title: Calculating the Potential from the E-Field

text: I find that often times I'll be tripped up by questioning whether or not I can do something mathematically, and be unable to come up with a satisfying answer. This is, unfortunately, one of those times. I'm told: > A uniform electric field, $\vec{E} = E_0\hat{x}$. What is the potential, > expressed using cylindrical coordinates, $V(s,\phi,z)$? My first course of action is: We know... $$|r| = \sqrt{x^2 + y^2 + z^2} = \sqrt{x^2} = x = E_0$$ $$\tan^{-1}{\frac{y}{x}} = \theta = 0$$ $$z = z = 0$$ So the electric field only has a component in the $\hat{r}$ direction. Now, we know that $\vec{E} = -\nabla V(r, \phi, z) = - \frac{\partial V}{r} - \frac{1}{r}\frac{\partial V}{\theta} - \frac{\partial V}{z}$ So, I think "Oh. I just have to integrate." ... but over what? Do I integrate three times, once w.r.t. radius, then phi, then z? I'm pretty sure that won't give me the right answer. If I decide to express $\nabla V$ in terms of cartesian coordinates, I get $- \nabla V(x,y,z) = E_0 \hat{x}$ ... but the question still remains. I feel like this is definitely the _easy_ part of the problem, and I can often do the more complicated parts—it's just small things like this often throw me off. How would I go about extracting the potential from either of those equations? I know I have to integrate, but... where?


title: Who was the R. Dolen behind Dolen-Horn-Schmid duality?

text: Can you provide biographical information for R. Dolen, known for Dolen-Horn- Schmid duality?


title: Is $\langle\psi_1|p\psi_1\rangle$ necessarily 0 for eigenstates?

text: Is $\langle\psi_1|p\psi_1\rangle$ necessarily 0 for harmonic oscillator eigenstates? If $\Psi(x,t)= c_0\psi_0(x)e^{-iE_0t/\hbar}+c_1\psi_1(x)e^{-iE_1t/\hbar}$, is the following true? Where $p$ is the momentum operator, $$\langle\psi_1|p\psi_1\rangle = \langle\psi_0|p\psi_0\rangle=0.$$ Any hint would be appreciated.


title: alternatives to supersymmetry and Coleman-Mandule theorem

text: Humour me for a minute here and let's imagine that all interesting and plausible supersymmetry models have been "cornered" out by the experimental data; what sort of alternatives are there for having quantum field theories with Poincare symmetry that are allowed to have nontrivial internal symmetries? i.e: that the Coleman-Mandula theorem does not apply? What other assumptions of the theorem can be relaxed or dropped, and still leave us with workable QFT? Are we forced to drop full Lorentz-Poincare symmetry or will the theorem still apply with slight violations of that symmetry?


title: Tracking down the locality assumption in a CHSH inequality derivation

text: CHSH inequality requires both _locality_ and _realism_. I will equate here realism with counterfactual definiteness. Now counterfactual definiteness tells us that given two different measurements on the same object, described by random variables $C_1$ and $C_2$, there exists a joint probability distribution for $C_1$ and $C_2$ (this is not always the case, search for the marginal problem and indeed we know that outcomes of measurements of noncommuting observables do not posses a joint probability distribution). Now if we can assume the existence of a joint probability distribution, then expectation values $E(C_1) + E(C_2)$ may be joined together to have $E(C_1 + C_2)$. So suppose that we now have four random variables $A_1$ and $B_1$ local to Alice and $A_2$ and $B_2$ for Bob, which can take values $\pm 1$. The expression in the CHSH inequality is $$ |E(A_1 A_2) + E(A_1 B_2) + E(B_1 A_2) - E(B_1 B_2)| $$ Now if we can assume realism (counterfactual definiteness), there exists a joint probability distribution for the outcomes of all four random variables, and we can join the expectation values in the above together to get $$ \Bigl\lvert E\bigl(A_1 (A_2 + B_2) + B_1 (A_2 -B_2)\bigr)\Bigr\rvert $$ So now we do the standard trick where either $A_2$ and $B_2$ are equal or they are opposite, so that the first term is either $\pm 2$ and same for the second term. Now to my question. I obviously used the realism assumption in the above. I assume locality means that the marginal distributions for $A_1$ and $B_1$ are independent of the choice of random variables Bob makes, $A_2$ or $B_2$ (but I otherwise allow for the correlation of outcomes, as long as it's independent of measurement choice, as there may be hidden variables that were encoded at the source of the state that produce correlations). Where did I use the locality assumption in this derivation? I would like if you either point out precisely where this assumption is needed in this calculation or argue that it is not needed with a convincing justification. **Edit:** Given the answer below, it seems likely that the locality assumption may have been used at the point where we take $E(C_1) + E(C_2) = E(C_1 + C_2)$. However, it is still not understood why realism is not sufficient to reach this conclusion without the assumption of locality. As stated above, the question is why is the above reasoning not correct and locality assumption is needed, or alternatively I am also open to the idea that it might not be needed. **Edit 2:** I have received a satisfactory answer to this question over at MathOverflow from @Steven Landsburg. > The existence of a joint probability distribution for all four variables is > definitely sufficient to get the inequality. The existence of joint > probability distributions for (A1,B1) and (A2,B2) (i.e. "realism") is > definitely _not_ sufficient to imply a joint probability distribution for > all four. Locality is a sufficient additional assumption. He also shows there that counterfactual definiteness for local outcomes is not a sufficient condition to obtain the CHSH inequality.


title: Faddeev-Popov ghost propagator in canonical quantization

text: Obtaining the propagator for the Faddeev-Popov (FP) ghosts from the path integral language is straightforward. It is simply $$\langle T(c(x) \bar c(y))\rangle~=~\int\frac{d^4 p}{(2\pi)^4}\frac{i e^{-ip.(x-y)}}{p^2-\xi m^2+i\epsilon}.$$ But I am unable to properly derive it following canonical quantization route. The problem is that for anti-commuting fields, the time-ordered product is defined as $$\langle T(c(x) \bar c(y))\rangle=\theta(x^0-y^0)\langle c(x) \bar c(y)\rangle-\theta(y^0-x^0)\langle \bar c(y) c(x)\rangle$$ with a _minus_ sign between the two terms. This minus sign is preventing me from closing the contours in the correct way to obtain the expression in my first equation. The only way I can save this is to say that FP ghosts are special, and their time-ordered product is defined with a plus sign instead of the minus sign. Is this legitimate? What is the right way to get the Ghost propagator following canonical quantization route?


title: Electric scooter acceleration

text: I tested two electric scooters with almost same power (1500 W), but very different performances: A "Zem Star 45" (1500 W, 60 Nm) can reach 50 km/h in "a few seconds" (not measured actually), An "Ingaeta Ingo" (1500 W, ??? Nm) took more than 10 seconds to reach 50 km/h. How much does torque affect acceleration performance? Given two identical scooters (same weight, same engine power, same wheels size), rgardless of wheels and air friction, how can I calculate acceleration given the torque on a flat street?


title: A force applied to an inclined plane

text: Below are two drawn diagrams of the problem with some explanation. Note: I think my notation for the force that the block exerts on the plane in the horizontal direction is not the standard notation. I chose $F_{bx}$ but I think it might have been better to say something like $F_{px}$. Hopefully this isn't a problem. ![System diagram](http://i.stack.imgur.com/faCGy.png) ![Worked block diagram](http://i.stack.imgur.com/PwdPq.png) So I've uploaded pictures of the problem, but I'll describe it here too. A block rests on a plane with incline $\theta$. The plane rests on the ground and both the plane and block have mass $m$. There is no friction on any surface. A force is applied to the plane in the horizontal direction with magnitude $F_{a}$. The plane and block accelerate with respect to the ground but are stationary with respect to each other. Solve for the acceleration in terms of $m,g,\theta,$ and $F_{a}$. I'm uncertain of how to deal with how the force of the block acts on the plane. Gravity acts downward, so part of me thinks that the block just acts downward on the plane but I'm pretty sure that's wrong. My other hypothesis is that the incline re-directs the force of gravity so that it acts perpendicular to the plane. In that case, call the force that the block exerts on the plane $\vec{F}_{p}$. Does that mean that $|\vec{F}_{p}|$ is the weight of the block? Or is the vertical component of $\vec{F}_{p}$ equal to the weight of the block? And more generally, I have I basically set up the problem correctly?


title: What papers detail the early research on heavy fermion superconductors?

text: Can someone point me to the papers detailing when/where/how heavy fermion superconductors were first synthesized, tested and documented?


title: Free fall of two spheres made of same materials, different masses, with air resistance

text: Two people falling with the same parachute will gain different speeds if their masses are different. The upward air drag will needed to be bigger for a heavier person, since gravitational force is bigger in his/hers case.   In parachute case, the shape is almost the same. But what about two iron balls, one with weight 100kg, and other 1kg. The densities of two balls are the same, but the area of a bigger ball is larger. Does this increase in area of bigger ball compensates for bigger mass, do the two balls still fall with different speeds?


title: Behavior of wave at boundary

text: Firstly, how is "density" of wave formulated as? Secondly, when wave, travelling in less density medium, meets with the boundary that has higher density medium next to, reflected parts of the wave changes phase by 180 degrees. If then, how does the amplitude change?


title: How do I join two co-axial cables?

text: I'm trying to join two ends of co-axial cables. I've managed to connect the two ends of copper wires but still the signal response is very low. When I try to stretch the cable signal gets good. Why does this occur? Is there a reason some sort of special procedure is needed to join the two ends of co-axial cables?


title: what determines the direction of a single photon

text: Photon has momentum and energy.assume i am creating a nuclear fusion by fusing hydrogen nuclei into helium and a photon is created.Now which direction does it(single photon) moves or travel?can we direct that single photon to any specific direction? i am not a physics student,but please help me understand.


title: Is eternal inflation Lorentz invariant?

text: Start without general relativity. Consider a metastable vacuum over good ol'-fashioned Minkowski space. It decays. A bubble forms and the domain wall expands. The domain wall is timelike, and accelerates, driven by the difference in pressure between both phases. Its speed approaches the speed of light asymptotically. Far away, it looks like an expanding light cone, but it stops looking so around the moment of decay. Roughly speaking, this breaks Lorentz covariance, and this picks out an approximate frame of reference. It might be argued that the metastable phase prior to decay is Lorentz invariant. If so, there is only one Lorentz invariant measure over the space of future- oriented unit 4-velocity vectors up to a multiplicative factor. This space has infinite measure, and the typical decay velocity of a decay would have to be infinite. Nonsense. The point to note is we have to specify the initial conditions of the metastable vacuum. That can't be infinitely in the past, as the lifetime, while exponentially long, is still finite. A Lorentz invariant initial boundary can only be an expanding light cone, or a hyperboloid. However, both have infinite volumes, and to regulate, we need to cut off these surfaces at some finite radius. This radius has to be less than the decay lifetime. This cutoff roughly picks out a frame of reference. At any rate, affine parameters on light cones are tricky. Boost them enough, and you can get an exponential change in the affine parameter, and in this boosted frame, the likelihood of crossing a bubble nucleation approaches one in probability. Turn on general relativity. Does eternal inflation break Lorentz invariance?


title: Ensemble average of product of spin operators?

text: How do you evaluate the canonical ensemble average of a product of spins, e.g.: $$[S_zS_x]$$ Where: $$S_x = \frac{\hbar}{2} \begin{pmatrix} 0 & 1\\\ 1 & 0\\\ \end{pmatrix}$$ $$S_y = \frac{\hbar}{2} \begin{pmatrix} 0 & -i\\\ i & 0\\\ \end{pmatrix}$$ $$S_z = \frac{\hbar}{2} \begin{pmatrix} 1 & 0\\\ 0 & -1\\\ \end{pmatrix}$$ The matrices are too simple, but then how is the ensemble average of the resultant matrix defined? EDIT: I knew it is about the density matrix, but I guess I was confuseing this with something else. Thanks anyway.


title: Spin angular momentum of a system of particles : Is there any energy associated with it?

text: Consider a system of point particles , where the mass of particle $i$ is $μ_i$ and its position vector is $\vec{r}_i$. Let $\vec{r}_\text{cm}$ is the position vector of the center of mass of the system. Considering the system from a reference frame attached to the center of mass, the system may have a spin about the center of mass and it is given by the spin angular momentum $\vec{L}_{spin}$. It is given by the expression $$\vec{L}_{spin} = \sum_i \mu_i \Bigl[(\vec{r}_i - \vec{r}_\text{cm}) \times (\dot{\vec{r}}_i - \dot{\vec{r}}_\text{cm}) \Bigr]$$ The rate of change of this spin angular momentum is the total torque acting on the system about the center of mass in the center of mass reference frame. My question is, is there any (spin kinetic (may be)) energy associated with the spin angular momentum in the center of mass reference frame ? How is it defined ?


title: Mechanics of a disk

text: I have an interesting problem. A disk travels in air at velocity v1. It travels in glass (assuming that the disk can enter the glass without breaking it) at velocity v2. v1 is greater than v2. The disk has radius r. Lets say that it hits the glass at a particular angle a. Then the part of the disk that enters the glass will travel at v2, the part outside the glass will still travel at v1. This will cause the disk to deviate from its initial path and bend inwards. The opposite would happen when the disc exits the glass on the other side. What will be the total angle that the disk will deviate from its initial path? What would be a good way to approach solving this question?


title: If time isn't continuous, what is the best-known upper bound on the length of time intervals?

text: There have been several questions about whether time is continuous or not and it seems like the answer isn't currently known. I know quantum mechanics treats time as continuous and any mathematics that involves integrating over some time interval treats time as continuous too. Surely though there are experiments that are quite sensitive to discrete time with large intervals. It seems the shortest laser pulse so far is only 67 attoseconds ($67 \times 10^{-18}\: \mathrm{s}$) but wouldn't this experiment actually constrain time intervals to much less than that? Are short laser pulses even a good experiment to determine if time is discrete or not? **So, assuming time isn't continuous, what is the best-known upper-bound on time intervals? Also, which experiments have done the best to constrain how non-continuous time could be?**


title: intrinsic concentration of charge carriers

text: In case of semiconductor the intrinsic charge carrier concentration N, always remains constant. This explains the decrease in number of holes when donor impurities are added to an intrinsic semiconductor and the decrease in number of free electrons when the no. of holes when acceptor impurities are added . Is there any explanation for the constancy of this intrinsic charge carrier concentration?


title: Ricci tensor of direct product of manifolds

text: Imagine I have a (Lorentzian) manifold with a metric $\left[ {\begin{array}{cc} g_{\mu\nu} &0\\\ 0&g_{mn}\\\ \end{array} } \right]$ Will the Ricci tensor be also block diagonal without any mixing of the two manifolds?


title: What is the name for the whistling "musical" sounds that change stepwise in pitch when a hollow tube is spun like a lasso?

text: You have likely heard those sounds, science museums sometimes sell Flexible plastic tubes you can whirl like a lasso. The air rushing by the end of the tube causes these sounds, which are admitted in discrete "notes". What physical process causes that strange segmenting of the sound into a discrete "scale"? Sounds almost pentatonic. Know a scientific name for that effect?


title: How does Aquarium Aeration work on the Aquatic Habitat aboard the International Space Station?

text: The most recent set of experiments sent to the ISS included 32 fish which are going to be studied in a space aquarium called the Aquatic Habitat. On Earth, an air pump pushes air which flows out of an airstone and travels, in the form of bubbles, up to the surface of the tank. What happens in space? How do they aerate the Aquatic Habitat?


title: From how high could have Felix Baumgartner jumped without disintegrating like a shooting star?

text: Today Felix Baumgartner jumped from 39 kilometres high and reached the earth safely. Just considering friction, from how high can a human jump? I expect that from a certain height, he would have reached a speed so high that he would have burnt when entering dense atmosphere? Some meteoroids disintegrate when they reach the atmosphere, other don't... so I am not sure what would happen for a 100kg human+spacesuit.   Maybe he could jump from arbitrarily high? (notwithstanding digestion/food/drinks, gravity from other stars/planets, lifespan, cold)


title: Ideal four-level Gain Medium (or just any old gain medium)

text: What exactly is happening in a laser's gain cell when the irradiance is greater than the saturation irradiance? Also can someone offer a clear conceptual treatment of the gain-coefficient and its connection to the effective pump rate densities. I apologize if my question is vague, I'm just having trouble seeing their connection. Thanks


title: What theory is the person in this song talking about?

text: What he is talking about sounds interesting to me, and I would like to know more. But I need to know the name first. http://www.youtube.com/watch?v=Ogn5FrVGtAg&hd=1


title: How can I show that an arbitrary wavefunction in a 1D SHO is periodic in time?

text: I want to show that an arbitrary wavefunction $f$ in a one dimensional harmonic potential reproduces itself after a period T up to a phase factor: $f(x,t+T)=Af(x,t)$, $|A|=1$ I am not sure if this is the correct way to begin, i googled the following, I do not know how to obtain it algebraically: $$f(x,t)= B\exp\biggl(\frac{-x^{2}}{2a^{2}}-iw_{0}\frac{t}{2}\biggr)$$ Then $t\rightarrow t+T$ gives: $$\begin{align}f(x,t+T) &= B\exp\biggl(\frac{-x^{2}}{2a^{2}}-iw_{0}\frac{t+T}{2}\biggr)\\\ &= B\exp\biggl(\frac{-x^{2}}{2}-iw_{0}\frac{t}{2}\biggr)\exp\biggl(-iw_{0}\frac{T}{2}\biggr)\\\ &= \exp\biggl(-iw_{0}\frac{T}{2}\biggr)f(x,t)\\\ &= Af(x,t) \Rightarrow A=e^{-iw_{0}\frac{T}{2}}, |A|=1\end{align}$$ Does anybody see if this is correct so far? Thank you.


title: what's the physical meaning of vector Laplacian of Electric field intensity?

text: Could someone explain to me the physical meaning of vector Laplacian of Electric field intensity? Where vector Laplacian means: $$\nabla^2 \mathbf{E} = \nabla(\nabla \cdot \mathbf{E}) - \nabla \times (\nabla \times \mathbf{E})$$ Thanks!


title: Is Force equal to components in different dimensions of Force or distance of those components

text: I'm trying to understand to find my homework problem. Its a simple concept of finding force. I wonder if you have two formulas then is the total Force = force from x + force from y. Or $\sqrt{forceX^2+forceY^2}$. And why. It would make sense to me for it to be the sum of two forces if its not so why not?? I have the actual example: 3.00-kg object is moving in a plane, with its x and y coordinates given by $x=5t^2 - 1$ and $y = 3t^3+2$, where x and y are in meters and t is in seconds. Find the magnitude of the net force acting on this object at $t = 2.00 s$ so I either get 66 by addition of the forces. Or 46.861 by distance formula.. Which one is correct and why. (feel free to use the example i provided or just explain the concept without it. I just wonder which way of finding force is correct and why)


title: Earth's Magnetic Field, Human Brain and Sleep

text: I remember reading somewhere: > when you sleep in a way subjecting your body to cut the geomagnetic field at > right angles, you become highly emotional whereas when it is Parallel, it > cools your mind. Meaning, it's better to sleep with your body/bed aligned in north-south direction. On the contrary, there is a local belief saying "do not sleep with your head towards north direction" there's a mythological story to back it up too! "Birth of Ganesh".   * Earth's Magnetic field is not that strong enough to influence our emotions. Or does it?   * Can this theory be applied in the presence of stronger magnetic fields around? Are there any such fields in day to day presence? just curious to know.


title: What physical phenomenon occured in this experiment?

text: I noticed a strange behavior of my experiment. I have a glass container with a fully saturated solution of CuSO4, which I use to grow CuSO4 crystals, by waiting for the water to evaporate. Recently I noticed that crystals were build above water surface and outside of the container. After some time I noticed that half of my solution ended up on the table and floor creating a big blue pool. It looked almost like the behavior of a super-fluid, which can creep out of its container. I removed the crystals using a paper towel, but after some time the same thing happened. Once the crystals go over the edge and below the water line on the other side, the fluid goes outside quite quickly. Just like if you put an inverted U-pipe into a container. Once you force the liquid to start going, it will continue to flow by itself. ![CuSo4 creeping out of a container](http://i.stack.imgur.com/juqYQ.jpg) Can you explain what physical phenomenon(s) is/are happening here? Is it some sort of capillary action?


title: What does a galaxy orbit?

text: Moons orbit planets, planets orbit stars, and stars orbit the center of a galaxy. So, my question is what does a galaxy orbit? The center of universe? (I know that the universe has no center)


title: Supergravity solution, metric for the total space, and connection

text: In supergravity solutions, one sometimes encounters the case where the manifold may be a bundle over some base space, and one has to write down the explicit metric regarding such bundle. I would like to illustrate this with the following simplified example: As is well known that the three sphere $S^3$ may be viewed as a $S^1$ bundle over base space $S^2$. And it is more interesting, but likewise confusing to me that the metric on $S^3$ may be written as: \begin{equation} ds^2=(d\psi−\cos\theta\ d\phi)^2+d\theta^2+\sin^2\theta\ d\phi^2 \end{equation} here $\cos\theta d\phi$ is exactly the Ricci form connection on the base space $S^2$. It seems here that the non-triviality of the fibre bundle can be seen from the metric, constrasting to the idea that the bundle is always locally trivial. Thus my question is (1) given a specific bundle how to construct the metric on the total space, and how the connection of the base might come into play? (2) given a specific metric, how could we tell if the underlying manifold exhibits a bundle structure? It would be very illustrative if examples may be given in the meantime (for example the Hopf fibration above). It would also be great if reference on the related topic can be provided. I have been searching online, but when come to bundles most reference tends to fall into topological discussions.


title: Convention about delta x in Waves Interference

text: ![enter image description here](http://i.stack.imgur.com/S5YRd.png) They seem to adopt a convention that delta x must always be positive. Otherwise, there is no way to know what the phase difference of the sources actually means. Does a 90° difference mean that source two has a 90° difference compared to source one, or does it have a -90° difference? So, they seem to adopt that convention that delta x must always be positive, which is the only way we could plug and chug an equation and be certain of the answer. Am I right?


title: Determine if an object is in orbit

text: I am planning on doing a space simulation program with full physics and as part of a hack to avoid too many issues with number precision I was going to switch orbiting objects to a calculated orbit (rather than rely on accurate gravity effects to keep it in perfect orbit). For this, I need to know if an object is in orbit at the moment and if it is, switch it across to using the formula in place of the less accurate application of gravity every frame. How would I go about determining if an object is in orbit assuming I know its velocity, distance from centre of planet and the planet's mass. Obviously I'm not looking for someone to do my work for me, but I have been unable to find any reading material online.


title: If I lift a submerged hose over 35 feet above the water's surface, what's in the top foot?

text: I've been arguing with a friend about the whole "perfect vacuum" concept. He and I agree that the most powerful vacuum pump in the world couldn't pump water more than ~34 feet above the surface of a regular elevation lake. I changed the problem slightly, and I'm not sure what would happen.   1. Submerge a hose such that it was completely filled with water   2. Cap off the top end   3. Start pulling it out of the water by a crane   4. Make sure the bottom end stays submerged If you pulled the hose 40 feet out of the water, what would be in the top 6 feet of the hose? A "perfect vacuum"? Gaseous water? Liquid water?


title: Vortex street and Reynolds number

text: Its been stated in Wikipedia regarding Reynolds number(Re) that "laminar flow occurs when Re<2300 and turbulent flow occurs when Re>4000. In another wiki file related to Kármán vortex street it has been stated that "A vortex street will only be observed over a given range of Reynolds numbers (Re), typically above a limiting Re value of about 90. According to Wiki "The term Kármán vortex street (or a von Kármán vortex street) is used in fluid dynamics to describe a repeating pattern of swirling vortices caused by the unsteady separation of flow of a fluid over bluff bodies".Doesn't this "unsteady" refer to turbulence? If so then how is it possible for vortices to have so low value of Re? Is 'more the turbulence higher is the vortex effect' a right statement?


title: Fundamentals of Physics vs Classical Mechanics book and physics undergraduate course

text: There are books like Halliday Fundamental of Physics and there are books titled Classical Mechanics. To me, these two types of books seem to cover the same thing. Is there any difference? (If anyone needs specific example, Halliday vs. Classical Mechanics by John R. Taylor.) Edit: What would be the third year and fourth year undergraduate subject in physics?


title: About solitons, what is the difference between kinks and vortices?

text: I am reading papers about solitons for my small reports, and i could not understand its physical meaning in detail. I know soliton is solitary wave which behaves like particle. And many text they talk about kink, vortices, strings, monopoles, dyons....... I read that solutions of Sine-Gordon equation are kink solutions, and the many solitons govern by the non-linear differential equations (such as KdV) **_What is a Kink in both physical sense? And what is the difference between kinks, vortices and strings?_** From the context of solitons(kinks, vortices, strings, monopoles, dyons ) they are in same section, but i could not understand **_what are the difference between them_**. And lastly, **_what is the difference between solitons and instanton?_**


title: Why is the AdS/CFT approach to superconductors rarely cited in condensed matter publications?

text: Let me put things into perspective by comparing with other applications of string theory. Nowadays review papers written by cosmologists about inflation models often discuss string theory scenarios such as brane inflation, among other non-stringy models. Also you often hear about heavy-ion theorists and experimentalists talking about the AdS/CFT bound on QGP viscosity. However, the AdS/CFT approach to high-Tc superconductors, while hailed as a major new achievement of string theory and heavily cited within string theory circles, seems to have received little citation from the condensed matter community. Hardly any physicist with a traditional background in high-Tc superconductors have made references to AdS/CFT in their publications. I find this situation to be a bit peculiar. Why do you think this is the case? Does this situation have any implication for the long-term promise of this research field? While being a complete outsider, I'd like to venture some guesses to this question. Please tell me whether they sound sensible: 1) Condensed matter physicists already had several theories that more or less explain the essential features of high-Tc, so AdS/CFT is not seen as a gift from God. What they really want is a "recipe" for room-temperature superconductors, but AdS/CFT hasn't provided this. 2) AdS/CFT currently only deals with toy models rather than realistic condensed matter systems, so people whose research interest lies firmly in real-world phenomenology see AdS/CFT as worthless. 3) Due to very different training, condensed matter physicists simply don't understand this research program. What do you think?


title: Statics question: When do I need to find support reactions?

text: We're finding forces (and whether they're in tension or compression) using the Method of Joints and Method of Sections. I don't understand why sometimes it's necessary to find the support reactions, but sometimes it's not. Please help. Thank you!


title: What is the most energy efficient way to boil an egg?

text: Starting with a pot of cold tap water, I want to cook a hard-boiled egg using the minimum amount of energy. Is it more energy efficient to bring a pot to boil first and then put the egg in it, or to put the egg in the pot of cold water first and let it heat up with the water?


title: Can be the ionosphere described as an elecro-magnetic field?

text: I read that the ionosphere can be defined as plasma.   * it has its plasma frequency   * it has its Debye length   * it is overall neutral As it is overall neutral it **does not produce any electric field**. * * * However, as far as I know, a **force field** _is an area in which a particle is an object of force_. A signal (or an electron) that travels through the ionosphere changes its speed and trajectory and these changes are caused by the charged particles in the ionosphere, thus, **the forces are of electro-magnetic nature**. Maybe I should ask: _can be a plasma defined as a electro-magnetic field_ , and at which conditions?


title: Deriving the Hamiltonian density for a free scalar field

text: I'm working through my old notes on QFT (cf. Ref 1) and I'm not quite sure how to approach the derivation of the Hamiltonian density for a free scalar field (question 2.3 on page 19) and the subsequent derivation of the components of the energy-momentum four-vector in terms of the new field operators. Could you point me in the right direction? $${\mathcal{H}~=~\frac{1}{2}\left[\left(\partial_t\phi\right)^{2}+\left(\nabla\phi\right)^{2}+m^{2}\phi^{2}\right]\textrm{.}}$$ References:   1. M Dasgupta, _AN INTRODUCTION TO QUANTUM FIELD THEORY,_ Lecture presented at the School for Experimental High Energy Physics Students Somerville College, Oxford, September 2009.


title: How do I describe two entangled electrons in the same state except for a different spin

text: I am trying to formulate the wave function that describes two entangled electons having the same position but opposite spin. According to the Pauli exclusion principle this should be possible. And when their spin would be equal, the wave function would cancel out and become 0. I tried: $\psi({r}_1,r_2,s_1,s_2)=\psi_a (r_1, s_1) \psi_b (r_2, s_2) - \psi_b (r_1, s_1) \psi_a (r_2, s_2)$. With $r_1=r_2=: r$ and $s_2=s_1 =: s$ (equal spin states) this becomes: $\psi({r}_1,r_2,s_1,s_2)=\psi_a (r, s) \psi_b (r, s) - \psi_b (r, s) \psi_a (r, s)=0$. With $r_1=r_2=: r$ and $s_2=-s_1 =: -s$ (opposite spin states) this becomes: $\psi({r}_1,r_2,s_1,s_2)=\psi_a (r, s) \psi_b (r, -s) - \psi_b (r, s) \psi_a (r, -s)$. But via the spin property: $\psi (r, -s)= -\psi (r, s)$ I get $0$ again:   $\psi({r}_1,r_2,s_1,s_2)= -\psi_a (r, s) \psi_b (r, s) + \psi_b (r, s) \psi_a (r, s) = 0$. What is/are the mistake(s)?


title: Virtual particles and physical laws

text: Recently, I was reading about Hawking Radiation in _A Brief History of Time_. It says that at no point can all the fields be zero and so there's nothing like empty space(quantum fluctuation etc.). Now, the reason mentioned was that virtual(force-carrier) particles cannot have both a precise rate of change and a precise position(Uncertainty Principle). So, my question is : This video says that virtual particles don't follow normal physical laws. So, how can we say that they obey the uncertainty principle?


title: Does the foam on top of boiling maple sap affect the rate of evaporation?

text: This is a serious question from someone engaged in evaporating large quantities of water to turn sap into syrup at this time of year. Probably some background will help. When sap boils vigorously it creates quite a bit of foam, which will overflow the evaporator (incidentally filling the building with a pleasing maple caramel smell as it burns on the side of the evaporator). When the foam gets too high we touch it with a bit of lard and the foam level drops (surface tension - I know). However, it is tempting for me to give a good swipe so that the foam almost disappears (instead of just dropping). The old-timers however contend that I should just reduce the foam to the point where it isn't overflowing any more. They say that it will take longer to boil away the water if I eliminate the foam. I fail to see how the foam will improve evaporation (although it seems to me that it might slow it down). Edit: by request ( @georg ) , a link to the evaporator in question https://sites.google.com/site/lindsayssugarbush/_/rsrc/1240515239201/Home/2005-03-30 --12-25-21.jpg


title: How to explain independence of momentum and energy conservation in elementary terms?

text: I'm trying to explain to someone learning elementary physics (16 year old) that linear momentum and energy are conserved independently. I'm not a professional physicist and haven't tried to explain this stuff for years, and I can't think of any convincing elementary argument to show that this is the case. Does anyone know of an elementary approach to this ? (i.e. one that does _not_ contain the expressions "Lagrangian" and "Noether's Theorem")


title: What properties do you need for building a tower?

text: When I was a boy I used to daydream about building a tower so tall that the top of it would ~~stick out of the top of Earth's atmosphere~~ project into near space. There would perhaps be a zero gravity area in the penthouse where my friends and I could bounce around and play space versions of various earth-based games and sports in most excellent zero-g conditions. Much to my continued disappointment and despite all the technological advances of the last thirty or so years, no one has built such a structure. Can anyone explain the physical limitations/constraints that are preventing someone from realising my fantasy of a 'Space Tower'? UPDATE: This Kickstarter Project seems to be pretty confident ...


title: How coordinate system shifting is related to similarity transformations?

text: I know that coordinate system shifting can be represented using matrices. But how exactly are similarity transformations related to coordinate shifts ?


title: Where's earths death bulge, destroying everything in it's path?

text: I was watching a BBC documentary on space last night. It was talking about gravity, and it said that the reason we only ever see one side of the moon, is because the earths gravity is strong enough to actually stretch the moon into a longer shape, creating a 7m bulge on the moon. This bulge travelled around the moon. It was described on the documentary as something along the lines of, if you were there, you would see the ground make a 7m high wave as it rotated. The bulge then acted a brake, and the moons spin slowly ground to a halt until where we are today when it's spin is very very slow so it looks like we only see one side. So where is this giant death bulge tidal wave on earth? Shouldn't we been seeing the ground rising up a few meters in relation to the suns position, demolishing and killing everything in its path? I assume the documentary grossly simplified what actually happened, or I misunderstood! I'm a physics noob, please go easy on me, chances are I'm not going to understand any formulas that use anything but addition or subtraction.


title: Coercivity of a ferromagnetic material?

text: I understand that coercivity is the field/force required to demagnetize/magnetize a ferromagnetic material. What if we had two opposite magnetic fields of different strengths values H acting on the same ferromagnet what would happen? **H1** = 50 Oe **H2** = 10 Oe The coercivity of the material is 0.5 Oe, what would happen? Will it stay magnetized to the stronger field(H1)? Or **magnetization = 0**?


title: Equation 7.22 in Peskin & Schroeder: writing the Fourier transform of a two-point function as a series of 1PI diagrams

text: In Peskin and Schroeder's QFT book, on page 219, there is the following equation:![enter image description here](http://i.stack.imgur.com/nC46a.png) The heading to the equation is: "The Fourier transform of the two-point function can now be written as". Could someone help me formally prove this formula? I don't really get why the perturbation expansion decomposes into 1PI's.


title: If only one slit is observed in the Double Slit experiment, will the unobserved slit produce an interference pattern?

text: I am having a difficult time solving this. Say that electrons are emitted from a source S at a very slow rate. If both slits S1 and S2 are observed, we would have roughly 50% probability of detecting an electron at one of the two slits. The interference pattern is lost and the intensity distribution will appear as the sum of two individual sources: I = I1 + I2. But what if only one slit (S1) is observed? The observed slit (S1) will appear to produce a normal distribution, but what about the unobserved slit? This experiment has been performed with individual electrons, so we know that if both S1 and S2 are unobserved the intensity distribution contains an oscillating term for _each_ electron. Does concluding that an electron must have passed through the unobserved slit count as an observation, and therefore destroy the interference pattern? Edit: changed the source to electrons


title: Why is acceleration expressed as m/s/s?

text: I'm a philosophy student (I, regrettably, don't know calculus or much physics). Last year I spent some time learning how work, power, speed, velocity, energy, force, and acceleration relate. But I was never able to fit my understanding of acceleration into my understanding of the world. I think my biggest challenge was understanding how you can have (m/s)/s. I.e. the 'per-second per-second' part doesn't make sense to me — and how that represents an increase in speed. Does it express, _every second a thing travels X meters/second faster_? That's my best guess, but I see a lot of problems with that guess, so I presume that it's incorrect. In words, why is acceleration expressed as (m/s)/s? How does that expression relate (if at all) to the everyday notion of acceleration?


title: Infinite well particle subject to additional time dep. potential

text: I am asked to find the wavefunction of the particle in a well subject to an additional potential $$V(x,t)=\frac{\pi x \hbar}{L}\delta(t).$$ I have already solved that $$\psi(x,t)=\exp\left(\frac{-i}{\hbar}\int^t_0H(t')dt'\right)\psi(x,0).$$ I understand this equation, but I am unsure of how to treat the delta potential because 0 is not included. I was thinking of integrating from -$\epsilon$ to $ \epsilon$. And we assume that it is in ground state for t<0\. I assume this means $$\psi(x,0)=\sqrt{2/L}\sin{(\pi x/L}).$$ Any input is appreciated! (I saw some post regarding perturbation theory, but we have not covered anything like that)


title: Power required to keep a freezer at constant temperature

text: Alright, I'm completely stuck on this and so is everyone else I've asked. A freezer has a temperature Tc = -23℃. The air in the kitchen has a temperature Th = 27℃. The freezer in not perfectly insulated and heat leaks through the walls of the freezer at a rate of 50 W. Find the power of the motor that is needed to maintain the temperature in the freezer. The answer to this question is 10 W. I don't understand how this is possible - 50 W is literally J/s in heat. To keep the temperature the same, the amount of heat in the freezer has to be the same. If the motor only removes heat at a rate of 10 J/s, then 40 J is being added to the fridge every second, and the temperature changes in response. Is there something I'm missing?


title: Under what conditions is a vector-spinor gamma trace free

text: I came upon the concept of irreducible vector-spinors while trying to simplify an expression involving the gravitino field. It is claimed that an _irredicible_ vector-spinor is gamma-traceless, i.e. $$ \Gamma^M \psi_{\alpha M} = 0.$$ Are there conditions on the irreducibility besides the above equation? When can I use the relation? Does it follow from more general group-theoretic relations or is it a rehash of the equations of motion?


title: When is this integral zero?

text: I have a particle with total energy $E$ confined in a potential $$U(x) = -\frac{\cos^4x}{2} - m \cos x - f \sin x. $$ The constants $f$ and $m$ are both in the range (-2,2). The energy is such that the particle is in a bound state, with turning points $x_1$ and $x_2$ on either side of the origin. I want to make the following statement: > If and only if $f = 0$, $$\int_{x_1}^{x_2} \frac{\sin x dx}{\sqrt{E - U(x)}} > = 0.$$ One direction of the implication is simple. If $f=0$, the potential is even, and $x_1 = -x_2$. Since the integrand is odd, this integral obviously vanishes. But I'm struggling with showing that the integral is zero only if $f = 0$. Is this true? My numerical solutions to the equations of motion seem to indicate so. EDIT: I evaluated the integral numerically in Mathematica with $E = -2,~m = 2$, and for a wide range of values of $f$. The plot below shows the value of the integral on the vertical axis, and the input $f$ value on the horizontal. It looks like any proof relying on monotonicity will not work. ![m = 2, E = -2](http://i.stack.imgur.com/Wyu4b.png)


title: Is there a physical system whose phase space is the torus?

text: **NOTE.** This is not a question about mathematics and in particular it's not a question about whether one can endow the torus with a symplectic structure. In an answer to the question What kind of manifold can be the phase space of a Hamiltonian system? I claimed that there exist (in a mathematical sense), Hamiltonian systems on the torus (and in fact on higher genus surfaces as well). However, when pressed to come up with a physical system in the real world (even an idealized one) whose dynamics could be modeled as a Hamiltonian system on the torus, I could not think of one. Does such a system exist? I would even be satisfied with a non-classical system which can somehow effectively be described by a Hamiltonian system on the torus, although I'm not sure that the OP of the other question I linked to above would be.


title: Relationship between power and max. speed

text: I'm talking about the maximum speed if let's say I have a car with the power $P = 1000 \text{W}$ and a force of friction of $5 \mbox{N}$ acting in the opposite direction. After some googling I found that the maximum speed is given by $P=Fv$, where $P$ is the power, $F$ is the force, and $v$ is the velocity. I understand that $W= Fs$ and that $P = W/t$ and $s/t$ is $v$, so yes I understand where the equation comes from, however wouldn't this be the average speed and not the maximum speed? And the force of friction is not the force that's doing the work, so why is it used in the equation? I hope I've made my question clear enough, thank you in advance!


title: How to exchange light and matter?

text: A while ago an experiment demonstrated that it is possible to stop a light pulse in a supercooled sodium cloud, store the data contained within it, and totally extinguish it, only to reincarnate the pulse in another cloud two- tenths of a millimeter away. ![light](http://i.stack.imgur.com/MzZxz.gif) Most of the papers discussing the details of this experiment are behind paywalls. I understand the setup that the light pulse can be revived, and its information transferred between the two clouds of sodium atoms, by converting the original optical pulse into a traveling matter wave which is an exact matter copy of the original pulse, traveling at a leisurely 200 meters per hour. The matter pulse is readily converted back into light when it enters the second of the supercooled clouds -- known as Bose-Einstein condensates -- and is illuminated with a control laser. The key as I understand it is the Bose-Einstein condensates (BEC) become phase-locked. The light drives a controllable number of the condensate's roughly 1.8 million sodium atoms to enter into quantum superposition states with a lower-energy component that stays put and a higher-energy component that travels between the two BECs. The amplitude and phase of the light pulse stopped and extinguished in the first cloud are imprinted in this traveling component and transferred to the second cloud, where the recaptured information can recreate the original light pulse. How does matter become "phased locked" to light and how do all these atoms contain the "information" of the light to reproduce it? And what is this talk of separating lower/higher energy components?


title: Concentration of photons: How many can you slow down in a given unit of space?

text: Being able to slow down light to the speed of a bicicle as shown in this vid: http://www.youtube.com/watch?v=EK6HxdUQm5s Led to a question: Given a unit of space e.g. 1cm^3, and then you slowed down light as it enters that unit of space, how many photons can you fit in that space? Is there a maximum photon concentration limit? ![enter image description here](http://i.stack.imgur.com/a0bo8.png)


title: Some questions about the edge states for time-reversal invariant topological superconductors?

text: Stimulated by my some recent calculations on edge states(ES) for time-reversal invariant(TRI) topological superconductors(TS) as well as many questions concerning the "edge states" in Physics StackExchange(e.g. Is edge state of topological insulator really robust？, Chiral edge state as topological properity of bulk state, What is the mathematical reason for topological edge states?), I get some questions about the gapless ES for TS and I'm puzzled by these problems. I studied two examples, say **A** and **B** , of TRI triplet p-wave supercondutors with full gap both on a 2D lattice, and my calculations of the $Z_2$ bulk invarints show that they are both _odd_ numbers, by definition, both **A** and **B** are TRI TS. Then I numerically calculated the ES for **A** and **B** both on stripy geometries with open boundary conditions(OBC) along one direction, now here come the puzzles: For **A,** no matter how I change the width of the strip, no matter how I change the shapes of the two edges, I just **can not** find the gapless ES "crossing" at the TRI momentum points( $k=0,\pi$ in my calculations), so my **first question is:** Is the _existence_ of gapless ES "crossing" at the TRI momentum points really a **necessary condition** for TRI TS ? Now I personally tend to think it's just a **sufficient not necessary condition** for TRI TS. Moreover, the so called _bulk-boundary correspondence_ can not be proved very generally(Counterexamples to the bulk-boundary correspondence (topological insulators)), so I think there may exist some special examples violating this. Regarding **B** , I fortunately found _one_ pair of gapless ES "crossing" at the TRI momentum $k=\pi$(or $-\pi$). But according to the numerical results, the dispersion near $k=\pi$ seems more like quadratic$(\propto k^2)$ rather than the usual _linear one_ . So my **second question is:** Is there any possibilty of the _quadratic_ dispersion of gapless ES for TRI TS ? Thanks in advance.


title: what is the magnetic quadrupole operator?

text: To find magnetic or electrical moments in quantum theory we must calculate the expectation value of an appropriate operator. the dipoles operator are similar and is easy to find but the magnetic quarupole moment, I couldn't find.


title: Can we measure "wavefunction" of quantum particles?

text: We know that there is uncertainty principle, so question: can we ever measure wavefunction of particles? I do not think this is possible, but I am not sure. I guess that everything is probabilistic. (that's why, I believe, we have sigma level when we say we discovered some particles...)


title: Combined speed of Earth in Space is 1.5 million mph - how much slower is time for Earthlings as a result?

text: Theres a problem for intergalactic astronauts which is finding their way back to Earth. Combining all the rotational speeds, we are spinning and orbiting the sun, in our solar system which is spinning in the Milky Way, which is spinning in a large cluster, which in turn spins in a super cluster. Combining all the speed of the rotational forces equals 1.5 million mph. Said Intergalactic Astronaut would have a job to find where the Earth is if they flew into space and out of the super cluster. But my question is this, the astronaut HAS managed to find a ship to leave the Earth, the solar system, the Milky Way, the Large cluster, the Super cluster and is at relative to us, is motionless in space, observing us. How much faster is time for the astronaut? Or how much slower is time for us on Earth, given the speed of Earth and gravitational effect on time.


title: What is the magnetic quadrupole moment of a nucleus in cylindrical coordinates?

text: What is the magnetic quadruple moment of a nuclei in cylindrical coordinates? The quadrupole moment of a nucleus is zero in spherical coordinates but in the cylindrical coordinates it can't be vanished. In cylindrical coordinates we are not able to use `jlms` and its very difficult to solve the integral as the expectation value of the magnetic quadrupole operator.


title: Why speed of Mangalyan is constant?

text: I have learned in school regarding the relationship between speed,acceleration and distance is $$v^2-u^2=2as $$ The initial velocity of India's Mars Orbiter Mission, MOM, is 32 km/s source and now the speed of MOM is 34.99 km/s source If we assume the acceleration is constant and put the values in the above formula then the final speed is never 34.99 km/s. So I wanted to know why MOM does not follow the above rule.


title: two distant galaxies seen from earth

text: From the Earth If we observe two galaxies that are diametrically opposed and each 1000ly far from the earth. the separation distance between the galaxies will be 20000ly? really question is: if the galaxies were separated from Earth 10Gly. then the separation distance between galaxies would 20Gyr?


title: Good reading on the Keldysh formalism

text: I'd like some suggestions for good reading materials on the Keldysh formalism in a condensed matter context. I'm familiar with the imaginary time, coherent state, and path integral formalisms, but lately I've been seeing Keldysh more and more in papers. My understanding is that it is superior to the imaginary time formalism at least in that one can evaluate non-equilibrium expectations.


title: Finite difference formulation of the heat equation with thermal conductivity in 1D

text: This may seem trivial, but I'm having some trouble deriving the finite difference form of the heat equation with a thermal conductivity function $a(x)$ depending on $x$: $$\frac{\partial u(x, t)}{\partial t} = \frac{\partial }{\partial x}[a(x)\frac{\partial u(x, t)}{\partial x}]$$ I find some trouble deriving the right hand side. Let $h$ be the space step, and $x_{i−1}=x_i−h$ and $x_{i+1}=x_i+h$. I want to evaluate the right hand side on node $i$. If I take centered differences I get: $$\begin{align*}\left\\{\frac{\partial}{\partial x}\left[ a(x)\frac{\partial \, u(x, t)}{\partial x}\right]\right\\}_i &= \frac{\left[a(x)\frac{\partial\, u(x, t)}{\partial x}\right]_{i+1/2} - \left[a(x)\frac{\partial \, u(x, t)}{\partial x}\right]_{i-1/2}}{h} \\\ &= \frac{a_{i+1/2}\left[\frac{u_{i+1}-u_{i}}{h}\right] - a_{i-1/2}\left[\frac{u_{i}-u_{i-1}}{h}\right]}{h} \end{align*}$$ Is this ok? The conductivities should be in between the nodes I'm calculating? (intuitively it seems this is reasonable). Is there a way, for example by using forward or backward differences, to use conductivity nodes that overlap with the nodes I'm calculating (I calculate only the interger indexed nodes)? Can I mix a centered difference approach for one derivative and a backward or forward approach for the other one? I would really appreciate any hint. Thanks in advance, Federico


title: How did enough material from other dying stars accumulate to start our sun and planets?

text: How far apart do scientists estimate was/were the dying star(s) that supplied the elements that comprise our sun, planet, and us? With stars so far apart and expansion of space (as I understand it) carrying things further away still, it would seem to be a low probability occurrence for sufficient quantities of elements blown out from across many light years to accumulate to birth a star system. And another related question, it would seem that when a star went supernova and blew heavier elements in all directions, that would result in mass densities for future star nurseries far away from the dead parent that were much lower than what the parent had, so won't favorable conditions for star births monotonically decrease?


title: Theory that gets rid of gravitional wave

text: Is there any theory that gets rid of gravitational wave and still matches with all correct predictions made by standard physics theories? (e.g. General Relativity)


title: Basic vertical projectile motion

text: A ball is thrown from the top of a tower with a certain initial speed. It's downward acceleration is g. But why? the initial force makes no difference? and, if not, if I get a revolver and shoot down, the bullet's downward acceleration equals g too?


When a Hamiltonian operator apply to a wavefunction, how could we write the hamiltonian as, $$H \psi = (E_n-\hbar \omega_0) \psi \ \ ? $$ > Is this because $E_n= H+ \hbar \omega_0$? where $\omega_0$ is the angular frequency.


title: Modelling of nuclear motions (Classification) after invoking the BO approximation

text: I know that after invoking the Born-Oppenheimer approximation, the nuclei will move on the adiabatic potential provided by the electronic energy (also called potential energy surface (PES)). Such nuclear motions, can be described following quantum-mechanical, classical or hybrids methods on a fitted potential energy surface (PES) or when we want to avoid any analytical representation or interpolation of the PES, we can resort in “on the fly” calculation of the potential. My question is, what are those quantum, classical or hybrid theories/methods that are in connection with an statistical description? and how can i make a general classification o each of them?, at least the most important methods.


title: Can a single atom explode?

text: Does the definition of an atomic explosion require the interaction of two or more atoms, or can a single atom be the source of an explosion? Another way of phrasing the same question. Can there be an atomic explosion involving a single atom without any other atom involved in the start or chain reaction of the explosion.


title: Can someone explain effects of time dilation on expansion of universe

text: I'm not a physics guy, but this is something I have always wondered about. If a clock flown by an object of high mass runs slower, what is the effect of this in regards to the density of the universe. It seems to me that as the universe expands, it would lose density and therefore clocks would run faster as the universe expands. Is this correct? And if so, could this create the perception of an accelerating expansion in a hypothetical universe of constant expansion. I want to understand the things I believe in, and I believe in the big bang. It seems that if the universe were a singularity at some point that a nearby clock would be running infinitely slow. Would the first few seconds of the big bang be perceived as slow, maybe from a distant view, despite much happening in the first few seconds? (I'm not sure how the human brain perceives time). Forgive my complete lack of understanding. I'm just curious if and how this is accounted for in the models.


Grilling in the park today I started to wonder just how implausible a portable battery-powered grill would be. There are of course a lot of parameters in this question, so let's narrow it down for now: How much energy does it take to cook a steak? To be a bit more specific let's say a 2.5cm thick steak to a core temperature of 63°C. And since we probably need one more measurement, let's say it's circular with a radius of 5cm. Everything from numbers for cylindrical steaks in a vacuum to precisely the numbers that are needed to get an actual tasty steak would be interesting.


title: Finding the Shape of a Hanging Massive Wire

text: This came from a physics wave book. I have a static, massive wire on the x-axis, with a y-displacement due to a force per unit length $F_y$. I start with the equation $F_y = T_0\frac{\partial^2(\eta)}{\partial(x)^2}$, where $\eta$ is a small displacement in the y direction and $T_0$, with units of force, is the tension supporting the wire. I need to find how twire is shaped, so I need to integrate both sides twice with respect to x. $$F_y\iint{{dx}^2} = T_0\iint{\frac{\partial^2(\eta)}{\partial(x)^2}{dx^2}}$$ Since these are both indefinite integrals this is rather easy and I get $F_y x^2 = T_o\eta +Cx+D$. But if I were to attempt definite integrals, how exactly would I proceed? If both are definite... $$F_y\int_{x_1}^{x_2}\int_{x_1}^{x_2}{{dx}^2} = T_0\int_{x_1}^{x_2}\int_{x_1}^{x_2}\frac{\partial^2(\eta)}{\partial(x)^2}{dx^2}$$ Then this is how I think the fundamental theorem of calculus would apply $$F_y(x_2-x_1)\int_{x_1}^{x_2}{dx} = T_0\int_{x_1}^{x_2}\left(\frac{\partial(\eta(x_2))}{\partial(x)}-\frac{\partial(\eta(x_1))}{\partial(x)}\right){dx}$$ But this seems very wrong. Having $\eta$ as a function of only the endpoints seems odd. Intuitively I would indefinite integrate this once, then definite integrate the second time to limit it. How do you apply the F.T.C here? Also I've seen from other questions that the curve is not in fact parabolic as my first equation suggests, what assumption am I making that threw this off? Any help is appreciated, I feel this hit's on a key concept I missed in calculus.


title: Blackbody and standing waves

text: I'm reading articles about black body radiation and why classical mechanics fails to explain it. My question is: Why do EM waves have to be standing wave in a cavity?


title: Spring oscillations and waves

text: Consider a block of mass $m$ attached to a spring. Let it oscillate at a frequency $f$. Now each part of the spring is in SHM. so this means a wave is propagating through this spring.bCan this wave be reflected at the fixed end of the spring resulting in the formation of standing waves?


title: A Derivation of Ehrenfest's Theorem in a particular case

text: What are the missing lines in the integration? $$\frac{\text d \langle {p} \rangle}{ \text{d} t} $$ $$= \frac{\text d}{\text d t} \int\limits_{-\infty}^{\infty} \Psi^* \left( \frac{\hbar}{i}\frac{\partial}{\partial x} \right) \Psi ~\text d x$$ $$= \frac{\hbar}{i}\int\frac{\partial }{\partial t} \left(\Psi^*\frac{\partial\Psi}{\partial x}\right)~\text d x $$ $$= \frac{\hbar}{i}\int \frac{\partial \Psi^*}{\partial t}\frac{\partial \Psi}{\partial x} +\Psi^*\frac{\partial }{\partial x}\frac{\partial \Psi }{ \partial t} ~\text {d} x$$ $$= \frac{\hbar}{i}\int\left( -\frac{i\hbar}{2m}\frac{\partial^2 \Psi^*}{\partial x^2}+\frac{i}{\hbar} V\Psi^*\right)\frac{\partial \Psi}{\partial x}+\Psi^* \frac{\partial}{\partial x} \left(\frac{i\hbar}{2m} \frac{\partial^2 \Psi }{\partial x^2} -\frac{i}{\hbar}V\Psi \right)~\text d x$$ $$=\int\left(V\Psi^*-\frac{\hbar^2}{2m}\frac{\partial^2 \Psi^*}{\partial x^2}\right) \frac{\partial \Psi}{\partial x}+\Psi^*\frac{\partial}{\partial x}\left(\frac{\hbar^2}{2m}\frac{\partial^2 \Psi}{\partial x^2}-V\Psi\right)\text d x$$ $$=\left. \left(V\Psi^*-\frac{\hbar^2}{2m}\frac{\partial^2 \Psi^*}{ \partial x^2} \right) \Psi \right|_{-\infty}^{\infty}-\int\left(\frac{\partial}{\partial x} (V\Psi^*)-\frac{\hbar^2}{2m}\frac{\partial^3 \Psi^*}{ \partial x^3}\right)\Psi \text d x+ \left.\Psi^*\left(\frac{\hbar^2}{2m}\frac{\partial^2 \Psi}{\partial x^2} - V\Psi\right)\right|_{-\infty}^\infty-\int\frac{\partial\Psi^*}{\partial x} \left( \frac{\hbar^2}{2m}\frac{\partial ^2 \Psi}{\partial x^2} - V \Psi \right)\text d x$$ $$=0 + \int\left(\frac{\hbar^2}{2m}\frac{\partial^3 \Psi^*}{ \partial x^3}-\frac{\partial}{\partial x} (V\Psi^*)\right) \Psi \text d x+0+\int\frac{\partial\Psi^*}{\partial x}\left(V\Psi - \frac{\hbar^2}{2m}\frac{\partial^2 \Psi}{\partial x^2}\right) \text d x$$ $$=\int\frac{\hbar^2}{2m} \left(\frac{\partial^3\Psi^*}{\partial x^3}\Psi -\frac{\partial\Psi^*}{\partial x} \frac{\partial^2\Psi}{\partial x^2}\right)+\frac{\partial\Psi^*}{\partial x}(V\Psi )-\frac{\partial}{\partial x}(V\Psi^*)\Psi\text d x $$ $$\vdots$$ $$=\int \frac{\hbar^2}{2m}\left( \Psi^* \frac{\partial^3 \Psi}{\partial x^3} -\frac {\partial^2 \Psi^*}{\partial x^2} \frac{\partial \Psi}{\partial x} \right)+\left( V \Psi^* \frac{\partial \Psi}{\partial x}-\Psi^* \frac{\partial}{\partial x} (V \Psi) \right)~\text d x$$ $$\vdots $$ $$= \int \left( V \Psi^* \frac{\partial \Psi}{\partial x}- \Psi^* \frac{\partial V}{\partial x} \Psi-\Psi^*V \frac{\partial \Psi}{\partial x} \right)~\text d x$$ $$= \int\limits_{-\infty}^{\infty} -\Psi^* \frac{\partial V}{\partial x} \Psi ~\text {d} x$$ $$=\left\langle - \frac{ \partial V }{\partial x} \right\rangle $$


title: Does the phase space (configuration and momentum space) of particles have a Euclidean norm? Does it have a useful meaning of "distance"?

text: Often in engineering physics, different vector spaces are used to visualize the trajectories (evolution) of systems. An example being the 6n dimensional phase space of n particles. It is not very clear to me if this space is Euclidean or has a meaning of distance. I understand the 2-norm when the physical quantities are the same (e.g., only positions or only momenta) but not when a vector has both. However even this seems to be a problem with some other spaces. For example if I consider the vector space of chemical composition of chemical system (mole numbers of all chemical species). I don't understand what "length" of a vector would signify? Thanks


title: 2 Time Dimensions

text: Our universe has 4 dimensions, 3 spacial and 1 temporal. We can add another spacial dimension and use math to explore it. I cannot think of any reason why we cannot add add another temporal dimension in the same way. So, what would a universe with 2 temporal dimensions be like? These are some thought I had about physics in 2 temporal dimensions. With one spacial dimension, forwards is the only movement possible. With two spacial dimensions, movement in many different ways is possible. It would follow then that with two temporal dimensions time travel in many different 'directions' would be possible. The formula for velocity is displacement/time. With two times, how would velocity be effected? What would happen to light, which must travel at a constant speed? Frankly, I have no idea a universe with 2 times is like, but I very much would like to know.


title: Conjectures that have been disproved with extremely large counterexamples

text: I would like to migrate this Math Question into physics. The question is:   * Are there conjectures in Physics which have been disproved with extremely large counterexamples? If yes, i would like to know some of them.


title: Do singularities have a "real" as opposed to mathematical or idealized existence?

text: I was thinking of, for example a Schwarzchild metric at r=0, i.e. the gravitational singularity, a point of infinite density. I realise that there are different types of singularities--timelike, spacelike, co-ordinate singularities etc. In a short discussion with Lubos, I was a bit surprised when I assumed they are idealized and I believe he feels they exist. I am not a string theorist, so am not familiar with how singularities are dealt with in it. In GR, I know the Penrose-Hawking singularity theorems, but I also know that Hawking has introduced his no-boundary, imaginary time model for the Big Bang, eliminating the need for that singularity. Are cosmic strings and other topological defects singularities or approximations of them (if they exist). In what sense does a singularity exist in our universe? --as a real entity, as a mathematical or asymptotic idealization, as a pathology in equations to be renormalized or otherwise ignored, as not real as in LQG, or as real in Max Tegmark's over-the-top "all mathematical structures are real"?


title: Laser frequency tripling

text: I've heard that common way to get 354nm laser is to triple 1064nm Nd:YAG one. But how to make non-linear crystal to do tripling instead of doubling? What are the best crystals for tripling and what's the efficiency?


title: Defining a Riemannian manifold - made easy?

text: In the context of GTR spacetime, I'm trying to get the basic idea of a Riemannian manifold clear in my mind. Apologies for the longwindedness. Question 1. Is this a reasonable, simplified summary of the steps needed to define such a manifold?   1. Start with an amorphous collection of points.   2. Find the minimum number of parameters needed to locate each point in space. That number will be the dimension of the manifold.   3. Construct a coordinate system to locate each point in space. I may need more than one coordinate system to to cover the whole manifold. I need a “good” coordinate system so that each point is uniquely described.   4. Decide on a metric that gives the infinitesimal distance between two adjacent point. The metric need not be the same all over the manifold (as $g_{\mu\upsilon}$ isn't in spacetime). In that case I would have a metric tensor field defined over the manifold.   5. The manifold has to be continuous and differentiable (not quite sure what this means, but I assume it's the obvious - I have to be able to differentiate it everywhere).   6. I can now start constructing scalar fields, covariant and contravariant vectors, tensors etc. Question 2. Assuming I do all the above with a flat plane (a table top for example), using Cartesian coordinates. Can I impose any kind of metric to my new “table top” manifold? Could I just make a weird one up: $$\left(\begin{array}{cc} x^{2} & \sin xy\\\ x+y & 13y \end{array}\right)$$ for example, instead of the Euclidean $$\left(\begin{array}{cc} 1 & 0\\\ 0 & 1 \end{array}\right)$$ Thank you.


title: Does car lose kinetic energy when turning?

text: I am writing simple car simulation. Assume non friction, then in straight line the car doesn't lose speed. But what if the car is turning, there should be some kinetic energy loses to change the direction of the car, so I should subtract some speed based on changing direction, isn't it right? Deeper explanation is welcome.


Can a functional derivative be calculated if we have a function of more than one variable? The functional derivative of, for example, $F[b(x)]=e^{\int_0^{x'} dx a(x,y) b(x)}$ is \begin{equation} \frac{\delta F[b(x)]}{\delta b(z)} = a(z,y) e^{\int_0^{x'} dx a(x,y) b(x)} \end{equation} But what about if there it's a functional of more than one variable? - $F[b(x,k)]=e^{\int_0^{x'} \int_0^p dx dk a(x,y,k) b(x,k) }$? Can we write the following? \begin{equation} \frac{\delta F[b(x,k)]}{\delta b(z,k)} = a(z,y,k)e^{\int_0^{x'} \int_0^p dx dk a(x,y,k) b(x,k )} \end{equation} **Edit:** I just want to note that I've been told that you **cannot** do this by a lecturer. Therefore, is the answer is yes (or no!), please give some details! (I wasn't convinced that he was correct).


title: How can I actually use the BB84 protocol?

text: I have read how the BB84 protocol could work. However, I couldn't see a way to implement it in the laboratory, so that I could make a demonstration in the university's quantum optics laboratory. From what I have read, I think that a way to demonstrate this is to somehow shoot only one photon at a time, each time with a prepared setting for the sender and receiver polarization, but it seems very unpractical.


title: Schiff Equation in a polar orbit

text: I am aware that the formula for Schiff Equation in used to determine frame- dragging is: $$\boldsymbol{\Omega} = \frac{GI}{c^2r^3}\left( \frac{3(\boldsymbol{\omega}\cdot \boldsymbol{r})\boldsymbol{r}}{r^2}-\boldsymbol{\omega} \right).$$ But why does this reduce to: $$\boldsymbol{\Omega} = \frac{GI\boldsymbol{\omega}}{c^2r^3}\frac{\int_0^{2\pi}(3\cos^2\theta - 1)\text{d}\theta}{\int_0^{2\pi}\text{d}\theta} = \frac{GI\boldsymbol{\omega}}{2c^2r^3}$$ in a polar orbit?


title: Alejandro Rivero's correspondence: diquarks and mesons as superpartners of quarks and leptons

text: The idea of “hadronic supersymmetry” originated in the mid-1960s and derives from the observation that baryons and mesons have similar Regge slopes, as if antiquarks and diquarks are superpartners. This is most efficiently explained by supposing a QCD string model of baryons in which a diquark substitutes for an antiquark at one end of the string, but there are superstring models in which the baryon-meson relationship is a genuine supersymmetry. In 2005, Alejandro Rivero proposed that mesons are to _leptons_ as diquarks are to quarks. If the top quark is excluded from consideration, as too heavy to hadronize, then one has enough quark-antiquark pairings to match the electromagnetic charges of all the leptons, with a few leftover u,c pairings with charge ±4/3. The construction is a little complicated, but the possibility that supersymmetry is already right in front of us is so amazing that it just cries out to be investigated. I have an ongoing discussion with Rivero, about whether his correspondence might be realized in a preon model, or a "partially composite" model, or a sophisticated string-theory construction. Meanwhile, I'd like to know, **is there some reason why this can't be realized?** The closest thing to such a reason that I've seen is found in comment #11 in that thread: "For a SUSY theory, not just the spectrum must be supersymmetric, but also the interactions between the particles." So far, all Rivero has is the observation that some quantum numbers of diquarks/mesons can be correlated with some quantum numbers of fundamental quarks/leptons, just as if they were elements of a single superfield. I suspect (but I'm such a susy newbie that I can't even confirm this) that you could realize the correspondence in a QCD-like supersymmetric theory, if it was first expressed in terms of meson and diquark variables, and then supersymmetry was completely broken (see comments #40 through #44), but that this would somehow be trivial. Nonetheless, I think the possibility that supersymmetry is already right in front of us would be important enough, that someone should investigate even such a "trivial" realization of Rivero's correspondence, with a view to understanding (i) whether such a "hard" form of susy breaking might realistically occur (ii) how it would affect the various roles and problems associated with supersymmetry in contemporary physical thought (protect the Higgs mass, supply dark matter candidates; technical model-building issues like "the supersymmetric flavor problem, the gaugino mass problem, the supersymmetric CP problem, and the mu-problem"...).


title: Easy proof of Noether's theorem?

text: Where could I find an easy proof of Noether's theorem? I mean I know that the variation must be $ 0=\delta S = (EULER-LAGRANGE)+ (CONSERVED\, \, \, CURRENT) $ for the case of a particle $q(t)$. I know how to obtain it by I have doubts for the case of fields $ \phi (x) $, any hint? I have checked several books but I cannot find any easy proof of Noether's theorem anywhere; they use too complicated methods.


title: Can there exist a Wave which changes the quantum states of particles?

text: i'm a high school student and i was reading about electromagnetic waves and how they transport energy and that the electric and magnetic fields sustain each other. I have also read about longitudinal waves which travel due to contractions and rarefactions and also transport energy by changing the relative positions of the particles of the medium. So,i was wondering whether there could exist a wave which travels 'through' an atom or sub-atomic particle which would traverse by changing the quantum states of the said particles? For example, a particle may oscillate between different quantum states and the particles surrounding it may do so too but in such a way that a 'wave' appears to pass through them.


title: Is there a way to increase the speed of light itself?

text: When light enters materials it slows down due to its refractive index (due to absorbing and re-emission of photons). But, is there a way to increase the speed of light itself? Can there be some material which would increase rather than decrease the speed of light?


title: Homework-Problem solved using Force but unsolved using Energy considerations

text: > A chain of mass $M$ and length $l$ is suspended vertically with its lowest > end touching a table. The chain is released and falls onto the table. What > is the force exerted by falling part of the chain on the table?(Neglect the > size of individual links) Here's what I did. Momentum of a size $dx$ of chain, which was initially at a height $x$ from the bottom of the chain, when it is just touching the table is given by $$p = M\sqrt{2gx}\frac{dx}{l}$$ This momentum is lost in time $$t = \frac{dx}{\sqrt{2gx}}$$ Force exerted by table on chain and chain on table is given by $$F = \frac{p}{t} = 2Mg\frac{x}{l}$$ As it turns out, this is the **right answer** But, the author asks to solve this using Work-Energy Theorem. Here's what I did: Work done by table on chain: $$F.dx$$ Energy of part of chain lost to this work done: $$Mgx\frac{dx}{l}$$ Equating the above two equations, I get $$F = Mg\frac{x}{l}$$ This is off by the right answer by a factor of two. **What goes wrong when I use Work-Energy Theorem, although I get the right answer when I use momentum?**


title: Two cylinders with same volume but different dimensions

text: If there are two cylinders (A and B) both with the same volume. B's radius is half of A's, so the length of B must be 4 ($2^2$) times that of A. The uncertainty for the radius of A is the same as the for the radius of B, and the uncertainty is the same for the lengths. But the uncertainty of the volume isn't necessarily the same as for the radius) Which cylinder A or B would have the greatest uncertainty for volume? Or would they be the same because as the percentage uncertainty for radius for B is larger than A but the percentage uncertainty for length B is less that for A? Do the uncertainties need to be know?


title: Flammability and statistical mechanics

text: I am wondering to what extent the flammability can be predicted from the statistical properties of an ensemble. Given the partition function of an ensemble, can we in principle predict this property? Is there a chance flammability is a classical property of matter?


title: Measure density with the help of buoyancy

text: I am trying to derive a formula to calculate the density of a irregulary shaped object. I can measure the ( **false** ) weight of the object in pure air (of known density), and the ( **false** ) weight of the object in water (of known density). I cannot measure (directly) the volume of the object, nor its real weight (in vaccum). I tried this approach: $$G_{air} = G_{vaccum} - B_{air} $$ $$G_{water} = G_{vaccum} - B_{water} $$ where $B$ is the buoyant force. Solving both equations for $G_{vaccum}$ gives $$G_{vaccum} = G_{air} + B_{air}$$ $$G_{vaccum} = G_{water} + B_{water}$$ so $$G_{air} + B_{air} = G_{water} + B_{water}$$ Because $B_{air} = \rho_{air} V g$ resp. $B_{water} = \rho_{water} V g$ and $$V = \frac{m_{obj}}{\rho_{obj}}$$ I can write: $$ G_{air} - G_{water} = B_{water} - B_{air}$$ $$ G_{air} - G_{water} = Vg \left(\rho_{water} - \rho_{air}\right)$$ $$ \frac{G_{air} - G_{water}}{\rho_{water} - \rho_{air}} = \frac{m_{obj}g}{\rho_{obj}} $$ And this is where I am stuck, as I cannot measure the real weight of the object $m_{obj}$. It feels like I am missing something here ... So, how do I solve this equation to finally get an equation for $\rho_{obj}$? Tell me if I forgot something I could measure. * * * **Edit:** Of course I can assume that the influence of the buoyant force in air is negligible and write: $$ m_{obj}g \approx G_{air} $$ Then I can also assume that the density of air is much less than the density of water and write: $$ \rho_{water} - \rho_{air} \approx \rho_{water} $$ Using those two assumpions I could write: $$ \rho_{obj} = \rho_{water} \frac{G_{air}}{G_{air} - G_{water}} $$ Can I only assume the first simplification and not the second and get a better result? And still: How do I solve the problem without those simplifications?


title: Higgs field in space around us?

text: Is the higgs field in space around us? I understand it as that the higgs field has a constant value on every space time point, is that right? And this value is the vacuum expectation value. This value is responsible for giving mass to the elementary particles, like the fermions, right? In the Lagrangian for e fermion there is a term $g \phi_0 \bar \psi \psi$. $g$ is the Yukawa coupling constant and $\phi_0$ is the vacuum expectation value, right?


title: Why is the projective symmetry group (PSG) called projective?

text: As discussed by Prof.Wen in the context of the quantum orders of spin liquids, **PSG** is defined as all the transformations that leave the mean-field ansatz invariant, **IGG** is the so-called _invariant gauge group_ formed by all the gauge-transformations that leave the mean-field ansatz invariant, and **SG** denotes the usual symmetry group (e.g., lattice space symmetry, time-reversal symmetry, etc), and these groups are related as follows **SG** = **PSG/IGG** , where **SG** can be viewed as the _quotient group_. However, in math, the name of _projective group_ is usually referred to the _quotient group_ , like the so-called _projective special unitary group_ $PSU(2)=SU(2)/Z_2$, and here $PSU(2)$ is in fact the group $SO(3)$. So physically why we call the **PSG** _projective_ rather than the **SG**? Thank you very much.


title: Does brightness of the light remains constant with the variation of the distance with respect to the source in vaccum?

text: I have this question going in my mind from many days, i.e why brightness of light emitted from any light source around us decreases with distance? The brightness of light from tube light, streetlight, etc or any light from any source around us, decreases with the distance. As my teacher has taught me that light is a form of radiation, I thought this decrease in brightness may be due to absorption of radiation in the medium. I don't know whether it is correct or wrong. If it is wrong, please explain _why there will be decrease in brightness of light with increase in distance from the light source?_   If we assume that decrease in brightness of light is due to absorption of radiation in the medium, then _in vaccum (where we can assume no energy dissipation), would there be no decrease in brightness of light with respect to distance from the light source?_   ![enter image description here](http://i.stack.imgur.com/x6nhs.jpg)


title: To what extent correlation functions determines the theory (and lagranian)

text: In other words, does a finite set correlation functions sufficient to determine a theory? Is there a chance correlation functions are more fundamental then the lagrangian?


title: Magnetic Field Given Magnetization

text: We are working on problem 6.12 from Griffiths Electrodynamics. It says that we have an infinitely long cylinder of frozen in magnetization of $M=ks\hat z$. We are trying to find the field. We have found: $$J_b=\nabla \times M=-k\hat{\phi}$$ $$K_b=M \times \hat n=kR\hat {\phi}$$ Then we are stuck on how to find the field. We know to use an amperian loop, but we are lost on what bounds to integrate and how to get a result.


title: normalizing a wavefunction

text: I have a homework problem that I can't get started on, below is the first bit. I feel like I should just be able to integrate to find $C$ but I get a divergent integral. Can someone give me a hint as to where to go here? > A particle of mass m is in a one-dimensional inﬁnite square well, with $U = > 0$ for $0 < x < a$ and $U = ∞$ otherwise. Its energy eigenstates have > energies $E_n = (\hbar πn)^ 2/2ma^2$ for positive integer $n.$ Consider a > normalized wavefunction of the particle at time $t = 0$ $$ψ(x,0) = Cx(a − > x).$$ Determine the real constant $C$.


title: 4 of Einstein equations without 2nd order time derivative

text: This question is related to my previous one and it was a homework problem and was due two weeks ago. _Problem_ ：prove that four of Einsteins' equations $$ G_{0\nu} = 8\pi T_{0\nu} $$ have to 2nd order time derivative and thus serve as constraint equations. I found a nice argument in Carroll's lecture notes, $$ \partial_0 G^{0\nu} =-\partial_i G^{i\nu} -\Gamma^{\mu}_{\mu\lambda} G^{\lambda \nu} -\Gamma^{\nu}_{\mu\lambda} G^{\mu \lambda} $$ since there are at most 2nd order time derivative on the RHS, $G^{0\nu}$ has at most 1st order time derivative. But $G_{0\nu}$ is not proportional to $G^{0\nu}$ for a general metric. In fact $G_{0\nu} = G^{\mu\lambda}g_{0\mu} g_{\lambda \mu}$ contains ingredients other than $G^{0\mu}$. Of course we can use $G_{\mu\nu}{}^{;\nu}=0$ to prove that there is not 2nd order $\partial^0$ in $G_{0\nu}$. But now "spatial derivative" here is a mixed derivative, $\partial^i = g^{i\mu} \partial_\mu$. Two such spatial derivative could contain 2nd order time derivative. I suspect that only the $G^{0\nu}$ version is true. However a talk here use $G_{0\nu}$ instead of $G^{0\nu}$ for the constraint equations, where they didn't provide a proof. Now let me compute it explicitly in the explicit form, while terms containing at most 1st order time derivative will be dropped. So the sign $\sim$ really means both sides have the same term of 2nd order time derivative. $$R_{0\nu} = R^{\alpha}{}_{0\alpha\nu} \sim \Gamma^{\alpha}{}_{0\nu,\alpha}-\Gamma^{\alpha}{}_{0\alpha,\nu} \sim \Gamma^{0}{}_{0\nu,0} -\Gamma^{\alpha}{}_{0\alpha,\nu}$$ $$R \sim \Gamma^{\alpha\beta}{}_{\alpha,\beta} - \Gamma^{\alpha\beta}{}_{\beta,\alpha} \sim \Gamma^{\alpha 0}{}_{\alpha,0} - \Gamma^{0\alpha}{}_{\alpha,0}$$ $$G_{0\nu} =\Gamma^{0}{}_{0\nu,0} -\Gamma^{\alpha}{}_{0\alpha,\nu} - \frac{1}{2}g_{0\nu}(\Gamma^{\alpha 0}{}_{\alpha,0} - \Gamma^{0\alpha}{}_{\alpha,0}) $$ First evaluate things in the parenthesis, namely the scalar curvature, $$\Gamma^{\alpha 0}{}_{\alpha 0} = (g^{0\beta} \Gamma^{\alpha}{}_{\beta\alpha})_{,0} = (g^{0\beta} \ln(|g|^{\frac{1}{2}})_{,\beta})_{,0} \sim g^{00} \ln(|g|^{\frac{1}{2}})_{,00} $$ $$\Gamma^{\alpha}{}_{0\alpha} = g^{\alpha\beta} \Gamma^{0}{}_{\alpha\beta}=\frac{1}{2}g^{\alpha\beta} g^{0\mu}(g_{\mu\alpha,\beta} + g_{\mu\beta,\alpha} - g_{\alpha\beta, \mu} ) = -g^{\alpha\beta} g^{0\mu}{}_{,\beta}g_{\mu\alpha} - \frac{1}{2} g^{\alpha\beta} g^{0\mu} g_{\alpha\beta,\mu} = -g^{0\mu}{}_{,\mu} -g^{0\mu}\ln(|g|^{\frac{1}{2}})_{,\mu} \sim -g^{00}_{,0} - g^{00}\ln(|g|^{\frac{1}{2}})_{,0} $$ and thus, $$R \sim 2g^{00}\ln(|g|^{\frac{1}{2}})_{,00} + g^{00}{}_{,00} $$ Now for Ricci tensor, $$\Gamma^{0}{}_{0\nu} = \frac{1}{2}g^{0\mu}(g_{\mu 0, \nu} + g_{\mu \nu, 0} - g_{0\nu, \mu} ) \sim - \frac{1}{2}g^{0\mu}{}_{,0}( g_{\mu 0} \delta_{\nu 0} + g_{\mu\nu} ) -\frac{1}{2} g^{00}g_{0\nu,0} $$ $$\Gamma^{\alpha}{}_{0\alpha,\nu} \sim \delta_{0\nu} \ln(|g|^{\frac{1}{2}})_{,00}$$ Hence, $$R_{0\nu} \sim - \frac{1}{2}g^{0\mu}{}_{,00}( g_{\mu 0} \delta_{\nu 0} + g_{\mu\nu} ) -\frac{1}{2} g^{00}g_{0\nu,00} \- \delta_{0\nu} \ln(|g|^{\frac{1}{2}})_{,00} $$ Finally the Einstein tensor $$G_{0\nu} \sim - \frac{1}{2}g^{0\mu}{}_{,00}( g_{\mu 0} \delta_{\nu 0} + g_{\mu\nu} ) -\frac{1}{2} g^{00}g_{0\nu,00} \- \delta_{0\nu} \ln(|g|^{\frac{1}{2}})_{,00} - g_{0\nu} g^{00}\ln(|g|^{\frac{1}{2}})_{,00} -\frac{1}{2}g_{0\nu} g^{00}{}_{,00}$$ Maybe there are some subtle mistakes in the calculation(please point them out), but I can't simply it any further and at along show it $\sim 0$. **My questions**   1. Is the "$G_{0\nu}$" version of the statement right?   2. Is it possible to prove it using the trick that similar to that of Carroll's?   3. Any mistake I made in the explicit calculation? Or any magic identity could make it $\sim 0$


title: Higher order covariant Lagrangian

text: I'm in search of examples of Lagrangian, which are at least second order in the derivatives and are covariant, preferable for field theories. Up to now I could only find first-order (such at Klein-Gordon-Lagrangian) or non-covariant (e.g. KdV) ones. Also some pointers to the literature about general properties of such systems are welcome. Thanks


title: Electron Spin Resonance and Free electrons

text: When performing an experiment to observe electron spin resonance, we use DPPH molecules as they contain an unpaired electron on one of the N atoms. My question is, why cant free electrons be used in this experiment? What is wrong with a beam of electrons or a metal? Any help would be greatly appreciated.


title: Can the fuzzball conjecture be applied to microscopically explain the entropy of a region beyond the gravitational observer horizon?

text: In this article discussing this and related papers, it is explained among other things, how the neighborhood of an observer's worldline can be approximated by a region of Minkowsky spacetime. If I understand this right (corrections of confused fluff and misunderstandings are highly welcome), a coordinate transformation which depends on the observer's current location $p_0$ in the classical backround spacetime, to a free falling local Lorentz frame is applied. In this reference frame, local coordinates ($\tau$, $\theta$, $\phi$) together with a parameter $\lambda$ (which describes the location on the observer's worldline?) can be used. As $\lambda$ deviates too mach from $\lambda(p_0)$, the local proper acceleration $\sqrt{a_{\mu}a^{\mu}}$ becames large and approaches the string scale (is this because flat Minkowsky space is only locally valid?) and stringy effects kick in. The authors postulate that at these points (called the gravitational observer horizon) some microscopic degrees of freedom have to exist that give rise to the Beckenstein-Hawking entropy describing the entropy contained in spacetime beyond the gravitational observer horizon (?). This is quite a long text to introduce my question, which simply is: Can these microstates be described by the fuzzball conjecture or what are they assumed to "look" like?


title: Black & Scholes and the Quantum Mechanics

text: I am interested in the link between the Black & Scholes equation and quantum mechanics. I start from the Black & Scholes PDE $$ \frac{\partial C}{\partial t} = -\frac{1}{2}\sigma^2 S^2 \frac{\partial^2C}{\partial S^2} -rS\frac{\partial C}{\partial S}+rC $$ with the boundary condition $$C(T,S(T)) = \left(S(T)-K\right)_+.$$ Performing the change of variables $q=\ln(S)$ this equation rewrites $$ \frac{\partial C}{\partial t} = H_{BS}C $$ with the Black & Scholes Hamiltonian given by $$H_{BS} = -\frac{\sigma^2}{2} \frac{\partial^2}{\partial q^2}+\left(\frac{1}{2}\sigma^2 - r\right)\frac{\partial}{\partial q} +r.$$ Now I compare this equation with the Schrödinger equation for the free particle of mass $m$ : $$i\hbar \frac{d\psi(t)}{dt} = H_0\psi(t),\quad \psi(0)=\psi$$ with the Hamiltonian (in the coordinate representation) given by $$H_0 = -\frac{\hbar^2}{2m} \frac{d^2}{dq^2}.$$ My problem comes from the fact that the various references I am reading for the moment explain that the two models are equivalent up to some changes of variables (namely $\hbar=1$, $m=1/\sigma^2$ and the physical time $t$ replaced by the Euclidean time $-it$). However, their justifications for the presence of the terms $$\left(\frac{1}{2}\sigma^2 - r\right)\frac{\partial}{\partial q} +r$$ in the Hamiltonian seem very suspicious to me. One of them tells that these terms are "a (velocity-dependent) potential". Another one tells this term is not a problem since it can be easily removed is we choose a frame moving with the particle. I have actually some difficulties to justify why, even with this term, we can say that the Black & Scholes system is equivalent to the one coming from the quantum free particle. I don't like this potential argument, since (for me) a potential should be a function of $q$ (so it would be ok for example for the $+r$ term) but not depending on a derivative. Could you give me your thoughts on this problem?


title: Wave function normalization

text: A book by C. J. Ballhausen led me to believe that a quick way to check that I performed step operators properly was by observing that the "wave function should appear normalized," but I have found some issue applying this in practice and believe it is due to my misunderstanding of the underlying physics; I'm trying to understand what C.J.B. meant by that and if it applies in my case. In his case he was observing two equivalent electrons. Let's say they are two equivalent $p$ electrons for the sake of example. (He was actually considering $d$ electrons and I can provide the specifics of that if it will help.) There are several unperturbed functions which can be described by symbols such as $(1^+ 0^+)$, in which case the first number means $m_l$ of the 1st 2p electron, the next symbol indicates its spin and so forth. For the term $ ^1 D : M_L = 2, M_S=0: (1^+ 1^-) $ is an eigenfunction of $p^2$ configuration that is known. Using a step down operator on the angular momentum gives: $M_L=1 : (2)^{(-1/2)} [ (1^+ 0^-) - (1^- 0^+) ]$. Here I get the impression that what we observe appears normalized because squaring the coefficients gives unity. I realize one could in principle perform $\int \psi^* \psi d \tau$, but I do not suspect that is what he means. Now if we apply the step operator again we get $M_L=0: (6)^{(-1/2)} [ (1^+ -1^-) - (1^- -1^+) + 2(0^+ 0^-) ]$. Here the square of the coefficients is decidedly unity. His examples given in the book also happen to go to unity; is this just coincidence or is it going to always be true? My specific example is as follows, starting with a $d^3$ configuration: $$\psi(L,M_L,S,M_S)=\psi(5,5,\frac{1}{2},\frac{1}{2})=(2^+,2^-,1^+)$$ Applying the lowering operators gives $\sqrt{10} \psi(5,4,\frac{1}{2},\frac{1}{2}) = -\sqrt{4} (2^+,1^+,1^+) - \sqrt{4} (2^+,1^+,1^-) + \sqrt{6}(2^+,2^-,0^+)$ and the coefficients go to unity as expected. Above you will notice that the ordering in the first and third terms has changed and an odd permutation brings about a change in sign. Also notice the first term must be equal to zero by Pauli Principle. Dividing out we get, $$\psi(5,4,\frac{1}{2},\frac{1}{2}) = \sqrt{3/5} (2^+,2^-,0^+) - \sqrt{2/5} (2^+,1^+,1^-)$$ You'll notice that the coefficients squared sum up to 1, so all appears normalized and well. Now we apply the lowering operator again to give $\sqrt{(L-M_L+1)(L+M_L)} = \sqrt{(5-4+1)(5+4)} = \sqrt{(2)(9)} = \sqrt{18}$ times the function for $M_L=3$ $^1 H$. Applying to the RHS using $\sqrt{(l-m_l+1)(l+m_l)}$. We are working with $d$ orbitals, therefore $l=2$. So for the case of $m_l=2$ we get $\sqrt{(2-2+1)(2+2)}=\sqrt{4}$ and for $m_l=1$ we get $\sqrt{(2-1+1)(2+1)} = \sqrt{6}$ and finally for $m_l=0$ we get $\sqrt{(2-0+1)(2+0)}=\sqrt{(3)(2)} = \sqrt{6}$. Applying this gives $$ \sqrt{18} \psi(5,3) = \sqrt{3/5} [ \sqrt{4} (1^+, 2^-, 0^+) + \sqrt{4} (2^+,1^-,0^+) + \sqrt{6} (2^+, 2^-, -1^+)] $$   $ - \sqrt{2/5} [ \sqrt{4} (1^+,1^+,1^-) + \sqrt{6} (2^+,0^+, 1^-) + \sqrt{6} (2^+,1^+,0^-) ]$ Simplification results in: $$ \sqrt{18} \psi(5,3) = \sqrt{12/5} (1^+, 2^-, 0^+) + \sqrt{12/5} (2^+,1^-,0^+) + \sqrt{18/5} (2^+, 2^-, -1^+)$$   $ - \sqrt{8/5}(1^+,1^+,1^-) - \sqrt{12/5} (2^+,0^+, 1^-) - \sqrt{12/5}(2^+,1^+,0^-) $   The fourth term cannot exist by the Pauli Principle, so we have instead, $$ \sqrt{18} \psi(5,3) = \sqrt{12/5} (1^+, 2^-, 0^+) + \sqrt{12/5} (2^+,1^-,0^+)$$   $ + \sqrt{18/5} (2^+, 2^-, -1^+) - \sqrt{12/5} (2^+,0^+, 1^-) - \sqrt{12/5}(2^+,1^+,0^-) $   Now we need to fix the ordering of the first term and the fourth term to give, $$ \sqrt{18} \psi(5,3) = -\sqrt{12/5} (2^-, 1^ +, 0^+) + \sqrt{12/5} (2^+,1^-,0^+) + \sqrt{18/5} (2^+, 2^-, -1^+)$$   $ + \sqrt{12/5} (2^+,1^-, 0^+) - \sqrt{12/5}(2^+,1^+,0-) $   Now we divide thru by $\sqrt{18} $ the like terms yielding, $$ \psi(5,3) = -\sqrt{12/90} (2^-, 1^ +, 0^+) + \sqrt{12/90} (2^+,1^-,0^+) + \sqrt{18/90} (2^+, 2^-, -1^+)$$   $+ \sqrt{12/90} (2^+,1^-, 0^+) - \sqrt{12/90}(2^+,1^+,0^-) $   Our problem is that 12/90 + 12/90 +18/90 + 12/90 +12/90 =11/15, instead of 15/15. I'm sure my mistake is stupid somewhere, can somebody point out where I've gone wrong?


title: Is it possible to detect if everyting in the Universe is changing size?

text: This may be a trivial question, but I cannot find a good answer to it. What would happen if the size of everything in the Universe is multiplied by some constant factor at the same time, let's say everything doubles its size for instance. People double their size, houses double their size, streets, towns, Earth... up to the whole known Universe. The relative proportions and distance between objects would remain... If we measure our own size with a meter, the reading would be the usual one. From current theories, it this something that can or cannot happen? that can or cannot be detected? Would that change our perception of constants like c, gravity, or temperature? Why? (in very simple terms...) Is there any reference, any previous study of this problem? * * * (This part has been added/edited to improve the question) From this question, I went to this book by Benjamin Crowell, following Ben Crowell advice. This section describe in better words my idea. Citation: > 8.2.6 Observability of expansion > > [...] To organize our thoughts, let's consider the following hypotheses: > >   * The distance between one galaxy and another increases at the rate given > by a(t) (assuming the galaxies are sufficiently distant from one another > that they are not gravitationally bound within the same galactic cluster, > supercluster, etc.). >   * The wavelength of a photon increases according to a(t) as it travels > cosmological distances. >   * The size of the solar system increases at this rate as well (i.e., > gravitationally bound systems get bigger, including the earth and the Milky > Way). >   * The size of Brooklyn increases at this rate (i.e., electromagnetically > bound systems get bigger). >   * The size of a helium nucleus increases at this rate (i.e., systems bound > by the strong nuclear force get bigger). > > > We can imagine that: > >   * All the above hypotheses are true. >   * [...] > The author then propose to look at the first claim, all hypotheses are true... > If all five hypotheses were true, the expansion would be undetectable, > because all available meter-sticks would be expanding together. Likewise if > no sizes were increasing, there would be nothing to detect. These two > possibilities are really the same cosmology, described in two different > coordinate systems. But the Ricci and Einstein tensors were carefully > constructed so as to be intrinsic. The fact that the expansion affects the > Einstein tensor shows that it cannot interpreted as a mere coordinate > expansion. Specifically, suppose someone tells you that the FRW metric can > be made into a flat metric by a change of coordinates. (I have come across > this claim on internet forums.) The linear structure of the tensor > transformation equations guarantees that a nonzero tensor can never be made > into a zero tensor by a change of coordinates. Since the Einstein tensor is > nonzero for an FRW metric, and zero for a flat metric, the claim is false. The author says this claim is impossible, the demonstration is not understandable for me as I'm not an expert in the domain, and don't know what are FRW or Ricci and Einstein tensors.


title: Does it matter how you order your tug-of-war participants?

text: In a tug-of-war match today, my summer camp students were very concerned about putting the biggest people at the back of the rope. Is there any advantage to this strategy?


title: What is a simple physics application of the reciprocal of a square root?

text: I learned yesterday that the inverse of square roots is used to calculate the vectors of surface normals in 3d graphics. It seems like such a mind- bogglingly simple idea, and it leads me to wonder if it hasn't found use elsewhere. Is there anywhere in physics where one might need to calculate the reciprocal of a number's square root?