} = h, \quad \left(1-\frac{2m}{c^2r}\right) \frac{dt}{d\sigma}= k$$</p>
<p>respectively, for constants $k$ and $h$."</p>
<p>I believe it is a simple question, as no steps are give, but I am unable to get it. I tried everything I could, substitutions et al, but to no avail. So please entertain this silly question. Thank you very much... </p>
<p>EDIT Please show the integration, I dont want to confirm the validity of the answers by working backwards, but I want to establish them.
I want to see the actual integration. If I am right, then the second term($2r\frac{dr}{d\sigma}\frac{d\psi}{d\sigma}$) of the first equation yields the desired result which is $r^2\frac{d\psi}{d\sigma}$ when integrated wrt $d\sigma$. It means that the integral of $r^2\frac{d^2\psi}{d\sigma^2}$ is $0$...I think the integral simplifies to $\frac{d}{d\sigma}\int{r^2d\psi}$, and to get the required answer, it should be zero or a constant.....But HOW??? PLease help....</p> | 6,003 | [
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<p>I am making a speech about <a href="http://www.google.com/search?q=miku+hatsune+3d+hologram" rel="nofollow">Miku Hatsune 3D hologram</a>, and I would like to know how they are made! I love Miku and Len...(drools) please help by explaining!</p> | 6,004 | [
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<p>When applying a force outside of the center of mass of the body, the body will get both linear and angular momentum. Right?</p>
<p>Does the linear velocity from this force equal to the linear velocity from the same force, which is applied on the center of mass?</p>
<p>$$v_{\text{center}} = v_{\text{non-center}}\quad\text{? (no vectors)}$$</p>
<p>If it does: How is it that the energy applied to the system is equal? In one we have both rotation and linear movement, and in the other just linear movement.</p> | 6,005 | [
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<p>I've got a question about derivation of RG equations made in "Gauge theory of elementary particles" by <a href="http://rads.stackoverflow.com/amzn/click/0198519613" rel="nofollow">Cheng & Li</a>. At page 79 (page 97 in Russian edition) they represent bare parameters ($\mu_0, \, \lambda_0, \, \phi_0$) as Laurent series (equations (3.80)-(3.82)), where coefficients $a_r,\,b_r,\,c_r$ depend only on $\lambda$. This is the point which I really don't get. I mean, why do these parameters depend only on $\lambda$ but not on $\mu$ or $m$? </p> | 6,006 | [
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<p>A theory among scientists says that quantum fluctuations caused the big bang and created the universe. This seems plausible to me. </p>
<p>What I can't grasp yet is how a quantum fluctuation can even start without an existing universe. Aren't all physical laws created with the universe? I understand that there is no notion of "before" with respect to time, however the big bang is theorised to have occurred, but for that to occur there must have first existed <em>something</em> right?</p>
<p>I wonder also, if there was a more <em>nothingness</em> instead of vacuum before the universe existed and how a quantum fluctuation could have started really from <em>ex nihilo</em> instead of a vacuum.</p> | 6,007 | [
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-0.0278152022510767
] |
<p>For a spinor on curved spacetime, $D_\mu$ is the covariant derivative for fermionic fields is
$$D_\mu = \partial_\mu - \frac{i}{4} \omega_{\mu}^{ab} \sigma_{ab}$$
where $\omega_\mu^{ab}$ are the spin connection.
And the transformation of spin connection is very similar to gauge field.
So is there any relationship between them. If there is any good textbook or reference containing this area, please cite it. Thanks!</p> | 865 | [
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<p>When you energise a taut string, the following resonant modes of vibration occur:</p>
<p><img src="http://i.stack.imgur.com/pA0iS.jpg" alt="enter image description here"></p>
<p>Plotting on the frequency domain, you can see their corresponding frequencies:</p>
<p><img src="http://i.stack.imgur.com/tPSI5.jpg" alt="enter image description here"></p>
<p>But what is the underlying physical principle? Why does this happen?</p>
<p>Is there any way of explaining it that could be understood by a smart 15 year old?</p>
<p>EDIT: I'm going to give my best attempt so far. Here goes:</p>
<ul>
<li><p>We can start with sympathetic resonance.
Sounding a particular frequency, a pure sinewave.
And noticing the string resonates sympathetically at the frequency of each harmonic. Say this is explained and understood.</p></li>
<li><p>Now imagine that plucking a string is equivalent to a burst of white noise, which contains frequencies all the way through the spectrum. This could be approached backwards, by starting with random frequencies and noticing that resultant wave produced looks like white noise.</p></li>
</ul>
<p>If the above is scientifically correct, then it restricts the domain of the question.</p>
<p>I would really like to be able to understand it scientifically and also be able to explain it intuitively.</p>
<p>PS Images from <a href="http://www.embedded.com/design/real-world-applications/4428811/2/Building-an-electronic-guitar-digital-sound-synthesizer-using-a-programmable-SoC" rel="nofollow">http://www.embedded.com/design/real-world-applications/4428811/2/Building-an-electronic-guitar-digital-sound-synthesizer-using-a-programmable-SoC</a></p> | 6,008 | [
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<p>Sorry for my ignorance, but does the result of <a href="http://en.wikipedia.org/wiki/Michelson%E2%80%93Morley_experiment" rel="nofollow">Michelson-Morley experiment</a> have some explanation? Is there some reason why light speed in vacuum is maximum or we just find it by experiments?</p> | 6,009 | [
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<p>The integral,</p>
<p>$$ \iint_{\mathbb{R}^{2+}}\frac{xy}{1+x+y} \mathrm{d}y \, \mathrm{d}x$$</p>
<p>possesses an overlapping divergence when $ x \to \infty $ and $ y \to \infty $. However, under a change of variables to polar coordinates,</p>
<p>$$ \int_{0}^{\infty}dr\int_{0}^{2\pi}du \frac{r^{3}\sin(u)\cos(u)}{1+r(\cos(u)+\sin(u))} $$</p>
<p>If we integrate over the angular variables numerically we are left with a sum of one dimensional divergent integrals, $$ \int_{0}^{\infty}\frac{ar^{3}}{1+br}dr $$</p>
<p>Therefore the overlapping UV divergence is gone. Can be this done to every UV divergent multiple-loop integral?</p> | 6,010 | [
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<p>Here is the question as given in my textbook:</p>
<blockquote>
<p>Find the distance of the object from a concave mirror of focal length 10 cm so that the image size is 4 times the size of the object.</p>
</blockquote>
<p>The solution in my textbook has the following data stated:</p>
<p>$u=-x$ as it is assumed that the object is real.</p>
<p>$v=-4x$ as it is assumed in case 1 that a real image will be formed and $|\frac{v}{u}|=|m|=4$</p>
<p>So now I am not able to understand why the image distance $v$ is taken $-4x$. In the question it is given that the <em>object size</em> is magnified 4 times, and not the distance of the of the object from the mirror. </p>
<p>And the magnification formula is $\frac{-v}{u}=m$, so why does the solution include the modulus of the formula?
Am I missing something in analysing the solution?</p> | 6,011 | [
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