title: Gas Laws And Adiabatic Process

text: Air at 20 degrees Celsius is compressed adiabatically from 1 bar to 10 bar, what will its temperature be? With $$P_1 = 1,$$ $$P_2 = 10$$ $$T_1 = 293K,$$ $$T_2 = unknown$$ using $$\dfrac{P_1}{P_2}=\dfrac{T_1}{T_2}$$ my solution was $$\dfrac{1}{10}=\dfrac{293}{T_2}$$ giving $$T_2 = 2930$$ My physics tutor said this is wrong and I should use $P_1V_1^\lambda=P_2V_2^\lambda$ and then use $\dfrac{V_1}{V_2}=\dfrac{T_1}{T_2}$ to find the temperature. The only problem I now have is that $V_1 = V_2 = unknown$ which is making me think that he may have forgot to add that information to the question. I would like to know if there is anyway to solve this (I do not want the answer just a point in the right direction). thanks in advance!


title: Why are the third generation superpartners lighter than the other sfermions in MSUGRA

text: In the MSUGRA breaking scenario, the stop particle typically appears at energies reachable at the LHC. Other sfermions, notably the partners of up, down, strange and charm are assumed to be degenerate in mass, and also heavier than the stop. Something similar holds for the stau. Why is the third generation different in MSUGRA (not degenerate as the first two), and why is the mass hierarchy inverted wrt. the Standard Model sector (3rd generation sparticles lighter)? (I guess these features are not neccessarily specific to MSUGRA, but might apply to more general models as well.)


title: Time dilation when falling into black hole

text: I know that if one astronaut falls into a black hole, then a distant observer will see him take an infinite amount of time to reach the event horizon (provided the observer can see light of arbitrarily large wavelengths). But the falling astronaut will only take a finite amount of time to reach the horizon. My question is: What will the falling astronaut see if he "looks backwards" while falling. Will he see the distant observer growing old and all the stars dying by the time he reaches the horizon ?


title: How fat is Feynman’s photon?

text: According to my calculations, it is a lot skinnier than Airy’s photon, but still a whole lot fatter than a straight line. So, how does a photon get from point A to Point B? The ray optics approximation treats a photon as an infinitely skinny straight line. Feynman, on the other hand, says a photon takes every possible path to get from Point A to point B, traversing the entire universe including all of space-time in the process. That’s very fat. However, Feynman also says the photon interferes with itself enough to wipe out almost all traces of its passing except for a very narrow tube surrounding that classical straight line. My question is, “How fat is that tube?” Even though this is not a homework problem, I will show my first try. Considering only self interference, paths that differ from the shortest one by more than a half or quarter wavelength will be wiped out by interference. Those within say a tenth of a wavelength will mostly interfere constructively. So the tube will be an ellipsoid of paths from A to B that is defined by the length approximately equal to the distance from A to B plus one-fifth of a wavelength. (Replace “one-fifth” with any similar factor of your choosing, if you want to.) The width of this ellipse is the square root of twice its length (times that order one constructive interference factor), which is the distance from A to B expressed in photon wavelengths. So the “width” or “fatness” of the photon increases with the square root of the distance from the source, unlike Airy’s refraction/diffraction disk, which increases linearly with the distance from the last obstruction. Is this right? This makes for some fairly fat photons. For instance using visible light, approximately 2 10^6 wavelengths per meter, if A and B are a meter apart, the width is 1000 wavelengths or about a millimeter. For light from the moon, it is about 20 meters wide. From the nearest star, about 200 kilometers wide. From across the universe, about 10 million kilometers, or about 30 light seconds wide. I have not yet found any references to this ellipse other than Feynman’s own brief reference to an ellipse in the Feynman Lectures on Physics Volume 1, chapter 26. Is there any experimental evidence that would support or refute this calculation? Of course, there are other causes of beam spreading, including misalignment, diffraction and the uncertainty principle. At least diffraction and misalignment could dominate the Feynman broadening discussed above. (I am going to promote this experimental issue to a separate question.)


title: EQUAL TIME commutation relations

text: Why is equal time commutation relation used in canonical quantization of free fields?


title: How much force is in a keystroke? (estimated, of course)?

text: I'm a software developer, and I need to calculate the estimated amount of force expended typing stored text. Preferrably in some interesting way. (i.e. the force exerted on keys thus far is enough to push a car 5 miles) (or: equivalent to 100 kg of TNT) Assumptions:   * We're not going to worry about deletes or moving the cursor or anything, I'm just counting characters of stored text.   * I don't really care if the space requires more force or not, this is more of a "fun fact" than anything.   * From what I've found online, **the mean force required for a keystroke is about 12.9N** (source).   * Hundreds of millions of characters have been typed. Questions   * What is a good way to make this something people can relate to?   * How can I calculate it? Thank you all in advance for your valuable time and input. EDIT: I thought my original post would make this pretty clear: I realize I only have the _force_ required to push each keystroke. I'm looking for a way to demonstrate that force _applied_ to something to help people quantify it, hypothetical energy in terms people can understand.


title: Laplacian of $1/r^2$ (context: electromagnetism and poisson equation)

text: We know that a point charge $q$ located at the origin $r=0$ produces a potential $\sim \frac{q}{r}$, and this is consistent with the fact that the Laplacian of $\frac{q}{r}$ is $$\nabla^2\frac{q}{r}=-4\pi q\cdot \delta^3(\vec{r}).$$ My question is, what is the Laplacian of $\frac{1}{r^2}$ (at the origin!)? Is there a charge distribution that would cause this potential?


title: Has quantum entanglement been demonstrated to be able to take place over infinite distances?

text: In my poor understanding of quantum physics, quantum entanglement means that certain properties of one of two 'entangled' quantum particles can lead to change over infinitely large distances when the other particles' properties are changed. Disregarding this already mind-boggling event taking place over say 10 meters distance; how have physicists been able to demonstrate, beyond reasonable doubt, that this can take place over infinitely large distances? For instance: have they done some of these tests between ISS and Earth perhaps? How can they be so sure?


title: Connection between a simple matter wave and Heisenberg's uncertainty relation

text: When looking at the wave function of a particle, I usually prefer to write $$ \Psi(x,t) = A \exp(i(kx - \omega t)) $$ since it reminds me of classical waves for which I have an intuition ($k$ tells me how it moves through space ($x$) and $\omega$ tells me how it moves through time ($t$), roughly speaking). However, I noticed that you can translate this from the $(k,\omega)$ into the $(p,E)$ space by extracting $\hbar$: $$ \Psi(x,t) = A \exp(i(px - Et)/\hbar) $$ When looking at this representation, I couldn't help but be remembered of the uncertainty relations $$ \Delta p \Delta x \geq \hbar\\\ \Delta E \Delta t \geq \hbar $$ This _cannot_ be a coincidence, but the derivation of these relations (as described e.g. on Wikipedia) are simply over my head and my textbook merely motivates them with an argument based on wave packets. **What is the connection here? Is there an intuitive explanation for the recurrence of both uncertainty relations in the wave function of a matter particle?**


title: Gauss's Law for a Uniformly Charged Solid Sphere

text: ![enter image description here](http://i.stack.imgur.com/s8VHn.jpg) We want to calculate $\vec{E}$ at a distance $r$ from the center $O$ of a spherical polar coordinate system. Let the point on the Gaussian surface at which we want to calculate $\vec{E}$ is $P=P(r, \theta, \phi)$. Applying Gauss's Law we get, $$ \oint_s \vec{E}(r, \theta, \phi) \cdot \hat{n}(r, \theta, \phi) \, ds = \oint_s E(r, \theta, \phi)\, ds = \iiint E(r, \theta, \phi)\, r^2 \sin\theta \, d\theta \, d\phi = \frac{Q_{enc}}{\epsilon_0}.$$ At this stage how do I know that $E$ depends only on $r$; i.e., $E=E(r)$? **Is it possible to give a mathematical proof that $E=E(r)$?**


title: Stealing heat, or not?

text: Say I hook a 1KW steam engine to the steam heat in my apartment, and generate 1KW of electricity (the engine is 1KW mechanical, not 1KW thermal) from it, so I don't have to pay ConEd. That means I am getting 1KW less heating power for the same amount of steam. Does it take the landlord's heating plant 1KW more to keep my room at the same temperature?


title: Tidal force on far side

text: I have a question about tidal forces on the far side of a body experiencing gravitational attraction from another body. Let's assume we have two spherical bodies $A$ and $B$ whose centers are $D$ apart, and who have radiuses $R_{A}$ and $R_{B}$, much smaller than $D$. Gravitation force has a law in $1/distance^{2}$. On the line $AB$, a mass $m$ at the point point of $B$ closest to $A$ experiences a pull towards the center of $A$ of magnitude $Km/(R_{B}-D)^{2}$, and the point farthest from $A$ experiences a pull of magnitude $Km/(R_{B}+D)^{2}$. Note that I did not mention the direction of the force at that second point. It seems that the centers of mass being all on one side of that point, the force should point towards the center of $A$ on the line $AB$. However, the tidal bulge on this "far side" suggests that some force (?) is pulling on matter _away_ from the center $A$. How can we explain this tidal bulge on the far side? - I am specifically interested in a clear derivation using classical mechanics.


Suppose a light wave with wavelength 3m. What happens if one tries to contain that wave within a 1m container? If I'm going about this entirely the wrong way or have wrong conceptions about light (which might be the case because I'm not a professional physicist), please tell me that instead.


title: What is imaginary time?

text: I am not professional physicist; but I am curious about Stephen Hawking's "imaginary time". It would be better to elaborate exactly what it is. I am not confused because of the word "imaginary" but I find it confusing to imagine a two dimensional "plane time". If we express time in a plane instead of a one dimensional axis, then what does the movement of an observer along the imaginary axis signify physically?


title: Origin of field deduced from potential

text: > Related: Tubelights+power lines pictures? I would've edited this into the above question, but I realized that there' enough to it to qualify as a new one. Plus this seems to be a confusion of many people (including me). I've already been able to explain part of this, so I'll put that explanation first just to make the question clearer. The original question is as follows: In a circuit, you can have potential, but what is the origin of the field associated with it? Conducting wires have no net field, just some tiny opposing fields popping in and out of existence which facilitate current. Explaining this for a simple capacitor is easy: With a charged (parallel plate) capacitor, one can see that we have a field between the plates but not in the rest of the circuit (for simplicity we can assume that the capacitor is just being shorted. So, traversing a different path in the line integral for potential gives a zero p.d, clearly a contradiction. After thinking about it, I realized that there is a significant fringe field near the terminals of the capacitor, which contributes heavily to the potential. SO the rest of the wire is equipotential without a field, and we have a p.d. only at the terminals due to the fringe fields. IMO there will be similar fringe fields that explain where the battery's potential comes from. Now, I can't seem to get a similar explanation for current through a resistor. I fail to see any field being formed. And my main issue is this: Let's say we set up a power line parallel to the ground. It may carry AC or DC current. Either way, it has a p.d with the ground at every point in time, which may vary. P.d. $\implies$ field, but I don't see any. In the AC situation, it _could_ be from EMI, though I doubt that EMI is strong enough to produce the required field strength. In DC, @akhmateli mentioned charge being distributed on the wire surface, but I doubt that as well. So where does the field come from? Is my explanation for a capacitor correct? Oh, and in your explanation, I'd prefer no "this comes from the potential" or "these charges move due to the p.d.". I want one that talks about fields and charges only. I've had too many explanations of this which talk about "the field comes from the potential", which is IMO cheating.


title: Sign of Velocity for a Falling Object

text: I'm working on a homework problem in Mathematica. We have to graph the height and the velocity of a function given an initial height and initial velocity. However, when I do the graph for the velocity with an initial velocity of 0 and an initial height of 100, the entirety of the velocity part of the graph is negative. Shouldn't the velocity be increasing as the object falls? ![](http://s17.postimage.org/b7r7rhfqn/Screen_shot_2012_02_28_at_9_23_46_PM.png) That is the graph and functions I am using. I have a feeling either something is typed wrong that I have overlooked in my numerous attempts to find it, or that there's something I'm just not understanding about what the graph is actually telling me.


title: N=2 Dualities; k-differentials on the riemann sphere and a spectral curve

text: Currently I am working on my masters thesis about dualities in QFT and their geometric realizations. As of now, I am trying to understand the article 'N=2 Dualities" by Davide Gaiotto. On the internet I found some exercises related to the article (http://www.sns.ias.edu/pitp2/2010files/Gaiotto-Problems.pdf). My questions are about some of these exercises. I will shortly summarize the exercise and then put my question forward. The full exercise is reachable via the link above. Exercise 1: We first look at degree k meromorphic differentials with poles of order $k$ at $n$ points $z_i$ on the Riemann sphere: $\phi_k(z)=F(z)dz^k$. Here $F(z)$ is a rational function on the complex plane. If we want to know the behaviour of $\phi_k(z)$ at $\infty$ we change coordinates to $z'=1/z$ under which the k-differential transforms as $\phi_k(z')=F(1/z')(-dz'/(z')^2)^k$. Furthermore, these differentials are required to have fixed residues $\alpha_i$ on each $z_i$. The question then is how big the dimension is of the space of these k-differentials. First of all it is unclear to me what precisely is meant with $$\phi_k(z) \approx \frac{\alpha_i}{(z-z_i)^k} dz^k +...$$. My interpretation is that $\phi_k(z)$ may be written as a fraction of two polynomials $f(z)/g(z)$ where $g(z)$ has $k^{th}$ order zeroes at $n$ points $z_i$ and $f(z)$ has $k(n-2)$ zeroes (to get the correct degree of the divisor of a k-differential on the Riemann Sphere, namely $-2k$). The zeroes of $f(z)$ we may choose freely (as long as we satisfy the fixed residues $\alpha_i$). First I attempted to solve this with Riemann Roch. This led me to a counting of $k(n-2)+1$ free parameters, however this doesn't account for the fixed residues I think. Then, with fixed residues, I reasoned it should be $(k-1)(n-2)$ by counting the free parameters for a k-differential. For n k'th order poles one has $(n-2)k$ zeroes to freely choose (in order that the degree of the divisor of the k-differential is $-2k$) and one constant $c$ multiplying $f(z)$ . However, for fixed residues, one has to subtract $n-1$ parameters (not $n$ since the residues sum to zero), which leads to the total of $(k-1)(n-2)$ free parameters. T his would also be the dimension of the vector space of k-differentials with n k'th order poles with fixed residues, since we can look at all linearly independent $F(z)'s$, ie different degrees of the polynomial $f(z)$ which may look like $\prod^l_{i=1}c(z-u_i)$ for $l\in {1,2,..,n}$, $u_i$ a zero and $c$ a constant. Does anybody know if this counting and way of looking at the $F_i(z)$'s is correct? 3i)I guess my problems with this question depend very much on the definitions in question 1. I tried to solve a simple example with $k=2$ and $n=3$: $$x^2 + F_1(z) x + F_2(z) = 0$$ with $$F_1(z)=\frac{c(z-u)}{(z-z_1)(z-z_2)(z-z_3)}$$ and $$F_2(z)=\frac{d(z-v)(z-w)}{(z-z_1)^2(z-z_2)^2(z-z_3)^2}$$ According to my calculation in 1ii) it follows that only $w$ is a free parameter in this equation; $c$ and $u$ are completely determined by the fixed residues of $F_1$ and $d$ and $v$ are determined by the fixed residues of $F_2$ (and are functions of $w$). EDIT: When I try to solve this equation (with a change of variables to $y = x(z-z_1)(z-z_2)(z-z_3))$ with Mathematica, the expressions become very complicated and it does not follow that that $\frac{\partial\lambda}{\partial v}$ is a holomorphic one form on the curve, where $\lambda = xdz$ as asked in exercise 3ii. The way to see this, I think, is that the residue of $\lambda$ at the points $z_i$ seems not to be independent of the parameters $v,w$. Does someone has an idea what mistakes I am making? Thanks, Sam


title: Thermodynamic cycles, when is the work negative/positive?

text: ever since I begun calculating thermodynamical cycles, I've had problems with determining the sign of the work along a particular bit of the cycle. Of course, I guess that an arbitrary cycle is 'bendy' and the sign of the work differential depends strongly on the coordinates, but usually the cycles I deal with consist of a couple of 'parts', for instance, adiabatics, isothermals, isochorics, etc. And whenever asked to calculate the total work done in a cycle (for instance to find the efficiency), I just kind of guessed the sign and managed to get by, but now when trying to understand this on a deeper level, this is coming back to haunt me. So, suppose I have a thermodynamic cycle like this: LINK Where: (1)-adiabatic (2)-isobaric (3)-isothermal (4)-isochoric How do I know the sign of the work along each of these paths?


title: Is a venturi pump feasible at very low pressure?

text: I need assistance (or experience feedback) in estimating if a Venturi pump would work at low pressure. A boiler under vacuum (10% atmosphere) will produce steam (at a given speed depending on the boiler power) and the steam flow will be used to suck water at the same temperature (of boiling water under this vacuum, about 50 degrees C) and pressure in a Venturi style setup. From the wiki articles I read about the Venturi tube, the usual equations giving the pressure difference between the constriction and the larger tube are for incompressible fluids. I do worry that 10% ATM steam would be very compressible though. So in short I need an equation for tying up the tube diameters, the steam feed speed, and the suction generated under such conditions


title: SUSY, ways to boost Chargino/Neutralino production?

text: Does anybody know a good reference that works out the equations for the Chargino/Neutralino production cross section in SUSY? I'm trying to understand if there are any tricks for boosting the production cross section. So far, I have just been testing using Prospino and discovered a couple patterns which seem counter-intuitive. For one, the production cross section seems to increase as I increase the masses of the squarks. This is strange because I would expect there to also be a contribution from strong production of chargino/neutralino so lower squark masses should boost the production. What am I missing?


title: Sign of the hopping integral in tight binding model

text: The Hamiltonian of tight binding model reads $H=-|t|\sum\limits_{<i,j>}c_i^{\dagger}c_j+h.c.$, why is there a negative sign in the hopping term?


title: Which pion is mediator in nucleon-nucleon interaction

text: In nucleon-nucleon interactions of n-n, p-p, n-p how do you determine which pion is the mediator?


title: Special conditions at layer F2 ionosphere

text: ![enter image description here](http://i.stack.imgur.com/H8qNp.png) 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. So, I wanted to know why is the recombination rate in F2 layer very low.


title: Can a blackhole eat a blackhole?

text: 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. 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. 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? What would happen if two blackholes suck on eachother?


title: Confused about unit of kilowatt hours

text: 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. 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?


title: Cut angle for BBO type I SPDC

text: 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. 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. Besides, I use a pulse laser rather than a continuous laser. Is this matters? Regards


title: Information scrambling and Hawking non-thermal radiation states

text: 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? 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. a reference to a recent TRF post about thermal and pure states of black holes being macroscopically indistinguishable from the outside


title: Path to obtain the shortest traveling time

text: Asume we have a particle sitting at the point A(0,0) in a gravitational field. (g=9.81) It is going to move along some path to the point B(a,b) Where a>0 and b<0\. What is the curve the particle needs to be moving at to arrive at B in the least time possible? (Fastest curve between two points. ) ![Example image,, various possible paths.](http://i.stack.imgur.com/D9fRA.png)


title: Multipolar expansion profile of Hawking radiation on Kerr black holes

text: 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. According to this answer, 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 Thoughts?


title: Help me get out of a speeding ticket

text: 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!


title: Geometry topics in physics

text: 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. 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. 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...)? Thank you for any answer!


title: Specify the Stress Energy Tensor and Calculate the Curvature

text: 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. **Edit: 2013 December 19th** I have found this article 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?


title: Spring problem?

text: I came across this problem in physics "Physics for Scientists and Engineers with Modern Physics by Serway" > _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?_ Why the answer is $4A$ instead of $2A$?


title: What is the reasoning behind the Hill Sphere?

text: According to Wikipedia, 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. 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? 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. 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? The answer to this would also eventually clarify why the earth orbits around the sun and not the center of the milky way.


title: Add air resistance to projectile motion

text: I am given an initial x and y position and initial velocity and I was asked to graph the trajectory in 1 second intervals. This is what I have so far: If $x_0 = 1, v_{0x} = 70, y_0 = 0, v_{0y} = 80, a_x = 0, a_y = -9.8$, and time will be 0,1,2,3... and so on. Using these equations on every second you can find the plot will be a bell shaped with the highest point being ~ 325 m at about 600 seconds: $$ x = x0 + (v_{0x})t + 1/2((a_x)t^2) $$ $$ y = y0 + (v_{0y})t + 1/2((a_y)t^2) $$ Usually in physics, we are taught in perfect condition with no air resistance. But what if there was air resistance? How would it affect this problem? How can i add it to my calculations and see what the difference is?


title: Why isn't it $E \approx 27.642 \times mc^2$?

text: 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? Why can so many equations be expressed so neatly with small natural numbers while recycling a relatively small set of physical and mathematical constants? For example, why is mass–energy equivalence describable by the equation $E = mc^2$ and not something like $E \approx 27.642 \times mc^2$? 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 $t' \approx 672.097 \times 10^{-4} \times \sqrt{\frac{t}{1 - \frac{v^2}{c^2}}}$. ... and so forth. * * * I'm not well educated on matters of physics and so I feel a bit sheepish asking this. 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.


title: Maximum voltage and charge a capacitor can store

text: 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. 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. Is there another equation I should be using or am I missing something here?


title: Can an object be deconstructed on an atomic Level?

text: 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. If you have any suggested readings on the topic that would be greatly appreciated. Q: Can an object be broken up to be just separate atoms?


title: If I charge a battery using a much higher amperage, can it explode?

text: 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? 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? 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.


title: Behavior of Pion-Mediated Nuclear Force before Electroweak Symmetry Breaking?

text: 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?


title: Flory-Huggins ternary phase diagram with a neutral component

text: 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. Thanks for any help. Best, Jd


title: What's the difference between t-channel and s-channel in particle physics

text: As the Feynman diagram shows above. Does the s-channel and t-channel stands for exactly same reaction or they have big difference? Many thanks in advance enter image description here ![enter image description here](http://i.stack.imgur.com/VG6GR.png)


title: Which particles can go right through the atom?

text: Can elementary particles (like the electron, photon, or neutrino ) go through an atom (not the nucleus)?


title: Intuitive explanation of why momentum is the Fourier transform variable of position?

text: Does anyone have a (semi-)intuitive explanation of why momentum is the Fourier transform variable of position? (By semi-intuitive I mean, I already have intuition on Fourier transform between time/frequency domains in general, but I don't see why momentum would be the Fourier transform variable of position. E.g. I'd expect it to be a derivative instead.)


title: If you were able to get rid of Hydrogen from a weather balloon as it were rising and expanding would the weather balloon rise further than usual?

text: 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?


title: Static equilibrium question

text: 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 **not** 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 :) This problem is from this book (8th edition), and instead of copying the entire problem and the image, I took a picture of everything: ![enter image description here](http://i.stack.imgur.com/A4mJ2.jpg) 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. First, I drew this simple diagram with all the forces applied on the body: ![enter image description here](http://i.stack.imgur.com/ZxDbe.jpg) Then I wrote this simple set of equations: ![enter image description here](http://i.stack.imgur.com/qb28n.jpg) If I solve this to get the maximum mass M that can be hanged, I get this result: m + M = m/2 + M 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 _normal_ for some reason). However, the result I'm getting does not make any sense, so there must be an error somewhere!


title: Is the earth expanding?

text: I recently saw this video on youtube: http://www.youtube.com/watch?v=oJfBSc6e7QQ 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? [could someone please tag this, I don't know what category this should come under]


title: Solving electromagnetic vector field using the Lagrangian

text: Given an action of the form \begin{equation}S=-\frac{1}{4}\int d^4x\eta^{\mu\nu}\eta^{\lambda\rho}F_{\mu\lambda}F_{\nu\rho}\end{equation} 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: \begin{equation}ds^2=a^2(\eta)[d\eta^2-(dx^i)^2]\end{equation} I would like to solve for $A_{\mu}$, and standard solution is \begin{equation}A_{\mu}^{(\alpha)}=e_{\mu}^{(\alpha)}e^{ik_\nu x^\nu}.\end{equation} I am interested in knowing how to derive this result. My approach is first write the Lagrangian from action and use EL eq \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} My main problem is mathematical difficulty in evaluating the EL eq. Can anyone please help me on this?


title: Linear/ non linear Scalar field theory

text: 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?


title: Does measurement, quantum in particular, always increase the total entropy?

text: 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. 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". **So, the question is:** 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?


title: Sailing against the wind. Is this a fair model?

text: I was ruminating the explanations about how boats can sail against the wind (or "into the wind"), and wondered if one could devise a simple mechanical model without hydrodynamics involved. 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. 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". Is this true, and is this a fair model of what happens when "sailing against the wind"? ![enter image description here](http://i.stack.imgur.com/ChO9f.png)


title: What compact central objects younger than the crab pulsar have been seen?

text: 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?


title: Why does a bucket of Water weigh more than a bucket of Air?

text: 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?


title: Scalar top quark (stop) pair production

text: A rather simple question: Starting from an electrically neutral state, pairs of top quarks are produced as top and anti-top, and denoted as $t\bar t$. 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. 1, 2), rather than $\tilde t \bar{\tilde t}$. Why this notational difference? What is $\tilde t^*$?


title: How do micro-channel plates detect neutral molecules?

text: 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?


title: How to determine Center of Gravity?

text: I came across this question while having conversation with one person. We know that Center of Gravity 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?


title: How to calculate the force acting on a magnet placed in a magnetic field?

text: The force on a small element (of length dl) of a current carrying wire, place in a magnetic field **B** can be calculated using the following equation (which is simply an application of the Lorentz force equation): d **F** = i **dl** X **B** My question is, how does one calculate the force acting on a small magnetized volume d **V** (say magnetized in the z direction with magnetization **M** or remanent flux density **Br** ) when the same is placed in a magnetic field **B**.


title: What makes some laser beams visible and other laser beams invisible?

text: What makes the beam of some lasers: 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) 2-invisible? such as the ones used in pointers for presentations


title: Deriving an Expression for Entropy

text: How to derive an expression for entropy in form of $S = \ln \Omega$ from the form $\displaystyle{S = - \sum_i \; p_i \ln p_i}$ ? That is the last formula taken as a definition of entropy. Just a reference will do. The backwards derivation (probably with some assumptions) is given in Landau and Lifshitz.


title: Electron shell bombardment

text: 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?


title: Surface tension of N non-mixing fluids

text: I am a mathematician, not a physicist, so please be gentle with me if I write something wrong. 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. There are three factors I have in mind:   * the interaction of $F_i$ and $F_j$ with $i\neq j$ by their surface tension;   * the interaction between $F_i$ and the boundary $\partial \Omega$ of the container;   * the action of gravity on each $F_i$. I have two questions: >   1. Is there a relation of the form > $\sigma_{ij}+\sigma_{kl}=\sigma_{ik}+\sigma_{jl}$ (scalar or vectorial) > between the surface tensions? > >   2. Are there any references or monographs which provide a good > introduction to this study? I'm interested especially in surface tensions. > >


title: high spin atoms SU(2) representation

text: 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. Why not $SU(N)$?


title: What is the best tool for simulating Vacuum and Fluids together?

text: 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. I have tried RealFlow but it doesn't support Vacuum. Any body knows any software for this?


title: Solving inclined plane using diff. frame

text: 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. 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. ![Frictionless inclined plane](http://i.stack.imgur.com/qlEuM.png)


title: Einstein tensor in Friedmann equations : where is the missing $c^2$?

text: 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. In all the following $\rho$ is the **mass density** and not the energy density $\rho_{E}=\rho c^2$ If we look at the wikipedia French page concerning the Friedmann equations, according to the demonstration of the last paragraph we have : The Einstein field equation : $G_{\mu\nu} = \frac{8\pi G}{c^4}T_{\mu\nu}$ The Einstein tensor : $G_{\mu\nu} = \begin{pmatrix} G_{00}&0&0&0 \\\ 0&G_{ij}&0&0 \\\ 0&0&G_{ij}&0 \\\ 0&0&0&G_{ij} \end{pmatrix}$ The Energy-Momentum tensor : $T_{\mu\nu} = \begin{pmatrix} T_{00}&0&0&0 \\\ 0&T_{ij}&0&0 \\\ 0&0&T_{ij}&0 \\\ 0&0&0&T_{ij} \end{pmatrix}$ with : $G_{00} = 3H^2+3\frac{k}{a^2}c^2$ $G_{ij} = -\left(3\frac{H^2}{c^2}+2\frac{\dot{H}}{c^2}+\frac{k}{a^2}\right)$ $T_{00} = \rho c^2$ $T_{ij} = -P$ 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. **My question are** : 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 ? EDIT : Maybe I've found something. At the beginning of the demonstration, the author say that the metric is of the form : $ds^2=c^2dt-a^2\gamma_{ij} dx^i dx^j$ where $\gamma_{ij}$ depends on the coordinates choice. This formula seems ok to me. But then he writes that : $g_{00} = c^2$ $g_{ij} = -a^2\gamma_{ij} $ 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 ?


title: Does recharging a battery at a lower temperature lower its internal resistance?

text: 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?


title: Why is technetium unstable?

text: Is there a simple account of why technetium is unstable? From the Isotopes section of Wikipedia's article on Technetium: > 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. 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)? 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: > 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. 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 _so_ unstable that all of its isotopes are essentially absent naturally, compared for instance to uranium-235? (This question is a specific case of an earlier question on synthetic isotopes.)


title: If a photon is a boson and has spin 1, shouldn't it have 3 spin orientations since spin 1 is a triplet?

text: I've gotten used to the fact that a spin can be described by its total spin and its $z$-component. And I've learned that a particle (really, anything) with spin 1 forms a triplet with three possible states for the $z$-component: +1, 0, and -1. Now my world is shattering since I keep hearing that a photon has indeed spin 1, but only two spin orientations: +1 and -1. This sounds like it clashes with what I have learned. What am I missing?


title: Why is the $S_{z} =0$ state forbidden for photons?

text: If photons are spin-1 bosons, then doesn't quantum mechanics imply that the allowed values for the z-component of spin (in units of $\hbar$) are -1, 0, and 1? Why then in practice do we only use the $\pm 1$ states? I have been told that this is directly related to the two polarizations of the photon. This seems to be more of a classical argument however, arising from the fact that Maxwell's equations do not permit longitudinal EM waves in a vacuum. I have also heard that it is related to the fact that photons have no rest mass, although I understand far less of this reasoning. What I'm looking for are elaborations on these two arguments (if I have them correct), and perhaps an argument as to how these two are equivalent (if one exists).


title: Normalising a wave function in parts?

text: 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.


title: Hooke's Law and the shape of coils

text: I've learned in school that the force in a coil is $F=kx$, linear on how much the coil is stretched. Two questions:   1. 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?   2. Can we deduce the value of $k$ based on the geometry of the coil? 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 Hooke's Law.


title: Mass distribution driven by gravitational field

text: In a gravitational field, should the mass distributions always behave well?


title: How will you fix the direction of the lines of force due to circular current?

text: I have tried to learn it myself but I was unable to understand. Can you please explain it in detaails with simple example. Thanks


title: Resonance in a gravitational field?

text: 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?


title: Computer game with quantum optics/ information

text: Is there a computer game using principles of quantum optics or quantum information? By game I don't mean just a simulation or an interactive course, but something that can be played in an enjoyable way. By 'principles' I mean actual rules, not just things bearing the same name or vaguely related. As a reference, let me use Chromatron \- 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).


title: Could the unobservable universe have curvature (be spherical)?

text: 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?


title: Conformal QFTs for D > 2

text: Which conformal QFTs do we know for spacetime dimension d > 2? I know that for D = 4 we have N = 4 SYM and some N = 2 supersymmetric Yang- Mills + matter models. > 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?


title: Generalized Complex Geometry and Theoretical Physics

text: I have been wondering about some of the different uses of Generalized Complex Geometry (GCG) in Physics. Without going into mathematical detail (see Gualtieri's thesis 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 Courant Bracket. The first hints of the necessity of GCGs in Physics came up in a famous paper by Gates, Hull and Roc̆ek, 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 _torsionful connections_. 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. 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 _Sasaki-Einstein_ manifold. As such, we start out with a metric of the form, $ 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) $ where $g_{C(Y^5)} = dr^2 + r^2 g_{Y^5}$ (the _metric cone_ 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 $C(Y^5)$. Related to this is the result of Graña, et. al which can be poorly paraphrased as: > 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 $A = 2\Delta + \phi/2$ 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. Thanks!


title: String-theoretic significance of extended CFT

text: 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). 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, > What is the signficance of **extended** SCFTs in this context? 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?


title: How can I describe an equation for multiple objects over time?

text: Lets say i have $n$ different objects that effects each other only by the classic gravity force. I have their initial locations, masses and velocity's: $$ x_1(0),\cdots,x_n(0) $$ $$ m_1,\cdots,m_n $$ $$ v_1(0),\cdots,v_n(0) $$ Is it possible to describe each object location with an equation? $$ x_i(t)= ? $$ I tried doing this : $$ x_i(t) = x_i(0)+\int_0^t v_i(t') dt' $$ $$ v_i(t')=v_i(0)+\int_0^{t'} \frac{1}{m_i} \sum\limits_{j=1}^n \frac{Gm_im_j}{r_{i,j}(t'')^2} dt'' $$ 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...


title: Thought experiment regarding perpetual motion machines and paradoxes

text: 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. 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. 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.


title: Mixed-Alloy materials

text: Short version: 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? Long Version: I was reading more into the designations of various metals and starting thinking about something. 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). 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?


title: How to numerically solve a complex equation?

text: 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..


title: What is the difference between atomic limit and hard core boson limit?

text: 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?


Here is an extremely naive question: Why would the apple fall under the tree? 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:   1. 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 $h=\frac{1}{2}gt^{2}$.    2. 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.    3. 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.    4. 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 _deviate from_ 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.    5. 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?   6. 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.  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.


title: How to think about antiparticles in KG equation?

text: I am a beginner to study QFT and have a problem. 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. 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. So, how to explain the antiparticles in KG equation?


title: Why must allowable physical laws have reversibility?

text: 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 _one_ state $S_0\in \mathcal{M}$ such that $S_0\mapsto S$ in the evolution of the system. 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?


title: Hawking evaporation is due to NEGATIVE mass?

text: 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!


title: Understanding the algebra associated with an implicit potential

text: In the paper here(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 : $$ U(z(r)) = \frac{h_1z(r)+h_0(1-z(r))-fz(r)(1-z(r))}{R(z)} - \frac{1}{2}\left\\{z,r\right\\}$$ 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 Schwarzian derivative. 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 shape invariance 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?


title: Microsecond trading with neutrinos

text: 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. 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. 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? Addendum - Over what distances can you reasonably detect a neutrino beam?


title: Did classical applications of density functional theory precede its use as an electronic structure method?

text: Density Functional Theory (DFT) is usually considered an electronic structure method, however a paper by Argaman and Makov highlights the applicability of the DFT formalism to classical systems, such as classical fluid density. This presentation by Roundy et al. calculates water properties using a classical approach, with an eye to combining it with KS DFT for calculating solvent effects. Not being too clear on the history of DFT, _was DFT invented for electronic structure problems first_ , and its applicability to classical problems incidental, or was it formulated for classical systems first and then adapted to the quantum many body problem?


title: Huge buildings affect Earth's rotation?

text: 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?


title: Some questions on Conformal Field Theory, Current algebras and the Sugawara construction

text: 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: Beginners questions concerning Conformal Field Theory 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. Now again some questions again turned up:   1. 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...)    2. 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.    3. 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.


title: Why is it when you microwave cold coffee and then add milk it creates a foam head?

text: 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.


title: Moment of weight of leaning beam

text: ![enter image description here](http://i.stack.imgur.com/KYj8v.png) Having trouble with a FBD moment. The problem is: > _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._ > > _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._ 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. 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? For reference, this is his equation of moment equilibrium about A: $$M_a = N_b \ell - m g \frac{\ell}{2} + C = 0.$$


Why does a wavefunction collapse 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.


title: Calculate Hamiltonian from Lagrangian for electromagnetic field

text: 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. **Edit:** 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" $\mathcal{H}$.


title: Transition Between Position and Momentum Basis

text: I'm having some trouble following pages 55-56 of Sakurai's _Modern Quantum Mechanics_. We're trying to transfer from position space into momentum space. Here's a quote: > 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 **transformation function** 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 $$\langle x' > | p' \rangle = N \exp \left( \frac{ip'x'}{\hbar} \right)$$ 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. What I'm thinking now is to rearange it like so: $$\int \langle x^{'} |p^{'}\rangle dx' = \int -\frac{i\hbar}{p'} \frac{\partial}{\partial x^{'}} \langle x^{'} | p^{'}\rangle dx'$$ But it doesn't seem to lead anywhere. What am I missing?


title: A question on lowering the total spin

text: 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?


title: Why does one experience a short pull in the wrong direction when a vehicle stops?

text: 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? * * * 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 Robert's answer. Also, as Gerard suggested the release of the brakes and some other friction effects might play a role. So let's be more precise with the question: > Which effect **dominates** the wrong-pull-effect? And thus, who can reduce > it most: > >   * the traveler >   * the driver >   * the vehicle designer? > * * * **edit** Let's make this more interesting: I'm setting up a bounty of ~~50~~ 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 ~~200~~ 150 for what I subjectively judge to be the best answer describing either:   * an accomplished experiment (some video or reproducibility should be included)   * a numerical simulation   * a rigorous theoretical description **update** since I like both the suggestions of QH7 and Georg, I decided to put up a second bounty of 50 (thus reducing the second bounty to 150 however)


title: Is there a distinguished reference system, after all?

text: 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. 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? 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)?


title: Solving the diffusion equation

text: 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 $$ \frac{\partial C}{\partial t} = D \frac{\partial^2 C}{\partial x^2}, $$ how can this be solved to get $$ C = \frac{A}{\sqrt{t}} \mathrm{e}^{-x^2 / 4 D t} ?$$


title: Modeling the formation of a stellar system and matter accretion

text: 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. Being more of a programming background than a physicist one, I find some difficulties trying to find all the forces in interaction. 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. 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. I'm not sure how to approach it and I'm fairly confused. How electric force is involved in my scenario? How can I account for it in my calculations? Specifically, in a cloud of hydrogen particles in space, what is happening?


title: Acoustic wave equation for a closed sphere

text: 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? 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).


title: How would a Lagrangian be used to recover the Schrodinger equation?

text: 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?


title: Do electrons have definite and single value of momentum and position?

text: 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 _measure_ the exact position and momentum or is it actually impossible for an electron to _have_ a single position and momentum? 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) Edit: Also, does $\hbar$ in the uncertainty principle equation, $\Delta x \Delta p = \frac{\hbar}{2} $ arise from Planck length ( **From my understanding, the impossibility to measure the exact position and momentum arises from the quantization of length and time**. Is my understanding correct?)


title: Are the polarization field and electric field in LIH dielectric in dynamic equilibrium?

text: If we know that the Polarization P in LIH dielectrics is proportional to the net field inside the dielectric according to: P = ε0χeE.....(1) And we know that D = εE........(2) 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? 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?


title: What is the Riemann curvature tensor contracted with the metric tensor?

text: Can the Ricci curvature tensor be obtained by a 'double contraction' of the Riemann curvature tensor? For example $R_{\mu\nu}=g^{\sigma\rho}R_{\sigma\mu\rho\nu}$.


title: Differences between strong, weak, and micro lensing distinct or subtle?

text: 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 this question). 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? 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).


title: What's the validity of the thermodynamic definition of entropy?

text: Thermodynamic Entropy Variation is defined as $$\Delta S = \int_i^f \frac{dQ}{T},$$ where $i$ and $f$ are the initial and final states of the process. My question is: **does this equation apply to quasi-static irreversible processes, or only to reversible processes?** 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).


title: How long would it take me to travel to a distant star?

text: Suppose I wanted to travel to one of the recently discovered potentially Earth-like planets such as Kepler 186f that is 490 light years away. Assuming I had a powerful rocket and enough fuel, how long would it take me?


title: Special relativity allows arbitrarily low travel times between two locations

text: I wish I had a good way of illustrating this, but anyway, doesn't the following travel strategy allow you to get anywhere in arbitrarily little time? You're at rest at the origin of space-time, and you'd like to meet up with an object some distance away in the positive x-axis - in other words, you'd like to reach the vertical world line of that object. You begin traveling towards the object at t = 0 in the following 3 phase trip: **I**. You accelerate nearly to the speed of light relative to the stationary frame of the origin and the object you're trying to reach. **II**. You level out your speed and coast. **III**. You decelerate right before you reach the object. As long as the observer travels on a space-time path that's always nearly lightlike, the proper time of his journey will be almost zero. Not that he sees a finite amount of distance go by him in zero time - the proper distance between the endpoints of his nearly lightlike journey is also nearly zero, so the speed of light is never exceeded by objects passed by the traveller. I realize that the travel time in the stationary reference frame is finite - it's the same as that of a light ray, and that during the acceleration phase ( **I**.), the traveller will see all of this finite time go by at his destination, but _he'll_ never experience that amount of time going by, right? If the above reasoning is correct, then there isn't really a limit on how far you can travel in a human lifetime, right? People seem to think that the finite maximum speed c is a death knell for, say, trips to Andromeda (and beyond). But if the above is correct, then the only limit on short travel times is the energy required to accelerate (a limit which exists in a Newtonian world as well). In short, isn't it possible (in principle) to go arbitrarily far in a human lifetime?


title: How much time passed for the passenger traveling with at speed-of-light spaceship?

text: Let's suppose we have a spaceship with the exact speed of light. If a traveller takes this spaceship to go to proxima centauri (approximately 4 years light away from Earth) and come back, we (as observers on Earth) will see the ship coming back after approximately 8 years. But how much time would have passed for the traveller on the ship? How can be this calculated with a formula?


title: Double slit experiment and indirect measurements

text: 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?


title: Finding kinetic friction coefficient

text: > 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. > > a. prove $\frac{t_{down}^2}{t_{up}^2}=\frac{a_{up}}{a_{down}}$. > > b. suppose time of getting up the incline is half of the time needed for > getting down it, find $\mu_k$ With `a`, I had no problem proving it, but with `b` I have a problem. Using `a` 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 $$F=-30m\sin\alpha+50\mu_{k}m\cos\alpha$$ $$\beta=-30\sin\alpha+50\mu_{k}\cos\alpha-10\sin\alpha-10\mu_{k}\cos\alpha=-40\sin\alpha+40\mu_{k}\cos\alpha\\\\\gamma=10\left(\sin\alpha-\mu_{k}\cos\alpha\right)$$ 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?


title: is frictional force right or wrong

text: 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!


title: Why is the observable universe so big?

text: The observable universe is approximately 13.7 billion years old. But yet it is 80 billion light years across. Isn't this a contradiction?


title: Are we travelling at a speed close to the light speed?

text: This is a question which has been puzzling me quite a while. There was a news report saying that the farthest galaxy spotted so far is at 13.3 billion light-years away, and it is said that the galaxy can show us how the universe at about 500 million years old was looked like. It is commonly accepted that the universe was started from a single point some 13.8 billion years ago, and continuously expanding since then. Therefore, the new discovered farthest known galaxy should be much closer to us, or, at least to the materials forming our galaxy, at 13.3 billion years ago. It implies, if the universe has been expanding in a constant speed, then we must be traveling away from the farthest known galaxy at 13.3/13.8, or 96%, of the light speed! If the universe's expanding speed is not constant (it is said that the universe's expanding is accelerating due to the dark energy), then we must traveling away from the farthest known galaxy at an even greater speed. But we know, it is not possible. So, what is wrong here? Is my calculation (96% c) wrong, or the assertion of that the farthest known galaxy reflecting the universe in 13.3 billion years ago wrong?


title: How is it possible for you to be at the centre of the universe wherever you are?

text: I saw on Richard Hammond Builds A Universe on BBC2 a few days ago that you are always at the center of the universe wherever you are. Surely this is illogical, because you could never get to the edge of the universe. If I am in the center, then what about that object 10 meters away. Is that at the centre too? How is this possible? What if I am driving for example. Does this mean that I am moving the universe while I stay still? Thanks


title: How far away is Cosmic Background Radiation?

text: I've been reading about Cosmic Background Radiation (CMBR) lately. My understanding so far is that this is the "Noise"/"Flash" from the big bang. Assuming that information is correct, how far away is it traveling in order to reach us? The way it was described to me makes it sound as if it should be coming from the "end" of the universe. Or is it omnipresent?


title: Does the universe have a center?

text: If the big bang was the birth of everything, and the big bang was an event in the sense that it had a location and a time (time 0), wouldn't that mean that our universe has a center? _Where_ was the big bang? _What_ is there now? Are we moving away from the center still? Are these even valid questions?


title: Speed of light's relation with the expansion of the universe

text: If nothing can travel at the speed of light and light cannot escape the universe, then the rate of expansion of the universe should be equal to the speed of light, why it is not?