title: How do we know what type of gauge field to add to a theory?

text: I've been watching Leonard Susskind's particle physics lectures and in one lecture, he discusses a very simple gauge theory. We have a complex scalar field $\phi(x)$ with Lagrangian $$\mathscr{L} = \partial_\mu \phi^* \partial^\mu \phi - V(\phi^* \phi)$$ and we want to do the gauge transformation $\phi(x) \to e^{i\theta(x)} \phi(x)$. But the derivative term in the Lagrangian is not invariant if $\theta$ varies from place to place, so we add a new vector field $A_\mu$, and define it to transform like $$A_\mu \to A_\mu + \partial_\mu \theta$$ under the gauge transformation. Then we change the Lagrangian to use covariant derivatives $D_\mu = \partial_\mu + iA_\mu$ instead of ordinary ones; now when the gauge transform is done all the derivatives of $\theta$ cancel out, and the result is invariant. My question is: was there any freedom in choosing to add a vector field, with that specific gauge transformation law? Are there any other ways we could get gauge invariance here—by using a different transformation law, or with a different type of field (say, a scalar, tensor, or spinor) with some other transformation law? More generally, how can we tell what type of field is needed and what its gauge transformation law should be, to get some particular Lagrangian to be invariant under some particular gauge transformation?


title: Gauge symmetries and elementary particles

text: The Weinberg-Witten theorem (disclaimer: I don't know this wikipedia entry) is usually mentioned as the reason why gravitons may not be composite particles. I do understand the proof of the theorem, but not the previous conclusion. The theorem states that in an interacting and Poincare invariant quantum field theory, there are not massless, spin-2 particles unless there exists a gauge symmetry — which makes the energy-momentum tensor non-covariant (actually covariant up to a gauge transformation) under Lorentz transformations in the Fock space. So the immediate conclusion of the theorem is that the existence of a massless, spin-2 particle (like a graviton) requires linearized diffeomorphisms. My question is: why do linearized diffeomorphisms imply that gravitons are elementary particles? Or, more in general, why the particle corresponding to a gauge field must be elementary (I know that a gauge symmetry must be exact, but why this implies that the corresponding particle must be elementary?). Thank you.


title: Difference between weight of water and pressure of water

text: _(I didn't even have a basic formal education in physics. I'm learning through the internet out of my own interest, so if there are any silly mistakes, kindly bear with and guide me through.)_ Everywhere, everyone is saying that pressure will same be for a given height. How is it possible? When volume changes, doesn't pressure change in calculations? [Pressure will be 2.5 bars approximately (including atmospheric pressure) for a height of 15 metres from ground level. I checked on many sites that 2.5 bars equals 2.5 kg/cm².] Now let's take **base area as** **2000 cm² and height of 15 m as constant** in three scenarios: EACH SCENARIO IS INDEPENDENT AND TUBES ARE SEPARATE, I.E NOT CONNECTED WITH EACH OTHER In **scenario 1** a straight vertical tube from ground level will have capacity of 3000 litres with a weight of 2.5 kg per cm² [3000 litres / 2000 cm² = 1.5 kg/cm² + 1 kg/cm² atmospheric pressure]. In **scenario 2** the tube becomes narrow from the base area, resulting in a volume of 2000 litres. In **scenario 3** the tube becomes wide from the base area, resulting in a volume of 4000 litres Question: When they say pressure remains the same at a given height, does it mean that weight will be 2.5 kg/cm² in all 3 scenarios, where volume is 3000, 2000, or 4000 litres (base area 2000 cm², height 15 m constant)? How is it possible? Where am I wrong?


title: Noncanonical commutation relation and noncanonical wave mechanics

text: Consider noncanonical operators $\hat{x}_1,\hat{x}_2,\hat{p}_1,\hat{p}_2$ satisying the following condition in the $q_1,q_2$ - basis ($\psi=\langle q_1,q_2|\Psi\rangle$)(similar to wave mechanics): $$\hat{x}_1\psi=q_1\psi-ia\hbar\frac{\partial}{\partial q_2}\psi$$ $$\hat{x}_2\psi=q_2\psi-ib\hbar\frac{\partial}{\partial q_1}\psi$$ $$\hat{p}_1\psi=-i\hbar\frac{\partial}{\partial q_1}\psi+cq_2\psi$$ $$\hat{p}_2\psi=-i\hbar\frac{\partial}{\partial q_2}\psi+dq_1\psi+i\hbar\gamma\frac{\partial}{\partial q_1}\psi$$ The commutation relations matrices are (similar to Hilbert-space quantum mechanics): $[x_i,x_j]=i\hbar\left(\begin{array}{cc} 0 & (b-a) \\\ (a-b) & 0 \end{array}\right)$   $[p_i,p_j]=i\hbar\left(\begin{array}{cc} 0 & (c-d) \\\ (d-c) & 0 \end{array}\right)$ $[x_i,p_j]=i\hbar\left(\begin{array}{cc} 1 -ac & -\gamma \\\ 0 & 1-bd \end{array}\right)$ Because $\hat{x}_1,\hat{x}_2$ are not commute so I have to introduce 2 other parameter $q_1,q_2$. My question is: is it true or is it not true that all wave mechanics theories with the same commutation relations will predict the same results?


title: How might inducted current/ voltage in multiple coil layers connected in parallel behave?

text: This is a branch from this question.   * In the event that two or more layers are connected in parallel - would the energy created across each layer likely be significantly less? In the event of (say) five layers of coiling connecting, each with differing currents (or voltages), how might the resultant current be combined once reunited (presuming a similar wire is receiving the energy)?  Would the variances result in feedback from the coils with most induced energy into the coils with least induced energy (at the point of uniting)? Are there further aspects that you would consider?


title: Is frequency quantized?

text: I'm aware that there're some questions posted here with respect to this subject on this site, but I still want to make sure, is frequency quantized? Do very fine discontinuities exist in a continuous spectrum like the black body spectrum? The quantization of photon energies


title: Why does gravitational collapse occur suddenly in a supernova progenitor?

text: I was reading the Wikipedia article on Supernovae, and it says that one of the reasons why a supernova occurs is due to sudden gravitational collapse when the core of the star has little fusable material. But why is this collapse sudden? Shouldnt the collapse be gradual as the fusable material gradually decreases? Why is the sudden collapse like driving off a cliff? (if I've interpreted the article rightly.)


title: Variational wavefunctions and "spread" of potential in quantum mechanics

text: A particle in a box has an energy that decreases with the size of the box. In the general case, it is often said that a variational solution for a "narrow and deep" potential is higher in energy than a variational solution for a "wider and shallower" potential, by appealing to the particle in a box as a special case. The former wavefunction is said to be "more confined" in the former than the latter, and the act of being given more space to spread out is said to lower the energy variationally. Question: are there _rigorous mathematical statements_ of this argument?


title: Long wave EM Attenuation in liquid

text: I want to transmit a long wave (electromagnetic wave) through a crude oil line using the pipe as the ground. I am just wondering what the attenuation will be and how best to set up this test. The essence of the test is to verify that 100 kHz electromagnetic wave can pass thru oil and determine the degree of attenuation.


title: How to determine the sign of the s-wave scattering length?

text: I guess it is relatively easy to determine the magnitude of the scattering length $a$. We just need to measure the scattering cross section. In this way, we can determine the value of $a^2$. But how to determine its sign?


title: Stir plate magnetic field RPM relative to stir bar

text: Observing the use of a variable speed stir plate prompted some discussion about the stir plate motor RPM vs the stir bar RPM and we did not have any physicists handy. The stir plate has a beaker filled with water on it and a stir bar inside the beaker. The stir plate measures RPM based on the RPM of the motor with attached magnet. We noticed that if the stir plate RPM setting is rapidly increased or set to a high setting the stir bar seems to "decouple" from the magnetic field and flop around in the beaker like a fish out of water. Here we assumed the polarity of the stir bar reversed relative to the stir plate magnet due to friction of the liquid causing the stir bar to lag behind the stir plate. 1) Is our assumption about the decoupling correct? 2) Is there a technical term for this decoupling? The main question is: 3) Due to friction can the RPM of the stir plate exceed the RPM of the stir bar without the magnetic field decoupling? We are wondering if there could be a substantial difference in revolutions between the stir plate and stir bar over a couple of minutes with a low viscosity liquid. "Substantial" being the stir plate running at 100 RPM and the stir bar running at 80, all without decoupling.


title: The mass of the photon will decrease with reduced velocity?

text: Since photon has zero rest mass, it will has zero mass when the velocity is zero. We are now being able to slower down the photon in experiment. Is the mass of photon reduced with the decreased velocity. Or it will remain the same, but it will decreased to zero only when the velocity is zero? The deeper question behind it would be that what does it mean for the observer if the object is under observation is in another medium with different speed of light.


title: Is enthalpy defined under non-isobaric conditions?

text: My question is this, Does Enthalpy have a meaning under non-isobaric conditions? Is its existence as a property of a system independent of whether the system is under isobaric condition or not? Edit:- I wanted to know if enthalpy as a property of a system is valid only if from its creation to current state, it undergoes only isobaric processes.


title: Vector Potential and Gauge Invariance in Quantum Mechanics

text: In classical electromagnetism, we are allowed to use gauge invariance through the argument that the only physical observable fields are the $E$-field and the $B$-field. So in that sense the scalar potential $\phi$ and the vector potential $A$ are to be defined up to an arbitrary constant as long as they give the same E and $B$-field. But as we know from the Aharonov-Bohm effect, it turned out in quantum mechanics, the $A$-field is also observable. So, by right, we are not allowed to use gauge invariance in quantum mechanics, right? Or is there something wrong with my reasoning?


title: Can Rydberg states exist within the bulk of a metal?

text: I am aware that outer shell electrons in rubidium atoms in an optical lattice can be excited to Rydberg levels, in which the electrons orbit well beyond the atoms to which they are bound. Is this something that can happen within the bulk of a metal as well? If this is not something that has been experimentally examined, assume there are one or more straightforward mechanisms that could provide the requisite excitation--naturally occurring alpha or beta emitters, for example. If Rydberg levels within the bulk of a metal are not possible, why is this the case? If they are possible, how would the electron charge density be affected? In the non-Rydberg case, I understand the d-electron density will be quite low in the interstitial regions. Would this change in the Rydberg case? How long would the excited state last? How would the blockade effect factor in? Can you recommend a suitable approach for modeling the charge density? (It seems a 1996 paper did some calculations that are relevant here, which I will take a look at. I am still interested in any information that people can provide.)


title: What does "all future lies within the event horizon" mean?

text: I was trying to find an answer as to why light does not escape black holes and I stumbled upon this Phys.SE question. In the answer it said that: > "Since all future lies within the event horizon, photons are trapped inside > the black hole." But what does that mean? It sounds to me that if light were to escape a black hole it has to travel back in time. And this leads to another question, why does the future lie within the event horizon? This could make sense to me if there was another reason as to why light can't escape a black hole once it has passed through the event horizon, and as a result of that, the future lies within the event horizon. But that just makes the answer to the linked question a circular proof, it is the same as saying that light can not escape a black hole because of X reason and that is the reason why the future lies within the event horizon. And the future lies within the event horizon because of X reason. So is there an "X" reason, if so, what is it?


title: Simulating the effects of external forces on a small population

text: I am looking to simulate the effects of external forces on a small population but I am having difficulty figuring out how I should go about doing such a simulation. My simulation consists of having small "atom" like objects that jostle about randomly and depending on the type of external force, the "atoms" will react differently. For example, if the atoms are introduced to a high viscosity fluid then "atoms" will slow down. So, how would I go about doing this? Thanks! EDIT: I would like to add on that atoms and high viscosity fluid is merely an example. I am looking to model how say, people, would react to the introduction of money given people have some characteristics and money has some characteristics that would influence those characteristics.


title: Interpretation of the off-diagonal terms of the conductivity tensor

text: Say we have the electrical conductivity tensor expressed as a 3x3 matrix. I've seen that if it's cubic material then the conductivity tensor reduces to just the diagonal terms and these are equal, meaning that the conductivity is isotropic. However, if some of the off-diagonal terms are not zero, what direction do they correspond to?


What is the definition of a timelike and spacelike singularity? Trying to find, but haven't yet, what the definitions are.


title: How do we know that there isn't a classical solution to the measurement problem/Quantum Mechanical uncertainty?

text: It was mentioned to me that it can be shown that there is no classical explanation for the uncertainty in Quantum Mechanics -- i.e. that there are no hidden workings that we have just not yet seen, which could be explained classically and would explain the probabilistic nature of Quantum events in a 'deterministic' fashion. Can someone explain how this is known please?


title: Are "timed" measurements actually revealing error-distributions of the measurement apparati?

text: A thought experiment: Given some object moving (swinging) from left to right and back with constant velocity, imagine a camera set up to take a picture of the scene at a fixed interval so that we can expect the object to be in the center. Now "the perfect"/expected scenario would be: in every picture taken the object is found at the same place. In reality what we probably get is a distribution: in some pictures the object more or less slightly left or right of the center even though in most pictures the object is really centered. That's quite resembling to the position measurements we do at small levels which come up with "distributions" and "probabilities" for positional (and other) information of atoms or electrons for example. Now, why are those distributions believed to be properties of the measured entities? Why can't they "simply" be explained by inexact timing of the camera triggers in the example above? The results would be the same, right? Or would they? I'm a layman so most probably I missed something profound and would like to know what ;) Thanks in advance Cheers


title: Naked singularity and null coordinates

text: I'm trying to understand the notion of a naked singularity on a more mathematical level (intuitively, it's a singularity "one can see and poke with a stick", but I'm having troubles on how to actually show it). Based on what (little) is written in Choquet-Bruhat's, a naked singularity is the one for which we can extend the outgoing time-like geodesics to infinity. Now, I was wondering, assuming I had a _given solution_ , how would I "test" the nakedness of the singularity? A natural thing to do would be to write the solution in some null coordinates, but what then? How to do I actually combine it with the (rather abstract) definition of a naked singularity?


title: How can I find the temperature of this system?

text: > A system was given a small amount of thermal energy dE, and its number of > states G grew by 25%. How can I find the system temperature? The system contains gas particles, I know that $dE << U$ I thought about using the entropy $\sigma$ = $ln(g)$ when $g$ is the number of states $\frac{1}{T} = K_b \frac{d\sigma}{dU}$ when $dU = dE $ and the entropy $\sigma$ is $ ln(g)$ so $d\sigma = ln(1.25g)-ln(1g) = ln(1.25)$ we get - $ T = \frac{dE}{K_b\cdot ln(1.25)}$ and the Bolztmann constant units are $J/K$ so it turns out okay units wise, but apparently its not the correct answer. Have I messed up something?


title: Operators in quantum mechanics

text: 1. According to the Quantum Mechanics, can we write $\langle q|p\rangle = e^{ipq}$?   2. If so then how?   3. And if we transfer to integrate formulation then how it will look like?


When is spacetime homogenous and isotropic? For example, some metric $g_{\mu \nu}$ is homogeneous and isotropic. We now construct effective metric $$n_{\mu \nu} ~\rightarrow~ g_{\mu \nu} + f(x^\mu)v_\mu v_\nu.$$ where $v_\mu$ is 4-vector, and $f$ is scalar. Is the effective metric $n_{\mu \nu}$ homogeneous and isotropic as well?


title: Can we live forever by taking advantage of the time dilation effect?

text: I have been reading about Time dilation and twin paradox in Relativity. So,I thought that if It can be the difference of 10 yrs between the Girl in space and on Earth, then why can't anyone not age rather. Is it possible? [I'm 14 and new in the field of modern physics, so pls correct if I'm wrong. I regard All of You as teachers.]


title: Einstein's Postulates: Simultaneity

text: Okay, I still don't get the solution (which I will lay out) to the following problem: Suppose that A', B', and C' are at rest in frame S', which moves with respect to S at speed v in the positive x-direction. Let B' be located exactly midway between A' and C'. At t'= 0, a light flash occurs at B' and expands outward as a spherical wave. I know that according to an observer in S', the wave fronts arrive simultaneously at A' and C'. I also know that they are not simultaneous in S frame. Now I find the time difference between the events as recorded by an observer in S. Let distance from B' to C' and B' to A' be L. Then the time difference between the events as seen by frame S is: $$\delta T = T(B' \to C') - T(B' \to A') = L/(c-v) - L/(c+v).$$ My question is, does this not contradict Einstein's Second Postulate? I thought the speed of light to any observer is always $c$? So why in the deltaT equation can we write $c-v$ and $c+v$? Shouldn't the speed of light be $c$ to any observer? cheers


title: There can be nothing with zero mass

text: Zero mass has no meaning and is an outcome of 2nd postulate of STR .The STR can be developed without this postulate and thus the postulate is redundant besides being contradictory to 1st postulate.The "c" appearing in STR is the limiting highest speed which can never be achieved by any material particle by any means.Instead of accepting the occult Zero mass to photon we should design experiments to: 1- establish the mechanism by which the photons are accelerated close to the limiting speed 2- determine the rest mass of photon The absurdity of assigning Zero mass to photon is apparent through following cases: 1- Thought experiment of inelastic collision of electron and photon 2- Bending of the trajectory of photons due to gravitational pull by massive bodies


title: Symmetry of Euler-Lagrange equations and conservation laws

text: 1. Continuous symmetry of the action implies a conservation law, but what if equations of motion have a continuous symmetry? Does it imply a conservation law?    2. Also is symmetry of equations of motion also a symmetry of the action?  I assume that the equations of motion I am talking about follow from the action, i.e. the equations of motion are Euler-Lagrange equations. Related examples could help and references also.


title: Are there any materials that change the electromagnetic wavelength of visible light?

text: What i mean is for example exclusively red light comes in but gets reflected or refracted as blue or xray wavelength or whatever. Does this happen at all?


title: Why are some things attracted to you but others repelled by you in rotating reference frames?

text: Note that my understanding of general-relativity is rudimentary. If I understand right, it means that basically any reference frame can be considered stationary, but there may be random gravitational forces going in any which way. A though experiment. It is the year 2100, and to celebrate, they are going to shoot an astronaut into SPACE. And by an astronaut, they mean you. Also the cannon is shaped like a shotgun (for whatever reason), and will even spin you. They fire you, and the acceleration causes you to go unconscious. You wake up. The earth and any massive objects are far away. Also, everything is spinning. You at first think that this is because of motion sickness, but you then realize that everything is actually spinning around you. It is great feeling to think everything revolves around you. You then realize that everything is just going around a line that goes through you. Still cool. You conclude that there must be a force throughout the entire universe that pushes everything to this line. You decide to pull out a camera to take a selfie to show your friends when you get home. You accidentally let go, and the camera stars spiraling away from you. (Also note the cameras weight is negligible, so you do not question why this isn't an equal opposite force, and from earth's perspective, you won't loss any angular momentum.) This is odd, because the camera was still, but then something caused it to spiral away from you, where as everything else seems attracted to you (or that line.) **So in summary** , if from the earth a rotating object releases a projectile (which isn't massive enough to affect the angular momentum in a noticeable way.) Why from the objects reference frame would the object fly away while everything else just orbits a line through it? Or do I just grossly misunderstand gr. Update: I want to understand what forces are present according to the astronaut's reference frame. Are the forces going towards the line, away, or something else. I do realize it is related to Mach's Principle. On later thought, this is basically trying to figure out exactly how Mach's Principle is applied. Stars out there affect us down here, but how?


title: The Equation of State for a Degenerate Fermi gas

text: I have read in Chandrasekhar's paper _The highly collapsed configurations of a stellar mass_ Appendix I the equation of a degenerate Fermi gas as follows: $$n=\frac{8\pi}{h^3}\int^{p_0}_0 p^2dp$$ and $$P=\frac{8\pi}{3h^3}\int^{p_0}_0 p^3\frac{dE}{dp}dp$$ I have derived myself the proper energy density: $$\rho=\frac{8\pi}{h^3}\int^{p_0}_0\sqrt{(pc)^2+(mc^2)^2} p^2dp$$ where $n$ is the number density of electrons and $P$ is the pressure. $E$ is the kinetic energy of a free electron. I have no problem understanding the first equation for $n$ but I am confused with the second one. Could anyone please explain to how to obtain the equation for $P$ and tell me if my energy density expression is correct?


title: Are magnetic hysteresis losses relevant to alternating currents flowing in a wire?

text: Say we have an AC in a magnetically lossy material, like iron. Because of iron's relatively high permeability, skin effect will be more pronounced than it is in say, copper, so this iron wire isn't so great a conductor, practically speaking. However, skin effect arises due to eddy currents, which are themselves due to the time varying magnetic field due to the time varying current. So, this suggests there is a magnetic field at work, and there's the potential for magnetic hysteresis to be an additional source of loss in the material. Yet, I'm not sure if this is true or not, if somehow the geometry of the conductor and the fields around it make this a non-issue. I would think that if a time-varying magnetic field exists inside the iron, then there would be hysteresis losses. Is this true? Does such a field exist? To be clear: the issue is with iron _as an electric conductor_ , not as part of a magnetic circuit such as a transformer core as is the more common application.


title: Push a box in a plane with friction. How to deal with the rotation?

text: Suppose I have a box (say, length-1m, width-1m, height-0.5m) on the plane with friction. I can apply a horizontal force in on the surface of the box. If the force doesn't pass through the center of the box, then the box will rotate. My question is, how to solve the angular velocity, angular acceleration and the center of rotation? The difficult thing in this problem is that we need to consider the torque of the friction. In my opinion, the friction should be distributed in the bottom surface of the box. So in order to get its torque, we need to do the integral within the bottom surface. My idea is: move all the forces to the center of mass, and add the corresponding torques to keep the equivalence before and after moving the force. So the box's movement can be divided into two parts: (1) the translation of the center of mass (2) the rotation around the center of mass. I'm not sure about my idea. The problem is that the box doesn't necessarily rotate around its C.O.M, right? Another question is about the direction of the friction. Suppose that the friction is distributed on the bottom surface. If the box rotates, then the movement of the point on bottom surface is a superposition of translation and rotation, which means the absolute velocities in different positions on the bottom surface are different. Because we know friction always has the opposite direction to the relative movement, does that mean the frictions in different positions on the bottom surface have different orientations? Who can give me the right approach to think about this problem? Thank you very much!


title: Which theory predicts that dubbed tires have more friction?

text: The physical reasoning I suppose could be that more contact areas mean more and higher friction. But is there an actual formula or a more mathematical explanation? Application for bicycling was my question http://bicycles.stackexchange.com/questions/14319/could-i-switch-tires-on-an- mtb-to-tires-with-less-friction-if-i-mostly-use-the-m


title: What are the "loopholes" in past Bell's theorem experiments?

text: I am intrigued by the following Phys.org article: > Researchers began using photons in 1980s to test Bell's theory and determine > if Einstein's reasoning is right or wrong. Since then, researchers have used > various quantum states to test the theory but continued to have loopholes in > their methods, therefore falling short of a definitive result. Luo said the > new collaboration would, for the first time, be using several different > quantum systems—including photons, ions, quantum dots and solid-state > ensembles—to test the theory across large distances and hopefully eliminate > all possible loopholes, he said. --Physics researchers join effort to finally complete quantum theory I am familiar with Bell's inequalities, and I would like to know: how did the experimental results fall short? Why is this time going to be different?


title: Probability current density : Isn't there something wrong with this proof?

text: Is there something wrong with the following proof (see below)? To me, it seems like the third line should show $$\frac{dP_{ab}}{dt}=-\int_a^b \frac{\partial}{\partial t}J(x,t)dx$$ Am I missing something obvious? * * * ![enter image description here](http://i.stack.imgur.com/9L5fT.jpg)


title: Resistance at Ohm's Law

text: Today I learnt about Ohm's Law(I had some basic knowledge earlier).When I came home I searched in the internet and found that at Ohm's Law, resistance should not change because then the circuit will not obey Ohm's Law.I googled about this "problem" but I found nothing. My physic professor says that resistance can change at Ohm's Law. I don't know which one to trust. Can anyone help me ?


title: Lagrange L4 L5 points and perifocal plane

text: I have 2 satellites at the L4 and L5 points and these are _watching_ an object. Each satellite provides the angle to the object from its own position from a line parallel to the $\text{x-axis}$ of Earths perifocal plane (assume that the orbit of earth is circular and system Sun-Earth). My questions are:   1. Using earths perifocal frame, the positions of the satellites are: $r=[r\cos f,r\sin f,0]$ ?   2. If i have the L4 and L5 positions (angles for example $(55,45)$) and after some period the angles are $(58,40)$ how can I find the coordinates of the object?


title: A formal procedure for thermodynamic relations

text: This is my third time taking a thermodynamics course (two in undergrad, one in grad), and I've finally become frustrated enough about something to post on here. A lot of thermodynamic questions want you to find some quantity, given something else. Typically, the problem will be to relate volume, temperature, or pressure measurements (along with a volumetric equation of state) to other thermodynamic variables. The math is very easy to verify, once you see it written down. The problem is that no textbook I've ever used has given a definitive procedure for going about deriving these things. For instance, a recent problem is: Determine $C_p(P, T) - C_v(P, T)$ for such- and-such equation of state. My typical method of solving this is going to the table of Maxwell relations and plugging in and substituting things until I get what the problem is looking for. But this leads to a lot of trails to nowhere and is time-consuming. How do I know in advance that taking a partial derivative with respect to entropy is going to give me something useful in a chain of transformations? My question is: Given an arbitrary function of two thermodynamic variables, what _procedure_ is used for relating it to two other thermodynamic variables? (i.e. $f(\frac{\partial H}{\partial T}, S)$ -> $g(P, T)$)


title: Hermitian operator ?? is this possible

text: we know that the operator $ H= - \hbar ^{2} \frac{d^{2}}{dx^{2}}+ V(x) $ is hermitian isn't it ?? however what would happen if the potential were still real but it depends on the Wave function, for example huypothetically $ V(x) = |\Psi (x)|^{2} $ or $ V(x) = arg \Psi (x) $ since the functions $ |x| $ and $ arg (x+iy) $ are ALWAYS real, then the hamiltonian with potentials (1) and (2) should be Hermitian but they depend on the solution so i am not sure about the Hermiticity of a Hamiltonian in the form $ H =p^{2}+ |\Psi (x) |^{2} $ i believe that a real potential makes the Hamiltonian hermitian even in the case that we do not exactly know what potential is


title: According to General Relativity, Does The Past "Exist"?

text: I'm curious about just what is meant by time being another dimension, like the three (observable) spatial dimensions. Does this imply, according to General Relativity, that the past and the future already "exist" and that we're just moving through it, as we move through spatial dimensions? Or, is time being a dimension just a mathematical construct allowing us to calculate time dilation effects? Do we even know the answer to this question? Intuitively, it would seem to me that the past doesn't "exist" except in our memories and that the future "hasn't happened yet" so that all that exists is the present... **Clarification** I suppose I should clarify what I mean by "exists". The three spatial dimensions definitely exist. You can travel through them and they are not just a bit of abstract mathematics. What you can't do is travel faster than C through space. But, _if you could_ you know when you stop, there would be something there. Now, if you were to take the hypothetical example of time travel, there are two distinct questions here that follow this same line of reasoning. For the present, we don't have the technology to travel through time and it seems like an extremely difficult thing to do. But _if you could_ would there be something there to travel to? So, even if it is impossible to travel through time, our theories could still inform us whether the past and future exist in the same sense as the present does along the above lines. This was **not** supposed to be a time travel question... Just a question about the fundamental nature of spacetime: all "presents" existing, or only one "present", and what our current theories tell us or don't tell us about the answer to that question. **Further Clarification** So, as one of the answer's helpfully pointed out, Relativity certainly states there is no "privileged present", or a present that nature prefers and that all other points in time are measured against. I understood this before I asked the question. But, you could still ask the question of whether in an observer's own frame of reference, does his own past and future exist? To illustrate: Imagine instead of three spatial dimensions, we had only two. When space would stretch, it would expand or contract a grid drawn on the "paper" universe's spacetime. But, the "paper" would not move out of it's plane (NOT like a trampoline). Time would be the third dimension, with infinitesimal space sheets stacked one after another. A stretching of time would mean pulling the sheet in or out like a trampoline. So, in this illustrative universe, if the past and future exist, there are a nearly infinite number of sheets of present slices in array from the universe's birth till it's death. If only the present exists, there is only ever one slice and it changes over time. Can we say which view is correct in a concrete way, not just in terms of the math? (If you haven't noticed, I'm not a math guy, but I am a picture guy)


title: Fan blade physics

text: Why does a table fan eventually stop circulating air? I noticed this with every fan I ever used. The fan will work great for about 2 hours blowing air at full speed, then I notice that, even though the blades are still spinning, the amount of air output has been greatly reduced to the point that I can barley feel any air coming off the blades. If I turn it off for a couple hours it will still go through the same cycle. Why does this happen? Everything works fine, just no air circulates out.


title: Current state of the existence of absolute space debate

text: Samir Okasha writes about the controversy concerning the existence of absolute space in Philosophy of Science: A Very Short Introduction (2002): > One of the intriguing things about the absolute/relational controversy is > that it refuses to go away. Newton’s account of space was intimately bound > up with his physics, and Leibniz’s views were a direct reaction to Newton’s. > So one might think that the advances in physics since the 17th century would > have resolved the issue by now. But this has not happened. Although it was > once widely held that Einstein’s theory of relativity had decided the issue > in favour of Leibniz, this view has increasingly come under attack in recent > years. More than 300 years after the original Newton/Leibniz debate, the > controversy rages on. Unfortunately this is the last paragraph in the chapter about this topic... which is a little bit frustrating. So my question is:   What is the current state of the debate (this was written more than 10 years ago)? Is it still not settled and what are the main strands and results of research on that topic?


title: correct formula for Mass Defect / Binding Energy?

text: I'm a web developer and I have to change an online course. The course teaches Advanced Nuclear Theory. In the 'Mass Defect and Binding Energy' chapter, it has this formula: $ \Delta M = Z(m_p) + N(m_n) - \frac{A}{Z}(M)$ . In the formula above, the letter 'p' and 'n' are written in subscript, and for the 'AZM' part, the letter A (written in superscript) is above the letter Z (which is in subscript). Above this equation, this is written               1/1(p) + 1/0(n) -> 2/1(D)          mass of proton = 1.007272 u     mass of neutron = 1.008665 u     mass of deuterium nuclear = 2.013553 u      . Below the equation which I wrote above (delta M = Z(mp) + N(mn) - A/Z(M)), the worker / user is asked to fill in the blank:               ∆M = __ x __ + __ x __ - __          B.E. of nucleaus = 931.5 x __ MeV          When the neutron and proton combine, the nuclear force pulling them together does __ MeV of work.      Now, can anyone tell me what the correct answers are to the blanks? I'm just a web developer and I have to write a program which checks that the user filled in the correct answers, but I do not know the correct answers. I have some knowledge of physics and I'm guessing mp = mass of proton, mn = mass of neutron and then M = mass of deuterium nucleus, but what about the other blanks?


title: How did the scientific community receive Einstein's theories when he published them?

text: By now, we have had multiple indications through observations and experiments that Einstein's theories on general and special relativity are correct. We recently had our second observation of gravitational waves (predicted by Einstein), we haven't found anything yet which goes faster than light (like Einstein said was impossible) and we know that time dilation through speed is correct due to the practical applications in GPS satellites. But when Einstein proposed these theories a century ago, what was the reaction of the scientific community? A quick Google search indicates he was quite new to the scientific scene at the time. Were people skeptical of the theories proposed by this new scientist or was there a general "That explains a lot" moment? Compared to Newtonian physics, this seems like a radical change.


title: Tension and Newton's Laws

text: I was wondering if tension on both ends of a rope is still the same, even if there is objects with different weights on both sides of the rope.


title: Issues with Celestron 130 SLT and shaking

text: I have a Celestron 130 SLT reflector. I am having some issues with it and I hoped you all could help me. I'm looking for some low tech and inexpensive solutions.   1. My telescope shakes like crazy. I understand this is probably the cheap aluminum mount. As much as I would love to buy a wooden mount, they are out of my price range right now. Is there a low tech solution for this problem? Would a heavy pendulum hung from the center solve this problem to some extent like it does for swaying buildings? Would it put to much unnecessary strain on the mount?   2. The focuser knob on my eyepiece seem to tight. Any time I try to focus the eyepiece it torques the telescope out of position leaving me with nothing to focus on. Once I let go of the knob I have to wait for like 10 seconds before the telescope stops shaking enough that I can check the focus again. I was thinking of some WD40 to loosen it up, but then I thought it might get on the optics. Is there a better solution? Can I loosen the focuser knob up a bit? Can I use graphite on it?   3. I received a digital camera mount for my telescope this Christmas. When I set my camera on a 10 second delay (to wait for the shaking to calm down) it takes pretty good pictures. I've been using a high speed night mode on my camera and that seems to do pretty well for the moon, but when I take pics of Jupiter and Saturn they seem over exposed and out of focus. Does anyone have any tips on setting I might try on the camera that might yield better pictures?


title: What will we see between the CMB and the current oldest object seen?

text: The cosmic background radiation (CMB) is estimated to be from 13.7 billion years ago (BYA), and very shortly after the big bang compared to that time frame. The oldest coherent objects we've detected are around 500 million years after the big bang, making them 13.2 BYA. If I understand these stories correctly, that means that we have not yet detected any radiation that can be clearly identified to be emitted from a source between these two times (otherwise NASA would announce an _even older_ discovery). I don't think that that is implying that there's nothing there to see, although it's obviously the case that it is beyond our current technology to see in that window. The severe redshift and small solid angle are two complicating factors, another is that most galaxies were only beginning to for around 13.2 BYA. Regarding astronomical objects of ages between 13.2 and 13.7 BYA:   * **What is there to see?** Obviously, the 13.7 BYA point has the problem of near homogeneity but what about the times in-between? Are there cosmic structures that we could see there but no where else? Things that predate proto-galaxies? After the CMB, did matter stop radiating? Are there other near homogenous emissions we could look for? Does that make any sense? What is an accurate characterization of that time frame? Does physics make good predictions for astronomers to chase? What discoveries in that time frame are next?   * **What telescopes will see them?** The record holding earliest protogalaxies I reference are highly red-shifted light and the CMB is $cm$ wavelength. Would new discoveries in this time frame mostly be in-between those two wavelengths? Are there promising telescopes that might open up new areas of study here?


title: New Type of Type Ia supernova. Implications to Dark Energy measurement?

text: As I understand it, we used Type Ia supernovas to determine that there is some sort of repulsive force (which we call dark energy) making things fly away from us at an accelerated rate. The reason we could use this measurement is because a Type Ia always happens when a white dwarf reaches 1.4 solar masses. According to this new research, it's possible that some white dwarfs can actually reach more than 1.4 solar masses. As Dr. Plait explains > So what you get is a white dwarf with more than enough mass to explode, but > its spin prevents the supernova from occurring. For a while, that is. > Various factors slow the star down over time (for example, a magnetic field > will accelerate particles in the stellar wind, acting a bit like a parachute > dragging on the white dwarf). At some point — and this may take a billion > years — the white dwarf slows to the point where centrifugal force can no > longer win the fight against gravity. Fusion of the material begins, and > BANG! Supernova. Which would mean that a white dwarf of 1.5 solar masses could be causing a supernova. What implications does this have on the method we used to determine the effect of dark energy?


title: Gravity of a disc

text: What proof is there that the center of gravity of a disc is at its center when the gravitational field is not uniform, such as objects close to the edge of the disc, or lying within the body of the disc?


title: True Ground State Population of Ideal Bose-Einstein Condensate at Critical Temperature

text: I'm supposed to demonstrate that although we make the assumption in an ideal BEC that the ground state population follows $N_0 = N\left[1-\left(\frac{T}{T_c}\right)^{3/2}\right]$ in reality the true ground state population at $T = T_c$ is not zero, and is $N_0 = \sqrt{\frac{2\pi^{1/2}N}{\zeta(3/2)}} + \mathcal{o}(\sqrt{N})$. The hint included is to use $N = \frac{e^{\beta \mu}}{1 - e^{\beta\mu}} + \frac{V}{\lambda_D^3}g_{3/2}\left(e^{\beta\mu}\right)$. I've been racking my brain trying to figure this out. I assumed I was supposed to consider the limit of small temperature above the critical temperature. I set $g_{3/2}\left(e^{\beta\mu}\right) \to g_{3/2}(1) = \zeta(3/2)$ and I tried expanding the exponents to first order in $\beta \mu$ for small values of $\mu$ but I can't seem to figure out the correct expression. I'm using Landau and found an expression for $\mu$ which was determined by expanding about $T_c$ but it didn't seem to be terribly useful to me here either... Any ideas?


title: Why do ice cubes come out easier from top trays?

text: This is my "hey, I've noticed that too!" question for the week. If you stack two plastic ice cube trays with water in them in a freezer, the resulting ice cubes in the top tray will usually come out cleanly and easily, while the those in the bottom tray will stick and crack. (Try it!) So... why? * * * _Summarized answer, 2013-01-12_ Ice expands as it freezes. So, if freezing starts at the _top surface_ of the water, the liquid water below it allows the cube to edge gently upwards as it freezes, without much stress on it. This same upwards edging effect also keeps the ice from every gaining a solid hold on the entire surface of the tray. Result: The top surface of the ice cube stays flat, and the cube comes out of the tray easily without cracking. Conversely, if freezing starts from below at the _tray surface_ , the freezing water has the time and opportunity to lock itself tightly to all parts of that surface. But as the ice continues to grow inward, it tries to expand from all sides at once into the internal space. This creates a lot of stress in the ice, and causes the cube to form a peak at the top due to the internally expanding ice having no place else to go. So, since the ice has both had ample opportunity to freeze tightly to all of the tray surface, and because it was stressed internally as the freezing progressed, the ice cube becomes a disaster when you try to get it out. Much of it sticks to the tray, and the parts that do come out are often cracked. How is all of this connected to whether a tray is stacked on top or bottom? It's because stacked trays almost always impedes the cold air of the freezer from getting between the trays. That causes the stacked trays to behave largely like a single unit, with freezing occurring from the _outside_ of that two-tray unit. The top tray therefore freezes mostly from the top down, producing ice cubes that have flat surfaces and come out easily. The bottom tray freezes mostly from the bottom up through the surface of the tray, and so ends up producing peaked ice cubes that stick to the tray and crack when you try to remove them. (See the next addendum for some actual experiments you can do to try to confirm this.) (Ron Maimon's was the first one to figure this out; see his accepted answer for details.) * * * _Addendum 2012-12-04:_ Here's an experimental prediction based on Ron Maimon's answer:   1. If the ice cubes have flat tops, they will come out easily. Conversely:   2. If the ice cubes have central peaks, they will crack badly when you remove them. See the answer summary above for why. Based on the answer, this experiment should make ice cubes come out easier: Tape an empty tray to the bottom of the tray with water in it. This will create an air-tight insulating air pocket below the filled tray, and so should result in flat-topped ice that comes out of the top tray easily. You can also try for he opposite result by taping a tray _upside down_ over the top of the tray with water in it. When it is done freezing, the ice cubes should be strongly peaked and should stick and crack when you try to remove them. So, anyone interested: Get crackin'!


title: How to derive energy expressions thinking of it as a conserved quantity only?

text: By now I understand that "energy is a conserved quantity" and that's all we need to know. Then, the idea of work comes as the change in kinectic energy of a system and we realise that having energy is being able to do work, but still remembering that the most basic and important fact about energy is it's conservation. Now, my doubt is, there are many expressions for the total energy of a system, depending on the situation. For example, in classical mechanics the total energy of a system composed by a particle with trajectory $\gamma : \mathbb{R}\to\mathbb{R}^3$ is: $$E(t) = K(t)+U(\gamma(t))$$ where $K(t) = m|\gamma\dot(t)|^2/2$ and $U(\gamma(t))$ is the potential energy along the trajectory. Now, how one might deduce that the conserved quantity is $E(t)$ and even further deduce that $K(t)$ must have this form? I mean, starting just with the assumption that energy is a conserved quantity and using Noether's theorem to grant this quantity does exits, how one can show it has this specific form and find the formulas for it? Thanks very much in advance.


title: Equilibrium of Force Systems including Torque

text: please help me to solve this problem.. ![enter image description here](http://i.stack.imgur.com/2z1Gz.png) this is only the #3 on my homework and the only thing i didn't know here is how to calculate the tension T. please teach me how to solve the tension here because the future problems that will be given to us is related in finding the tension or compression. thank you!


title: Regarding the centrifugal force in a 3-body system

text: In our lecture about general mechanics, we have treated the simplified 3-body system consisting of the Sun, the Earth and the Moon. We assumed that the moon does not have any influence on the Sun or the Earth and wanted to find its equation of motion. Let us consider the frame of reference of the Earth as illustrated in this sketch: ![](http://i.stack.imgur.com/dsMIr.png) Denote by $M$ the Sun's mass and by $m$ the Earth's mass. In the lecture notes, it is stated that gravitational and centrifugal force of the Sun and the Earth have to cancel each other which leads to $$G \frac{mM}{R^2} = \frac{mM}{M+m}R\omega^2 \approx mR\omega^2.$$ I don't really see what is happening here. The LHS is just the gravitational force, which is clear. Which centrifugal force does this have to be equal to? If we look at the centrifugal force of the Earth, if I'm not mistaken, it should be zero, since in general it is expressed by $-m \omega \wedge (w \wedge y)$ and $y = 0$. However, if we look at the Sun's centrifugal force, then the mass should be $M$ and not $m$. What centrifugal force are we looking at exactly? Why are we plugging in $\frac{mM}{m+M}$ for the mass in the general expression?


title: Does gravitational lensing violate Fermat's Principle that light must travel in straight lines?

text: Does bending of light due to warping of space violate Fermat's Principle or is it that in the principle light goes in a straight line with respect to space (taking space as the reference) and in Relativity that reference itself is bending and so does the light?


title: General relativity: Induced metric and Killing vector fields

text: Assume that in spacetime ($M,g_{ab}$) there is a hypersurface generated by a set of independent one-parameter transformations acting on one single point, the generators of these transformations being Killing vector fields in $M$. Questions: Are they Killing vector fields in the hypersurface mentioned above, possessing the induced metric of $g_{ab}$? And how to prove?


title: Adjoint of Gamma Matrices - Dirac

text: I just started to learn how to quantise Dirac field. Meanwhile, as we can write the Dirac equation in terms of gamma matrices : $$ (i\hbar\gamma^\mu\partial_\mu - m)\psi = 0 $$ where $\gamma_\mu$ matrices obey Clifford algebra $$ \\{\gamma^\mu,\gamma^\nu\\} = \eta^{\mu\nu} $$ Now I just came across that the adjoint of the gamma matrices can be written as : $$ \gamma^{\mu\dagger} = \gamma^0\gamma^\mu\gamma^0 $$ I already checked this question, but it doesn't suggest a way to prove it in a representation independent manner. Also the Wikipedia page suggests that gamma matrices are chosen such that they satisfy the above relations, since they are arbitrary upto a similarity transformation. So is that the way this identity comes, or is there something else ?


title: Does van Eck phreaking really work, or is it an urban myth?

text: Van Eck phreaking, the ability to reconstruct distally the text on a CRT or LCD screen using the leaking em from the target computer, was in the news about five to ten years ago. It is talked about as if it is easily done, but I see little hard confirmation. The NSA has something called project TEMPEST to study this, but I think most of it is classified. Does anyone know how good van Eck phreaking is, or is it mostly hot air? This question also leads to attempts by neuroscientists to read brain states to control movement etc (ie flying a plane by thought, using brainwave em to control machines (paraplegics etc)...some progress has been made.


title: Can the work done between two non-equilibrium states be calculated?

text: The work done during a process between two equilibrium states can be described by thermodynamics. Even when process itself is out of equilibrium, the thermodynamic laws can still be used, though calculating the work is much more difficult. But if the initial or final states, or both, are not in equilibrium, can the work done in driving the system from one to the other be calculated? How? **Edit:** @Roy @genneth @Marek I do mean far out of equilibrium. There wasn't a nonequilibrium tag, and I don't have enough reputation points to create one! I can't seem to post comments today, so I'll write this here instead: I'm familiar with Jarzynski's equality and while it is very useful, it is still only valid between equilibrium initial and final states (though the final state being in equilibrium can be relaxed). I'd like to know if there's any way work (or heat) can be defined or calculated when the initial and final states are out of equilibrium, possibly very out of equilibrium. **Edit2:** @Roy I'd like to know if it's possible in a system with an initial state which is not in equilibrium which is then driven to a final value which is still not in equilibrium. I don't want to make any other assumptions if possible. So I don't necessarily expect local equilibrium to hold, although I'd still be interested to know if work between the 2 nonequilibrium states can be found in that case.


title: Can entanglement with an inaccessible system be useful?

text: Quantum phenomena in bipartite pure state systems like teleportation are pretty well understood. What I'm interested in is the following situation: Alice, Bob and Charlie hold some general tripartite pure state $\rho_{ABC}$. As usual, Alice and Bob are trying to perform some task, and are allowed only LOCC. Charlie's subsystem, however, is completely "inaccessible", by which I mean that Charlie can't do anything at all- no local operations and no classical communication with either of the other two. His role is solely to hold onto his part of the system. So my question is, is there any situation where Alice and Bob, with full knowledge of the tripartite state, would be able to perform some task 'better' (e.g. with a greater probability of success) with this setup than they would if they just traced out Charlie's system and were left with $\rho_{AB}$? If the answer is yes, then I'd be interested to know of any general results in this area.


title: Does Saturation velocity in semiconductors have a relation with the wavelength in which the peak in the absorption spectrum occurs?

text: > Saturation velocity is the maximum velocity a charge carrier in a > semiconductor, generally an electron, attains in the presence of very high > electric fields. (source) I want to know if the absorption peak can be calculated with this parameter when a semiconductor like GaAs is under illumination.


title: Do atoms and molecules affect light rays

text: Molecules of air are all around us all the time. If so, during daylight do rays from the sun diffract as it passes through molecules in the air? and if so is this diffraction negligible to be noticed? plus does this affect anything? Molecules move at high speeds and at random directions so their diffraction effect must be low, however as their are an infinite number of various air molecules in the atmosphere certainly there must be _some sort of effect_ Also how and why is the skies blue **fixed**? _(molecules don't stay at the same place they move randomly)_ and why isn't the air below the stratosphere seem blue? are there any required criteria's for the diffraction by molecules to be noticeable?


title: Equivalent spring-constant for infinite square grid of springs

text: Consider an infinite square grid, where each side of a square is a spring following Hooke's law, with spring constant $k$. What is the relation between the force and displacement between two points? If they are proportional, what is the equivalent spring constant between the origin and the point $(x,y)$ (integers) ? **Edit 1** : I also want to know this: Suppose you make the springs so small that this can be treated as a continuous sheet, at what speed will a wave propagate? Assuming a wave starting as an initial displacement perpendicular to the sheet. Given some initial state, is there an equation for the time-evolution of the continuous sheet? **Edit 2** : Suppose there is a mass at every node, and its $(x,y)$-coordinates is fixed, it only vibrates out of the plane. Consider that we take the continuous limit, such that we get a 2D membrane of mass density $\mu$.   1. Is the membrane isotropic?   2. Suppose we use another tiling (like hexagonal) before taking the continuous limit, will this sheet behaves the same way?   3. If not, but they are both isotropic, how does one characterize their difference, can they be made to behave the same way by changing the spring constant $k$?   4. What is the equation of motion for the square sheet with spring constant $k$?   5. What is the equation of motion for the square sheet if the springs obey a generalized Force law, $F=kx^n$, where $n$ is a variable.   6. What is the equation of motion for a 3D cubic grid? I am particularly interested in answers to 1., 2. and 3. I dont expect anyone to answer all these and will also accept an answer which does not explain anything but simply provides a good reference.


title: How long a straw could Superman use?

text: To suck water through a straw, you create a partial vacuum in your lungs. Water rises through the straw until the pressure in the straw at the water level equals atmospheric pressure. This corresponds to drinking water through a straw about ten meters long at maximum. By taping several straws together, a friend and I drank through a $3.07m$ straw. I think we may have had some leaking preventing us going higher. Also, we were about to empty the red cup into the straw completely. ![alt text](http://i.stack.imgur.com/VCLN5.jpg) My question is about what would happen if Superman were to drink through a straw by creating a complete vacuum in the straw. The water would rise to ten meters in the steady state, but if he created the vacuum suddenly, would the water's inertia carry it higher? What would the motion of water up the straw be? What is the highest height he could drink from? Ignore thermodynamic effects like evaporation and assume the straw is stationary relative to the water and that there is no friction.


title: How large is the smallest object that can be detected at a given wavelength?

text: What is the cross section of the smallest object that can be detected with say visible light ($\lambda$ ~380 - 750 nm) or X - band radar ($\lambda$ ~20 - 50 mm). Does the object need to have one side larger then $\frac{\lambda}{2}$ or something like that to get a reflection?


title: Cheetah prosthesis efficiency

text: I want to compare the much-talked about Cheeta running prosthesis to a the normal running process in terms of force and energy, but I don't know where to start. How would you start the comparison? A mathematical model would be much appreciated.


title: Oscillate a swing without touching the ground

text: Is is possible to set a swing into oscillations without touching the ground? This occurred to me while watching the second pirates movie. There is a scene where the ship's crew is suspended in a cage from a bridge in between two cliffs. They escape by swinging the cage towards one of the cliff. Is that even possible? **Update:** From the answers, it is clear that it is possible to make the swing oscillate. Assuming the model Mark has proposed would there be limits to much you can swing? Is it possible to quantify it?


title: Why isn't $F = \frac{\partial \mathcal{L}}{\partial q}$?

text: If momentum is, $$p = \frac{\partial \mathcal{L}}{\partial \dot{q}}$$ and force is, $$ F = \frac{dp}{dt}$$ and by Euler-Langrange equations, $$ \frac{d}{dt}\frac{\partial \mathcal{L}}{\partial \dot{q}} = \frac{\partial \mathcal{L}}{\partial q}$$ then why isn't force, $$ F = \frac{\partial \mathcal{L}}{\partial q} $$ Am I getting my definition of momentum wrong? Or am I wrong about force being the time derivative of momentum?


title: Simple pendulum. quick question

text: ![enter image description here](http://i.stack.imgur.com/8AfGU.jpg) I was trying to find an equation to find $T$ and $\omega$ for a simple pendulum when in an elevator while the elevator is accelerating. One scenario is when it accelerates in the positive up direction and another in the negative down direction, and another when it accelerations at $g$ which is also negative. $$\begin{aligned} T =& \frac{2 \pi}{\omega} \\\ \omega =& \frac{2 \pi}{T} \\\ T =& 2 \pi \sqrt{L/g}. \end{aligned}$$ $L$ is the string length. Is this right?


title: What's with Mandelstam's argument that only linear regge trajectories are stable?

text: While thinking about how to answer a "describe string theory" question, I remembered an old argument of Stanley Mandelstam's that linear Regge trajectories implies stability. I never fully understood the exact argument, or its limitations. (The article is "Dual-Resonance Models" from 1974, sciencedirect.com/science/article/pii/0370157374900349, thanks for digging it up) Here is the argument as I remember it: consider the Regge trajectory function $\alpha(s)$, and expand it in a dispersion relation with two subtractions: $$ \alpha(s) = b + as + {1\over i\pi} \int_0^\infty {\mathrm{Im}(\alpha(s'))\over s-s'} ds'$$ Where a,b are constants. Mandelstam says that the imaginary part of $\alpha(s)$ is a measure of some sort of instability or decay of the string states, so if the string resonances are exactly stable, then the imaginary part is zero, and the trajectory is linear. This argument bugged me for these reasons   * It seems to work with no subtractions, with one subtraction, with two subtractions, etc. Can you conclude that exactly constant, exactly linear, exactly quadratic Regge trajectories are also stable? Maybe exactly constant trajectories at special values can be interpreted as free stable point particles of a given spin and mass, but what are quadratic trajectories exactly? Does he mean to argue that nearly linear trajectories are necessarily stable when they are exactly linear? What is the precise conclusion?   * The Regge trajectory function appears in an exponent in the scattering amplitude, so you have to take a log to extract it. Why you don't get a cut contribution from taking the log of negative values which has nothing to do with physics, just from the Regge scattering ansatz?    * Actual scattering amplitudes are combinations of different trajectory contributions, so why does each individual trajectory have to be separately analytic, with singularities that are determined by physics? Is this a heuristic assumption?   * Ok, even if you don't have cuts from the log, the expansion suggests that an imaginary part of the trajectory function, like the imaginary part of a two-point function, has some sort of physical interpretation that allows you to interpret it immediately as some sort of decay rate. What is that physical interpretation? Does it work far away from linear trajectories?   * How does this argument translate to modern string theory? I have never seen this kind of argument anywhere else. Mandelstam is still around, but I don't know him, and I think that someone else who is better versed in Regge theory than I am will know the answer immediately, because Mandelstam states it in two lines, without explanation, so it must be obvious. Later Comment: Having received a short cryptic answer from Mandelstam regarding this, I started thinking about it again, and I gave a partial answer below. But I am not really satisfied with it yet. But one of my complaints, though, that separating out the individual trajectory and expanding it by analytic dispersion relations might fail, is not really serious. The trajectory exchange amplitude is the sum of the resonances on the trajectory, so it is always separately analytic. Why it's log should also be analytic to justify the dispersion expansion is probably equally easy to see, but I don't see it


title: Air vs Glass refraction coefficient

text: I have a rather strange question, let's say I can play with the temperature and atmospheric pressure conditions in a sealed room. What will I need to do to make the air's refraction coefficient **greater then** the glass' coefficient? ![enter image description here](http://i.stack.imgur.com/CsjOu.gif)


title: Estimation of accuracy of measurements?

text: When stating an estimation of the accuracy of my measurements (of electric current, in this case) would I state that it is equal to the residual error, or the limit of reading? (in this case, ±0.2 or ±0.01). Why? This is for pretty simple high school physics - so nothing too complex. Thanks!


title: Schematic expression of the Schrodinger equation

text: it would be great if someone could help me understand the following quote regarding wavefunctions :) > "$$\psi(x)=\sum_n C_nu_n(x)+\int dE C(E)u_E(x)$$ The expression is schematic > because we have not specified exactly which form of the continuum > eigenfunctions we are supposed to use." Why is it "schematic"? What would the expression look like when we _have_ specified the "forms" of the eigenfunctions? Thanks! :)


title: Why is the magnetic field inside a solenoid constant?

text: Why is the magnetic field along the length of a solenoid constant? (preferably in relatively simple terms) Thanks!


title: Is this Landau's other critical phenomena mistake?

text: There was an old argument by Landau that while the liquid gas transition can have a critical point, the solid-liquid transition cannot. This argument says that the solid breaks translational symmetry, and it is impossible to do this in a second order transition. But this argument is subtly false. Second order transitions break symmetries, which can be discrete, like in the Ising model, or continuous, like in the x-y model. The reason Landau said it is because it is hard to imagine breaking all the translational and rotational symmetries all at once to make a second order liquid-solid point. But nowadays we know about nematics, and we can imagine the following chain of second-order transitions: fluid (I)-> fluid with broken x-y-z rotational symmetry with a z-directional order (II) -> fluid broken translational symmetry in the same direction -> broken x-y direction rotational symmetry in the y-direction -> broken y- direction translational symmetry -> broken x-direction translational symmetry Each of these transitions can be second order, and together, they can make a solid from a fluid. the question is, how badly does Landau's argument fail.   * Are there any two phases which cannot be linked by a second order phase transition?   * Are there always parameters (perhaps impossible to vary in a physical system) which will allow the second order points to be reached?   * Is it possible to make the second order transitions collide by varying other parameters, to bring them to one critical point (in the example, a critical point between fluid and solid).   * Do these critical points exist in any system?


title: "Seeing" the solution

text: I often hear people say that so-and-so [some famous physicist] can **_see_** the solution without calculation. What exactly does this mean? Is it an intuition? But how does one have intuition when it comes to say QM? How might one develop such an intuition?


title: Temperature rate of change inside a cylinder

text: I was wondering about this today, mainly for baking a cylindrical cake. Say you have a solid cylinder which is at temperature `T0` inside. Say you then move the cylinder to a warm location (like an oven) which is at temperature `T1`. ![Solid Cylinder](http://i.stack.imgur.com/4hE0W.png) **Is it possible to achieve a rate equation for`dT/dt=f(t)` where `T` is the temperature exactly in the centre of the cylinder and `t` is time?** And therefore work out the time it would take this location to reach a temperature `T2` somewhere between `T0` and `T1`. To keep it as general as possible I would prefer to assume the cylinder is height `h` and radius `r`. I've been trying to use the thermal diffusivity equation: ![Diffusivity Eqn](http://i.stack.imgur.com/XZa4O.png) But my working gets messy quickly, I figured as this is quite a simple shape it should be less complicated than some similar examples I've found. * * * EDIT: So using my notes from a differential equations course a few years back I managed to do the following: ![Working](http://i.stack.imgur.com/0YeHf.png) But I'm stuck here..


title: Terminal velocity for falling in a shaft

text: One falls slower in a mine shaft than in free air. This is due to collisions with the walls. How should one model the terminal velocity in the presence of such collisions?


title: Simple models that exhibit topological phase transitions

text: There are a number of physical systems with phases described by topologically protected invariants (fractional quantum Hall, topological insulators) but what are the _simplest_ mathematical models that exhibit topological phases? Is the toric code as simple as we can go? **Edit:** Just to be clear, I'm talking about phases meaning _states of matter,_ and not just the geometric phase that a wavefunction will pick up under parallel transport in a nontrivial configuration space. I'm looking for simple models where one can make a phase diagram, and as a function of the available couplings there is a change in some topological property of the system. (For example, in the XY magnet, neglecting bound vortex-antivortex pair formation, there is an instability at finite temperature to creation a single vortex, which is topologically distinct from the vortex-free state.)


title: Electrostatics:Basic question on electric current

text: I've got two questions for you.   1. Electric current is the flow of electrons across a conductor.Why has it always got to do with electrons and not with protons? (I know neutrons are not eligible for electricity as they carry no charge.)   2. Sometimes my computer starts properly after I disconnect all the cables and press and hold the power button for a few seconds and then re-connect all the cables (same thing with my modem). What's going on? I've heard that some amount of left over current get's drained. But where do they go? How did that excess current get accumulated in the first place?


title: Expectation of dipole is vector in x direction

text: So I am asked to find a state $|\Psi(t)\rangle$ in terms of the hydrogen wave functions, such that the expectation of the dipole operator -$e\hat r$ is a vector in the x direction. I am not completely sure how to approach this. I am having a hard time visualing what this is trying to say in a sense of wavefunctions. I know what a dipole is, but I would expect the dipole moment for hydrogen states to be a vector in two directions. Any ideas on where to start?


title: How exactly do protons and electrons interact with each other?

text: How do these particles exchange information about charge and position between themselves, even though there's mostly empty space between them? Also what happens if a free electron passes closer to a nucleus of a hydrogen atom than its own electron? My random guesses so far:   1. Proton won't react because it's somehow locked with its existing electron   2. Proton will catch it and release the previous one    3. Proton will catch it and keep both of them with half the force   4. Existing electron will simply repel incoming one   5. Something else?


title: Transformer and magnetic field/flux through the iron core?

text: I am familiar with how a transformer works. This is the diagram that I found on the wikipedia page:![Mangetic flux in a transformer](http://i.stack.imgur.com/9ccpO.png) The current through the red solenoid varies with time, which creates a time- dependent magnetic flux (green) that also goes through the blue solenoid therefore inducing an electromotive force. Both solenoids are wrapped around a magnetic object so that this "carries" the magnetic flux created _inside_ the solenoid (which is much bigger than the one outside of it) around. I am not quite sure why the magnetic flux bends in the way displayed in the diagram. The magnetic material is ferromagnetic right? So shouldn't the magnetic field point produced inside the red solenoid force all the other domains to align with it? In other words shouldn't the magnetic field point in the same direction everywhere in the magnetic material (as I have tried to illustrate below) ?![enter image description here](http://i.stack.imgur.com/vNeEM.png)


title: Are there any musical instruments which use resonance tubes with two closed ends?

text: Many musical instruments use resonance tubes with one closed end - all brass instruments (I think), clarinet, etc. There are also instruments where both ends are open (flute, pipes) Are there any which use two closed ends, that is, a resonance tube with nodes at both ends? I am aware of the problems which would exist for such an instrument - essentially not be very loud - but I am just curious if anyone knows of any.


title: What causes the Fermi Tail in Photoelectric effect?

text: I did a Photoelectric effect experiment, with this setup (schematic):   ![enter image description here](http://i.stack.imgur.com/tGhRb.jpg)   **Figure 1 Scheme of the setup**      The kathode is made of Potassium. Light is passing through the monochromator and on the kathode. Which releases electrons out of the kathode. In this experiment, I tried to find the voltage that is needed to stop the electrons from moving, in other words; change the voltage until there is not current. The current was measured by a Keithly Electrometer and resulted into this: ![enter image description here](http://i.stack.imgur.com/w6Z1C.jpg)   **Figure 2 Measurements with different wave lenghts**      You would expect that it would be lineair. But as you can see, it isn't. This effect is called the "Fermi Tail". But why is there a so called fermi tail? What is causing this effect?


title: Excited Energy levels of Hydrogen vs Solids

text: My question has to do with Excited Energy Levels. I keep reading and learning that all objects/most solids emit infrared radiation. But to emit radiation, an object's particles must first enter a higher energy orbital (electrons) which will then immediately lead to the atom returning to its ground state, releasing the energy of the level jump from ground state to level 2 in eV in the form of a photon wave. BUT! When I looked for information about energy levels of elements, I could only find hydrogen, and when I plugged in the energy for: l=hc/E where l is wavelength, h is Plancks constant, c is the speed of light, and E is energy of the photon. The wavelength came up to be 1.121x10^-7! That is in the ultraviolet range! That means that it takes a ton of EM Energy to emit a photon wave! This doesn't make sense unless solids require much less energy to move up orbitals. Is this true? What happened to Infrared or Radio? And how do I find out about energy levels of solid elements and understand the data?


title: Uncertainty Definition QM

text: On my introductory course in Quantum Mechanics, the uncertainty of an operator $A$ in the state $\psi$ is defined by $$(\Delta A)^2_{\psi}=\langle(A-\langle A \rangle_{\psi})^2\rangle _{\psi}$$ I'm having trouble extracting meaning from the R.H.S. If $A$ is a self-adjoint operator then if the underlying hilbert space has finite dimension, $A$ is a matrix. However, $\langle A\rangle_{\psi}$ is a number. How could one be subtracted from the other. I think my lack of understanding stems from my lack of knowledge about operators. For example, I know that there is something called the momentum operator $p$, but to my knowledge it is not a matrix; $p=-i\hbar\frac{\partial}{\partial x}$. Can anyone clarify?


title: 2-body interaction energy of 3 particles

text: If only two-body interaction is considered then what's the energy if I put three particles on one site? Assume delta-interaction and interaction strength is proportional to scattering length $a_F$ which depends on the total spin of the two-particle state. For two particle case, the interaction is simply $\hat U |F,M\rangle = U_F |F,M\rangle $ Suppose the three particles are fermions, then I will take a completely anti- symmetrized spin wave function. But now, how to express the interaction energy in terms of two-body scattering lengths?


title: Magnetic Pole Emission from Pulsars

text: I've been reading up on some pulsar emission theory (admittedly written in the 70's and 80's, but I figured that's a good place to start), namely this review by Curtis Michel as well as the book _Black Holes, White Dwarves and Neutron Stars_ \- Shapiro and Teukolsky. They both mention that a pulsar predominantly emits radiation from it's magnetic poles, even though it has a dipolar magnetic field. In the case of a charged plasma-filled magnetosphere, I understand that the charged particles (electrons/positrons) are pulled out from the surface due to the large electric field, and this causes them to radiate in the direction of the magnetic pole, but why don't they radiate while circulating around the magnetic field lines? Surely since the field lines are curving themselves, they will provide acceleration to the charged particles causing them to radiate? Or am I missing something?


title: Would a laser with four possible energy levels be better than three?

text: I'm wondering about achieving population inversion for a laser. I learned that without an active medium, it's not possible to create a laser with only two energy levels, but it would be possible with three. What is the advantage of having four levels then?


title: Water vs Milkshake being sucked through a straw

text: Consider water in a glass being sucked through a straw. The water rises up in the straw because of a pressure gradient introduced by the sucking action. Now, change the liquid from water to something thicker like a thick milkshake (higher viscosity and higher density). It would need a greater amount of suction to raise the level of the slushy/milkshake in the straw to that achieved by using water. How should I explain this ? Can this be explained with the Bond number (gravity/capillarity ratio) or should I explain this phenomena based on the Capillary number (viscosity/capillarity ratio)? Can/Should the velocity of the liquid flow in the straw be calculated with the Hagen-Poisseuille's equation?


title: Young experiment: square of classical real wave function

text: I can't understand why the sum of two real waves result in a time dependent wave, but not so for the complex waves. In details, I can't get this passage on p.38-39 in A.C. Phillips, _Introduction to Quantum Mechanics:_ $$\tag{3.9}\Psi ~=~ A\cos (kR_1 - \omega t) + A\cos (kR_2 - \omega t)$$ $$\Downarrow$$ $$\tag{3.10}\Psi^2 ~=~ 2A^2\cos^2\left(\frac{k(R_1-R_2)}{2}\right)\cos^2\omega t.$$ This is driving me crazy!


title: In what way do Cooper pairs of electrons bond and stay bonded in superconductors?

text: I understand how electrons initially move into another's vicinity, but nowhere can I find a fathomable answer to this. Also, does the pairs forming 'a condensate' mean a Bose-Einstein condensate?


title: Will observers moving on a sphere experience time dilation?

text: A single source of light exists at a fixed point in space relative to two observers. The two observers move on the surface of a shell with a fixed radius with the light source at its centre. They move along a circular path on the sphere. They move at different velocities u and v. Will they experience time dilation relative to each other? If relativity says that there will be time dilation on the shell of a sphere with a fixed radius with a light source at its centre, then there is a problem. Let's say the light blinks once every 10 secs. Then that light will reach all observers on the shell at the same time no matter what their relative velocity to each other. All observers would agree the light blinks once every 10 secs, no matter what their velocity is as long as they are on the shell.


title: What is a photon's speed inside a dieletric?

text: We know that EM waves are slowed down in a dielectric. But at what speed does the photons that make up the wave travel? Do they always travel at the speed $c$, but colliding/being absorbed and re- emitted by the molecules in the medium causes the photons' average speed to be reduced? Or does the electromagnetic field created by the polarization in the material somehow affects the photons at a quantum level to give them mass? If they do acquire mass, would the formulae $E = \hbar \omega$ and $E=\gamma m c^2$ work? I suppose it's not absurd that an electromagnetic field gives a particle a mass – there is, after all, the famous Higgs field – but what is special about the electric field produced by a polarized dielectric? For example, the electric field of a point charge doesn't slow light down.


title: Does a material exist that reduces a magnetic field without being affected by the magnetic field itself?

text: Consider a common bar magnet, `magnet 1` resting on a surface with its `North` pole facing up. Suspended distance $y$ above it (supported side-to-side by a plastic tube) is a second, smaller bar magnet, `magnet 2`, with its `North` pole facing down. The magnetic forces between them exceed the force of gravity, and keep `magnet 2` suspended. Consider some material, `material-X` that is moving towards the gap between the two magnets at an initial velocity $v$. Does a material, `material-X` exist that would reduce the distance $y$ between the two magnets, and pass through the gap without changing velocity $v$? ![3D mockup](http://i.stack.imgur.com/oUy1a.png)


title: What electric potential is found over individual resistors connected in series to an AC power supply?

text: Given a set of resistors connected in series to an AC power supply:   1. What formula governs the peak-to-peak voltage which will be measured when voltage is probed over individual resistors and sets of resistors in that circuit?   2. What is a good way to think about this construction so the answer to question #1 is obvious and intuitive? Intuitively, I would think that the measured voltages would be the same as in a DC circuit such that:               V_out = (R_partial/R_total) V_in      Where V_out is the voltage which is probed, R_partial is the total resistance of the sub-set of resistors which are being probed and R_total is the total resistance in the circuit. This formula makes sense intuitively because the current throughout a circuit of series components is everywhere constant and according to Ohm's law for a constant current, the higher a given resistance over an area the higher the voltage potential measured there. Is this correct for AC circuits as well?


title: if the universe is flat does it mean it exists only in our mind as math?

text: > My dad, who is a retired mathematician, has this attitude, which I think we > all have as kids, that ultimately reality is made of stuff. End of story. If > you look around yourself in the world, you don’t see anything that’s > mathematical, right? But physicists have discovered that all this stuff is > made out of elementary particles, like quarks and electrons. What properties > does an electron actually have? It has the properties -1, ½, and 1. These > are properties that we physicists have made up geeky names for, like > electric charge and spin and lepton number. But they’re just numbers! > They’re just mathematical properties! > > So all these building blocks of nature, these particles, actually have no > properties at all, except for mathematical properties. So in that sense, > they are purely mathematical objects. Classic materialism is dead: “stuff” > isn’t the end of the story. > > And the same deal with the fabric of what’s around us, space itself. It has > the property three, the number of dimensions. That’s a number. That’s a > mathematical property. All this space and the stuff in space is purely > mathematical with only mathematical properties. Max Tegmark - cosmologist


title: Time to establish saturated vapour pressure above liquid

text: Thought experiment - a liquid is in a closed container in equilibrium with its vapour, and then suddenly all the vapour is pumped away. Switch off the pump so that instantaneosuly there is no vapour present. How long does it take for the equilibrium vapour pressure to re-establish from the time the pumping is stopped?


title: Quasi-Degenerate Perturbation Theory

text: Is there any sort of objective rule on when to use quasi-degenerate perturbation theory vs degenerate vs non-degenerate?


title: Why Are Even and Odd Regge Trajectories Degenerate?

text: The Gribov-Froissart projection treats even angular momentum differently from odd angular momentum. But in QCD, I believe that the odd trajectories interpolate the even trajectories--- the two trajectories are degenerate. Is there a way of understanding why the even-odd trajectories are degenerate? Is is a symmetry argument? Can you find a natural system where they are not degenerate? A quick google search revealed this reference, which doesn't answer the question, but gives an experimental signature for the degeneracy: http://www.slac.stanford.edu/cgi-wrap/getdoc/slac-pub-0576.pdf , There is probably excellent data by now on this.


title: Does time actually exist or is there just a single stateful system being updated over time?

text: What I mean is, is there a past, future and present? or just a NOW which changes? I'm a computer programmer so the second alternative is from my programming experience. In this second alternative, I mean to ask, are we just in a stateful system like Conways game of life where there is just a NOW and then the system updates and then we are in a different NOW. In the conways game of life, there is never any past stored in the computer memory. In this case, the past is just an illusion from us recording events which have happened in our brain. There is a distinct difference between the two, if there is actually a past, future and present then there is opportunity for things in the future to affect things in the past or time-travel might be possible. However in a stateful system like the game of life, then that really cannot happen. Secondly, do any experiments confirm or deny either of these two options, is this already a decided problem in physics? Conways game of life * * * Updated: A system with a "real" (real as I describe) past, present and future could allow for travel in the time dimension just as you would travel in the x,y or z dimension. One person could travel slowly backwards in time to a point in the past, then move forward slowly and see events as they happen slowly. Not everyone is moving "forwards" together like in conways game of life. In a conways game of life, everyone is moving forwards. In a system with a real time dimension, potentially a person could make a machine to move backwards, which you cannot do in conways game of life, you cannot move backwards to the older state, you cannot see time in reverse as well which you would see if you moved backwards as you would with a film playing backwards or sound recording playing backwards. The rules of the conways game of life wouldn't permit travelling backwards, so there's no real time dimension to it, only a collective NOW. The distinction between a system with a real time dimension (as I describe) would be that you can move forwards or backwards in the time dimension. In a world without a real time dimension there is only conways game of life where everyone is moving forward and there is only a NOW at all times. Now you could say that the system which I described with a real time dimension could be simulated by a suitable conways game of life, a suitable rules based system, but I think that you could make a distinction between the two, just as you can determine there is a real x dimension in 3 dimension space which is not a result of a conways game of life in the x dimension. So my question is, is there a real time dimension or is there just a now that everyone is at all the time?


title: "Completeness" of eigenvectors in a complete, commuting set

text: This question was originally the one below dashed line. Now after further discussions, it has boiled down to this question: Is the following construction possible? Suppose we have a 3 dimensional ket space in which the basic vectors are simultaneous eigenvectors of two observables $\xi_{1}$ and $\xi_{2}$ such that they can be represented as $|a,b\rangle , |a,d\rangle , |c,b\rangle$ where 'a' and 'c' are eigenvalues of $\xi_{1}$ and 'b' and 'd' are eigenvalues of $\xi_{2}$. Clearly, $\xi_{1}$ and $\xi_{2}$ form a complete, commuting set. Or is it necessary that $|c,d\rangle$ eigenvector must exist? If this is the case then in 3D ket space, two observables can never form a complete, commuting set and the same thing can be generalized to any prime dimensional ket space. This leads to the conclusion that any prime dimensional ket space has either 1 observable having all distinct eigenvalues or it has p observables forming a complete commuting set each one having only one eigenvalue. And this thing is quite hard to believe unless there is a nice argument which can account for these weird conclusions coming from this. If the above construction holds and $|c,d\rangle$ can be missing then I would like to ask as to why do we take n-fold integrals while writing the representation of an inner product of a ket and a bra when the set of eigenvectors forming complete, commuting set has continuous eigenvalues? The argument against it being that as some eigenvectors can be missing, the limits of each integral are dependent on other integrals and hence the notion of n-fold integrals breaks down. Another possibility is that we consider those missing eigenvectors as 0 vectors with the breaking of integral at discontinuity points being understood. Is it so or is it the former way if the above construction is correct and valid? The text below the line and comments to the question may help in giving the context of the question more clearly. * * * On Page 65 of his book, Dirac argues that we can generalize the results mentioned previously by taking a $u-v$ fold integral of a particular inner product mentioned there. I have a problem that it can't be done as $\xi(v+1), \xi(v+2) ... \xi(u)$ may not be independent. And hence it's not necessary that for a fixed value of $\xi(v+1)$ there would exist eigenvectors as we vary other $\xi's$ and so the integral can't be formed as the limits of other integrals vary as we go on evaluating a particular integral. As a counter-example in a discrete case, consider two observables which form a complete set with eigenvectors as $|a,b>, |a,d> and |c,b>$. Here, there is no eigenvector $|c,d>$. I hope that I make myself clear that while passing on to continuous case, we might get same circumstance in which the notion of $u-v$ fold integral breaks down. May be I have largely wrong intuitive concepts as to when a multiple fold integral can be taken. If so, please rectify me. If the above counterexample is correct, then how would you justify the generalization that Dirac takes?


title: Modellng mechanical behavior of heat shrink film

text: Consider a heat shrink film (as used in shrink sleeves that decorate plastic or glass bottles). These materials are produced by blow extrusion. When the film is heated (hot steam, hot air or radiation) the film will retract to a smaller size, typically anisotropic (shrink more in one direction than in the other). Empirical data is available for unconstrained samples of film. However in real life situations there are constraints to the movement of the film and the system will look for a balance of thermal shrink and elastric stretch forces. What would be a practical model for the macroscopic mechanics of such film? This model should provide internal forces (shrink and stress forces) for a given temperature, strain and history of an small area of film. I wonder how related this is to visco-elasticity? VE can be modeled as a system of springs and dampers. Because of the damper(s) the forces also depends on the history. Does it make sense to model heat shrinking as visco- elasticity with temperature-dependent springs and dampers?


title: What exactly is the weak portion of the SM gauge group?

text: This Wikipedia article: http://en.wikipedia.org/wiki/Left%E2%80%93right_symmetry states that the weak part of the SM gauge group is not $SU(2)_L \times U(1)_Y$ but $ \frac{ SU(2)_L \times U(1)_Y}{\bf{Z}_2}$. If this is correct, what is the significance of the $\bf{Z}_2$ ?


title: Parity and Helicity of the Higgs Boson

text: I have been studying how the spin and parity of the new boson discovered at the LHC will be studied and have run into some confusion. The Standard Model Higgs is expected to be a scalar (i.e. have even parity with spin 0). My question is how is the intrinsic parity of a particle experimentally determined? Is it related to the helicity amplitudes? Why is parity even an important quantum number to determine?


title: What percentage of the sky is occluded by stars?

text: If you drew rays from the center of the earth out to infinity at every angle, what percentage of them would intersect a star? Extra details:   * Assume the rays are mathematical rays, or that they travel at infinite speeds.   * Even in an infinite universe, since the apparent magnitude of stars decreases with distance, it is conceivable that the area occluded is less than 1, i.e. that the total magnitude of the stars at a given distance approaches zero as the distance increases.   * Assume a snapshot of the universe as it currently exists.


title: Transmission of Gaussian Beam Through Graded-Index Slab

text: The $ABCD$ matrix of a glass graded-index slab with refractive index $n(y)=n_0(1-\frac{1}{2}\alpha^{2}y^{2})$ and length $d$ is $A=\cos(\alpha d)$, $B=\frac{1}{\alpha}\sin(\alpha d)$, $C=-\alpha \sin(\alpha d)$, $D=\cos(\alpha d)$ for paraxial rays along the z axis. Usually, $\alpha$ is chosen to be sufficiently small so that $\alpha^{2}y^{2} << 1$. A Gaussian beam of wavelength $\lambda_0$, waist radius $W_0$ in free space, and axis in the z direction enters the slab at its waist. How can I use the $ABCD$ law to get an expression for the beam width in the $y$ direction as a function of $d$?


title: does AdS/CFT implies that there is an CFT in physical horizons?

text: so my rough understanding is that the AdS/CFT duality is some sort of isomorphism between an N dimensional gravitational theory and a N-1 dimensional conformal field theory on the boundary. The boundary of our 3+1D spacetime is the union of the cosmological horizon plus the event or apparent horizons of black holes in it, which are 2+1D spaces So my question is: does this imply that there are some conformal fields evolving in those 2+1D physical boundaries at this very moment, that map one- on-one the evolution of the 3+1D spacetime? like some sort of crystal ball? can you influence what happens in the 3+1D space by affecting what happens in those 2+1D boundaries? I intentionally made this question as wild as possible so we mere non-stringy mortals can know where we stand with all this duality business


title: Relationship between boiling point and pressure

text: In the bubble chamber, liquid hydrogen is kept at high pressure. When a stream of charged particles are injected into the bubble chamber, pressure on the liquid hydrogen is released and bubbles of gas are produced around the charged particles. However my question is that:                  why would the boiling point of hydrogen decrease when pressure decreases?


title: Help with the Heisenberg relation in Gaussian wave

text: In short laserpulses there is a minimal product of the frequency width and the pulselength for Gaussian pulses $\tau \cdot \Delta\omega \geq4\ln2$ this is the fourier boundary. So I know it origins from the Heisenberg uncertainty principle, but how can I derive it?


title: Torque due to a magnetic field

text: We have two infinite perpendicular wires, each carrying a current "i" and the distance between them is l. If both the wires are free to move, how will they arrange themselves because of the torque exerted by the magnetic field produced by the two wires?


title: Is there a conveniently small unit of volume for talking about molecules?

text: When discussing energy transformations on the molecular scale, we usually use electronvolts as the energy unit. This is handy because chemical bond energies are a few electronvolts in magnitude. I had an idea to go through some thermodynamic tables and convert them into molecular-scale units (energy in $\mathrm{eV}$, entropy in bits, etc.) in order to get a feel for how chemistry works on the scale of single molecules. However, for talking about volume changes (that is, average volume change per molecule), I can't think of a convenient unit. All of the most commonly used volume units ($\mathrm{\mu L}$ etc.) are far too large, apart from Planck units, which are far too small. Even yoctolitres are a bit on the big side, and I've never heard of anyone tabulating values in $\mathrm{yL}$. Of course it wouldn't be hard to define a convenient unit, for example by deriving it from the ideal gas law at a standard temperature and pressure. But it would be better to use a unit that people already recognise. So my question is: is there a unit of volume in common use that is of a convenient size for thinking about single molecules? That means, I guess, that it should be between around $10^{-31}$ to $10^{-27}\;\mathrm{m^3}$, and preferably towards the smaller end of that.