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<p><img src="http://i.stack.imgur.com/H8qNp.png" alt="enter image description here"></p>
<p>I saw this graph about the electrons density in different altitudes and difference between night and day, the difference between the 2 electron densities (day and night) decreases till 300 Km (F2 layer) and then the difference increases again.</p>
<p>So, I wanted to know why is the recombination rate in F2 layer very low.</p> | 1,124 | [
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<p>I'm not a physicist and I do not understand maths. But I watch documentaries about "how it all began", "the big bang", "What is time", etc etc just really fascinating. </p>
<p>I was wondering if a blackhole would get in contact with another blackhole, which one would consume time and space? Because they have both infinite mass and gravity.</p>
<p>Will there be a point in space, right in the center between two blackholes, where time and space will move away from you in multiple direction? </p>
<p>What would happen if two blackholes suck on eachother?</p> | 1,125 | [
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<p>So I am a little confused on how to deal with the Kilowatt hours unit of power, I have only ever used Kilowatts and I have to design a residential fuel cell used as a backup generator for one day.</p>
<p>The average power consumption of a US household is 8,900 kW-hr per year and 25 kW-hr per day and approximate 1 kW-hr per hour. Does this mean that the power output of my fuel cell is 1 kW and if I wanted to use it for the entire day would it have to be designed to be 25 kW?</p> | 1,126 | [
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<p>I am using an old BBO crystal, which is not designed for spontaneous parametric down conversion, to see whether I can generate converted photons of 800nm from 400nm. I used a photon avalanche detector and I observed some signal of 800 but I couldn't observe them directly or by a CCD.</p>
<p>I would like to know which is the cut angle designed for Type I BBO phase matched SPDC to understand why I couldn't observe the ring of SPDC.</p>
<p>Besides, I use a pulse laser rather than a continuous laser. Is this matters?</p>
<p>Regards</p> | 1,127 | [
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<p>Could a very small black hole where half of its entropy has been radiated, emit Hawking radiation that is macroscopically distinct from being thermal? i.e: not a black body radiator. Or would the scrambling property mean that the initial states that produce the non-thermal Hawking radiation are micro states that are macroscopically indistinguishable from micro states that lead to Hawking radiation that is nearly thermal?</p>
<p>For the sake of the question, if two microscopic states are macroscopically indistinguishable, it should be assumed that preparing the state in one or the other is physically impossible, even if it is possible in principle.</p>
<p>a <a href="http://motls.blogspot.com/2012/12/hawking-radiation-pure-and-thermal.html" rel="nofollow">reference</a> to a recent TRF post about thermal and pure states of black holes being macroscopically indistinguishable from the outside</p> | 1,128 | [
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<blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="http://physics.stackexchange.com/questions/17524/path-to-obtain-the-shortest-traveling-time">Path to obtain the shortest traveling time</a> </p>
</blockquote>
<p>I've been told that if one would want to make a ramp to get a ball from point A to a lower point B (at a certain horizontal distance from A), the best shape to make this ramp in - by which I mean the ramp that makes the ball reach point B the fastest - would be a cycloid path. Why is this?</p> | 6 | [
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<p>I would be very curious if Kerr black holes emit Hawking radiation at the same temperature in the equatorial bulges and in their polar regions. I've been looking some reference for this for a couple of months now but i haven't been successful yet.</p>
<p>According to <a href="http://physics.stackexchange.com/a/46830/955">this answer</a>, Too great Hawking radiation could thwart mass from being feed into a black hole, so this begs the question: Could possibly a (very small) Kerr black hole be able to accept infalling matter if it falls along the rotational axis? Possibly, since 4D Kerr black holes angular momentum are bounded by extremality condition, the possible difference in temperature between polar and equatorial regions will not be wide enough to be interesting (possibly not bigger than an order of magnitude), but it would still be interesting to know how much is the temperature ratio. It could very well be zero, or more intriguingly, could depend on the black hole mass. But in any case, i doubt that the ergosphere region will not interact nontrivially to boost equatorial Hawking radiation to some degree</p>
<p>Thoughts?</p> | 1,129 | [
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<p>I was driving uphill from a complete stop for a distance of .4 miles estimated to take 1 minute in a navigation app. I was pulled over right after cresting the hill. The cop had me on radar going 53mph. What do I need to know to see if it was possible to be going that fast uphill ? The grade of the incline? The 0 to 60 time capability of my car? I appreciate any help I can get!</p> | 1,130 | [
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<p>I'd like to learn modern physics at an advanced level, but since I've no access to university, I'm self-teaching, and appeal to the Internet for information about what to study and how.</p>
<p>Currently, I'm trying to form a strong mathematical background. Until now, I've studied basic Linear Algebra and Affine/Euclidean/Projective geometry (in a linear algebra approach), because these are the first topics studied by Physics undergraduates in my country (Italy). But since I've virtually no time limit, instead of progressing to Real Analysis, I'd like to deepen the knowledge of what I've already studied.</p>
<p>What are according to you, of the topics I've listed, the most important for Physics, or the most important for future geometrical studies (which, I suppose, should be differential/algebraic geometry, topology...)?</p>
<p>Thank you for any answer!</p> | 1,131 | [
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<p>I have a simple question about general relativity and the Einstein field equations, I wonder if you can specify the stress energy tensor, i.e. specify some mass distribution in space and then calculate the curvature to later find equations of motions etc, instead of starting out with how the geomerty would look. I am quite new to general relativity and so I am bound to have misconceptions. </p>
<p><strong>Edit: 2013 December 19th</strong>
I have found this <a href="http://mathreview.uwaterloo.ca/archive/voli/2/olsthoorn.pdf" rel="nofollow">article</a> which at page 10, Chapter 5, section 5.2 does something simillar to what I meant, apperently there is a general from for the Stress energy tensor (for what is known as a perfect fluid(?)), and from it they derive something simillar to the second component in the normal schwarzschild metric i.e $$A(r)=(1-\frac{2U}{r})^{-1}$$ where $U$ is the energy. I do have one remaining question, the name of the general form of the stress energy tensor confuses me somewhat, "perfect fluid" is it just its name, and is it still fully capabable of describing the stress energy tensor in general relativity?</p> | 1,132 | [
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<p>I came across this problem in physics "Physics for Scientists and Engineers
with Modern Physics by Serway"</p>
<blockquote>
<p><em>A block on the end of a spring is pulled to position $x = A$ and released from rest. In one full cycle of its motion, through what total distance does it travel?</em></p>
</blockquote>
<p>Why the answer is $4A$ instead of $2A$?</p> | 1,133 | [
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<p>According to <a href="http://en.wikipedia.org/wiki/Hill_sphere" rel="nofollow">Wikipedia</a>, Hill Sphere is : the volume of space around an object where the gravity of that object dominates over the gravity of a more massive but distant object around which the first object orbits.</p>
<p>True as this may be, it just mathematically supports a phenomenon that has been observed but it does not give reason or logic as to why does this happen in the first place. I mean why should the gravity of a less massive object dominate the gravity of a more massive one?</p>
<p>I wasn't aware of the Hill Sphere until recently when I was trying to visualize the orbits of different celestial bodies. The Hill Sphere comes closest to explaining why the moon orbits the Earth, more than it orbits the Sun and why the Earth orbits the sun, more than it orbits the center of our galaxy. By this logic all celestial bodies within the Gravitational pull of the center of our galaxy should directly be orbiting the center.</p>
<p>My argument is that if the Hill sphere of the Sun is as large as the solar system itself, any object within this sphere should be orbiting the sun. Why was the moon caught into the earth's gravitational pull in the first place when it had a much stronger pull from the sun? </p>
<p>The answer to this would also eventually clarify why the earth orbits around the sun and not the center of the milky way.</p> | 1,134 | [
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<p>If you have a projectile with these variables.
$x_0 = 1v_{0x} = 70, y_0 = 0, v_{0y} = 80, a_x = 0, a_y = -9.8$
I know how to plot these points with this equation.
$$ x = x0 + (v_{0x})t + 1/2((a_x)t^2) $$
$$ y = y0 + (v_{0y})t + 1/2((a_y)t^2) $$</p>
<p>I want to add air resistance to this problem and i know its a sphere, so the drag coefficient is 0.47, and lets say the area is 0.5. I use this equation to find the resistance.
$$K = 1/2*C_p*A_p$$ where $C_p$ is the drag coefficient and $A_p$ is the area of the sphere.
I then try to find the velocity of x and y by using these equations.
$$F_dx = KV^2_x$$
$$F_dy = KV^2_y$$
I then plug these in back into my initial x and y equations
$$ x = x0 + (v_{0x})t - 1/2((F_dx/m +a_x)t^2) $$
$$ y = y0 + (v_{0y})t - 1/2((F_dy/m +a_y)t^2) $$</p>
<p>I am having a hard time getting the right numbers and pictures when i use these equations. Am i doing something wrong here? Will someone please help me.
I would really appreciate any help.</p> | 7 | [
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<p>Sorry for the strange question, but why is it that many of the most important physical equations don't have ugly numbers (i.e., "arbitrary" irrational factors) to line up both sides? </p>
<p>Why can so many equations be expressed so neatly with small natural numbers while recycling a relatively small set of physical and mathematical constants?</p>
<p>For example, why is mass–energy equivalence describable by the equation $E = mc^2$ and not something like $E \approx 27.642 \times mc^2$?</p>
<p>Why is time dilation describable by something as neat as $t' = \sqrt{\frac{t}{1 - \frac{v^2}{c^2}}}$ and not something ugly like $t' \approx 672.097 \times 10^{-4} \times \sqrt{\frac{t}{1 - \frac{v^2}{c^2}}}$.</p>
<p>... and so forth.</p>
<hr>
<p>I'm not well educated on matters of physics and so I feel a bit sheepish asking this. </p>
<p>Likewise I'm not sure if this is a more philosophical question or one that permits a concrete answer ... or perhaps even the premise of the question itself is flawed ... so I would gratefully consider anything that sheds light on the nature of the question itself as an answer.</p> | 1,135 | [
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<p>I'm given a parallel plate capacitor with plate separation of 4mm and plate area 20cm^2. I've calculated the capacitance to be 4.425 x 10^-10 F. I'm asked to also find the maximum voltage and charge the capacitor can store.</p>
<p>If I'm using the right equations then the voltage should be V = Ed and the charge should be Q = CV. I'm confused because I can't calculate the voltage without the charge and I don't believe I can calculate the electric field with the given information.</p>
<p>Is there another equation I should be using or am I missing something here?</p> | 1,136 | [
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<p>Can a machine deconstruct objects on an atomic level. But is that possible? Not the machine per say but the simple (not that simple) act of what it does. Ex.Taking a broken computer and separating the different atoms so it would be just a group of atoms instead of a computer. </p>
<p>If you have any suggested readings on the topic that would be greatly appreciated. </p>
<p>Q: Can an object be broken up to be just separate atoms? </p> | 1,137 | [
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<p>If I have a 12V 4Ah lead acid battery and use a battery charger that, let's say for example, can charge 10A, 50A, or 100A. If I theoretically turned it to 100A will the battery explode?</p>
<p>I understand that when you use a higher amperage the battery will charge quickly but due to resistance and flow of ions a lot more heat will be generated, so will this heat cause an explosion..or perhaps just a bursting of that battery spewing boiling acid?</p>
<p>And no I am not trying this in real life..I just recall seeing the scene in the Amazing Spider-Man 2 when Parker is trying to build his web shooters to be able to resist large amounts of electricity yet they keep exploding.</p> | 1,138 | [
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<p>When Chiral Symmetry was exact, as it was before EWSB due to the lack of mass terms for quarks, would the residual strong force have infinite range? Related to this, does the Negative Beta Function apply for the Nuclear Force, or does it only affect the Strong Force within nucleons?</p> | 1,139 | [
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<p>I am searching the literature for the Flory-Huggins phase diagram with the following components : polymer, solvent, and a third component that does not interact with the other components (just entropy effects). It must have been done but I can't find the special case in which the third component is not interacting. I am interested in the entropy effect of the third component.</p>
<p>Thanks for any help.</p>
<p>Best,
Jd</p> | 1,140 | [
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<p>As the Feynman diagram shows above. Does the s-channel and t-channel stands for exactly same reaction or they have big difference?</p>
<p>Many thanks in advance
enter image description here</p>
<p><img src="http://i.stack.imgur.com/VG6GR.png" alt="enter image description here"></p> | 1,141 | [
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<p>Can elementary particles (like the electron, photon, or neutrino ) go through an atom (not the nucleus)?</p> | 1,142 | [
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<p>We know that if we take Fourier transform of momentum we go to position space. But why Fourier transform only.(credit_ Abh Gupta)</p> | 8 | [
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<p>I know that the weather balloon will eventually be stopped because of the atmosphere no longer being buoyant but would the balloon be able to go farther than it regularly would?</p> | 1,143 | [
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<p>I was doing some static equilibrium problems and I came across this problem which should be easy to solve, but is posing quite a challenge. I want to point out that this is <strong>not</strong> homework, just plain old studying. By the way, I don't know how to format mathematical equations and I think writing them out here without any formatting is a mess, so I took phtographs of the relevant bits. If any brave soul feels up to the task, I'd greatly appreciate it :)</p>
<p>This problem is from this <a href="http://rads.stackoverflow.com/amzn/click/1133954057" rel="nofollow">book (8th edition)</a>, and instead of copying the entire problem and the image, I took a picture of everything:</p>
<p><img src="http://i.stack.imgur.com/A4mJ2.jpg" alt="enter image description here"></p>
<p>So, I know that the sum of forces on the x axis, the sum of forces on the y axis and the torque considered from any origin must all equal 0 for equilibrium to happen.</p>
<p>First, I drew this simple diagram with all the forces applied on the body:</p>
<p><img src="http://i.stack.imgur.com/ZxDbe.jpg" alt="enter image description here"></p>
<p>Then I wrote this simple set of equations:</p>
<p><img src="http://i.stack.imgur.com/qb28n.jpg" alt="enter image description here"></p>
<p>If I solve this to get the maximum mass M that can be hanged, I get this result:</p>
<p>m + M = m/2 + M</p>
<p>Which is nonsense! This cannot be correct... So, I went back to the drawing board trying to figure out what was wrong. The only thing I could think of was that I was assuming the normal force R was perpendicular to the ground, which I think it is (it's called <em>normal</em> for some reason). However, the result I'm getting does not make any sense, so there must be an error somewhere!</p> | 1,144 | [
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<p>I recently saw this video on youtube:</p>
<p><a href="http://www.youtube.com/watch?v=oJfBSc6e7QQ" rel="nofollow">http://www.youtube.com/watch?v=oJfBSc6e7QQ</a></p>
<p>and I don't know what to make of it. It seems as if the theory has enough evidence to be correct but where would all the water have appeared from? Would that much water have appeared over 60 million years? Also what would cause it to expand. The video suggests that since the time of dinosaurs the earths size has doubled in volume, how much of this is and can be true?</p>
<p>[could someone please tag this, I don't know what category this should come under]</p> | 1,145 | [
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<p>Given an action of the form </p>
<p>\begin{equation}S=-\frac{1}{4}\int d^4x\eta^{\mu\nu}\eta^{\lambda\rho}F_{\mu\lambda}F_{\nu\rho}\end{equation}</p>
<p>where $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$, $\eta_{\mu\nu}=g_{\mu\nu}/a^2(\eta)$, where $g_{\mu\nu}$ is given by the line element:</p>
<p>\begin{equation}ds^2=a^2(\eta)[d\eta^2-(dx^i)^2]\end{equation}</p>
<p>I would like to solve for $A_{\mu}$, and standard solution is </p>
<p>\begin{equation}A_{\mu}^{(\alpha)}=e_{\mu}^{(\alpha)}e^{ik_\nu x^\nu}.\end{equation} </p>
<p>I am interested in knowing how to derive this result.</p>
<p>My approach is first write the Lagrangian from action and use EL eq</p>
<p>\begin{equation}\frac{\partial \mathcal{L}}{\partial A_{\mu}}-\frac{d}{d x^{\nu}}\frac{\partial \mathcal{L}}{\partial(\partial_{\nu}A_{\mu})}=0\end{equation}</p>
<p>My main problem is mathematical difficulty in evaluating the EL eq. Can anyone please help me on this? </p> | 1,146 | [
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<p>How do I understand that the action for the free relativistic scalar field theory is non linear? What will be the associated interaction potential of that equation?</p> | 1,147 | [
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<p>Measurement of a quantum observable (in an appropriate, old-fashioned sense) necessarily involves coupling to a system with a macroscopically large number of degrees of freedom.
Entanglement with this "apparatus" takes care of the decoherence. It is often said (I can provide references upon request) that the remaining problem is the one of "selection", and this is the point where one invariably invokes something philosophically radical, like many-worlds interpretation.</p>
<p>In the above (pretty standard) context, I am trying to make sense of the following observation. Looking at the measuring system from a statistical mechanics point of view, it seems that triggering a particular macroscopic outcome requires spontaneous symmetry breaking via a (thermodynamically) irreversible transition of the "apparatus" from a metastable to a higher entropy final state. My attitude is that "statistical mechanics point of view" is not far from "decoherent large quantum system".</p>
<p><strong>So, the question is:</strong></p>
<p>Is it fair to say that statistical irreversibility ("the second law") and quantum measurement irreversibility (the "wave-function collapse") are necessarily linked? Can this link be made more concrete (e.g., traced in details in a particular model)? Can you give references to approaches to the measurement problem that explore this connection? </p> | 1,148 | [
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] |
<p>I was ruminating the explanations about how boats can sail against the wind (or "<a href="http://www.ksl.com/?sid=19781767" rel="nofollow">into the wind</a>"), and wondered if one could devise a simple mechanical model without hydrodynamics involved. </p>
<p>Imagine a cart (in red in the figure) that is allowed to move along a straight rail (NE-SW orientation). It has a vertical mast at the center, and we attach to it a panel (blue), like a rigid sail, that is kept at a fixed angle with respect to the rail. The only propulsion is to be extracted from a stream of green balls that are thrown, from the east, and bump against the panel (assume perfect elastic collisions). We want to move the card upwards, in the NE direction. </p>
<p>It would seem that this can be done, by placing the panel in an angle as in the figure, and that by mere mechanical arguments -transfer of momentum- the cart should move upward, "against the balls".</p>
<p>Is this true, and is this a fair model of what happens when "sailing against the wind"?</p>
<p><img src="http://i.stack.imgur.com/ChO9f.png" alt="enter image description here"> </p> | 1,149 | [
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<p>We see the crab pulsar, we don't see any compact remnant from Supernova 1987A.
I can't find any others, but I believe they exist. Help?</p> | 1,150 | [
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<p>Given the numerous chemical compounds found in dry air [compressed into a liquid] of a given volume [lets say $22.4$ L for simplicities sake] whose atomic weights far surpass that of water alone assuming water is $H_2 O$. Why does $22.4$ L of compressed liquid air weigh less than the same bucket of liquid water?</p> | 1,151 | [
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<p>A rather simple question:
Starting from an electrically neutral state, pairs of top quarks are produced as top and anti-top, and denoted as $t\bar t$. </p>
<p>Now the production of pairs of scalar top quarks, the supersymmetric partners of the top quarks, seems to be commonly denoted as $\tilde t \tilde t^*$ (e.g. <a href="http://arxiv.org/pdf/1206.3865v3.pdf" rel="nofollow" title="arXiv:1206.3865">1</a>, <a href="http://arxiv.org/pdf/1304.2411v2.pdf" rel="nofollow" title="arXiv:1304.2411">2</a>), rather than $\tilde t \bar{\tilde t}$. Why this notational difference? What is $\tilde t^*$?</p> | 1,152 | [
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<p>Micro-channel plate (MCP) detectors are used to detect photons, electrons or charged particles. But how can MCPs be used to detect neutral particles? In ion traps, the neutral molecules (after being irradiating with a laser, the ions fragment or lose electrons to produce neutral molecules or fragments) are detected with MCPs. I didn't understand the principle of how MCPs work for neutrals! Can anyone please explain this?</p> | 1,153 | [
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<p>I came across this question while having conversation with one person.
We know that <a href="http://en.wikipedia.org/wiki/Center_of_mass" rel="nofollow">Center of Gravity</a> of a solid cube is at the intersection of connecting the opposite vertex of the cube.
Suppose, you have a hollow cube, filled with water/air. Now, how do you find the center of gravity of that object?</p> | 1,154 | [
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<p>The force on a small element (of length dl) of a current carrying wire, place in a magnetic field <strong>B</strong> can be calculated using the following equation (which is simply an application of the Lorentz force equation):</p>
<p>d <strong>F</strong> = i <strong>dl</strong> X <strong>B</strong></p>
<p>My question is, how does one calculate the force acting on a small magnetized volume d <strong>V</strong> (say magnetized in the z direction with magnetization <strong>M</strong> or remanent flux density <strong>Br</strong>) when the same is placed in a magnetic field <strong>B</strong>.</p> | 1,155 | [
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<p>What makes the beam of some lasers: </p>
<p>1-visible?
such as the ones used in clubs or such as the laser pointers sold at amazon which if pointed to the sky look like a solid visible beam of light crossing the sky (it reminds me of the laser sword in star wars)</p>
<p>2-invisible?
such as the ones used in pointers for presentations</p> | 1,156 | [
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<p>How to derive an expression for entropy in form of</p>
<p>$S = \ln \Omega$</p>
<p>from the form</p>
<p>$\displaystyle{S = - \sum_i \; p_i \ln p_i}$ ?</p>
<p>That is the last formula taken as a definition of entropy.</p>
<p>Just a reference will do. The backwards derivation (probably with some assumptions) is given in Landau and Lifshitz.</p> | 1,157 | [
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<p>If you bombard an electron shell with a photon below the critical level to promote the electron to a higher state, will the shell absorb nothing and the photon get deflected with the same amount of frequency/energy that it came with initially?</p> | 1,158 | [
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<p>I am a mathematician, not a physicist, so please be gentle with me if I write something wrong.</p>
<p>Consider a bounded, regular container $\Omega$, which is filled with the fluids $F_1,...,F_N$ which do not mix (i.e. $\bigcup_{i=1}^N F_i=\Omega$ and $F_i\cap F_j=\emptyset, \forall i\neq j$). Between two adjacent fluids $F_i,F_j$ there is a surface tension $\sigma_{ij}$ (which is eventually zero if $F_i$ and $F_j$ are not adjacent). The problem I want to study is given $F_i$ with volume $V_i$ and density $\rho_i$ then what is the final state in which the fluids will arrive.</p>
<p>There are three factors I have in mind:</p>
<ul>
<li>the interaction of $F_i$ and $F_j$ with $i\neq j$ by their surface tension;</li>
<li>the interaction between $F_i$ and the boundary $\partial \Omega$ of the container;</li>
<li>the action of gravity on each $F_i$.</li>
</ul>
<p>I have two questions:</p>
<blockquote>
<ol>
<li><p>Is there a relation of the form $\sigma_{ij}+\sigma_{kl}=\sigma_{ik}+\sigma_{jl}$ (scalar or vectorial) between the surface tensions?</p></li>
<li><p>Are there any references or monographs which provide a good introduction to this study? I'm interested especially in surface tensions.</p></li>
</ol>
</blockquote> | 1,159 | [
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<p>I am very confused that some atoms called high spin or magnetic atoms have spin level more than $\frac{1}{2}$ but are still said to have $SU(2)$ symmetry.</p>
<p>Why not $SU(N)$?</p> | 1,160 | [
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<p>I require a software to simulate Fluid simulation with the capability of supporting vacuum simulation. My requirements are that all numbers must reflect their real counterparts almost exactly. For example I need to mix Fluid, Air and Vacuum.</p>
<p>I have tried RealFlow but it doesn't support Vacuum. </p>
<p>Any body knows any software for this?</p> | 1,161 | [
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<p>I was trying to solve frictionless inclined plane problem using a diff. frame as shown in the figure, and can't figure out that acceleration along the plane = g.sin(Θ), but I think it should be = g/sin(Θ) as according to the figure. </p>
<p>I have gone through numerous examples and theory regarding such cases, but all of them use the frame with x axis along the incline plane and y axis perpendicular to it.</p>
<p><img src="http://i.stack.imgur.com/qlEuM.png" alt="Frictionless inclined plane"></p> | 1,162 | [
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<p>I would like to demonstrate the several forms of the Friedmann equations WITH the $c^2$ factors. Everything is fine ... apart that I have a missing $c^2$ factor somewhere.</p>
<p>In all the following $\rho$ is the <strong>mass density</strong> and not the energy density $\rho_{E}=\rho c^2$</p>
<p>If we look at the wikipedia French page concerning the <a href="http://fr.wikipedia.org/wiki/%C3%89quations_de_Friedmann" rel="nofollow">Friedmann equations</a>, according to the demonstration of the last paragraph we have :</p>
<p>The Einstein field equation : $G_{\mu\nu} = \frac{8\pi G}{c^4}T_{\mu\nu}$</p>
<p>The Einstein tensor :
$G_{\mu\nu} =
\begin{pmatrix}
G_{00}&0&0&0 \\
0&G_{ij}&0&0 \\
0&0&G_{ij}&0 \\
0&0&0&G_{ij}
\end{pmatrix}$</p>
<p>The Energy-Momentum tensor :
$T_{\mu\nu} =
\begin{pmatrix}
T_{00}&0&0&0 \\
0&T_{ij}&0&0 \\
0&0&T_{ij}&0 \\
0&0&0&T_{ij}
\end{pmatrix}$</p>
<p>with :</p>
<p>$G_{00} = 3H^2+3\frac{k}{a^2}c^2$</p>
<p>$G_{ij} = -\left(3\frac{H^2}{c^2}+2\frac{\dot{H}}{c^2}+\frac{k}{a^2}\right)$</p>
<p>$T_{00} = \rho c^2$</p>
<p>$T_{ij} = -P$</p>
<p>But : $T_{00}$ and $T_{ij}$ have the same physical unit ($P$ and $\rho c^2$ are in $kg.m^{-1}.s^{-2}$) whereas $G_{00}$ and $G_{ij}$ does not have the same unit : in the first one we have $H^2$ and in the second one we have $\frac{H^2}{c^2}$ for example.</p>
<p><strong>My question are</strong> : is there a mistake in the french wikipedia demonstration ? Where is the missing $c^2$ ? Where can I find a good demonstration with the $c^2$ factors ?</p>
<p>EDIT :
Maybe I've found something. At the beginning of the demonstration, the author say that the metric is of the form :</p>
<p>$ds^2=c^2dt-a^2\gamma_{ij} dx^i dx^j$</p>
<p>where $\gamma_{ij}$ depends on the coordinates choice. This formula seems ok to me. </p>
<p>But then he writes that :</p>
<p>$g_{00} = c^2$</p>
<p>$g_{ij} = -a^2\gamma_{ij} $</p>
<p>I have a doubt on $g_{00}$ : is it equal to $c^2$ or to $1$ ? In fact, if we choose to write $g_{00} = c^2$, then $T_{00} = \rho c^4$ isn't it ?</p> | 1,163 | [
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<p>Does temperature affect the internal resistance of batteries? And does charging a "frozen" battery allow it to charge faster than a warm or room temperature battery?</p> | 1,164 | [
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<p>Is there a simple account of why technetium is unstable?</p>
<p>From the <a href="http://en.wikipedia.org/wiki/Technetium#Isotopes">Isotopes section</a> of <a href="http://en.wikipedia.org/wiki/Technetium">Wikipedia's article on Technetium</a>:</p>
<blockquote>
<p>Technetium, with atomic number (denoted Z) 43, is the lowest-numbered element in the periodic table that is exclusively radioactive. The second-lightest, exclusively radioactive element, promethium, has an atomic number of 61. Atomic nuclei with an odd number of protons are less stable than those with even numbers, even when the total number of nucleons (protons + neutrons) are even. Odd numbered elements therefore have fewer stable isotopes.</p>
</blockquote>
<p>It would seem that simply its atomic number is part of the reason why it is unstable, though this just pushes back the mystery back one step for me: why are nuclei with even atomic number more stable? And why then are all of the elements from 45 through 59 stable — notably including silver (Z=47) and iodine (Z=53) — not to mention higher odd-proton nuclei such as gold (Z=79)?</p>
<p>Even the most stable isotope of technetium has a half-life less than a hundredth that of uranium-235, which has a half life of 703.8Ma:</p>
<blockquote>
<p>The most stable radioactive isotopes are technetium-98 with a half-life of 4.2 million years (Ma), technetium-97 (half-life: 2.6 Ma) and technetium-99 (half-life: 211,000 years) [...] Technetium-99 (99Tc) is a major product of the fission of uranium-235 (235U), making it the most common and most readily available isotope of technetium. </p>
</blockquote>
<p>It's perhaps an unfair comparison, as uranium has an even atomic number (however that is suppose to help mitigate its instability); but it also has nearly twice the number of protons. This deepens the mystery for me. Even granted that Tc has no stable isotopes, how does it come to be <em>so</em> unstable that all of its isotopes are essentially absent naturally, compared for instance to uranium-235?</p>
<p>(This question is a specific case of an <a href="http://physics.stackexchange.com/q/40894/4976">earlier question on synthetic isotopes</a>.)</p> | 1,165 | [
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<p>Photon is a spin-1 particle. Were it massive, its spin projected along some direction would be either 1, -1, or 0. But photons can only be in an eigenstate of $S_z$ with eigenvalue $\pm 1$ (z as the momentum direction). I know this results from the transverse nature of EM waves, but how to derive this from the internal symmetry of photons? I read that the internal spacetime symmetry of massive particles are $O(3)$, and massless particles $E(2)$. But I can't find any references describing how $E(2)$ precludes the existence of photons with helicity 0.</p> | 836 | [
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<p>If we have the wave function $\psi_{100}(r,\theta,\phi)=R_{10}(r)Y_{00}(\theta,\phi)$ when we are normalising it we do the following:
$$1=\int| \psi_{100}(r,\theta,\phi)|^2sin(\theta) r^2drd\theta d\phi$$ but can we also normalise the individual parts separately i.e.
$$1=\int r^2|R_{10}(r)|^2 dr $$ and $$1=\int |Y_{00}(\theta,\phi)|^2 sin(\theta)d\theta d\phi $$ I ask this as this is what my textbook is doing and knowing integration it seems wrong! Please if this is right can you explain it, thanks.</p> | 1,166 | [
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<p>I've learned in school that the force in a coil is $F=kx$, linear on how much the coil is stretched. Two questions:</p>
<ol>
<li><p>Is it always linear for every shape of a coil? Does it remain linear if we increase the radius of the coil, increase the length between loops and so on?</p></li>
<li><p>Can we deduce the value of $k$ based on the geometry of the coil?</p></li>
</ol>
<p>The motivation is that I know how to deduce the inductance of a coil based on its shape, but I haven't seen that calculation done to deduce the $k$ in <a href="http://en.wikipedia.org/wiki/Hooke%27s_law" rel="nofollow">Hooke's Law</a>.</p> | 1,167 | [
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] |
<blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="http://physics.stackexchange.com/questions/5456/the-speed-of-gravity">The speed of gravity</a> </p>
</blockquote>
<p>It takes a long time for a radio signal to travel to the planet Mars.</p>
<p>What if we made a special type of radio that could detect small changes in gravity from a fall away object. For example, a satellite in orbit around Earth. That satellite had a device that allowed it to make small changes in it's gravity field. The detector orbiting Mars could detect these changes, and convert it into a communications signal.</p>
<p>Would these two devices be able to communicate with each other instantly using gravity? There by providing a real-time communications link between Earth and Mars?</p> | 4 | [
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<p>In a gravitational field, should the mass distributions always behave well?</p> | 1,168 | [
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<p>I have tried to learn it myself but I was unable to understand. Can you please explain it in detaails with simple example.</p>
<p>Thanks</p> | 1,169 | [
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<p>Assume that there are only well behaved functions as mass distributions, and there are no other forces except gravitation. Is it than possible to create an arrangement where a variation of a certain quantity (could be mass density or gravitational field or momentum) has a resonance?</p> | 1,170 | [
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<p>Is there a computer game using principles of quantum optics or quantum information?</p>
<p>By game I don't mean just a simulation or an interactive course, but something that can be played in an enjoyable way.</p>
<p>By 'principles' I mean actual rules, not just things bearing the same name or vaguely related.</p>
<p>As a reference, let me use <a href="http://silverspaceship.com/chromatron/" rel="nofollow">Chromatron</a> - it is a puzzle game using principles of geometrical optics and color mixing. (There are some non-physical or unrealistic elements, but most most of them - like mirrors, prisms, beamsplitters and filters - are based on real physics). </p> | 1,171 | [
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<p>What prevents the unobservable universe from being a continuously enlarging sphere due to the inflating bubble universes? Our observable universe being close to flat could support such a scenario?</p> | 1,172 | [
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<p>Which conformal QFTs do we know for spacetime dimension d > 2?</p>
<p>I know that for D = 4 we have N = 4 SYM and some N = 2 supersymmetric Yang-Mills + matter models.</p>
<blockquote>
<p>What is the complete list of such QFTs? Is it plausible we know all of them? In particular, what examples do we have with D = 3?</p>
</blockquote> | 1,173 | [
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<p>I have been wondering about some of the different uses of Generalized Complex Geometry (GCG) in Physics. Without going into mathematical detail (see <a href="http://arxiv.org/abs/math/0401221">Gualtieri's thesis</a> for reference), a Generalized Complex Geometry attempts to unify symplectic and complex geometry by considering the bundle $TM\oplus T^* M$ with its natural metric $\langle X+\xi, Y+\eta\rangle = \frac{1}{2} \left( \eta(X) + \xi(Y)\right)$ and the <a href="http://en.wikipedia.org/wiki/Courant_bracket">Courant Bracket</a>. </p>
<p>The first hints of the necessity of GCGs in Physics came up in a famous paper by <a href="http://www.sciencedirect.com/science/article/pii/0550321384905923">Gates, Hull and Roc̆ek</a>, in which they found an 'extra' supersymmetry in the $(2,2)$ supersymmetric model. This extra symmetry turns out to be related to specifying two (integrable) complex structures $J_1, J_2$ which in turn are covariantly constant under <em>torsionful connections</em>. This means that the manifold need not be Kähler (which is Hermitian and Torsion-free) and led Nigel Hitchin (and his students) to propose more general geometries that could be useful in physics.</p>
<p>More recently, a connection between GCGs and AdS/CFT has been discovered. Recall that in AdS/CFT, we consider a spacetime that is a warped product of $AdS_4$ and a 6-manifold. It turns out that it is natural to consider a 5-manifold $Y^5$ whose cone has some special geometry. If this geometry is Calabi-Yau then such a manifold is known as a <em>Sasaki-Einstein</em> manifold. As such, we start out with a metric of the form,</p>
<p>$ g_{ij} = g_{AdS_5} + g_{Y^5} = e^{2\Delta + \phi/2}r^2 \left(g_{\mathbb{R}^{1,3}} + r^{-4} g_{C(Y^5)} \right) $</p>
<p>where $g_{C(Y^5)} = dr^2 + r^2 g_{Y^5}$ (the <em>metric cone</em> of $Y^5$). If we want to obey $\mathcal{N}=1$ supersymmetry, we must enforce on the dilatino and gravitino which eventually leads to a condition on pure spinors. In Generalized Complex Geometry, $TM\oplus T^*M$ naturally acts as a Clifford Algebra on the Clifford Module $\wedge^{\bullet} T^*M$. It turns out that in this situation, we can represent the pure spinors over a Generalized Complex Manifold as the sum of differential forms of different degree (polyforms). As such GCGs can be good candidates for $C(Y^5)$. </p>
<p>Related to this is the result of <a href="http://iopscience.iop.org/1126-6708/2004/08/046">Graña, et. al</a> which can be poorly paraphrased as:</p>
<blockquote>
<p>All $\mathcal{N}=1$ solutions of IIB string theory are described by a
pair of pure spinors $\Omega_{\pm}$(up to $B$ transform) that satisfy
a pair of differential constaints, $d \Omega_+ = 0$, $d\Omega_- = dA \wedge \Omega_+ + \frac{i}{8}e^{3A}e^{-B}\star (F_1 - F_3 + F_5)$, where $F_k$ is the $k$-form flux and $A = 2\Delta + \phi/2$</p>
</blockquote>
<p>I was wondering if there were any other significant uses of GCGs in physics that I have not mentioned. I have seen a variety of papers that do mention GCGs, but outside of these examples, I have not been particularly compelled by their usage. </p>
<p>Thanks! </p> | 1,174 | [
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<p>Extended TQFT and CFT have been puzzling me for while. While I understand the mathematical motivation behind them, I don't quite understand the physical meaning. In particular, it's not clear to me to which extent these constructions produce more information (i.e. allow for several "extended" versions of the same "non-extended" theory) vs. restring the "allowed" theories (by ruling out theories that cannot be "extended).</p>
<p>The most bugging question, however, is in the context of string theory. In string theory we are supposed to consider the moduli space of all SCFTs, more precisely for type II we need all SCFTs with central charge 10, for type I we need BSCFTs with certain properties etc. The question is,</p>
<blockquote>
<p>What is the signficance of <strong>extended</strong> SCFTs in this context?</p>
</blockquote>
<p>Is there a reason that what we really need is the moduli space of extended theories? Is it rather that "extendable" theories play some special role within the larger moduli space?</p> | 1,175 | [
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<p>Lets say i have $n$ different objects that effects each other only by the classic gravity force.</p>
<p>I have their initial locations, masses and velocity's:</p>
<p>$$ x_1(0),\cdots,x_n(0) $$
$$ m_1,\cdots,m_n $$
$$ v_1(0),\cdots,v_n(0) $$ </p>
<p>Is it possible to describe each object location with an equation?</p>
<p>$$ x_i(t)= ? $$</p>
<p>I tried doing this :
$$ x_i(t) = x_i(0)+\int_0^t v_i(t') dt' $$
$$ v_i(t')=v_i(0)+\int_0^{t'} \frac{1}{m_i} \sum\limits_{j=1}^n \frac{Gm_im_j}{r_{i,j}(t'')^2} dt'' $$</p>
<p>Now I'm stuck because the distance function $ r_{i,j} $ is depends on $ x_i $ and $ x_j $ which makes the whole thing circular...</p> | 1,176 | [
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<p>A paradox (Ex.: if someone goes back in time to kill their own grandfather, then they won't exist, but then they wouldn't be able to go in time to kill their grandfather, but then they WOULD exist, but then they would go back in time...) is, really, the only "perpetual motion machine" (and I know I use that term loosely here) that works, since the conclusion of one argument causes the beginning of the argument to be false, which changes the conclusion, which changes the premises of the argument, ad infinitum.</p>
<p>Could this mean that if a fully verifiable paradox were ever found in action within the physical world, the properties of said paradox might be able to power an actual, physical perpetual motion machine? Perhaps the wave/particle duality of light might be considered such a paradox, or possibly the properties of quantum particles. If we could somehow put their inherent paradoxical qualities to practical use, we might have a solution to problems that, up until that point, have seemed to us to be, well, paradoxes.</p>
<p>DISCLAIMER: I am not an upper-level scientist; this was merely a thought experiment. In all honesty, I probably have a number of concepts flubbered around in my head, which why I'm asking here - the home of many more-capable/knowledgeable-than-me actual scientists.</p> | 1,177 | [
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] |
<p>Short version:</p>
<p>Is it a physical problem (crystal structure/grains/redox/etc.) or just a logistics problem (keeping the solutes from homogenizing, molten/solid/temperature related problems) that keeps us from commonly employing objects that have transitioning alloy compositions throughout the object? </p>
<p>Long Version:</p>
<p>I was reading more into the designations of various metals and starting thinking about something.</p>
<p>Is there a reason why we don't see materials where the alloy composition is (significantly) different in one part of the material than the other. For example, if a molten starts freezing and we introduce in more of other materials. Lets say we wanted to make a aluminum wing for a plane that was corrosion resistant but a bit stronger than our homogenous metal of choice (6061 for example). While making a 6061-aluminum wing if the exterior were to cool until it was no longer molten, could I not then inject some other solute into the molten still aluminum (for example copper/manganese/magnesium for 2024 aluminum). </p>
<p>My intuition tells em this happens to a small extent with various steels when we treat their surfaces in a coal-heated environment, since we introduce new carbon into the iron. The only problems that seem apparent to me is if there are significant changes in crystal structure between the two(or more) alloys, but even so wouldn't one expect a gradient between various alloy domains? Why isn't this something we see around more often?</p> | 1,178 | [
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<p>I want to know that if you are given a very complex equation g(x)=A(T). How could you solve for x, which is a function of variable T. To be more specific, I encounter a polylogarithmic function I need to solve numerically..</p> | 1,179 | [
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<p>In Bose-Hubbard model, we neglect the hopping term in atomic limit (U>>t) and the dominant interaction term result in a Mott Insulator phase. In hard core boson limit ($U \rightarrow \infty$), there is no double or higher occupancy are suppressed (n=0 or 1), so the interaction term in the Hamiltonian vanishes and the hopping term becomes dominant. Hard core boson limit and atomic limit are both in strongly interaction regime, but they seem have different results in Hamiltonian. Where am I wrong?</p> | 1,180 | [
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<p>Here is an extremely naive question: Why would the apple fall under the tree?</p>
<p>I am puzzled by this, because the conventional answer that the gravity between the apple and the earth pulling apple down is not satisfactory to me. My thought process goes as follows:</p>
<ol>
<li><p>We know that in an appropriate reference frame, we can view the apple falling down from the tree as free fall. Therefore we know $F=mg, a=g$ and the apple should fall to the ground in $\sqrt{\frac{2h}{g}}$ time period since $h=\frac{1}{2}gt^{2}$. </p></li>
<li><p>However, the picture is not so clear when we consider earth's rotation. For convenience I ignore the earth movement around the sun. We know that the centripedal force and the gravity are given by
$$
F_{1}=m\omega^{2}R, F_{2}=c\frac{mM}{R^{2}}
$$
where $c$ is some constant. Therefore, the reason the apple falls to the tree must be the gravity is much stronger than the centrialfugal force required when the apple rotates with the tree. If the centrialfugal force is equal to the gravity, than the apple should be staying in the same spot at the tree. If the centrialfugal force needed is greater than gravity, then the apple would not stay in the free and would flying away from the earth. </p></li>
<li><p>Now image a pear falling from the middle of the tree. From our everyday experience, the pear would fall in the same spot as the apple. However, since the pear is closer to the earth, the centripedal force it experiences is less, and the gravity it experiences is greater, too (Here we denote $R'$ for the pear's distance from the earth center, $m_{1}$ for its mass):
$$
P_{1}=m_{1}\omega^{2}R', P_{2}=c\frac{Mm_{1}}{R'^{2}}
$$
Therefore it is not difficult to see that the acceleration the apple and the pear experiences must be different because of the height. The pear must fall faster. However, since the apple and the pear moves from the same tree, they must have the same angular velocity. In particular when $R$ goes very large, the gravity would be too small and the object would fly away from earth. </p></li>
<li>But I feel this explanation is unclear. Now instead of using a constant $g$ denoting the acceleration the apple experiences, we have:
$$
g(R)=c\frac{M}{R^{2}}-\omega^{2}R
$$
Therefore the apple should somehow <em>deviate from</em> the tree. Intuitively, since at the bottom of the tree the apple would not move at all, and at very high the apple would flying away, at a middle height the apple should have a moderate but measurable deviation. </li>
<li>
My question is, how can we calculate the deviation from the height of tree exactly? The above calculation assumed the angular velocity of the earth is constant; in reality if the tree is tall enough, the angular velocity might be changing subtlely as well. But cast this aside for the moment. If we assume the earth is a sphere and we know $h, c, \omega, M$, etc, can we compute it? Should I expect that an apple falling from the empire state building would move to a different spot than an apple falling down from my hand?</li>
<li>
The problem is difficult to me because assume we know the position, velocity and the exterior force exerted on the apple at some moment:
$$
F_1-F_2=F(t_0), V=V(t_0), r=r(t_0)
$$
(at least we know when $t=0$), we would not know where the apple is at the next moment unless we do some calculation. At moment $t_{0}$ we know what the angular velocity is; but when the apples falls, its velocity changes. And its angular velocity $\omega=\frac{V}{R}$ would also changes. Therefore we would have to solve a differential equation (a non-linear second order ODE) to compute the answer. And the amount of deviation is simply unclear to me. </li>
</ol>
<p>Due to the extreme naive nature of the question, all answers are welcome. If I made some stupid mistake in the derivation, please do not hesitate to point it out. Maybe this strange phenomenon I thought would happen never happens in real life because I made some mistake. </p> | 1,181 | [
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<p>I am a beginner to study QFT and have a problem.</p>
<p>I know, in Dirac equation, thanking to the Pauli exclusion principle and believing that the vacuume is the state that all the negative energy states are occupied and all the positive energy states are empty, the antiparticles in Dirac equation are not occupied negative energy states, the positons just like holes in occupied states.</p>
<p>But in the KG equation, a equation for zero spin particles, there is not Pauli exclusion priciple and it is not a good idea that think the vacuume is the same as the case in Dirac equation because particles will jump from positive energy states to negative energy states and successively jump to the lower energy states. </p>
<p>So, how to explain the antiparticles in KG equation?</p> | 1,182 | [
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<p>I'm watching Susskind's video lectures and he says in the first lecture on classical mechanics that for a physical law to be allowable in classical mechanics it must be reversible, in the sense that for any given state $S\in \mathcal{M}$ where $\mathcal{M}$ is the configuration space there should be only <em>one</em> state $S_0\in \mathcal{M}$ such that $S_0\mapsto S$ in the evolution of the system.</p>
<p>Now, why is this? Why do we really need this reversibility? I can't understand what are the reasons for us to wish it from a physical law. What are the consequences of not having it?</p> | 1,183 | [
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<p>So I have seen an animation about Stephen Hawking (after his recent study state universes claim) that Hawking evaporation is due to negative mass; But how is this possible? I mean, there is no such thing as negative mass!</p> | 1,184 | [
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<p>In the paper <a href="http://arxiv.org/pdf/hep-th/9804191.pdf" rel="nofollow">here</a>(page 7-8) the authors make a claim that the Natanzon potential (an implicit potential) follows an $SO(2,2)$ algebra. This potential defined as :</p>
<p>$$ U(z(r)) = \frac{h_1z(r)+h_0(1-z(r))-fz(r)(1-z(r))}{R(z)} - \frac{1}{2}\left\{z,r\right\}$$</p>
<p>where $R(z(r)) = az^2+b_0z+c_0 = a(1-z)^2-b_1(1-z)+c_1$ and $f,h_o,h_1,a,b_0,b_1,c_0,c_1$ are constants and $\left\{z,r\right\}$ is the <a href="http://en.wikipedia.org/wiki/Schwartzian_derivative" rel="nofollow">Schwarzian derivative</a>. And the relationship between $z(r)(0<z<1)$and $r$ is governed by the differential equation :
$$\frac{dz}{dr}=\frac{2z(1-z)}{\sqrt{R(z)}}.$$
I find this result particularly impressive since the Nantanzon class of potentials hold special importance because of it's <a href="http://en.wikipedia.org/wiki/Supersymmetric_quantum_mechanics#Shape_Invariance" rel="nofollow">shape invariance</a> property and relevance in supersymmetric quantum mechanics and contains many well-known potentials, such as the generalized Coulomb, Poschl-Teller(type I and II),Rosen-Morse. And I was wondering if anyone knows how this result was obtained, how does one find an algebra for an implicit potential? Or any potential for that matter?</p> | 1,185 | [
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<p>The Spread Networks corporation recently laid down 825 miles of fiberoptic cable between New York and Chicago, stretching across Pennsylvania, for the sole purpose of reducing the latency of microsecond trades to less than 13.33 milliseconds (http://www.spreadnetworks.com/spread-networks/spread-solutions/dark-fiber-networks/overview). The lesson I would draw from this is that, in the near future, oil and natural gas extraction won't be the only lucrative use of ocean platforms. </p>
<p>So here's my question - since trades are occurring on the scale of tens to hundreds of microseconds, and considering the amount of money involved, can one use neutrino beams to beat the limitation due to having to travel the great-circle/orthodromic distance between two trading hubs? I'm imagining something similar to the MINOS detector (http://en.wikipedia.org/wiki/MINOS), where a neutron beam was generated at Fermilab in Batavia, Illinois, and detected ~735 km away, ~700 meters under the ground in a Northern Minnesota mine.</p>
<p>Is it possible to beat a signal traveling at the speed of light across the great-circle distance from, say, New York to Tokyo, using a neutron beam traveling the earth? Is it realistic to talk about generating these beams on a microsecond time-scale? </p>
<p>Addendum - Over what distances can you reasonably detect a neutrino beam? </p> | 1,186 | [
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<p><a href="http://en.wikipedia.org/wiki/Density_functional_theory" rel="nofollow">Density Functional Theory</a> (DFT) is usually considered an electronic structure method, however a paper by <a href="http://arxiv.org/abs/physics/9806013" rel="nofollow">Argaman and Makov</a> highlights the applicability of the DFT formalism to classical systems, such as classical fluid density. This <a href="http://www.physics.orst.edu/~roundyd/talks/research_seminar.pdf" rel="nofollow">presentation</a> by Roundy et al. calculates water properties using a classical approach, with an eye to combining it with KS DFT for calculating solvent effects.</p>
<p>Not being too clear on the history of DFT, <em>was DFT invented for electronic structure problems first</em>, and its applicability to classical problems incidental, or was it formulated for classical systems first and then adapted to the quantum many body problem?</p> | 1,187 | [
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<p>Does constructing huge buildings affect the rotation of the Earth, similar to skater whose angular rotation increases when her arms are closed comparatively than open?</p> | 1,188 | [
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<p>Since I don't know how to add another question to an already existing topic,
I'm opening a new thread. However I'm referring to:
<a href="http://physics.stackexchange.com/questions/8201/beginners-questions-concerning-conformal-field-theory">Beginners questions concerning Conformal Field Theory</a> </p>
<p>As noted, a few weeks ago I started reading about Conformal Field Theory.
I'm actually from a more mathematical background, however I'm not very familiar
with Quantum Field Theory. Though I'm quite familiar with Quantum Mechanics/Classical Mechanics. </p>
<p>Now again some questions again turned up: </p>
<ol>
<li><p>Think of a theory with an energy-momentum-tensor that is given
on the plane. Let's assume the most general form $T(z)=\sum z^{-n-2} L_{n}$ and
$L_{n} = \frac{1}{2 \pi i} \oint dz z^{n+1} T(z)$.
Now some of my reference (such as David Tong in the reference question above) point
out that $L_{0}$ generates scalings/rotations and $L_{1},L_{-1}$ generate translations.
So let's consider the example of a rotation. The generator of a rotation is $z \frac{\partial}{\partial z}$. Now in order to show that $L_{0}$ actualyy generates
this rotation one needs to show that $[L_{0},\phi]$ = $z \frac{\partial}{\partial z}$ \phi.
I've shown this for the example of the free boson, however I'm not 100% sure how to prove it in the general case. Can someone help me? (Maybe it's related to Operator Product Expansions...) </p></li>
<li><p>The second question goes a bit deeper into the theory.
It concern Current Algebras. I've read some articles on the Sugawara construction
and there Mr Sugawara proposes and Energy-Momentum-Tensor of the form
$T(z) = \gamma \sum_{a=1}^{dim g} : j^{a}(z) j^{a}(z):$ . However I don't really see
how he comes up with it or why this seems to be a "natural choice" of an Energy-Momentum tensor. I've heard that it includes the Energy Momentum Tensor of the free boson
(given by $T(z)=\partial_{z} \phi \partial_{z} \phi$) as a special case.
For me this is not so obvious. Can someone please explain to me how he comes up that
in an easy way. I don't think it's necessary to show me all the calculations.
Just the basic idea would be useful to get some intuition. </p></li>
<li><p>I'm having some troubles on understanding the intuition behind current algebras.
(I haven't read about WZW Models yet). The Virasoro algebra appeared to me in a kind
of natural way in the example of the free boson. The generalization is then pretty much
straight forward. However I don't have that kind of intuition for current algebras.
I've read that they provide some "additional symmetry structure" which reduces the number of possible correlation functions. But I don't know any details.
I'd be more than happy if someone could comment on that.</p></li>
</ol> | 1,189 | [
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<p>As compared to when the coffee is just hot from brewing. I suspect it has something to do with the way the microwaves are affecting the molecules of the coffee.</p> | 1,190 | [
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<p><img src="http://i.stack.imgur.com/KYj8v.png" alt="enter image description here"></p>
<p>Having trouble with a FBD moment. The problem is:</p>
<blockquote>
<p><em>How big can the force couple C (looks like a G in the pic) be in order for the disk to not spin? The disc has mass m and the beam also has mass m. All other variables are given.</em></p>
<p><em>Let point A = bottom the disc and point B = where the beam touches the ground. Also let $\ell = R/\tan(\alpha)$ = distance from point A to point B.</em> </p>
</blockquote>
<p>In the solution sheet, the professor stated that the counter-clockwise moment about point A due to the beam is $\frac{\ell}{2} m g$, implying that the center of mass of the beam is located halfway between A and B.</p>
<p>How can this be? If the beam was parallel to the ground, surely that would have been the case. But now it's tilted $\alpha$ degrees - should that not shift the center of gravity to the right?</p>
<p>For reference, this is his equation of moment equilibrium about A:</p>
<p>$$M_a = N_b \ell - m g \frac{\ell}{2} + C = 0.$$</p> | 1,191 | [
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] |
<p>Why does a <a href="http://en.wikipedia.org/wiki/Wave_function_collapse" rel="nofollow">wavefunction collapse</a> when observation takes place?
Can this question be explained in non mathematical terms? I have tried finding the answer but couldn't find a clear explanation.</p> | 1,192 | [
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<p>I am unable to derive the Hamiltonian for the electromagnetic field, starting out with the Lagrangian
$$
\mathcal{L}=-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}-\frac{1}{2}\partial_\nu A^\nu \partial_\mu A^\mu
$$
I found:
$$
\pi^\mu=F^{\mu 0}-g^{\mu 0}\partial_\nu A^\nu
$$
Now
$$
\mathcal{H}=\pi^\mu\partial_0 A_\mu-\mathcal{L}
$$
Computing this, I arrive at:
$$
\mathcal{H}=-\frac{1}{2}\left[\partial_0 A_\mu\partial_0 A^\mu+\partial_i A_\mu\partial_i A^\mu\right]+\frac{1}{2}\left[\partial_i A_i\partial_j A_j-\partial_j A_i\partial_i A_j\right]
$$
The right answer, according to my exercise-sheet, would be the first to terms. Unfortunately the last two terms do not cancel. I have spent hours on this exercise and I am pretty sure, that I did not commit any mistakes arriving at this result as I double checked several times. My question now is: did I start out right and am I using the right scheme? Is this in principle the way to derive the Hamiltonian, or is it easier to start out with a different Lagrangian maybe using a different gauge? Any other tips are of course also welcome. Maybe the last two terms do actually cancel and I simply don't realize it. Texing my full calculation would take a very long time so I am not going to post it, but as I said, it should be correct. But if everything hints at me having committed a mistake there, I will try again.</p>
<p><strong>Edit:</strong></p>
<p>After reading and thinking through Stephen Blake's answer, I realized that one can get rid of the last two terms in $H$, even though they do not vanish in $\mathcal{H}$. This is done by integrating the last term by parts and dropping the surface term, leaving $A_i\partial_j\partial_i A_j$. One can now proceed to combine the last two terms:
$$
\partial_i A_i\partial_j A_j+A_i\partial_j\partial_i A_j=\partial_i(A_i\partial_j A_j)
$$
This can be converted into a surface integral in $H$ which can be assumed to vanish, leaving us with the desired "effective" $\mathcal{H}$.</p> | 1,193 | [
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<p>I'm having some trouble following pages 55-56 of Sakurai's <em>Modern Quantum Mechanics</em>. </p>
<p>We're trying to transfer from position space into momentum space. Here's a quote:</p>
<blockquote>
<p>Let us now establish the connection between the $x$-representation and the $p$-representation. We recall that in the case of the discrete spectra, the change of basis from the old set $\{ | a^{'} \rangle \}$ to the new set $\{ | b^{'} \rangle \}$ is characterized by the transformation matrix (1.5.7). Likewise, we expect that the desired information is contained in $\langle x^{'} | p^{'} \rangle$, which is a function of $x^{'}$ and $p^{'}$, usually called the <strong>transformation function</strong> from the $x$-representation to the $p$-representation. To derive the explicit form of $\langle x^{'} | p^{'} \rangle$, first recall (1.7.17), letting $|\alpha \rangle$ be the momentum eigenket $|p^{'} \rangle$, we obtain $$\langle x^{'} |P|p^{'}\rangle = -i \hbar \frac{\partial}{\partial x^{'}} \langle x^{'} | p^{'}\rangle$$ or $$p'\langle x' |p^{'}\rangle = -i \hbar \frac{\partial}{\partial x^{'}} \langle x^{'} | p^{'}\rangle$$
The solution to this differential equation for $\langle x' | p' \rangle$ is $$\langle x' | p' \rangle = N \exp \left( \frac{ip'x'}{\hbar} \right)$$</p>
</blockquote>
<p>I'm not sure where our differential equation is coming from. We have our momentum operator $P$ in the position basis, acting on our eigenket $|p'\rangle$, I can't see how we can find explicitly what the inner product of the position bra $\langle x'|$ and the momentum ket $|p'\rangle$ is. </p>
<p>What I'm thinking now is to rearange it like so: $$\int \langle x^{'} |p^{'}\rangle dx' = \int -\frac{i\hbar}{p'} \frac{\partial}{\partial x^{'}} \langle x^{'} | p^{'}\rangle dx'$$</p>
<p>But it doesn't seem to lead anywhere. What am I missing? </p> | 1,194 | [
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<p>Is there a way to lower the total spin of the state and fixing the $S_z$ rather than lowering the $S_z$ by spin ladder operator? Or in other words, how to connect the $S=1$ state with $S=2$ or $S=0$ state? Is there such an operation though it might be unphysical?</p> | 1,195 | [
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<p>When you're in a train and it slows down, you experience the push forward from the deceleration which is no surprise since the force one experiences results from good old $F=m a$. However, the moment the train stops one is apparently pulled backwards. But is that a real physical effect or just the result from leaning backwards to compensate the deceleration and that force suddenly stopping?</p>
<hr>
<p>So far the answers basically agree that there are two spring forces involved, for one thing oneself as already guessed by me and for the other the vehicle itself as first suggested in <a href="http://physics.stackexchange.com/questions/629/why-does-one-experience-a-short-pull-in-the-wrong-direction-when-a-vehicle-stops/638#638">Robert's answer</a>. Also, as <a href="http://physics.stackexchange.com/questions/629/why-does-one-experience-a-short-pull-in-the-wrong-direction-when-a-vehicle-stops/670#670">Gerard suggested</a> the release of the brakes and some other friction effects might play a role. So let's be more precise with the question:</p>
<blockquote>
<p>Which effect <strong>dominates</strong> the wrong-pull-effect? And thus, who can reduce it most:</p>
<ul>
<li>the traveler</li>
<li>the driver</li>
<li>the vehicle designer?</li>
</ul>
</blockquote>
<hr>
<p><strong>edit</strong> Let's make this more interesting: I'm setting up a bounty of <s>50</s> 100 (see edit below) for devising an experiment to explain this effect or at least prove my explanation right/wrong, and by the end of this month I'll award a second bounty of <s>200</s> 150 for what I subjectively judge to be the best answer describing either:</p>
<ul>
<li>an accomplished experiment (some video or reproducibility should be included)</li>
<li>a numerical simulation</li>
<li>a rigorous theoretical description</li>
</ul>
<p><strong>update</strong> since I like both the suggestions of <a href="http://physics.stackexchange.com/questions/629/why-does-one-experience-a-short-pull-in-the-wrong-direction-when-a-vehicle-stops/5052#5052">QH7</a> and <a href="http://physics.stackexchange.com/questions/629/why-does-one-experience-a-short-pull-in-the-wrong-direction-when-a-vehicle-stops/4926#4926">Georg</a>, I decided to put up a second bounty of 50 (thus reducing the second bounty to 150 however)</p> | 1,196 | [
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<p>The equivalence principle, being the main postulate upon which the general relativity theory rests, basically states that all reference systems are equivalent, because pseudo forces can (locally) be interpreted as gravitational fields and it is therefore impossible for the local experimenter to decide whether he is moving, or being accelerated, or motionless. In other words: there is no distinguished, "motionless" reference system.</p>
<p>Question: doesn't the rotating water bucket (parabolic water surface) give us an indication of our rotational state? It would be a weird gravitational field indeed that causes my water to be pulled outward while causing the rest of the universe to rotate around me?</p>
<p>And doesn't the red/blue-shift of the microwave background (often dubbed "echo of the big bang") give us a clue of our translational motion within the universe (I read lately that they compensate the precision measurements of the background radiation by the motion of the solar system around the galactic center, obviously assuming that galactic center is "motionless" within the universe)?</p> | 1,197 | [
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<p>I am trying to clarify the relation between random walk and diffusion, and the source book proposes the following which I can't get. Starting from the diffusion equation</p>
<p>$$ \frac{\partial C}{\partial t} = D \frac{\partial^2 C}{\partial x^2}, $$</p>
<p>how can this be solved to get</p>
<p>$$ C = \frac{A}{\sqrt{t}} \mathrm{e}^{-x^2 / 4 D t} ?$$</p> | 1,198 | [
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<p>I am trying to figure out what do I need to know to properly simulate the creation of a solar system from a particle cloud with random distribution of hydrogen atoms.</p>
<p>Being more of a programming background than a physicist one, I find some difficulties trying to find all the forces in interaction.</p>
<p>As for now, I'm looking to simulate the accretion of matter in a random distribution of particles in a void space. I can successfully identify gravitational effects, and have included them as mutual interaction between every particle in my system. I am computing this by summing the effect of every other particle on each one iteratively, and applying the resulting acceleration to the said particle, and so on.</p>
<p>This approach is limited to small numbers of particles and it's fine for me now, because I don't go over 1000 particles per simulation. However, the trouble is that I also want to include other forces that may be playing a significant role in this process. I'm thinking specifically of electric interaction, as I am not yet in the step of simulating eventual nuclear interactions.</p>
<p>I'm not sure how to approach it and I'm fairly confused.</p>
<p>How electric force is involved in my scenario? How can I account for it in my calculations?</p>
<p>Specifically, in a cloud of hydrogen particles in space, what is happening?</p> | 1,199 | [
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<p>I am looking to model the nodal surfaces in a resonating closed sphere. The sound source is external. What sort of wave equation will reveal the spherical harmonics depending on the frequency, speed of sound, sphere diameter, possibly other relevant factors?</p>
<p>The purpose is to make a computer simulation before attempting to visualize the nodal surfaces using neutrally-buoyant particles within the sphere (the gas might be, for example, SF6). </p> | 1,200 | [
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<p>I heard that the Lagrangian is defined in the path integral formulation of quantum mechanics. How would the Lagrangian in this formulation be used to recover the Schrodinger equation that we normally use?</p> | 1,201 | [
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<p>Do electrons (individually) have definite and single value of momentum and position or do they simultaneously have multiple position (a spread) at a time?
In other words, according to the uncertainty principle, is it just impossible to <em>measure</em> the exact position and momentum or is it actually impossible for an electron to <em>have</em> a single position and momentum?</p>
<p>Consequently, assuming it's just impossible to measure the exact position and momentum, does this mean that in atomic orbitals the electrons actually do occupy single position at a certain time (instead of somehow having 'smeared out' multiple positions at a time).
How does velocity apply to atomic orbitals? Do electrons just randomly teleport according to the probability distribution or do they actually travel in 'normal' trajectories (going step by step instead of teleporting, for example footballs travel in step by step and don't randomly teleport AFAIK)</p>
<p>Edit: Also, does $\hbar$ in the uncertainty principle equation, $\Delta x \Delta p = \frac{\hbar}{2} $ arise from Planck length (<strong>From my understanding, the impossibility to measure the exact position and momentum arises from the quantization of length and time</strong>. Is my understanding correct?)</p> | 1,202 | [
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<p>If we know that the Polarization P in LIH dielectrics is proportional to the net field inside the dielectric according to:</p>
<p>P = ε0χeE.....(1)</p>
<p>And we know that</p>
<p>D = εE........(2)</p>
<p>Does it not follow that we can ascertain the polarization directly from the applied (free charge) field, since we can relate D to E, and then E to P?</p>
<p>The author of my electrodynamics text (Griffiths), says that we cannot. His explanation being that once we place dielectric in an external field E0, the material will polarize and create an opposing field to the applied field, which in turn modifies the polarization again, and this process repeats over and over. In actuality, are these two quantities (E and P) in some sort of dynamic equilibrium within the material? If so, how come (1) and (2) are valid?</p> | 1,203 | [
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<p>Can the <a href="http://en.wikipedia.org/wiki/Ricci_curvature" rel="nofollow">Ricci curvature tensor</a> be obtained by a 'double contraction' of the Riemann curvature tensor? For example</p>
<p>$R_{\mu\nu}=g^{\sigma\rho}R_{\sigma\mu\rho\nu}$.</p> | 1,204 | [
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<p>In gravitational lensing, there are three categories of lensing: strong, weak, and micro. As I understand it, strong lensing (just as the name implies) occurs when a source and a gravitational lens are relatively close by and the lens is strong, producing extreme distortions of the light from the source in phenomena such as Einstein rings, weak lensing produces a still distorted image of the source but not as distorted as in strong lensing, and microlensing produces a brightening of an object without any distortion (as described in <a href="http://physics.stackexchange.com/questions/59321/is-weak-lensing-the-statistical-effect-of-microlensing">this question</a>).</p>
<p>My question is, are these three fundamentally different phenomena (perhaps caused by separate terms or different limits in whatever equations govern gravitational lensing) or are they arbitrary classifications in a continuum of gravitational lensing effects, much as infrared, microwave, and radio are arbitrary classifications of the electromagnetic spectrum?</p>
<p>To repeat, my question is not as much about what the differences between these three classifications of gravitational lensing are as much as whether these differences create three largely distinct and independent phenomena or three classifications of the same phenomenon (such as the difference between a lake and a pond, no fundamental difference in the properties of each, just a size difference). </p> | 1,205 | [
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<p>Thermodynamic Entropy Variation is defined as
$$\Delta S = \int_i^f \frac{dQ}{T},$$</p>
<p>where $i$ and $f$ are the initial and final states of the process. </p>
<p>My question is: <strong>does this equation apply to quasi-static irreversible processes, or only to reversible processes?</strong></p>
<p>Obviously, it does not apply to processes that go through non-equilibrium states, since Temperature (or any state variable) is not even well defined in these states. But I'm unsure of whether it applies to irreversible processes that are quasi-static (and therefore don't go through non-equilibrium states).</p> | 1,206 | [
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<p>So, I'm traveling to another star 100 light years away in my spaceship. This ship has a solar sail pushed by a laser beamed from my home star system, so can achieve a velocity close to c. It's also got a robust parachute to slow down with.</p>
<p>I understand that if I measure light coming from my origin star, it will always still seem to be streaming past me at light speed (but will red shift as my speed increases). Light coming from my destination star also travels past me at light speed, and will become increasingly blue shifted as I gain speed.</p>
<p>I also understand that an observer checking on my speed at my origin or destination will always find it to be less than c.</p>
<p>However, <strong>will I perceive that in terms of the time it apparently takes me to reach my destination, my speed <em>was</em> greater than c?</strong> In other words, will it seem to take less than 100 years to reach the destination? 10 years on my watch, say. Or 1 year. Or a week?</p>
<p>Ie, as far as I'm concerned, while light keeps zipping past me at light speed, do I continue to accelerate unabated to an arbitrary apparent speed?</p>
<p>If not, how do I notice my continued acceleration being prevented?</p> | 9 | [
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<p>In the classic Young double slit experiment, with slits labeled as "A" and "B" and the detector screen "C", we put a detector with 100% accuracy (no particle can pass through the slit without the detector noticing) on slit B, leaving slit A unchecked. What kind of pattern should we expect on the detector C? Probably the right question is: knowing that a particle hasn't been through one of the slits makes the wavefunction collapse, precipitating in a state in which the particle passed through the other slit?</p> | 1,207 | [
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<blockquote>
<p>A box is thrown up an incline with degree $\alpha$. the kinetic-friction coefficient is $\mu_k$. the body returns back to its start point. </p>
<p>a. prove $\frac{t_{down}^2}{t_{up}^2}=\frac{a_{up}}{a_{down}}$.</p>
<p>b. suppose time of getting up the incline is half of the time needed for getting down it, find $\mu_k$</p>
</blockquote>
<p>With <code>a</code>, I had no problem proving it, but with <code>b</code> I have a problem. </p>
<p>Using <code>a</code> we get
$$a_{up}=-4\cdot a_{down}$$
and after substituting in Newton's second law,
$$ma=\sum F_x=F-mg\cos\alpha-f_k$$
I got
$$F=-30m\sin\alpha+50\mu_{k}m\cos\alpha$$ $$\beta=-30\sin\alpha+50\mu_{k}\cos\alpha-10\sin\alpha-10\mu_{k}\cos\alpha=-40\sin\alpha+40\mu_{k}\cos\alpha\\\gamma=10\left(\sin\alpha-\mu_{k}\cos\alpha\right)$$</p>
<p>We also know that (using $f_k$ from $\sum F_x$)
$$\mu_k=\frac{f_k}{N}=\frac{-30m\sin\alpha+50\mu_{k}m\cos\alpha-40m\sin\alpha+40m\mu_{k}\cos\alpha-mg\sin\alpha}{mg\cos\alpha}\\
=\frac{-80m\sin\alpha+90\mu_k\cos\alpha}{10m\cos\alpha}$$
and it follows that
$$\mu_k=tan\alpha \quad{\rm or}\quad \alpha=90^\circ$$
Both cases are impossible (in one $\mu_k$ cannot be found and in another the incline is just a wall). What am I doing wrong?</p> | 1,208 | [
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<p>an experiment to disprove the statement--"frictional force is irrespective of the surface area in contact."
take a x rs note. fold it in a half and put it in the pocket of a shirt. then invert the shirt. lets assume it doesn't fall. now, take it out, fold it again and repeat the experiment. after a certain no. of folds, we can see that the bank note falls off. in our experiments, we have changed nothing but the surface area. and the frictional force has changed. voila!</p> | 1,209 | [
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<p>According to Hubble observatory, the age of universe is 14 billion years. But, the distant galaxies are about 40 billion light years. How could that simply be possible? That means the information that we are receiving from those galaxies took place 40 billion yrs ago ?</p> | 10 | [
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<p>Suppose that we have a Lagrangian density like $$\mathcal L = -\frac{1}{4} \operatorname{tr} F_{\mu\nu}F^{\mu\nu} + \frac{\theta}{32\pi^2} \operatorname{tr} \big( \epsilon^{\mu\nu\rho\sigma} F_{\mu\nu}F_{\rho\sigma}\big) + \overline{\psi}\gamma^\mu D_\mu \psi$$
where $F_{\mu\nu}$ is the gauge field strength and $D_\mu$ the gauge covariant derivative, and $\psi$ is a fermion field. This Lagrangian is not $P$ conserving because of the $\theta$ term.</p>
<p>However if we redefine the fields $\psi \mapsto \exp(i\alpha \gamma_5)\psi$ we can make $\theta$ go away, by choosing $\alpha = \theta/2$ as per the Fujikawa method (described in [Weinberg], Chapter 22 or [Fujikawa]); this is due the path integral measure also transforming under the redifinition. With this redefinition of fields $\mathcal L$ is manifestly $P$ conserving. But surely I can't get more or less symmetry by redefining fields, so <strong>how should I understand that the $P$ symmetry is not manifest with the original definition of the fields?</strong></p>
<p>I suspect that the $P$ transformation too transforms the path integral measure, in a way that sends $\theta \mapsto -\theta$, but I do not know how to show this.</p>
<ul>
<li>[Weinberg] Weinberg, S. The Quantum Theory of Fields. 2: Modern Applications
(Cambridge, 2005).</li>
<li>[Fujikawa] Fujikawa, K. Path-Integral Measure for Gauge-Invariant Fermion Theories.
Phys. Rev. Lett. 42, 1195{1198 (18 Apr. 1979).</li>
</ul> | 1,210 | [
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<p>Let $n_F(\omega) = \large \frac{1}{e^{\beta (\omega)} + 1}$ be the Fermi function.</p>
<p>A fermionic reservoir correlation function is given by:</p>
<p>$$C_{12}(t) = \int_{-\infty}^{+\infty} d\omega~ \tag{5}J_R(\omega) \, n_F(\omega) \, e^{-i\omega t}$$</p>
<p>The Fermi function here is given in terms of Chebyshev polynomials.</p>
<p>The coefficients of the Chebyshev polynomials are given by:</p>
<p>$$c_k = \frac{2}{\pi}\int_{0}^{\pi}f(\cos\theta)\cos(k\theta) \;d\theta$$And the Fermi function itself is (Chebyshev approximated, if you will):</p>
<p>$$n'_F(x) = \sum_{k=0}^{n}\left[ \frac{2}{\pi}\int_{0}^{\pi}\frac{\cos k\theta \; d\theta}{e^{\beta(E_F-\cos\theta)}+1} \right] T_k(x) \tag{6}$$
Where $T_k(x)$ are the Chebyshev polynomials of the first kind. </p>
<p>Now, since the Fermi function here has no poles(as it's given in terms of Chebyshev interpolation polynomials which do not have any poles), the poles in the correlation function are only those of the Spectral Density: $$J_R(\omega) = \sum_{k=1}^{m}\frac{p_k}{4\Omega_k(\omega-\Omega_k)^2+\Gamma_k^2}\tag{7}$$</p>
<p>There's only pole at: $\omega = \Omega_k - i\Gamma_k=\Omega_k^-$, and the residue is $\left.\frac{1}{(\omega - \Omega_k) - i\Gamma_k}\right|_{\omega=\Omega_k^-} = \frac{1}{-2i\Gamma_k}$.</p>
<p>Or the residue of $J_R(\omega)$ at $\omega=\Omega_k^-$: </p>
<p>$$\mathop{\text{Res}}\limits_{\omega=\Omega_k^-} J_R(\omega) = \frac{p_k}{4 \Omega_k(-2i\Gamma_k)}\tag{8}$$</p>
<p><strong>My question is: How can the integral of the correlation function now be solved using the theorem of residues/Jordan's lemma? Is it still possible or another scheme should be employed?</strong></p>
<p>If the Fermi function was given in terms of Matsubara frequency sum, it would have had poles and then it's residues could be calculated. Now it does not, and I can't see how the integral can be solved now. If the Fermi function had poles, we could have said:</p>
<p>Noting that Poles of $J_R(\omega)$: $\Omega_k^-$, and the poles of $n_F(\omega)$: $\nu_{k'}^*$, we could have gotten:</p>
<p>$$C_{12}(t) = (-)(2i\pi) \left \lbrace \sum_{k=1}^m \mathop{\text{Res}}\limits_{\omega=\Omega_k^-}\left[ J_R(\omega) \right] n_F(\Omega_k^-)e^{-i\Omega_k^- t} \\+ \sum_{k'} \mathop{\text{Res}}\limits_{\omega=\nu_{k'}^*} \left[ n_F(\omega) \right] J_R(\nu_{k'}^*)e^{-i\nu_{k'}^* t} \right \rbrace\tag{9}$$</p>
<p>And then the residues could have been calculated. </p>
<p><em><strong>A bit different Version of the same problem(If you decide to answer, kindly answer this first):</em></strong></p>
<p>Here's a function:</p>
<p>$$C_{12}(t) = \int_{-\infty}^{+\infty} d\omega~ \tag{1}J_R(\omega)n_F(\omega)e^{-i\omega t}$$</p>
<p>Here's $n'_F(x)\approx n_F(\omega)$:</p>
<p>$$n'_F(x) = \sum_{k=0}^{n}\left[ \frac{2}{\pi}\int_{0}^{\pi}\frac{\cos k\theta \; d\theta}{e^{\beta(E_F-\cos\theta)}+1} \right] T_k(x) \tag{2}$$
Where $T_k(x)$ are the Chebyshev polynomials of the first kind.</p>
<p>Also, $n'_F(x)$ has no poles.</p>
<p>And,
$$J_R(\omega) = \sum_{k=1}^{m}\frac{p_k}{4\Omega_k(\omega-\Omega_k)^2+\Gamma_k^2}\tag{3}$$
Where $\Omega$, P and $\Gamma$ are only some numbers.</p>
<p><strong>Are the prerequisites of Jordan's Lemma fulfilled? That is, can equation (1) be written as equation (4) after inserting (2) and (3) in (1) and then applying Jordan's Lemma?</strong></p>
<p>$$C_{12}(t) = (-)(2i\pi) \left \lbrace \sum_{k=1}^m \mathop{\text{Res}}\limits_{\omega=\Omega_k^-}\left[ J_R(\omega) \right] n'_F(\Omega_k^-)e^{-i\Omega_k^- t}\right \rbrace\tag{4}$$</p> | 1,211 | [
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<p>I wondered whether the Fermi-Dirac Statistics describes the anti-fermion particles. Does it include the anti-particles?</p> | 1,212 | [
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<h2>Dilemma</h2>
<p>The uncertainty principle of energy and the 2nd law of thermodynamics don't add up : the uncertainty principle of energy says that </p>
<p>$\Delta \tau \cdot \Delta E \ge \frac{h}{4\pi} = \frac{\hbar}{2}$ </p>
<p>where $\Delta$ is the uncertainty in measurement.</p>
<p>Now lets consider a situation: lets say that an isolated system $A$ is in thermodynamic equilibrium. It has two particles $b$ and $c$ so that it is in the state of maximum possible entropy. To preserve the uncertainty principle, some net energy must flow from $b$ to $c$ or from $c$ to $b$, and that results in a non-equilibrium state, may be for a fraction of a fraction of a second. But the system must move from the state of maximum entropy (equilibrium) to the state of lesser entropy, which is a sure violation of the second law of thermodynamics. Does anyone have an explanation?</p> | 1,213 | [
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<p>My son is very keen on Astronomy but I don't know when there are meteor showers or something else that would be worth going out to the country to see. Last year we went to see a meteor shower, but I only knew about it because I'd heard it on the radio.</p>
<p>Which websites list events that we can either see with the naked eye, or with a decent pair of binoculars? (He doesn't have a telescope because the advice I found on the web was that decent binoculars are better than cheap telescopes.)</p> | 1,214 | [
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<p>it is widely known that elliptical galaxies have little or no gas, but how is this determined? What is the amount of gas? Is there a decent ratio of stellar mass to gas for ellipticals? How does this vary between normal ellipticals and those that were formed by mergers (or still look like merger remnants)? Thanks for your help!</p> | 1,215 | [
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<p>We have : $E=h/f$ </p>
<p>I realised that the problem what quanta solved was that $h/0$ equals infinity but energy can't be infinity. But when frequency is zero we haven't any energy to calculate - there is no real solution. But Planck came with another solution: The light is packages with different energy - so we always have integer energy score. Why solved he solved problem? Thank you a lot.</p> | 1,216 | [
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