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<p>I am struggling with the following affirmation found in Ryder's QFT book, <a href="http://books.google.com.br/books?id=nnuW_kVJ500C&pg=PA177&lpg=PA177&dq=%22a%20small%20negative%20imaginary%20part%20to%20the%20Hamiltonian%22&source=bl&ots=vpxodARK1V&sig=gFZ9WUfgKBMAPy4olDusYEyRdoU&hl=en&sa=X&ei=kkniT5OTC4Kg6QHwvaAI&redir_esc=y#v=onepage&q=%22a%20small%20negative%20imaginary%20part%20to%20the%20Hamiltonian%22&f=false" rel="nofollow">page 177</a>:</p>
<blockquote>
<p>instead of rotating the time axis as we have done, the ground state contribution may be isolated by adding a small negative imaginary part to the Hamiltonian</p>
</blockquote>
<p>The author refers to an effort to isolate the vacuum state in a sum over energy eigenstates:
$$\langle Q|e^{-i (T-t) H}|q\rangle = \sum \phi_n(q)
\phi^{*}_n(Q)\; e^{-i (T-t) E_n }$$
One option is to make time imaginary: $T \rightarrow \infty e^{- i \epsilon}$. Another, says the author, is to change the Hamiltonian by adding $-\frac{1}{2} i \epsilon q^2$. We would have then: $H^{\epsilon} = H -\frac{1}{2} i \epsilon q^2$, and: </p>
<p>$$\langle Q|e^{-i (T-t) H^{\epsilon}}|q\rangle = \sum \phi_n(q)
\phi^{*}_n(Q)\; e^{-i (T-t) E^{\epsilon}_n }$$</p>
<p>I´m guessing that you could treat this as a time independent perturbation, so that the first correction to energy is (lets call the new eigenvalues $E_n^{\epsilon}$):</p>
<p>$$E_n^{\epsilon} = E_n -\frac{1}{2} i \epsilon \langle E_n|q^2|E_n\rangle + ...$$</p>
<p>That makes the new eigenvalues imaginary, but that’s not enough. What we need to have the sum dominated by the ground state is for $Im[E_n^{\epsilon}]$ to be proportional to $E_n$, so that we have a $E_n$ factor in the non-oscillatory part of the exponential. </p>
<p>That means $\langle E_n|q^2|E_n\rangle$ should be proportional to $E_n$. It is true for an harmonic oscilator but, can we say that in general?</p>
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<p>Strings always have a dilaton in their spectrum. Its a scalar field (so presumably no spin), and so far a hypothetical particle. What is its physical significance?</p>
| 9,823 |
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<p>Einstein postulated that gravity bends the geometry of space-time then what does magnetism do in to the geometry of space-time, or is there even a correlation between space-time geometry and magnetism?</p>
| 9,824 |
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<p>How is theoretical computer science getting united with physics? Phenomena like Quantum Computing uses Quantum Mechanics to be able to compute things, how are computers helping not just to model our equations but actually predict new equations, helping us to see the computational aspect of nature and how various things are being looked at from new perspectives using Computer Science?</p>
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<p>I'm watching an episode of Mythbusters where they show aircraft saving 3-5% fuel when flying in a tight V formation. Interestingly, this also applies for the lead airplane. </p>
<p>How is that possible for the lead plane, too? Does the same apply for cars as well, or is it a flying phenomenon?</p>
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<p>Can the findings of the Physics Nobel Laureates of 2011, namely the overpowering existence of dark energy (vacuum energy) have any implications in the quest the combine Quantum Mechanics and General Relativity? Maybe toward a theory of Quantum Gravitation?</p>
| 9,827 |
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<p>Suppose a device or object were traveling at or near the speed of light, and fission or fusion occurred while in this state of motion, creating an enormous blast, what would occur? Would this cause an acceleration of the objects or particles immediately in front of this blast? Or would some other event occur that I am unfortunately unaware of?</p>
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<p>Looking at <a href="http://www.tcm.phy.cam.ac.uk/~bds10/phase/scaling.pdf" rel="nofollow">this paper on page 1</a> how is the first limit obtained? That is, if I have some homogeneous function $g_f(h/t^{\Delta})$, how does setting the gap exponent $\Delta$ to $3/2$ ensure that $$\lim_{x \to 0} g_f(x) = -1/u?$$</p>
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] |
<p>The potential for the Higgs field is standard a quartic one (Mexican hat). Is this done for simplicity or are there fundamental reasons for this choice? I can imagine further contributions to this potential without altering the essentials. But this may lead to differences in the derived particle masses.</p>
| 9,830 |
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] |
<p>About one year ago I attended a pretty interesting seminar of <a href="http://www.ge.infn.it/~zanghi/" rel="nofollow">Nino Zanghì</a> on the actual state of <a href="http://en.wikipedia.org/wiki/Bohmian_quantum_mechanics" rel="nofollow">Bohmian mechanics</a>.</p>
<p>Now, during my undergraduate studies, I didn't have the possibility to take a class in Bohmian Quantum Mechanics, and I guess in a few University it is taught, and in most of them is not even mentioned. Do you know what is the actual situation of the diffusion of this (in my opinion ambitious and beautiful) theory? And which are in your opinion the main reasons of this (beyond of course the success of Copenhagen/Bohr QM) ? </p>
| 9,831 |
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<p>If I was to pass a metal object through a magnetic field would there be any friction?</p>
| 9,832 |
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<p>I just learned some introductory quantum meachnics, but not statistical mechanics, so I am curious how partition functions would be used in the following case:</p>
<p>Suppose there are three particles in x-axis. These particles all have spin 1/2. Neighboring two particles interact with interaction energy of U. (particle A--particle B -- particle C, particle C and A do not (mutually) interact.)</p>
<p>In temperature T, what would partition function look like, what would be the meaning of this function?</p>
| 9,833 |
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<p>I'm just a student of grade 11 but, I was interested in knowing about Physics much deeper. In order to start my interest in Physics, I watched this video of Quantum Physics NOVA : <a href="http://www.youtube.com/watch?v=QVPIGtGcYE0" rel="nofollow">Quantum Physics(NOVA)</a></p>
<p>I have some questions in mind : </p>
<p>$\bullet$ Is <a href="http://en.wikipedia.org/wiki/Quantum_mechanics" rel="nofollow">Quantum Mechanics</a> also applicable for <strong>massive</strong> objects like Human Beings, Animals etc. (in comparison with the <strong>small atoms</strong>) ? </p>
<p>$\bullet$ How much does Quantum Mechanics affects the rules that we have studied yet, like in <strong>Newtonian Mechanics</strong> and <strong>Classical Mechanics</strong> ? </p>
<p>$\bullet$ Has Quantum Mechanics been successful in defining the movement of atoms and other negligibly small objects like sub-atomic particles etc. ? Or is there something yet to come...? </p>
<p>$\bullet$ While, I know that any theory is simply <em>NOT</em> based on <strong>calculations</strong> or only on <strong>experiments</strong>. Has Quantum Mechanics been able to define all the rules with the help of Experiments etc.? </p>
<p>$\bullet$ As a student, I'm basically confused that which theory is the <strong>most</strong> accurate and <strong>applicable</strong> in real life. So, which theory is the most accurate and applicable to the reality? </p>
<p>$\bullet$ After watching the video (linked at the start), I was not able to conclude whether Quantum Mechanics is based on <strong>accuracy</strong> or on <strong>probability</strong> ? </p>
<p>$\star$ I'm sorry, I understand that these questions are much broad and may be concluded as foolish or meaningless questions, but, as a student these were the questions which came in my mind. I will appreciate a lot if anyone can help me in clearing my doubts! </p>
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<p>Einstein said that gravity can be looked at as curvature in space- time and not as a force that is acting between bodies. (Actually what Einstein said was that gravity was curvature in space-time and not a force, but the question what gravity really is, is a philosophic question, not a physical one)</p>
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<p>I want to model liquid lead swirling in a sphere. This is connected to General Fusion’s fusion machine. <a href="http://www.ted.com/talks/michel_laberge_how_synchronized_hammer_strikes_could_generate_nuclear_fusion" rel="nofollow">A 55 million dollar, Jeff Bezos funded, 60 person company trying to change the world with cheap, clean, fusion energy.</a> Here is the problem:</p>
<p><img src="http://i.stack.imgur.com/mEHUf.png" alt="enter image description here"></p>
<p>I do not think there is an analytical solution (to navier-stokes) for this. I looked at Oseen-Lamb vortices and Rankine Vortices. Also, I don’t think GF published their solution. I want to answer the following questions (in this order):</p>
<ol>
<li>Is there a canned solution to this? (I don’t think so)</li>
<li>Which situation do you model? (Steady State)</li>
<li>What is the right coordinate system to use? </li>
<li>How do you write the boundary conditions for this?</li>
<li>Can you arrive at math that is solvable? </li>
<li>Does the centripetal force overcome the surface tension/inter molecular forces? </li>
<li>What is the shape of the air cavity?</li>
<li>What is the minimum speed needed to maintain the air cavity?</li>
<li>Do they continuously inject and pull during compression?</li>
</ol>
<p>===============================================</p>
<p>I am going try solving it. I will post what I have on here. Help appreciated.</p>
<p>=======</p>
<h1>Earlier version of this question:</h1>
<p>I am working through a problem, modeling liquid lead spinning in a 1 meter diameter sphere. Looking for some help. This is a model of General Fusion's machine. Here is a picture of what I am trying to model:</p>
<p><img src="http://i.stack.imgur.com/9wloy.png" alt="enter image description here"></p>
<p>The reactor is a liquid lithium/lead fluid being swirled around a steel chamber, with an air cavity in the center. 14 pistons strike anvils which sit in holes along it's outer walls. This creates a pressure wave which compresses a cavity in the center. At present this cavity is air filled. Here are the properties’ of the liquid lead (roughly):</p>
<p>• Density: 10,000 Kg/M^3
• Temperature: 673 Kelvin
• Viscosity: 0.18 N*S/M^2
• Lead Velocity (at Wall, Estimate): < 4 M/S
• Air cavity: 0.4 meters diameter</p>
<p>My gut tells me that air/lead surface tension will also be needed. What I want to understand is the shape of the cavity in the center. I imagine a vortex like water draining from a bathtub.</p>
<p><img src="http://i.stack.imgur.com/BLHQU.png" alt="enter image description here"></p>
<p>Except that this is drained from both top and bottom. My plan is to start with the two dimensional case, Navier-stokes equation, incompressable, steady state, in cylindrical coordinates:</p>
<p><img src="http://i.stack.imgur.com/xZpZP.png" alt="enter image description here"></p>
<p>By making this incompressable, can't I remove the other terms on the left hand side of the equation? Yes/No?</p>
<p>If so, I would continue to simplify.</p>
<p><img src="http://i.stack.imgur.com/zpaHC.png" alt="enter image description here"></p>
<p>I would start the process of separating the variables, but I need to deal with this pressure term. How do I eliminate it? Or do I try to solve Pressure as a function of Theta, R and Z? Any other issues you see?
Other options:
After solving this situation, I intend to add in the Z-direction, to see why fluid tilts as it spins around. I know that fluid moves fast as it gets closer to a drain. I also see that using Bernoullis' equation I can find that there is a pressure drop near the center of a vortex. If anyone has another approach to modeling this vortex, I would appreciate it.</p>
<p><img src="http://i.stack.imgur.com/dl2vK.png" alt="enter image description here"></p>
<h1>Would love for any assistance here.</h1>
<p>Edit: Looking at two option for modeling this vortex.
1. Rankine: The Rankine is a very simple model of the vortex. It has a center where the rotation rises linearly. It passes a critical radius, where there is the highest rotation. After this it decays 1/r. This is an analytical solution to the Navier-Stokes. I put this into excel and plotted it.</p>
<p><img src="http://i.stack.imgur.com/zTQns.png" alt="enter image description here"></p>
<ol>
<li>Lamb-Oseen: This is a Rankine vortex, which decays with time due to viscosity. I was thinking that this (running backwards) could be a way to estimate the starting of a vortex.</li>
</ol>
<p><img src="http://i.stack.imgur.com/fVLth.png" alt="enter image description here"></p>
<p>Both of these equations do not help me with the shape of the cone. Can anyone else recommend an analytical solution of the Navier-Stokes for this?</p>
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] |
<p><em>Why there is no stable nuclei with
$$A=5$$
in nuclide the chart and so in nature like we know it?</em></p>
| 9,837 |
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<p>I recently watched the documentary miniseries "How the Universe Works" and few things can't stop bothering me. I am not an astronomer nor a physicist so those may be dummy questions.
what I get know from the movie is cluster's stars are not orbiting but they are bond by the gravity so:</p>
<ul>
<li>Does the stars in clusters rotate?</li>
<li>Does cluster's stars have moons.If yes do they rotates/orbits or
are they "frozen" in space?.</li>
<li>What will happen if one of the stars blows up?</li>
<li>Does that structure attracts or repels space objects?</li>
</ul>
| 9,838 |
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<p>If a container filled with pure oxygen and fitted with leak-proof valves were to be dumped into the exosphere at night, would the contained oxygen condense?</p>
| 9,839 |
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<p>Thermal conductivity are often used for surfaces between the computer chip and the heat sink to increase heat transfer and they want high thermal conductivity to decrease the thermal resistance. By $$\Delta T=RQ$$ and $Q$ is constant by the chip. When we decrease R, and keep Q constant from the chip, we decrease $\Delta T$ between the chip and the heat sink.</p>
<p>This is where I get confused because I assumed we would want the temperature difference between the chip and heat sink to be as high as possible. If the two temperatures are close to each other then wouldn't that make the chip heat up and exceed the operating temperature? I thought we would want the heat sink to be much colder than the chip so the chip will cool.</p>
<p>Can someone clarify?</p>
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<p>The <a href="http://ncatlab.org/nlab/show/G2-MSSM" rel="nofollow">G2-MSSM</a> is supposed to be the low-energy theory of a hypothetical class of M-theory compactifications studied by some phenomenologists. It is just the <a href="http://en.wikipedia.org/wiki/Minimal_Supersymmetric_Standard_Model" rel="nofollow">MSSM</a>, but in a particular region of parameter space. The high-energy theory is M-theory compactified on a manifold with G2 holonomy. The MSSM fields come from a specific part of that manifold, while fields in other parts break supersymmetry and stabilize the extra dimensions. The "G2 region" of MSSM parameter space is singled out by a mixture of bottom-up arguments (phenomenological necessities) and top-down arguments (nature of the high-energy theory). </p>
<p><a href="http://arxiv.org/abs/1112.1059" rel="nofollow">Late in 2011</a>, Kane, Kumar, Lu and Zheng came out with a claim that the G2-MSSM implies a Higgs boson mass in the range 105-129 GeV. <a href="http://arxiv.org/abs/1211.2231" rel="nofollow">A year later</a>, this had become the claim that "the Higgs mass was predicted to be 126 +/- 2 GeV before the measurement". That looks more like a retrodiction to me. </p>
<p>Also, the abstract to the second paper states "The derivation has some assumptions not related to the Higgs mass, but involves no free parameters", which means there are no quantitative fudge factors; but perhaps those "assumptions" are acting as <em>qualitative</em> fudge factors. The argument is far from transparent, and one has to wonder whether these authors are retrospectively discovering the refinement of their qualitative assumptions that is needed to single out the desired mass. </p>
<p>But it's true that there were supersymmetric arguments for a Higgs mass in the mid-120s or less than 130, years before this. So let us at least suppose that Kane et al have produced a valid example of such an argument, made in an M-theory context. </p>
<p>However, unlike Higgs mass predictions based e.g. on metastability or near-criticality of the SM vacuum, this one comes packaged with supersymmetry. For example, we are <a href="http://www.newton.ac.uk/programmes/BSM/seminars/2012062510101.pdf" rel="nofollow">told to expect</a> particular signatures of gluino pair production at the LHC. And this brings me to my real question: </p>
<p><strong>How is the G2-MSSM doing, as one supersymmetric model among many, amid the ongoing falsification of <a href="http://arxiv.org/abs/1309.0528" rel="nofollow">"pre-LHC expectations"</a> regarding supersymmetry?</strong> </p>
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] |
<p>While the moon is certainly not a good reflector of solar radiation, surely the radiation it reflects back heats the Earth (even if it is a terribly small amount).</p>
<p>How would one go about calculating (or estimating) this heating contribution on a night with a Full Moon?</p>
| 9,842 |
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<p>I'm trying to explain the behaviour of a geostationary satellite using different frames of reference.</p>
<ol>
<li><p><strong>Inertial frame:</strong> The satellite has a circular motion with angular velocity $\omega$. The centripetal force $F$ required for this motion is created by the gravitational pull of Earth. Earth itself rotates around its axis with $\omega$, but that is irrelevant. <strong>OK</strong></p></li>
<li><p><strong>Rotating frame ($\omega$):</strong> The frame of reference is fixed to Earth. Everything appears stationary. Gravity is still present, which still acts on the satellite with force $F$. Due to the acceleration of our frame of reference we introduce a centrifugal force, which acts on the satellite with $-F$. The forces cancel out, so the satellite's lack of acceleration is explained. <strong>OK</strong></p></li>
<li><p><strong>Rotating frame ($2\omega$):</strong> This frame of reference rotates around Earth's axis with angular velocity $2\omega$. The satellite appears to have angular velocity $-\omega$. The centripetal force $F$ is provided by gravity. However, we have not yet accounted for the acceleration of our frame of reference! There should be a centrifugal force of $-2F$, meaning that the satellite should be accelerating away from Earth!<br>
<strong>Not OK</strong></p></li>
</ol>
<p>How do we explain case 3?</p>
| 9,843 |
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<p>In Randall Munroe's <em>What If?</em> He is calculating the <a href="http://what-if.xkcd.com/73/">Lethal Neutrinos</a> dose.</p>
<blockquote>
<p>If you observed a supernova from 1 AU away—and you somehow avoided being being incinerated, vaporized, and converted to some type of exotic plasma—even the flood of ghostly neutrinos would be dense enough to kill you.</p>
</blockquote>
<p><strong>How do I stay alive to be killed by neutrinos?</strong></p>
<p>Can I pick a large supernova or some other cosmic event, and hide behind a nearby neutron star?</p>
| 9,844 |
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<p>Imagine two locations with different amounts of gravity. I carry up a weight in low gravity, move it on this height over to the other place, and let it fall down there with higher gravity.</p>
<p><img src="http://i.stack.imgur.com/pWNNf.png" alt="experiment setup"></p>
<p>Wouldn't falling down release more energy than lifting up hast cost? If so, is it theoretically possible to generate such a transition between different levels of gravity near to each other?</p>
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<p>From my point of view, high gas pressure input creates air rotation inside the tube. Due to the rotation, the kinetic energy in the air change to internal energy. The internal energy release as heat energy. The heat energy goes out and the cold air stay. The cold air flows back to the other end to create the cold air flow.</p>
<p>If what I am thinking is right, I don't understand why the inner vortex tube is cold. From the circular motion point of view, air needs more kinetic energy to rotate in a smaller radius than in a larger radius. This could draw a conclusion that the air in the inner tube rotates slower than the air in outer tube. </p>
<p><strong>-HOWEVER</strong></p>
<p>I read the following from <a href="http://en.wikipedia.org/wiki/Vortex_tube" rel="nofollow">Wikipedia</a></p>
<p><em>What is usually agreed upon is that the air in the tube experiences mostly "solid body rotation", which means the rotation rate (angular velocity) of the inner gas is the same as that of the outer gas. This is different from what most consider standard vortex behavior — where inner fluid spins at a higher rate than outer fluid. The (mostly) solid body rotation is probably due to the long length of time during which each parcel of air remains in the vortex — allowing friction between the inner parcels and outer parcels to have a notable effect.</em></p>
<p>=====</p>
<p>How I understand is difference from Wikipedia. Do I make some conceptual mistake? Please help</p>
| 316 |
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<p>Explain the Physical implications behind the exchange antisymmetry condition of fermions. This condition forms the basis of the pauli principle but I can't find/understand what happens physically that requires then the presence of a minus sign upon particle interchange.</p>
| 9,846 |
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<blockquote>
<p>Material scientists have discovered a new fluid property called "radost" that is carried along with a fluid as it moves from one place to the next (just like a fluid's mass or momentum). Let $r(x,y,z,t)$ be the amount of radost/unit mass in a fluid. Let $\rho(x,y,z,t)$ be the mass density of the fluid. Let $\vec{v}(x,y,z,t)$ be the velocity vector of the fluid. Use the divergence theorem to derive a conservation law for radost.</p>
</blockquote>
<p>We did an example like this in class, but for conserving mass, so it was a little different. What we ended up with was the following expression $$\frac{\partial \rho}{\partial t}+\nabla \cdot (\rho \vec{v})=0$$</p>
<p>We started by writing, $$dM=\rho\, dV$$ Thus, $$M=\int_V \rho\, dV$$ Then we applied the divergence theorem and that was basically it.</p>
<p>I'm just kind of confused how to start this one.</p>
| 9,847 |
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<p>Are there any ideas of explaining the time dilatation as limits in "computing power"? What I mean is basically that the greater is a concentrated mass, the harder is to "compute" what happens in such system, because more data needs processing (because e.g. more near positioned mass –> more interactions). The same is for speed: if nature is "computing" then having to establish cause-chain of fast objects is simply more demanding – if the "nature" is somehow limited by a time herself (or maybe because greater velocity implies more interactions). This reminds me physical simulations in ODE, Bullet Physics Engine, or other similar package. If the simulation is too demanding, it will appear slow-motion, <em>but only for an outside spectator</em>.</p>
<p>This creates a possible verification experiment: two setups, each having properties, that make general relativity phenomena expected to occur. Then one of the setups should differ in such a way, that would make it more "difficult", more demanding in "complexity" for the suspected "computing machine", but neutral for general relativity theory.</p>
<p>The question is soft, inspired by many theories that try to explain various general questions in physics.</p>
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] |
<p>A unitary matrix U is a matrix such that the conjugate transpose of U, when multiplied on the right with U, yields identity. My question is, is it possible to obtain the transpose of any density matrix using some unitary operation? (I read somewhere that the transpose operation is an 'anti-unitary' operation, but I don't think that means that a unitary operation can NOT simulate a transpose operation).</p>
| 9,849 |
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<p>I am a bit confused by the description of Halbach arrays. It is said that the line of magnets aligned in certain way results in cancellation of magnetic field on one side of the array, and amplification on the other side.
And there is a schematic distribution of magnetic flux - <img src="http://i.stack.imgur.com/GqSeh.jpg" alt="The flux diagram of a Halbach array"></p>
<p>I don't understand what happens to the poles of magnets, or the direction of magnetic flux? Does it change its direction in each next or third element of array? Or is it possible to make something like a magnetic monopole, so that the direcition of magnetic flux will be uniform on one side of the array?</p>
<p>There are some other images showing different arrays with arrows which, I suppose, represent the direction of the magnetic flux.</p>
<p><img src="http://i.stack.imgur.com/JNCWX.png" alt="Diagram 1">
Diagram 1
<img src="http://i.stack.imgur.com/qSZFV.png" alt="Diagram 2">
Diagram 2</p>
<p>The Diagram 2 looks like a monopole, because there is the same direction of flux all the time?
Is it true, or am I misunderstanding something?</p>
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<p><a href="https://www.google.com/search?hl=en&gl=us&tbm=nws&q=tornado%20thiel">Peter Thiel just paid $300,000</a> to Canadian inventor <a href="http://vortexengine.ca/index.shtml">Louis Michaud</a> who is working to construct useful "man-made tornadoes" or "atmospheric <a href="http://en.wikipedia.org/wiki/Vortex_engine">vortex engines</a>" which could be components of future power plants.</p>
<p>Less ambitiously, they could replace chimneys and reduce the losses. More ambitiously, the surface-higher-atmosphere temperature difference could be enough to drive these vortices, i.e. give us energy almost from nothing.</p>
<p>See some <a href="http://motls.blogspot.com/2012/12/energy-from-man-made-tornadoes.html?m=1">very short introduction</a> of mine.</p>
<p>Questions:</p>
<ol>
<li><p>Isn't it a perpetual motion machine? If it isn't, where and how does it take energy for operation?</p></li>
<li><p>Is there a physical upper bound on the amount of energy one could construct in this way? Does the hypothetical engine reduce the lapse rate or otherwise modify the atmosphere?</p></li>
</ol>
| 9,851 |
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<p>I am interested in the linear absorption of 762nm light near a transition of molecular oxygen. I need to find some experimental numbers that will tell us how far the 762nm light will propagate before getting absorbed. Specifically, I want to know the e-folding length, $\gamma^{-1}$ (the length over which the intensity will drop by $e^{-1}$). I believe this is also called the optical depth when using Beer-Lambert law.. </p>
<p>My main problem is that I do not know the definitions of experimentally measured quantities and how they relate to the e-folding length. I was reading "Atmospheric Propagation of Radiation" by Frederick Smith and page 61 says that for the inverse wavelength $\lambda^{-1}=13120.909\text{ cm}^{-1}$ the Band Intensity is $1.95\times10^{-22} \text{ cm}$. In "Laser Remote Chemical Analysis" they call it the integrated band intensity for this line but with units of cm-molecule (basically the same thing). </p>
<p>Does anyone know how the band intensity relates to the e-folding or absorption length?</p>
<p>Our best guess based on physical and dimensional arguments is that the e-folding length will go like $\gamma^{-1} \propto 1 / (B N \Delta\lambda)$ where $B$ is the band intensity with units of $\text{cm}$, $N$ is the number density with units of $\text{cm}^{-3}$, and $\Delta\lambda$ is the linewidth of the transition with units of $\text{cm}$. </p>
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<p>In <a href="http://en.wikipedia.org/wiki/Mathematical_formulation_of_quantum_mechanics" rel="nofollow">Quantum Mechanics</a>, why is it that a self-adjoint operator is linked to an observable? What makes it measureable? And why isn't a non-Hermitian operator linked to an observable?</p>
<p>Also, what type of observables are we talking about here? Particles?</p>
| 9,853 |
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<p>Are neutrinos affected by gravity?</p>
<p>If not, could that be a plausible reason for a neutrino taking a shorter path than light, since light is affected by gravity?</p>
| 925 |
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-0.02298605442047119,
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<p><strong>I would like to know if any epistemologist or science popularizer has ever think about a better/simpler name for <em>exotic</em> or <em>non-baryonic cold dark matter</em> like <em>black matter</em>.</strong></p>
<p>That would be interesting in my opinion to stress: </p>
<ul>
<li>the change of paradigm that already occured from the hot dark matter cosmological model initiated by Zeldovitch to the cold dark matter model initiated by Peebles;</li>
<li>the paradigmatic shift (supersymmetry or some other hypothesis) that could hide behind the discovery of wimps or other cold dark matter.</li>
</ul>
<p>The <em>black matter</em> conundrum indeed could be comparable with former historical puzzles like black body radiation and black hole evaporation which brought new paradigms in physics (the first driven by experimental facts, the second by theoretical inquiry).</p>
<p><strong>Edit :</strong> </p>
<p><strong>I am looking for references (text form an epistemologist or science popularizer) in English rather than just opinions</strong> (to answer the legitimate commentary of @dmckee). My question was triggered by reading a very clear <a href="http://books.google.fr/books?id=nvUk1E37bu8C&pg=PA321&lpg=PA321&dq=fran%C3%A7ois%20bouchet%20cosmologie%20einstein%20aujourd%27hui&source=bl&ots=UKlbqNif2M&sig=-069pXGcQtnoAQVJ9HaHOdqPFAU&hl=fr&sa=X&ei=J36bUrfVFqTF0QXi2oCYAg&ved=0CDEQ6AEwAA#v=onepage&q=%20mati%C3%A8re%20noire&f=false" rel="nofollow">text</a> in French by <a href="http://en.wikipedia.org/wiki/Fran%C3%A7ois_Bouchet" rel="nofollow">François Bouchet</a> about Cosmology where I was surprised to find at some point the French word for <em>black matter</em> while most of the time the author uses the common term <em>dark matter</em> (in his case I guess the change of terminology is just a stylistic artifact). </p>
| 9,854 |
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<p>I know that static friction isn't the cause of deceleration of a rolling body. But if static friction is the only force in the horizontal direction, then shouldn't there be some acceleration produced in this direction? The body should decelerate or accelerate. So how come in this case, a force does not produce acceleration?</p>
<p>Please help! I have been pondering on this question for a long time.</p>
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<p>Imagine a ladder leaning against a wall. All surfaces are smooth. Hence the ladder will slip and fall. While falling it rotates because there are external torques acting on it. My question is about which axis does the ladder rotate?</p>
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] |
<p>We are mostly all familiar with a microscope, and know that it helps to see MICRO components, like stuff that is photolithographically etched on silicon semiconductor die.</p>
<p>(The latter can also be nano, but let's bide that.)</p>
<p>Anyways, we rarely hear of <a href="http://www.google.com/#q=nanoscope" rel="nofollow">nanoscopes</a>, i.e. a lens that can digitally reflect traces of nanometer objects, like quarks, hadrons, and valence electrons.</p>
<p>I want to purchase a good microscope, but that will only let me see cells, substrate micro-components, etc. I want to see the actual atoms that make up the silicon, the atoms that make up the doped logic gates in my tablet, the atoms that make up my arm, and the atoms in the air that surrounds our very demanding atmosphere of carbon dioxide!</p>
<p>So I ask again, with endless respect for science, do nanoscopes exist, and if they don't, how close can I get to the atoms, quarks, and hadrons, and how do I tell them all apart! </p>
| 9,857 |
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<p>A common analogy for gravity is the ball-on-a-rubber-sheet model. In this model, mass distorts spacetime and creates a 'valley' into which other mass can fall. Is this same principal valid for magnetic fields as well (proton-electron)? If so, then how is the repulsion effect modelled?</p>
<p>I ask because the underlying properties of both forces is very similar (inverse-square law). Thanks.</p>
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] |
<p>A spin is created in a superposition of up and down states. A magnet is moved very slowly, towards the spin. What is the work done by the magnet. It may be helpful to imagine that the magnet is connected to a spring that expands or contracts depending on work done by the magnet. </p>
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<p>Suppose I have something like $ f = g_{\mu \nu} x^{\mu} x^{\nu} $, where the Einstein summation convention is implied. Now suppose I want to to take the derivative $ \partial_{\mu}f = \frac{\partial f}{\partial x^{\mu}} $. How would I go about doing this? I figure it's not just going to be $ \partial_{\mu} f = g_{\mu \nu} x^{\nu} $.</p>
| 9,860 |
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<p>Is the following assertion sufficiently unique to merit a paper? <em>Every absolute conservation law implies a corresponding form of entanglement</em>, not just spin (angular momentum). Linear momentum conservation would for example entangle originating equipment with wave packets and help determine packet dispersion rates. Mass-energy entanglement would provide a different way to look at wave function "collapse," via nominally instantaneous exclusion of finding the same mass-energy at remote locations. I have no idea what $T_3$ would be. Even some types of approximate laws may have corresponding entanglements.</p>
| 9,861 |
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<p>Let's say I have two <a href="http://en.wikipedia.org/wiki/Probability_amplitude" rel="nofollow">probability amplitude</a> functions given by $\psi_1$ and $\psi_2$. That is, $\psi_i:\Sigma\rightarrow\mathbb{C}$ for some domain $\Sigma$ with $\int_\Sigma|\psi_i|^2=1$ for $i\in\{1,2\}$. Is there a canonical distance metric that can measure how "similar" these functions are?</p>
<p>I'm thinking of something similar to the <a href="http://en.wikipedia.org/wiki/Wasserstein_metric" rel="nofollow">Wasserstein</a> or "Earth Mover's" metric for probability distributions. The $L^2$ distance (subtract, square, and integrate) doesn't work here since $\psi$ and $c\psi$ are the "same" in a quantum-mechanical sense for all $c\in\mathbb{C}$ with $|c|=1$.</p>
<p>[This is a follow-on to my <a href="http://physics.stackexchange.com/questions/39015/mathematical-probabilistic-interepretation-of-probability-amplitude">other question</a>]</p>
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<p>It is well-known that quantum mechanics does not admit a noncontextual ontological model, and there are countless different proofs of it. I'm interested in the simplest proofs that can be cast as an inequality, and bonus points if there's a proof that a simpler one can't be found.</p>
<p>The definition of contextuality that I care about is the one by <a href="http://arxiv.org/abs/quant-ph/0406166" rel="nofollow">Spekkens</a>, that is: I don't care about determinism nor require that the proof of impossibility be specifically about measurement contextuality; failure at either measurement or preparation is fine. Spekkens himself provided very simple proofs for two-dimensional Hilbert spaces, but it is not clear to me how to cast his proofs as inequalities.</p>
<p>Also, it's well-known that unlike nonlocality, contextuality admits state-independent proofs. It would be nice to know the simplest one in this category as well.</p>
<p>Of course, "simplicity" is subjective, but I hope not enough to forbid a definite answer. If you want, a criterion could be: First, dimension of the Hilbert space needed to exhibit a contradiction. Second, number of measurements needed. Or maybe the product of these numbers.</p>
<p>My candidates are currently <a href="http://arxiv.org/abs/0706.0126" rel="nofollow">Klyachko's</a> 5-observable inequality, that is violated by 3-dimensional quantum systems, and <a href="http://arxiv.org/abs/1109.4396" rel="nofollow">Yu's</a> 13-observable inequality for 3 dimensions that is violated independently of the quantum state. I have no idea if these are the best, and I find it weird that I couldn't find an inequality violated for qubits.</p>
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<p>What is the most essential reason that actually leads to the quantization. I am reading the book on quantum mechanics by Griffiths. The quanta in the infinite potential well for e.g. arise due to the boundary conditions, and the quanta in harmonic oscillator arise due to the commutation relations of the ladder operators, which give energy eigenvalues differing by a multiple of $\hbar$. But what actually is the reason for the discreteness in quantum theory? Which postulate is responsible for that. I tried going backwards, but for me it somehow seems to come magically out of the mathematics.</p>
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] |
<p>Is the Higgs mechanism a fundamental interaction of the same standing as the strong, weak and electromagnetic interactions? If not, is it mediated by the weak interaction? It seems that all the massive particles participate in the weak interaction, and the massless ones do not, but this might just be a coincidence.</p>
| 9,865 |
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] |
<p>Standard Model is advanced (lorentz invariant) version of Quantum physics. It tried to include everything which came in the way while understanding quantum world. It even didn't bother to include even Higgs Boson which was hypothetical at that time. Did they never find gravitation in the way of other quantum interaction.</p>
<p><strong>Note:</strong> I know, there were many unsuccessful attempts to add gravitation with SM to make <em>Theory of Everything</em>. My question: Why didn't Standard Model keep gravitation as raw ingredients (with unresolved relationship with others)?</p>
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<p>According to the knowledge I have, there are routers, switches etc. Therefore, packets would have to be "measured" before continuing on. (If not, how will anyone know the damn IP address?)</p>
<p>But this effectively ends any advantage of quantum channel..</p>
<p>So how is quantum network working theoretically?</p>
| 9,867 |
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] |
<p>Rockets separate from the launch pad and supporting structures very early in flight. It seems like they should tip over once that happens. </p>
<p>Why don't they tip over? Is it due to a well designed center of gravity or do they somehow achieve aerodynamic stabilization?</p>
| 9,868 |
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<p>I'm preparing for entry into a Physics degree course, and was planning on a double major in Physics (with a specialization in Quantum Information later) but am now considering a minor in Astrophysics/Astronomy. How useful would Astrophysics be? I have an interest in that area also.</p>
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<p>For a pure state $\rho_{AB}$, the entropy of entanglement of subsystem $A$ is</p>
<p>\begin{equation}
S( \rho_A) = -tr (\rho_A \log \rho_A)
\end{equation}</p>
<p>where $\rho_A$ is the reduced density matrix of A. </p>
<p>For a single site of a spin chain, $\rho_A$ can be written in terms of single site correlation functions $\langle \sigma_l^\alpha \rangle$ where $\alpha = x,y,z$.</p>
<p>Are there any entanglement measures for mixed states that use only the same correlation functions, $\langle \sigma_l^\alpha \rangle$ where $\alpha = x,y,z$?</p>
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<p>The imprint of gravitational waves created shortly after the big bang may offer direct evidence for inflation theory, according to a discovery by the BICEP2 experiment at the South Pole and released today. What this means for those alternatives theories to inflation such as string gas cosmology and the ekpyrotic universe? What would happen with these two cases particularly?</p>
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<p>Honestly though, is the Earth considered air-tight in the sense that its gases don't escape? </p>
<p>I'm sure every physicist who reads this is going to tear their hairs out, but the extent of my knowledge in this area is that you need to travel a certain speed to break Earth's gravitational pull and that has me wondering how gases could escape.</p>
| 9,872 |
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<p>The recent results of the BICEP 2 experiment published on March 17th 2014, has generated a lot of media attention, with the general consensus being that "this is a major discovery" perhaps leading to a Nobel Prize for some. </p>
<p>But for those that are still skeptical, what are the competitors to this experiment, and when are their results expected to be published?</p>
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<p>I have seen people counter the problem of big box and small weighing machine by stepping on the machine while carrying the box in arms and subtract the body weight. Do you think it'll give accurate result (not counting negligible difference of course)? </p>
<p>It seems like a DUH kind of question, but I wanted to ask it anyway. </p>
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<p>I have 2 rigid-bodies (b1,b2) if i linked one to the other (as if they are conjoined together) , how to represent b1 effect on b2 and b2 effect on b1</p>
<p><img src="http://i.stack.imgur.com/E0XOV.png" alt="enter image description here"></p>
<p>Is there any LAW that affect the position/orientation of the other body ?</p>
<p>notes : </p>
<ul>
<li>i am using Quaternions for orientations</li>
<li>i don't want to treat them as one body</li>
<li>i have only primitive shapes (box,sphere,..) to link.</li>
</ul>
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<p>Yes, you all talk about neutrinos and spins, but I came out with this basic s**t :D</p>
<p>All of us learnt the basic equations of collisions, elastic (everything bounces and energy remains the same), or non-elastic (the classic example where objects remain stick together). What if objects don't remain together and some energy is lost (a fraction of the kinetic energy of course)? I tried to solve the maths and since I'm just a Java programmer, I realized that after 6 years away from University I can only do 2+2 and little more. Apart from joking, the equations come out as a system with an elliptic curve (for energys) and a line (for momentums). Anyway I'd like to get a solution that is good enough to do some analisys over initial speeds, energys and so on.</p>
<p>Please don't flame me, probably my english for physics is not exact, I studied physics in Italian and my everyday bread is IT.</p>
<p>Thanks</p>
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<p>If a body with mass $m$ begins at position $x_0$ with velocity $v_0$ and experiences a force that varies as a function of time $f(t)$ (and we ignore gravity, friction, and everything else that might complicate matters), then we can compute the position and velocity of the body at any time:</p>
<p>$$v(t) ~= ~\int\limits_0^t \frac{f(t)}{m}\mathrm{d}t+v_0, $$
and
$$x(t) = \int\limits_0^t\int\limits_0^t \frac{f(t)}{m}\mathrm{d}t+v_0\mathrm{d}t+x_0.$$</p>
<p>Now, if we have another body of the same mass that starts off at position $\hat{x}_0$ with velocity $\hat{v}_0$ and we want to apply a force, $\hat{f}(t)$, in order to match the first body's trajectory (position and velocity) as quickly as possible subject to the constraint that $|\hat{f}(t)|\le \mathrm{fmax}$.</p>
<p>What are the tools that I need to solve this?</p>
| 9,877 |
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<p>Suppose we have an asymptotically flat, globally hyperbolic spacetime $M$ endowed with two one-parameter isometry groups $\sigma_t$ and $\chi_{\phi}$ which commute (i.e. $\sigma_t \circ \chi_{\phi}= \chi_{\phi} \circ \sigma_t.$)</p>
<p>Assume moreover that the orbits of $\sigma_t$ are timelike curves generated by the Killing vector field $\xi^a.$ The orbits of $\chi_{\phi}$ are closed spacelike curves generated by the Killing field $\psi^a$.</p>
<p>In chapter 7.1, p 165 of Wald's GR text, he states that the asymptotic flatness of the spacetime implies that "there must be a rotation axis on which $\psi^a$ vanishes." Why must this be the case?</p>
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<p>If We put a small floating object (penoplast granule) on a surface of water it performs spiral like movements and eventually sticks to a side of a vial. </p>
<p>What physical forces are involved? I'm particularly interested what causes spiral like movements. Would be grateful if you can suggest also some literature.</p>
| 9,879 |
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<p>To find the Higgs boson, we had to build the biggest machine mankind has ever built: the LHC with a collision energy of up to 14 TeV. Inside the sun there is a huge pressure and temperature, but is the energy density high enough for Higgs bosons to be created?</p>
| 9,880 |
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] |
<p>Why is the Boltzmann distribution used over the Maxwell distribution in many cases such as the derivation of Plancks law of thermal radiation, derivation of Einstein A and B coefficients, Langevin theory of paramagnetism etc.?</p>
| 9,881 |
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<p>The circuit breaker at the electrical mains trips at home when there is a thunderstorm outside.
Why does this occur?</p>
| 9,882 |
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<p>I know it's not economically feasible, but is there a simple equation telling you the ellipse earth would follow if you theoretically launch all of the landfills, i.e. garbage, beyond our orbit, thus decreasing mass by X.</p>
| 9,883 |
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<p>Fairly straightforward question. If not, why not?</p>
<p>I suspect that if they do, it is not perceived due to the regions of highest dispersion being in one's region of lowest visual acuity.</p>
| 9,884 |
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] |
<p>Let the amount of energy in one pulse of (laser) light be $E$, and the wavelength be $\lambda$.</p>
<p>This pulse goes straight to the mirror, and it is reflected by the mirror.</p>
<p>Let the reflectivity of this mirror be $r_{\lambda}$.</p>
<p><img src="http://i.stack.imgur.com/APbUi.jpg" alt="enter image description here"></p>
<p><strong>Question.</strong> How can I get the formula of $r_{\lambda}$?</p>
<p>Any answers (or hints) will be appreciated, thank you.</p>
| 9,885 |
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<p>I'm looking at two papers in particular: <a href="http://www.wiley-vch.de/vch/journals/2231/pss40/167_177.pdf" rel="nofollow">A. L. Efros and B. I. Shklovskii, Critical Behaviour of Conductivity and Dielectric Constant near the Metal-Non-Metal Transition Threshold, Phys. Status Solidi B 76, 475 (1976)</a> and <a href="http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=1222023" rel="nofollow">Measurement of the conductivity exponent in random percolating networks of nanoscale bismuth clusters </a>.</p>
<p>In percolation theory, for concentrations $p$ higher than the critical concentration $p_c$, the conductivity $\sigma \propto (p-p_c)^t$ for $p>p_c$, where $t$ is 1.3 for 2 dimensions. It's often simplified so that the substrate is a perfect insulator, so you have $\sigma=0$ for $p<p_c$. So it should look pretty simple, something like:</p>
<p><img src="http://i.stack.imgur.com/VeCeo.png" alt="enter image description here"></p>
<p>My point is, concave up. However, looking at the two papers I mentioned above, they have these graphs:</p>
<p><img src="http://i.stack.imgur.com/FdxOx.png" alt="enter image description here">
with:<img src="http://i.stack.imgur.com/azYOy.png" alt="enter image description here"></p>
<p>and:</p>
<p><img src="http://i.stack.imgur.com/FiDoY.png" alt="enter image description here"></p>
<p>Both of which are concave <em>down</em> in the region $p>p_c$.</p>
<p>Is there something going on, or is this a series of errors?</p>
<p>I suspect the Efros paper may have just had an error, as the caption corresponding to the dotted line (1) points to the region of $x<x_c$, where they explicitly said that equation doesn't apply. It does have a very strange scale though.</p>
<p>I have no idea what's going on in the graph of the second paper. Is there any explanation besides an error?</p>
<p>Thank you!</p>
| 9,886 |
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<p>It seems to me that special relativity has a weird definition of the causality relation. In that theory, the only thing that matters is the space-time distance between events. But I don't think this is the case. It seems to me that if I am pressing a few keys on my keyboard and then causing this question to appear on your screen, my action has a greater locus of influence than just me sleeping in bed. So, what exactly is the definition of the causality relation?</p>
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<p>I've recently been looking into performance figures for various auto mobiles, and I see the terms <a href="http://en.wikipedia.org/wiki/Horsepower#Brake_horsepower" rel="nofollow">BHP</a> and <a href="http://en.wikipedia.org/wiki/Torque" rel="nofollow">Torque</a> used quite often, but I can't say I really understand the terms, or how they relate to engine size.</p>
<p><strong>Take for example the following specifications:</strong></p>
<blockquote>
<p>Ford Focus Zetec 1.6 (Petrol): BHP = 100, Torque = 110 lbs/ft</p>
<p>Ford Focus Zetec 1.6 (Diesel): BHP = 113, Torque = 210 lbs/ft</p>
<p>Subaru Impreza 2.0RX (Petrol): BHP = 148, Torque = 145 lbs/ft</p>
<p>Subaru Impreza 2.0RC (Diesel): BHP = 148, Torque = 258 lbs/ft</p>
<p>Subaru Impreza 2.5WRX(Petrol): BHP = 227, Torque = 236 lbs/ft</p>
</blockquote>
<p>Can someone shed some light (in layman's terms) what is the difference between BHP and Torque, and how they relate to one another?</p>
| 9,888 |
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] |
<p>How do scientists place satellites into orbit? How do they calculate the gravity acting on the spacecraft in order to generate exact opposite force by means of Kinetic energy to keep the satellite stable?</p>
| 9,889 |
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<p>I would like to know how to calculate the temperature change of a moving object. For example if I throw a ball that is warmer than the surrounding air, how much would it lose form it's temperature in every second? I searched for two days now, but I have only partial solutions :(. I would need a formula that takes in account the followings:</p>
<ul>
<li>The material of the ball (Iron, heat conductivity?)</li>
<li>Area of the ball</li>
<li>Speed of the ball (true airspeed)</li>
<li>Material of the surrounding material (air, heat conductivity?)</li>
<li>And ofc temperature of the ball and the surrounding material.</li>
<li>anything else I forgot and neccessary :):)</li>
</ul>
<p>It dosen't need to take in account friction generated heat and other things, I only need the temperature loss :)</p>
<p>BIG TANKS!</p>
| 9,890 |
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<p>Lets say we take the standard configuration when $x'y'$ is moving away from system $xy$ (image 1). By knowing that the phase is constant in all frames $\phi=\phi'$ we can derive the Lorenz transformations for a standard configuration.</p>
<p><img src="http://shrani.si/f/18/R5/2pJr7UXu/1.png" alt="image 1"></p>
<p>Derivation (using the parametrization):
\begin{align}
\phi &= \phi'\\
-\phi &= -\phi'\\
k \Delta r - \omega \Delta t &= k' \Delta r'- \omega'\Delta t'\\
[k_x , k_y , k_z][\Delta x , \Delta y , \Delta z] - \omega \Delta t &= [{k_x}'\! , {k_y}'\! , {k_z}'][\Delta x'\! , \Delta y'\! , \Delta z']\! - \!\omega'\Delta t'\\
k_x \Delta x + k_y \Delta y + k_z \Delta z - \omega \Delta t&= {k_x}'\Delta x' + {k_y}' \Delta y' + {k_z}' \Delta z'\! - \!\omega' \Delta t'\\
{k_x} \gamma \Bigl(\!\Delta x' + u\Delta t' \!\Bigl) + {k_y} \Delta y' + {k_z} \Delta z' - \omega \gamma \left(\Delta t' + \Delta x' \frac{u}{c^2}\right)&= ...\\
{k_x} \gamma \Delta x' + k_x \gamma u\Delta t' + {k_y} \Delta y' + {k_z} \Delta z' - \omega \gamma \Delta t' - \omega \gamma \Delta x' \frac{u}{c^2}&= ...\\
\gamma \Bigl(\!k_x - \omega \frac{u}{c^2}\! \Bigl) \Delta x' + k_y \Delta y' + k_z \Delta z' - \gamma \Bigl(\omega - {k_x} u \Bigl) \Delta t' &= k_x' \Delta x' + k_y' \Delta y' + k_z' \Delta z' - \omega' \Delta t'\\
\end{align}</p>
<p>Lorentz transformations and their inverses (are derived similarly):
\begin{align}
&\boxed{\omega' = \gamma\Bigl(\omega - {k_x} u \Bigl)} & &\boxed{\omega = \gamma\Bigl(\omega' + {k_x}' u \Bigl)}\\
&\boxed{k_x' = \gamma \Bigl(k_x - \omega \frac{u}{c^2} \Bigl)} & &\boxed{k_x = \gamma \Bigl(k_x' + \omega' \frac{u}{c^2} \Bigl)}\\
&\boxed{k_y' = k_y} & &\boxed{k_y = {k_y}'}\\
&\boxed{k_z' = k_z} & &\boxed{k_z = {k_z}'}
\end{align}</p>
<p>We can express Lorentz transformations and their inverse using some trigonometry ($k_x = k \cos{\xi} = \frac{\omega}{c} \cos{\xi}$, $k_y = k \sin{\xi} = \frac{\omega}{c} \sin{\xi}$ and $k_z = 0$) as:</p>
<p>\begin{align}
&\boxed{\omega' = \gamma \, \omega \! \Bigl(1 - \cos{\xi} \frac{u}{c} \Bigl)}& &\boxed{\omega = \gamma \, \omega' \! \Bigl(1 + \cos{\xi'}\frac{u}{c} \Bigl)}\\
&\boxed{k_x' = \gamma \, \frac{\omega}{c} \! \Bigl(\cos{\xi} - \frac{u}{c} \Bigl)}& &\boxed{k_x = \gamma \, \frac{\omega'}{c} \! \Bigl(\cos{\xi'} + \frac{u}{c} \Bigl)}\\
&\boxed{k_y' = \frac{\omega}{c} \sin{\xi}} & &\boxed{k_y = \frac{\omega'}{c} \sin{\xi'}}\\
&\boxed{k_z' = k_z} & &\boxed{k_z = {k_z}'}
\end{align}</p>
<hr>
<p>NOW CONSIDER A CASE:</p>
<blockquote>
<p>The spaceship is approaching Earth with a speed $0.6c$ under an angle
of $30^\circ$. What frequency does an observer on Earth measure if
spaceship is sending frequency $1.00\cdot10^9Hz$</p>
</blockquote>
<p><strong>Question 1:</strong></p>
<p><img src="http://shrani.si/f/e/v/4MUjJWsG/0108.png" alt="enter image description here"></p>
<p>If i draw the picture in black color (image 2) it occured to me that solving this case could be possible by simply using a relativistic Doppeler effect shift equation for 2 bodies which are closing in (in which i would use the $u_x = u \cdot \cos 30^\circ$).</p>
<p>$$\nu = \nu' \sqrt{\frac{c+u_x}{c-u_x}} \approx 1.78\cdot 10^8Hz$$</p>
<p><strong>Am i allowed to solve this case like this?</strong> </p>
<p><strong>Question 2:</strong></p>
<p>I wasnt so sure about the above solution, so i tried to get the similar situation to the one i had in image 1. I noticed that if i rotate coordinate systems (image 2 - systems which are colored in red) i get fairly similar configuration, with the $\xi$ and $u$ a bit different than the ones in image 1. <strong>I wonder how do the Lorentz transformation change? Can anyone tell me?</strong></p>
<hr>
<p>EDIT: </p>
<p>Our professor solved this like this:</p>
<p>\begin{align}
\gamma = \frac{1}{\sqrt{1-\frac{0.6c}{c}}} = 1.25
\end{align}</p>
<p>\begin{align}
\nu' &= \gamma \nu \left(1 - \cos\xi \frac{u}{c}\right)\\
\nu &=\frac{\nu'}{\gamma \left(1 - \cos\xi \frac{u}{c}\right)}\\
\nu &= \frac{1\cdot 10^9Hz}{1.25 \left(1 - \cos30^\circ 0.6\right)}\\
\nu &= 1.665\cdot 10^9 Hz
\end{align}</p>
<p><strong>I think he is wrong because he inserted a positive $u$. Should the $u$ be negative or not?</strong></p>
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<p>Sometimes I see comments about <a href="http://arxiv.org/abs/1209.5056" rel="nofollow">the big desert hypothesis</a> that I don't understand. For instance <a href="http://www.math.columbia.edu/~woit/wordpress/?p=5569" rel="nofollow">in a famous blog</a> :</p>
<blockquote>
<p><em>...This is based on a renormalization group calculation extrapolating
the Higgs effective potential to its value at energies many many
orders of magnitude above LHC energies. To believe the result you have
to believe that there is no new physics and we completely understand
everything exactly up to scales like the GUT or Planck scale. Fan of
the [Standard Model] that I am, that’s too much for even me to swallow as
plausible...</em></p>
</blockquote>
<p>and recently in an <a href="http://physics.stackexchange.com/q/65207/">interesting answer to another question</a> I asked on physics.stackexchange </p>
<blockquote>
<p><em>This is a very strong assumption (although certainly not unknown in
particle physics): They are assuming that there is no new physics
across 16 orders of magnitude.</em></p>
</blockquote>
<p>I wonder if this kind of feeling about this hypothesis is common because I've never found similar comments about the see-saw mechanism which, as far as I understand it of course, makes a connection between grand unification or Planck scale and neutrino sector energies (a much larger leap for a physicist I guess ;-)</p>
<p>As far as I am concerned I am a Standard Model (and effective (and noncommutative) theories) enthousiast! I imagine that its validity could go from $10^{-18}$eV (upper limit of the photon mass) to $10^{+12}$eV (LHC energy), these are 30 orders of magnitude on the energy scale. Then the 16 ones of the big desert hypothesis from TeV scale to Planck scale are just one giant step further, half long forward so to speak.
I could be wrong of course but very naively I take inspiration from the past when chemists were bold enough to imagine atoms and physicists like Rayleigh clever enough to <a href="http://thethoughtstash.wordpress.com/2012/11/17/estimating-the-size-of-an-atom/" rel="nofollow">evaluate molecular size</a> extrapolating the validity of euclidean 3D geometry from human scale (cubic centimeter) to oil molecule scale ($10^{-21}$? cubic centimeter)! </p>
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<p>I wonder how to properly write the motion equations for the inverted pendulum on a cart in case of overdamped dynamics. Imagine the system <a href="http://en.wikipedia.org/wiki/Inverted_pendulum#Inverted_pendulum_on_a_cart" rel="nofollow">illustrated in Wikipedia</a> placed in a liquid with high viscosity $\beta$. I completely understand how the system with the fixed pendulum base behaves. Let's start from the equation for the classic inverted pendulum: $$ml^2 \ddot \theta - \beta \dot \theta + mgl\sin\theta=0.$$ High values of $\beta$ <a href="http://www.physicsforums.com/showthread.php?t=337989" rel="nofollow">allow one to neglect the inertia term</a>, so that $$- \beta \dot \theta + mgl\sin\theta = 0.$$</p>
<p>However, as a person who is intrinsically bad at physics, I am greatly confused about what effect the motion of the cart has on the dynamics. The derivations <a href="http://physics.stackexchange.com/questions/35000/elementary-derivation-of-the-motion-equations-for-an-inverted-pendulum-on-a-cart">here</a> are for the classic, non-overdamped case, I guess that here everything should be much simpler. Any help is appreciated.</p>
<p>Update: Probably the equations I ask about should take the simplest possible form in the frame of reference centered at the moving base of the pendulum.</p>
| 9,893 |
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] |
<p>How will a light source with a specific $\lambda$ positioned right at the nth order maximum pointing towards a diffraction grating be affected? Will it come out "directed"?<img src="http://i.stack.imgur.com/hhWJn.png" alt="enter image description here"></p>
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<p>Is the double slit experiment an example of entanglement when it seems as if the photon is going through both slits? Or put another way, is it at this stage when we attempt measurement we see a photon on one side affect the photon on the other side? Do entangled particles have to be made first to show entanglement or is the double slit experiment in itself showing entanglement?
Also, what else in nature collapses the wave function?</p>
| 9,895 |
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<p>What form does the discharge stream of a propeller take in a Herschel-Bulkley fluid, chracterised by having a certain yield stress, shear between layers only occurs beyond that stress.</p>
<p>This situation is known from biogas-plants, where the slurry is often mixed by propellers - Almost always in cylindrical tanks, with the propellers mounted so that they create a horizontal circular flow.
Generally speaking, noone in the industry knows the viscosity of their slurry (measuring the flow-characteristics of liquid cow manure with gras in it is hard).</p>
<p>The following phenoma have been observed:</p>
<p>A long jet, accompanied by a flow of entrained liquid </p>
<p>A 'cavern' with a jet and recirculation to the back/suction side of the propeller, no gloabal flow beyond the cavern</p>
<p>A cavern, the propeller suck material on front an back and pushes it outward radially</p>
<p>It is also a known phenomena that propellers downstream of another propeller work better (don't create caverns). </p>
<p>We have serveral variables, like min. yield stress, viskosity, diameter of propeller, revolutions per time of propeller, form of propeller ( like steeper blades), power of propeller </p>
<p>I think that the relation 'higher yield stress => more likelyhood of forming a cavern' is intuitivly right.</p>
<p>What other quantitative relations like that are there?</p>
<p>What conditions make the third form of flow more or less likely?</p>
<p>The general (and plausible) attitiude seems to be that large diameter, low number of revolutions create the most global flow for the least power input. What limits are there to 'bigger, slower' from a fluid-mechanic standpoint?</p>
| 9,896 |
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<p>I am reading Leslie Ballentine's Quantum Mechanics, section 7.2, which is all about the explicit form of the Angular Momentum operators.</p>
<p>I understand how he gets the form for the single component state function, equation (7.18) which has the form $$\mathbf{R} \Psi(\mathbf{x}) = \Psi(R^{-1}(\mathbf{x})) $$ where $\mathbf{R}$ is given by $$ \mathbf{R}_n(\theta) = e^{i\theta \mathbf{\hat{n}} \cdot \mathbf{J}/\hbar} $$
He then identifies $\mathbf{J}$ with the orbital angular momentum operator $\mathbf{L}$. No problems there.</p>
<p>However, in the following section he claims that for a multicomponent state function we take the general form of (7.19) $$\mathbf{R} \begin{bmatrix} \Psi_1(\mathbf{x}) \\ \Psi_2(\mathbf{x}) \\ \vdots\end{bmatrix} = D \begin{bmatrix} \Psi_1(R^{-1}\mathbf{x}) \\ \Psi_2(R^{-1}\mathbf{x}) \\ \vdots\end{bmatrix}$$ where now we have, in addition to the coordinate transformation $R^{-1}(\mathbf{x})$, we also have a matrix $D$ that operates on the internal degrees of freedom --- which is to say it makes linear combinations of the components. Our $\mathbf{R}$ now takes the form (7.20) $$ \mathbf{R}_n(\theta) = e^{i\theta \mathbf{\hat{n}} \cdot \mathbf{L}/\hbar} D_n (\theta)$$</p>
<p>He then identifies $D$ with spin angular momentum so total angular momentum $\mathbf{J} = \mathbf{L} + \mathbf{S}$.</p>
<p>I still don't understand the reason why we need this $D$ matrix. Can someone explain to me what is going on here, specifically why the form (7.19) instead of (7.18)? Why does this matrix show up when we have a multicomponent state function? </p>
| 9,897 |
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<p>I have heard of the invariant quantity $\Delta s$ in relativity for which I have stumbled upon an equation $\Delta s^2 = \Delta x^2 - c \Delta t^2$. This reminds me of hyperbola, which has general equation like this one: </p>
<p>$$\frac{x^2}{a^2}-\frac{y^2}{b^2} = 1$$</p>
<p>But i already know that in relativity there must be 2 asymptotes which are $\perp$ to eachother, so i can make a conclusion that in equation above $a=b=1$ and i can reduce the equation to:</p>
<p>$$x^2 - y^2 = 1$$
$$x^2 - y^2 = 1^2$$</p>
<p>Because in Minkowski diagram axis $y$ is writen as $ct$ i can rewrite equation like writen below:</p>
<p>$$\boxed{x^2 - (ct)^2 = 1^2}$$</p>
<p>I draw above hyperbola together with the one below and i get image below:</p>
<p>$$\boxed{x^2 - (ct)^2 = 2^2}$$ </p>
<p><img src="http://i.stack.imgur.com/uR2ZG.png" alt="enter image description here"> </p>
<p>In the image i marked <strong>what i think is invariant quantity of relativity</strong> (but i am not sure) $\Delta s$, because from boxed equations i can clearly see universal form below which shows why hyperbolas intersect $x$ axis in points $\Delta s = 1$ and $\Delta s =2$.</p>
<p>$$\Delta s^2 = \Delta x^2 - (ct)^2$$</p>
<p><strong>Q1:</strong> I ve heard that $\Delta s$ was supposed to be invariant, but how is it invariant if its value is first 1 and then 2 for two of my hyperbolas in the picture? </p>
<p><strong>Q2:</strong> How can i show in my picture that this quantity is really invariant?</p>
<p><strong>Q3:</strong> It seems to me (but i am not sure) that $\varphi$ in my picture corresponds to $\beta = \frac{u}{c}$. Please correct me if i am wrong or tell me that i am right if i am.</p>
<p><strong>Q4:</strong> Does $\varphi$ in my picture correspond to $\kappa$ in this matrix? Is this some sort of rotational matrix? </p>
<p>$$
\begin{pmatrix}
\cosh{\kappa} & 0 & 0 & -\sinh{\kappa} \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
-\sinh{\kappa} & 0 & 0 & \cosh{\kappa} \\
\end{pmatrix}
$$</p>
<p><strong>EDITED PART:</strong></p>
<p><img src="http://i.stack.imgur.com/wmi0e.png" alt="enter image description here"></p>
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<p>Fundamental notions of QM have to do with observation, a major example being The Uncertainty Principle. </p>
<ol>
<li><p>What is the technical definition of an observation/<a href="http://en.wikipedia.org/wiki/Measurement_in_quantum_mechanics">measurement</a>? </p></li>
<li><p>If I look at a QM system, it will collapse. But how is that any different from a bunch of matter "looking" at the same system? </p></li>
<li><p>Can the system tell the difference between a person's eyes and the bunch of matter? </p></li>
<li><p>If not, how can the system remain QM? </p></li>
<li><p>Am I on the right track?</p></li>
</ol>
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<p>This question is directed mostly at people giving lectures on black holes, but input by other physicists or students is very much appreciated.</p>
<p><b>Do you know a good (home)-experiment with a black hole analog (such as water in a bathtub) that allows to discuss most of the pertinent concepts and features of a black hole?</b></p>
<p>To clarify what I mean by 'good' let me provide what I consider as a good experiment with a white hole analog (see exercise 2.2 in <a href="http://quark.itp.tuwien.ac.at/~grumil/pdf/ex2.pdf">this exercise sheet</a>), namely a hydraulic jump.</p>
<p>This experiment allows you to discuss in very simple terms what a white hole is (the white hole region is visibly distinct from the exterior because of their different water depths). It is also quite easy to show experimentally that you cannot send information into the white hole (in the form of shallow water wave excitations aka 'ripplons'). Moreover, the white hole exhibits some interesting features in the near horizon region (see the concentric rings in the picture on the back of the exercise sheet). Most importantly, this experiment can be done basically by anyone - there is no need for access to lasers, waveguides, Laval nozzles, supersonic flow etc.</p>
<p>The purpose of the hydraulic jump experiment is to provide a pedagogic introduction to white holes. I would like to have a comparable (non-Gedanken-)experiment available to introduce black holes. </p>
| 448 |
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<p>Back in the day I learned that a few special thermodynamical processes have special names. </p>
<p>For example, if one keeps $P$ constant, the process is called isobaric, if one keeps $T, V$ or $S$ constant, one gets, correspondingly, isothermic, isochoric or isentropic processes. Similarly, if one keeps $\dfrac{\mathrm{d} \ln P}{\mathrm{d} \ln \rho}$ constant during the process, it is called polytropic, and if $\delta Q = 0$ at any time, the process is called adiabatic. </p>
<p>Now, the question: what is the process called, if one keeps internal energy $U$ constant?</p>
| 9,899 |
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<p>I am pretty sure, that during last 60 years of well-funded research a lot of ideas on nuclear fusion were already tried and ruled out. </p>
<p>Is there some summary describing all of them and why they didn't work? (I am looking something going way beyond just Tokamak / Stellarator / Laser ICF / Z-Pinch).</p>
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<p>I wrote a first program that simulates a solar system. I was able to calculate the locations for every planet on its elliptical route for any given time.<br>
In a second program i managed to simulate newtonian gravitational behavior (n-body problem, time-step approach). </p>
<p>But i'm wondering how it is possible to:<br>
(1) find routes (different possibilities) from a given location/planet to another<br>
(2) choose the best route according to duration or fuel cosumption</p>
<p><strong>So where's a good place to start?</strong></p>
<p>To be more exact: It's not about writing another simulation, its about understanding the physics behind it! </p>
<p>Until now, i was not able to find any good resources on the internet.</p>
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<p>If gravity is this "unexplainable force" that pulls everything to the center of a planet or stellar remnant you stand upon, why doesn't gravity pull itself?</p>
<p>If gravity effects anything with energy, why doesn't gravity effect itself?</p>
<p>Gravity is energy, right?</p>
| 9,902 |
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<p>In quantum theory we have some principles that guides us, e.g. Pauli's principle. What I am after in this question is a list of fundamental results, be it equation or identities that must hold in a respectable QFT. I am thinking for instance of the Wards identities or Adler-Bell-Jackiw anomaly and similar things. </p>
<p>Please could you make a list of these types of identities and comment on how fundamental these are (say on a scale 1-3 normalized such that Pauli=1). </p>
<p>Also, please comment on how these identities arise and if there are any analogs of these in other theories (e.g. in the classical limit, or in String Theory or whatever) </p>
<p>Finally I would appreciate if you comment on and discuss what would happen in a theory where the pertinent identities are not satisfied. </p>
<p>Please don't close this question as being vague and not explicitly a solvable problem. In my opinion, these types of question are also needed here, apart from the usual "why is F=ma"-questions. </p>
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] |
<p>I read this book here: <a href="http://tiny.cc/Gravity-Atom-Myth" rel="nofollow">http://tiny.cc/Gravity-Atom-Myth</a></p>
<p>It claims that while gravity does affect individual atoms, it's not quantified like molecular mass due to lack of binding proteins which render air-oxygen trajectories improbable.</p>
| 9,904 |
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<p>My atomic physics lab is in a building that experiences huge swings in humidity levels during the year due to the <a href="http://en.wikipedia.org/wiki/North_American_Monsoon">monsoon season</a></p>
<p>Our building provides temperature, but not humidity control. Using just the building temperature control results in the following lab climates:</p>
<p>10 months out of the year, the room is at</p>
<p>T $\approx 24.4 ^oC$</p>
<p>Relative Humidity $< 10\%$</p>
<p>2 months out of the year, the lab is at</p>
<p>T $\approx 24.4^oC$</p>
<p>Relative Humidity $\approx 50\%$</p>
<p>This season variation necessitates significant recalibration twice per year at the beginning and end of the monsoon season. The sensitive components are mainly opto-mechanical.</p>
<p>The lab currently has a dehumidifier that is spec'd at 45 pints per day during the wet season. This specification indicates how much water the unit will remove from the air in a given day when the air is saturated with water (100% relative humidity). The problem with such a specification is that 100% relative humidity is a way different environment than 40% or 50% humidity. </p>
<p>On a wet day, this unit reduces the lab relative humidity by about 10% from 55% to 45%. This is still far from the lab's climate most of the year. It is a trade off, though, because it will also raise the lab temperature by about 1 degree C, which necessitates other recalibration. I am investigating options to further reduce the humidity.</p>
<p>The lab is approximately 5 meters X 10 meters X 3 meters in size. Most of the experiment is on a very full optics table that is 1.5 meters X 4 meters. There are lots of cables and water tubing that require access to the table, making climate isolation of the table difficult (although not impossible).</p>
<p>A few options under consideration are the following:</p>
<p><strong>1:</strong> Introduce an additional higher capacity dehumidifier</p>
<p>Pros:</p>
<ul>
<li>Fast and easy implementation</li>
</ul>
<p>Cons: </p>
<ul>
<li><p>it is unknown how efficient a dehumidifier will function when the relative humidity is only 45%. </p></li>
<li><p>Manufacturers do not specify how well the unit will work at low humidity levels, only at 80% +. </p></li>
</ul>
<p><strong>2:</strong> Fill sensitive areas with positive pressure Nitrogen</p>
<p>Pros:</p>
<ul>
<li>Excellent climate control</li>
<li>Minimal impact on room temperature</li>
</ul>
<p>Cons: </p>
<ul>
<li>requires significant reconfiguration of experimental setup. </li>
<li>Requires refilling Nitrogen tank frequently, a recurring cost.</li>
</ul>
<p><strong>3:</strong> Isolate experiment from lab climate using large plastic enclosures and recirculate air in this enclosure</p>
<p>Pros:</p>
<ul>
<li>Excellent environment isolation</li>
</ul>
<p>Cons:</p>
<ul>
<li>requires significant reconfiguration of laboratory and would likely restrict access to areas of the experiment. </li>
<li>It could also likely result in a temperature increase of the experiment area.</li>
</ul>
<p>Introducing an additional room dehumidifier would be the easiest option by far.</p>
<p><strong>So my Question is:</strong> does anyone know how efficient dehumidifiers works in dry environments? E.g., if I were to purchase an additional dehumidifier, would could I achieve a humidity level of less than 30% or does the humidity level asymptote off at some level due to a limit on the efficiency of dehumidifiers?</p>
<p>I realize an alternative would be to <em>humidify</em> the lab 10 months out of the year. However, having low humidity is extremely convenient for rapidly water-cooling components. During our wet season, our water-cooling results in considerable condensation on our components.</p>
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<p>I'm trying to solve a system of springs and masses that is confusing me.
First, the balls are all lined up linearly. Secondly, the ball in the middle has a smaller mass $m$ while the first and last balls have a larger mass of $M$.
The larger balls are each connected to the middle ball with a spring with a spring constant of $k$. They are assumed to move in the right direction and there is no additional external force. </p>
<pre><code> x_1 x_2 x_3
*--------*---------* ------> x
M k m k M
</code></pre>
<p>I'm trying to solve this system using the eigenvalue concept, but I'm having trouble. I've dealt with a two mass spring system, but never a three mass, so I'm a bit confused. </p>
<p>From what I gather, I just set the $\ddot{x}$ to $F/m$ and set up the matrix, much like for a two mass system. But I don't know how to do that with the third mass. Like for example, for $\ddot{x}_1$, I think that we have </p>
<p>$$0 -(k/m)(x_1-x_2) -(k/M)(x_2-x_3) $$</p>
<p>and for $\ddot{x}_2$</p>
<p>$$ 0 -(k/m)(x_2-x_1) -(k/M)(x_2-x_3)$$ </p>
<p>and for $\ddot{x}_3$</p>
<p>$$ 0+0-(k/M)(x_3-x_2)$$</p>
<p>but I really don't know if I'm right. I highly doubt it. So if some kind soul could tell me what I'm doing wrong/right and point me in the next direction, I would be forever grateful.</p>
| 9,906 |
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<p>If I poured water into my tea, would I see more or less of the bottom of the tea-cup?</p>
<p>Intuitively, there would be as many particles blocking as many photons, and so I'd see the bottom just as clearly as before.</p>
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<p>After playing a game called "Kerbal Space Program" I got interested in orbital mechanics and started messing with <a href="https://www.dropbox.com/s/ha8dlboo6pjgc7s/KSPequations.pdf">simplified calculations</a> to determine $\Delta v$ requirements. In which I compared two trajectories to get into orbit/land from orbit to see how much it would matter if you would burn vertical or horizontal (Hohmann) at the surface. Graphs of some results can be seen <a href="http://imgur.com/a/Ci5iL">here</a>.</p>
<p>However after this I also wanted to know how much time it would take to perform these maneuvers. I assumed that changes in velocity are instantaneous, so this would be equal to the time between the two changes in velocity. For the Hohmann-like-transfer it is simple, since you travel from periapsis tot apoapsis, which takes half of the orbital period (due to symmetry). But for the other transfer, which applies a vertical change in velocity at the surface, it is very difficult (if there is a sidereal rotational velocity). I tried solving it using <a href="http://en.wikipedia.org/wiki/Kepler%27s_laws_of_planetary_motion#Second_law">Kepler's second law</a>, given this formula:
$$
r=\frac{a(1-e^2)}{1+e\cos{\theta}}
$$
Since the area underneath the path from $\theta_0$ to $\theta_1$ can be calculated like this:
$$
A_{\theta_0,\theta_1}=\int_{\theta_0}^{\theta_1}{\frac{1}{2}r^2d\theta}
$$
Which can be used to determine the amount of time it would take, since the total area of the ellipse would take one orbital period $T=2\pi\sqrt{\frac{a^3}{\mu}}$:
$$
t_{\theta_0,\theta_1}=T\frac{\int_{\theta_0}^{\theta_1}{\frac{1}{2}r^2d\theta}}{\int_{0}^{2\pi}{\frac{1}{2}r^2d\theta}}=T\frac{\int_{\theta_0}^{\theta_1}{\frac{1}{2}r^2d\theta}}{\pi a^2\sqrt{1-e^2}}
$$
I am not that good at solving integrals, so I both tried to solve it with MATLAB and WolframAlpha, which gave me this as the result:
$$
\int{\frac{1}{2}r^2d\theta}=a^2\sqrt{1-e^2}\left(\tan^{-1}\left(\frac{\sqrt{1-e^2}\tan{\frac{\theta}{2}}}{1+e}\right)-\frac{e\sqrt{1-e^2}\sin{\theta}}{2(1+e\cos{\theta})}\right)
$$
So:
$$
t_{\theta_0,\theta_1}=\sqrt{\frac{a^3}{\mu}}\left[2\tan^{-1}\left(\frac{\sqrt{1-e^2}\tan{\frac{\theta}{2}}}{1+e}\right)-\frac{e\sqrt{1-e^2}\sin{\theta}}{1+e\cos{\theta}}\right]^{\theta_1}_{\theta_0}
$$
However when I tried to determine the integral with symbolic parameters and bounderies, MATLAB returned "Warning: Explicit integral could not be found." probably because there is not a general solution for both closed and open orbits. But if I use this formula and filled in the boundaries myself, it seemed that I got wrong results. However when I let MATLAB calculate the integral for given values of $a$, $e$, $\theta_0$ and $\theta_1$ I do get results which seem correct.</p>
<p>So is this a good approach to find a time dependency of an (elliptical) orbit. If so is there an (continuous) equation derivable from this. And what might be a better approach?</p>
<p><strong>EDIT:</strong> The unexpected results form this formula might be caused by the fact that $\tan^{-1}$ returns results from $-\frac{\pi}{2}$ to $\frac{\pi}{2}$. Since when I calculated the time from $\theta=0$ to $\theta=\pi$ I got results which agree with the conservation of energy and angular momentum.
For this is used the following equations to determine the time derivatives:
$$
\omega=\frac{\delta\theta}{\delta t}=\frac{\delta \theta / \delta \theta}{\delta t / \delta \theta}=\sqrt{\frac{\mu}{a^3\left(1-e^2\right)^3}}\left(1+e\cos{\theta}\right)^2
$$</p>
<p>$$
\dot{r}=\frac{\delta r}{\delta t}=\frac{\delta r / \delta\theta}{\delta t / \delta\theta}=\omega\frac{\delta r}{\delta\theta}=\omega\frac{ae(1-e^2)\sin{\theta}}{\left(1+e\cos{\theta}\right)^2}
$$
Which yields that the specific orbital energy $\epsilon$ and specific angular momentum $h$ are constant when substituting theses therms into there expressions:</p>
<p>$$
\epsilon=\frac{v^2}{2}-\frac{\mu}{r}=\frac{\dot{r}^2+\omega^2r^2}{2}-\frac{\mu}{r}=-\frac{\mu}{2a}
$$</p>
<p>$$
h=\omega r^2=\sqrt{a\mu\left(1-e^2\right)}
$$</p>
| 9,908 |
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<p>I have some questions about this exercise:</p>
<blockquote>
<p><em>In an horizontal plane, a $OA$ bar with mass $m$ and length $a$ moves, with another bar $AB$ (same mass, double length) attached in the point A.
In the point B, there is a force $F=K\frac {K} {r^2} \frac {B-O} {r}$
Find the equations of motion.</em></p>
</blockquote>
<p>To get the equations, I used the balance of angular momentum. But my problems started when I tried to get the inertia tensor (I know that just with the inertia momentum it would be enough, but I want to find the tensor in order to understand more the physics behind all of this).</p>
<p>The inertia tensor of the first bar $OA$ is easy, just use the Steiner's Theorem to move every inertia momentum of a bar in his center off mass, to a momentum in one of its extremes.</p>
<p>But When I try to get every inertia momentum of the second bar $AB$, some difficulties appear:</p>
<p>I use the formula $I_a=I_b + M(R\times n)^2$ With $I_b$ as the inertia momentum of the bar in its center of mass, $M$ the mass of the body, $R$ the distance between the origin and the center of mass, and $n$ as a normal vector parallel with the principal axis I want to calculate.</p>
<p>But when I use this formula, I get results of the momentum which depends on the module of $R$, which depends on time. ¿Is this ok?</p>
<p>I had the idea of make a "double Steiner", translating the inertia momentum's axis to the extreme of the bar, and then use it again with an axis which crosses the origin. That way, it wouldn't depend on time. ¿Is this possible? </p>
<p>Which of the both ways is the correct one?</p>
| 9,909 |
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] |
<p>In this question-</p>
<blockquote>
<p><em>A motorboat is racing towards north at 25km/h and the water current in that region is 10km/hr in the direction of 60 degree east of south. Find the resultant velocity of the boat.</em></p>
</blockquote>
<p>The first part is quite easy and we get 21.8 approx as the magnitude of the resultant.</p>
<p>My doubt is in the 2nd part of the question. How do we calculate the direction using the normal method. one way is to use the sine formula and say that
R/sin(a)=velocity of current/sin(alpha)
where a is angle between the vectors and alpha is the angle of resultant R with the north direction. this gives us the angle with the north that is 23.4 degrees which is correct but how does one use the normal tan(alpha) method which works in questions where alpha is less that 90 or equal to 90.</p>
<p>By the tan(alpha) method i am referring to this formula-
tan(alpha)=Asin(a)/(b+A*cos(a))
one of the book says that it is tan(alpha)=10sin120/(25+10cos120) which gives the correct answer also
but i cant understand how we get this. after making a parallelogram if the angle b/w vectors is less than 90 or 90 we just extend one of the sides and get 2 right angle triangles. in this case we get only one.please explain how do we do this.</p>
| 9,910 |
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<p>This is a question for anyone who is familiar with Di Francesco's book on Conformal Field theory. In particular, on P.108 when he is deriving the general form of the 2-point Schwinger function in two dimensions. He writes that the most general form of the tensor is $$S_{\mu \nu \rho \sigma} = (x^2)^{-4} \left\{ A_1 g_{\mu \nu} g_{\rho \sigma} (x^2)^2 + A_2 (g_{\mu \rho}g_{\nu \sigma} + g_{\mu \sigma}g_{\nu \rho})(x^2)^2 + A_3(g_{\mu \nu}x_{\rho}x_{\sigma} + g_{\rho \sigma}x_{\mu}x_{\nu})x^2 + A_4 x_{\mu}x_{\nu}x_{\rho}x_{\sigma}\right\}$$ This I understand and have obtained this result myself. What I don't understand however, is why he has neglected the following term since it seems to satisfy all the constraints presented on P.108: $$S_{\mu \nu \rho\sigma} = A_5 (x^2)^{-3} (g_{\mu \sigma} x_{\rho}x_{\nu} + g_{\mu \rho}x_{\sigma}x_{\nu} + g_{\nu \sigma}x_{\rho}x_{\mu} + g_{\nu \rho}x_{\sigma}x_{\mu})$$ <a href="http://physics.stackexchange.com/questions/134588/demostrating-possible-equivalence-of-two-tensors">In another thread I posted here</a>, I wondered whether this could be reduced to terms already present in the form Di Francesco gave, but I was quickly reassured this to not be the case. So, if anyone is familiar with his book and would be willing to clarify this it would be great. I asked a professor at my university and he was not sure either why it has been neglected, so I thought I would pose the question here.</p>
| 9,911 |
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<p>I read a book on pop sci book on quantum mechanics and the author said that electrons do not fall into the nucleus due to quantum mechanics- which principles suggest this (I think it was Heisenberg's Uncertainty and Pauli's Exclusion Principle) and why?</p>
<p>Also, I've heard that if Bohr's planetary model were correct, then electrons would lose energy/momentum and fall in- is this true and again, which physics principles say this?</p>
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<p>In comments to <a href="http://physics.stackexchange.com/q/135212/5739">a Phys.SE question</a>, it has been written: </p>
<blockquote>
<p><em>'Tunneling' is perfectly real, even in classical physics. [...] For sufficiently large temperatures this can put the system above a hump in its potential energy.</em></p>
</blockquote>
<p>and </p>
<blockquote>
<p><em>the only difference between the classical case and the quantum mechanical one is that classical physics is a random walk in real time, while QM is a random walk in imaginary time.</em></p>
</blockquote>
<p>I understand that in a system of particles with finite temperature some particles can overcome a potential barrier. That's how I interpret the first statement. I don't understand the business of "random walk in imaginary time". Can someone explain?</p>
<p><strong>Update</strong> </p>
<p>What I was originally looking for was 1.) classical system that can transport mass through a forbidden region and 2.) explanation of "random walk in imaginary time". So far, I don't see anything for question 1.), but I think I'll grok 2.) if I invest some time and energy.</p>
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<p>I am an newbie general relativistic learner and I learnt that gravity is bending of space-time and since objects move in straight-lines but since its curved they follow curved movement through space thus creating the effect we know as gravity.</p>
<p>That said, what if an object has velocity of 0 and since its not moving at all, (except in Time) why does the object fall or move in curved space-time (geodesics)? Is there external force that also pulls the object if so why and how does it work? </p>
<p>If the object moves in an curved space-time, why does an object just move in an normal space-time (not curved) rather than stay at velocity 0? </p>
<p>Is there any reason why this happens? </p>
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<p>I am doing some self-study on photonics and have encountered the following question:</p>
<blockquote>
<p><em>We know that amorphous electronic crystals such as <a href="http://en.wikipedia.org/wiki/Amorphous_silicon" rel="nofollow">amorphous silicon</a> have a bandgap. Can amorphous photonic crystals also have a bandgap? Roughly how large would the spacing, d, have to be for a bandgap centered around ~ 600 nm?</em></p>
</blockquote>
<p>Here is my solution:
Since $n\lambda=2dsin(\theta)$, if $\lambda=600 nm, \theta=\pi/4, n=1,$ then $d=4.24 \times 10^{-7}$.</p>
<p>Could somebody check over my work?</p>
| 9,915 |
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<p>In an experiment we placed a long dielectric cylindrical shell (cylindrical tube)(with inner and outer radii r1 and r2, respectively) in a homogenous field such that its axis was orthogonal to the field. The medium elsewhere has a dielectric constant of unity. </p>
<p>How can I calculate the potentials and fields everywhere?</p>
| 9,916 |
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<p>I'm working to solve the steady-state short circuit current of a solar cells, using the coupled continuity equations with a drift-diffusion expression and Poisson's equation:</p>
<pre><code>D[n[x, t], t] - (1 / q) D[Jn[x, t], x] == G - R;
D[p[x, t], t] + (1 / q) D[Jp[x, t], x] == G - R;
Jn[x, t] == q \[Mu]n (n F + (kb T / q) D[n[x, t], x]);
Jp[x, t] == q \[Mu]p (p F - (kb T / q) D[p[x, t], x]);
D[D[\[Phi][x, t], x],
x] == (q / (\[Epsilon]r \[Epsilon]0)) (p[x, t] - n[x, t]);
</code></pre>
<p>In particular, I'm following the work from a literature paper that (no surprise) leaves out many details. The paper says to first find the equilibrium solution by solving Poisson's equation numerically with the equilibrium expressions:</p>
<pre><code>nboltz[x] =
Nc Exp[(-kb T Log[Nc] eVperJ - q \[Phi][x]) / (kb T eVperJ)] ;
pboltz [x] =
Nv Exp[(-kb T Log[Nv] eVperJ + q \[Phi][x] ) / (kb T eVperJ)];
</code></pre>
<p>I can get this far using the boundary conditions below (for electric potential). However, it then says: "the steady-state solutions found using the time-evolution method of David's et al, which uses an additional equation for the time derivative of the electric field:</p>
<pre><code>D[F[x, t],
t] == - (1 / (\[Epsilon]r \[Epsilon]0)) (Jn[x, t] +
Jp[x, t] - (1 / L) Integrate[ Jn[x, t] + Jp[x, t], {x, 0, L}]);
</code></pre>
<p>In addition, I have the boundary conditions, </p>
<pre><code>p[0, t] = Nv
n[0, t] = Nc Exp[(Ev - Ec) / (kb T eVperJ)]
n[L, t] = Nc
p[L, t] = Nv Exp[(Ev - Ec) / (kb T eVperJ)]
</code></pre>
<p>Davids et al. says, "To find the steady state solution at an applied voltage bias, a time dependent potential ramp which stops at the desired voltage is applied to the right contact and the equations are integrated forward in time starting from thermal equilibrium until state state is reached . The position independence of the total particle current $J = J_n + J_p$ is used to verify that steady state has been reached."</p>
<p>Conveniently, I am looking at the short-circuit case where the applied voltage bias is 0. However, I don't understand how one can integrate a steady state solution forward in time. Isn't a steady state solution fundamentally unchanging with time? Moreover, if integrating forward in time is just to help the solver, I am still unsure of how to implement this (I'm working in Mathematica).</p>
<p>Hopefully this all formats well and thanks for the help!</p>
| 9,917 |
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] |
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