title: Section of Phase space

text: Suppose $\mathfrak{X}$ is the configuration space of some system on a Riemannian manifold $\mathcal{M}$. Then, the phase space is $\mathrm{T}^{*}(\mathfrak{X})$. What is a section of $\mathrm{T}^{*}(\mathfrak{X})$ as interpreted as a geometric structure on $\mathcal{M}$?


title: If a photon has no mass, how can it be attracted by the Sun?

text: I read that the photon doesn't have mass, but my teacher says that the photon has mass because the sun can attract it (like in the experiments to prove the theory of relativity). I think that there is another reason to explain that. How can I explain that the photon doesn't have mass and the sun attracts photons?


title: Difference between Urca process and $\beta$-decay in neutron stars cooling

text: I was reading about the Urca process and its importance in the cooling of astrophysical compact objects. Indeed, it is supposed to be one of the major contribution to the cooling of neutron stars (please, correct this sentence if it is not actually supported). Now, the general expression for the Urca process is: $B_1\rightarrow B_2 + l + \bar{\nu_1}$ Where $B_1$ and $B_2$ are baryons and $l$ is a lepton. This is, in my understanding, the general expression for $\beta$-decay: $n\rightarrow p + e^- + \bar{\nu_e}$ Since the neutron star is made mostly by neutrons, why is the Urca process indicated instead of the $\beta$-decay ? In other words, what else is going to contribute to neutrino emission and so to the neutron star cooling by neutrinos? Relating to this question, also muons can be produced, but this asks for more energetic neutrons, which is a limited condition, since the particles are tightly packed in the neutron star. So, what am I missing?


title: How thick (in molecules) would a piece of paper be if it was folded to the moon?

text: Apparently you can fold a piece of paper 42 times and reach the moon! 42 is the answer the everything after all! Oh, Douglas Adams, you clever son of a bitch. The reasoning behind this is that as you fold a paper, it becomes twice as thick. It's exponential and can be modeled by the following formula for a paper that is say- 0.001 cm thick. 0.001(2)^42 = space travel I want to test this. You have to figure that as you fold a paper and it becomes twice as thick, the area is also halfed. So, for a standard piece of A4 paper, being 8.5x11, would have an area of 92 in^2 or about 603 cm^2. The equivalent decrease in area can be modeled by this formula: 603(0.5)^42 Now, for the number of molecules in a sheet of paper. The number of molecules for a piece of paper consisting solely of cellulose (C6H10O5) strains can be found by using the molar mass of cellulose and the mass of the paper in question. So the calculations for a standard piece of A4 paper with a mass of 5 grams should look something like this: 5g × (1 mol C_6 H_10 O_5)/162g =3.09 × 〖10〗^(-2) mol damn stackexchange formatting And since a mole of cellulose contains 6.022 x 10^23 molecules, you would have about: 1.8586 x 10^22 molecules in that piece of paper This is where that math breaks down, I've tried to divide the number of molecules by the resulting area, but the answer doesn't seems quite right. :/


title: electron flow in capacitor with dielectric

text: if I make a circuit with a battery and a capacitor (with a dielectric inside), how it is possible to get a current in the circuit? If electrons go from one pole of the battery and they arrive to one plate of the capacitor they cannot flow to the other plate of the capacitor because the plates are separated by a dielectric and this dielectric doesn't have free charges, they are bound charges. How can I explain a net flow of electrons in this circuit? Thanks.


title: Was uncertainty principle inferred by Fourier analysis?

text: I would like to know: did Heisenberg chance upon his Uncertainty Principle by performing Fourier analysis of wavepackets, after assuming that electrons can be treated as wavepackets?


title: Integration question from book "e: The Story of a Number" by Eli Maor

text: In the book 'e: The Story of a Number', a derivation of a parachutist problem is given on pg. 109-110. A parachutist jumps from a plane and at $t=0$ opens his chute. At what speed will he reach the ground? $k$ is proportionality constant. $g$ is acceleration of gravity. $m$ is mass of parachutist. (This derivation assumes air resistance is linear with velocity.) I am able to follow the derivation up to equation 3. $$\tag{1}m\frac{dv}{dt} = mg - kv$$ $$\frac{dv}{dt} = g - av \tag{2}, \quad a = \frac{k}{m}$$ $$\frac{dv}{g-av} = dt \tag{3}$$ How does one integrate (3) to get the following? $$-\frac{1}{a}\ln|g-av| = t + c \tag{4}$$


title: Statements about gravitational potential energy contradicting intuition?

text: So in my textbook I am told: two masses have gravitational potential energy because work had to be done in order to move one the masses form a position very far away (lets say infinity) to the position near the other mass. Additionally, I found on a physics forum: "zero potential energy is at an infinite distance from the centre of a spherical object." I am lost. My intuition suggests that you have zero gravitational potential energy when the two masses are right next to each other! As gravity is attractive, you need to do work in fact get away, not to get closer! So when you are "infinitely" far away from another mass m (there exists nothing else other than m and you), you actually have very high gravitational potential energy (I know I am over-simplifying). What am I getting wrong?


title: How to cancel momentum without causing damage?

text: This is a fictional scenario I've designed and I'm trying to obtain a plausible logic for it to be somehow possible: I have a moving object, like a car or a person running, how can I stop it instantly without damaging it? I know that in order to stop it, I could apply an opposite force to cancel the momentum, but that would surely damage it? Is there any other way to cancel it?


title: What is the difference between thermal and infrared imaging?

text: Is there any sort of difference between thermal radiation and infrared radiation at least when detected by a camera? What are the differences between the two?


title: How does slow anti-hydrogen annihilate with normal matter in the lab?

text: In a recent article:   Physical Review A 83, 032903 (2011), A.Yu. Voronin, P.Froelich, V.V. Nesvizhevsky, _Antihydrogen Gravitational Quantum States_ the authors claim that anti-hydrogen has a lifetime of 0.1 seconds when placed in a container with metal walls. They say that the atom is kept from annihilating with the walls because of the Casimir-Polder effect. This raises in my mind the question of how it is that the atoms annihilate. My first guess is that the positron would annihilate against an electron, and then the proton would be attracted into the metal by the image charge effect. But this would seem to be in contradiction to the Casimir-Polder effect mentioned in the paper. So how does slow anti-hydrogen decay in the presence of normal matter?


title: estimating ion collisions from Vlasov equations

text: Suppose you have a distribution function $ f_{\alpha}( \vec{r} , \vec{p} , t)$ obtained from Vlasov equations for a certain $\alpha$ species, say some ions. Is there a rigorous way (in the domain of validity of the model, that is) to estimate a collision density function $ p^{collisions}_{\alpha}( \vec{r} , E_{CM} , t)$ , where $E_{CM}$ is a variable describing collision energy parameters? A link or reference would be great


title: infinite parallel conducting planes

text: two infinite parallel conducting planes grounded and are separated by a distance d. place a point charge "q" between the two planes, using the "green teoerma reciprocity" how I show that the total charge induced is "-q" by the product of the fraction of the total perpendicular distance between the plane and the load?


title: Amplitude of a Forced Harmonic Oscillator

text: For an assignment in one of my maths units at uni, I've been asked to derive and solve the differential equation of motion for a forced harmonic oscillator, with the forcing function having the form $F_0sin(\omega t)$, with some given properties (mass = 1kg, spring constant = 400N/m, amplitude of the sinusoidal driving force = 20N). I am reasonably confident of the solution I found; $$x(t) = -0.0245\cos(20t) + \frac{\omega}{\omega^2 - 400}\sin(20t) + \frac{20}{400 - \omega^2}\sin(wt) + 0.0245$$ Next the question tells us that the spring will fail if its extension exceeds 1 metre, and asks us what forcing frequencies will allow for safe oscillations. I formulated this condition as; $$1 \geq |-0.0245\cos(20t) + \frac{\omega}{\omega^2 - 400}\sin(20t) + \frac{20}{400 - \omega^2}\sin(wt)+0.0245|$$ Obviously this condition needs to hold for all t, so it seems to me that I need only look at the sum of the amplitudes, however $\cos(20t)$ and $\sin(20t)$ will never superimpose entirely constructively, and I cannot find an adequate way of expressing them as one trig function. How should I approach this?


title: Why didn't the control rods in Fukushima shut down the reactor?

text: With all the hubbub over the reactors in Japan and nuclear safety, I was wondering why the insertion of the control rods in Fukushima did not shut down the reactor? Shouldn't they slow the fission reaction so far that cooling is not needed? Why do they need to continue cooling the core when I would expect no more exothermic reaction to take place?


title: Earnshaw's theorem and springs

text: Earnshaw's theorem states that the Laplacian of the potential energy of a small charge moving through a landscape full of static negative and/or positive charges (and gravity) is zero. Thus you can't have stable levitation (electrostatic or magnetostatic) because that would require a "bowl" in the potential energy field (levitation uses forces such as diamagnetism to circumvent the theorem and allow objects float). What if the landscape charges are attached to springs and dampers so they move slightly when we move our charge? My intuition is that this would _always_ create a negative (mountaintop) Laplacian. If there was a case which didn't it would be exploited for the magnetic levitation since diamagnetism is very weak. Consider an infinite sheet of plane charges. Without springs the force on our + point charge is constant. But with springs, the sheet will bulge away (if +) or toward (if -) our +. Both cases create a negative Laplacian. Is a negative Laplacian true for any spring configuration? If so, is there a simple way of proving it?


title: Normalizing continuous eigenstates

text: As far as I understand, to normalize the eigenfunctions, corresponding to the continuous spectrum, we use Dirac delta function: $\langle \psi_\lambda \mid \psi_{\lambda'} \rangle = \delta(\lambda - \lambda')$ However, I don't get how does that work if the eigenvalue $\lambda$ is degenerate. This is the case for the free particle Hamiltonian $-\frac{d^2}{dx^2}$ (I am omitting the constants like $\hbar$ and $m$): for every energy value $E > 0$ there are two linearly independent wavefunctions $\psi_+(x) = e^{i \sqrt{E} x}$ and $\psi_-(x) = e^{-i \sqrt{E} x}$. Intuitively, they should be orthogonal in some sense, however, there is no way of getting zero in the right hand side of the Dirac orthonormality condition (since $\delta(\lambda - \lambda) = \delta(0)$). What am I missing?


title: What happens if a body free-falls at a certain speed?

text: It is known that a body falling to the ground is affected by gravity, and its velocity increases by 9.8 m/s per second. But when this body is falling, and it reaches the speed of 340 m/s (the speed of sound as a limit example), So its velocity will keep increasing to larger than the velocity of sound? If so it increases, and it reaches a velocity of half the speed of light, then we know that the mass of this body would increase as it's moving, and this increase in mass will not affect the velocity as we know. but if this mass keeps increasing, and velocity keeps increasing by 9.8 m/s/s. Special relativity predicts that it's impossible for objects that have mass to reach the speed of light. so this body will not continue its increase in speed and mass. so there must be a velocity and mass limit for this body. Or else this body's mass will be larger than the mass of the planet itself. and make a curvature in space-time fabric larger than that of the planet. So There have to be a relation between the velocity of this body, It's mass, the mass of the planet, and the relative velocity of the rotation of the planet, so we can conclude the speed limit that this body cannot exceed from them.


title: What mechanism produced the baryon excess in the universe?

text: A baryon is a composite subatomic particle made up of three quarks (as distinct from mesons, which comprise one quark and one antiquark). Baryons and mesons belong to the hadron family, which are the quark-based particles


title: decay time for a blast wave

text: The blast wave that is formed as a result of an explosion has some import parameters. One of them is the peak pressure which is basically the pressure measured when the blast wave arrives at a given location. It is the case that this peak pressure decreases as one moves further and further from the center of the explosion. Ignoring the actual mathematical form, this is plausible. Now the other parameter of interest is the decay time. It is the time it takes for the peak overpressure to decay to the atmospheric pressure at a fixed point in space. One could imagine an experimentalist putting a pressure gauge at some location. When the blast wave arrives, the pressure rises to the peak pressure and after some time $t_{d}$, it decays to atmospheric pressure. This $t_{d}$ is the decay time. From my readings so far, I don't seem to have an intuitive grasp of the decay time. The books I read suggest that the decay time increases as one moves away from the source. Could any body provide me with some simple explanation as to why this might happen?


title: water surface tension and buoyancy

text: When you put a little pin on the surface of water and it floats. Is this because of surface tension or buoyancy? Can somebody also draw a force diagram for me to explain how water surface tension support an object. And anybody has any advise for me that I can do any simple experiment to demonstrate water- surface tension.


title: Normalizing Wave Functions

text: We normalize the wave function to $1$, but couldn't we also normalize it to $-i$ as $(-i)^2=1$? Does this not work? Is it equivalent?


title: Einstein's special relativity beyond the speed of light

text: Could someone with access to this paper which claims to have new transformations between frames with relative motion faster than light which are supposedly consistent with special-relativity, say what the new transformations are and why they might make sense ?


title: How is the stickiness of a fluid measured?

text: How does one compare the "stickiness" of different substances on a certain surface? For example:   * Does glue A stick better to steel than glue B?    * Does water stick better to steel than oil?


title: Mass-Energy Equivalence

text: I've asked before, but I'm still confused as to what the mass energy equivalence implies. I've taken an introductory course in relativity, so I only covered special relativity. From what I gather, all things have a rest mass. As you increase your energy, your mass increases as well, and the rest mass behaves as some potential. But then when we talk about fusion and fission, we are splitting the atoms to generate energy-breaking apart the potential stored in forces between atomic components. This kind of makes sense to me. But then are _all_ energies and masses equivalent? An electron has a rest mass, is this considered energy as well? Is the mass stored in objects I hold due to the gravitational field simply too small for me to notice? Furthermore, if mass is energy, then why do nuclei weigh less? Is some of their mass going into the bonds? The fission and fusion also confuse me, both are building towards a more stable atom (32?) but somehow both paths generate energy.


title: A clock devoid of motion

text: My understanding is that every clock mechanism we have depends on motion of something w.r.t the observer. From atomic transitions to clockwork gears. So, does this property/constraint makes every clock available to use inherently susceptible to effects of relativistic motion? For example, if you travel at 99% of speed of light with a clock, then there'd be some particle in the clock whose motion, w.r.t you, is also in the same direction, and when added up, it'll surpass 100%. For above situation to be avoided, every particle that makes up the clock will have to be:   1. Stationary w.r.t observer, but if this happens, the (frozen) clock can't tell time to the observer.   2. Move in a direction other than that of observer's motion, If this happens, clock can't travel with observer. Please help me out of this confusion.


title: A question about vertex operator

text: (skip disclaimer) I have a question about writing raising and lowering operators in the Schroedinger basis in the section of vertex operator in Polchinski's string theory vol 1 p.68. It is given > $$ \alpha_n = - \frac{in}{(2 \alpha')^{1/2}} X_{-n} - i ( \frac{ \alpha'}{2} > )^{1/2} \frac{ \partial}{\partial X_n}, \,\,\,\, (2.8.25a) $$ $$ > \tilde{\alpha}_n = - \frac{in}{(2 \alpha')^{1/2}} X_{n} - i ( \frac{ > \alpha'}{2} )^{1/2} \frac{ \partial}{\partial X_{-n}}, \,\,\,\, (2.8.25b) $$ I cannot find a way to derive these equations, especially how to find an expression of $$\frac{ \partial}{\partial X_n}?$$ Although they are similar with the harmonic oscillator in quantum mechanics. My question is, how to derive Eqs. (2.8.25a) and (2.8.25b)?


title: Why are most metals gray/silver?

text: Why do most metals (iron, tin, aluminum, lead, zinc, tungsten, nickel, etc.) appear silver or gray in color? (What atomic characteristics determine the color?) What makes copper and gold have different colors?


title: How to show that value is conserved along geodesics?

text: Let's have the motion of charged particle in a field of Reissner-type black hole. The equation of motion looks like $$ \frac{d^{2}x^{\mu}}{d \tau^{2}} + \Gamma^{\mu}_{\nu \lambda}\frac{dx^{\nu}}{d \tau}\frac{dx^{\lambda}}{d \tau} = \frac{q}{m}F^{\mu}_{\quad \nu} \frac{dx^{\nu}}{d\tau}. $$ How to show that value $$ E = m\left( 1 - \frac{2M}{r} + \frac{Q^{2}}{r^{2}}\right)\frac{dt}{d\tau} + \frac{qQ}{r} $$ is conserved along geodesics, $u^{\mu}D_{\mu} E = 0$? The first step, maybe, must be the attention to find the $\frac{dt}{d\tau}$. But what to do after that? It doesn't seems like convolution $\varepsilon_{\mu}u^{\mu}$ or to $u_{\mu}u^{\mu}$, where $u^{\mu}$ is the 4-speed. So what to do next?


title: Can we create a magnet with only one Pole?

text: > **Possible Duplicate:**   >  How to make a monopole magnet? No matter how many times you cut a magnet, we always end up with 2 poles. Is there any possibility of creating a monopole magnet?


title: Observables still commute even if fields only anti-commute

text: In Peskin & Schroeder page 56, after introducing _anti_ commutation relations for the fields instead of commutation relations (in order to fix the negative energy problem as well as to have proper causality conditions stemming from particles propagating from y->x and anti-particles propagating from x->y), it is stated that even though now we have anti-commutation relations, any two observables evaluated outside of the lightcone should still commute (and not _anti_ commute) and thus our theory still preserves causality, in the sense that if two observables commute then measuring one does not affect the measurement of the other. The reason they give for this is that any observable is ultimately going to be composed out of an even number of fields. I have tried to put this claim into mathematical sense, by doing the following:   1. Assume $g(x-y,x-y)<0$ where $g$ is our metric.   2. Thus we know that $<0|\\{\Psi(x),\,\bar{\Psi}(y)\\}|0>=0$ from our previous calculations.   3. Define $O_1(x):=<0|\bar{\Psi}(x)A_1(x)\Psi(x)|0>$ and $O_2(x):=<0|\bar{\Psi}(x)A_2(x)\Psi(x)|0>$. $A_1$ and $A_2$ are two differential / spinor operators which are thus far arbitrary.   4. Try to show that $[O_1(x),\,O_2(y)]=0$, because $g(x-y,x-y)<0$. I have failed at showing 4. assuming 2. (I have tried to somehow make linear combinations of the commutator as anti-commutators of the fields and it didn't work out). My question is:   * Is this the most general formulation of the claim that one could make? If not, how would you better formulate it?   * Once we have the formulation we want, how to prove it assuming 1. thus 2.?


title: Solenoid core and magnetism

text: I understand the core of a solenoid can help increase magnetic field produced by a given current and every metal has a specified permeability $\mu$, but what if we had a hybrid core, say 50% iron ($\mu=200$) and 50% nickel ($\mu=100$)? If I want to find the total $\mu$ of the core, can I determine the $B$-field produced by a given current by multiplying each $\mu$ to the percentage and add up?


title: Simulations of Planetary Motions

text: I wrote a spreadsheet that simulates the trajectory of 3 planets in 2D space. The method is simple: for each moment in space, calculate the force felt and velocity of each planet, then for the next $\delta$t period, assume that the planet continues to move with the same acceleration. Then recalculate the position, velocity, and acceleration of each object. Proceeds until desired time frame is simulated. The computation works roughly correct, but there is a problem with it, that is: I can not produce closed curves typical seen in a planet's orbit. Regardless of what velocity and mass I chose, the orbit's radius always gradually increases. This is probably caused by the approximation method I chose. I am wondering, what are some improvements I can make to the numerical method?


title: Can visible light be emitted from a non-thermal source?

text: I was reading about thermal and non-thermal radiation and I was wondering if visible light can be emitted from a non-thermal source?


title: Why does the beam in a weighing balance get tilted proportional to the weights added to each pan?

text: I'm talking about a beam balance(a simple weighing balance with a beam and two pans hung on either side) As answered in a previous question, the beam comes back to the original position when one side is pushed down and released because the centre of gravity of the beam is a little below the axis of rotation. practically, when unequal weights are put to the pans, the beam gets tilted proportional to the weight added to each side. Yet, according to my knowledge, this cannot happen theoretically. for example, if a weight of M is added to the left pan , and a weight of m is added to the right pan, and M>m, and the length of the beam is 2l, the torques acting around the point of rotation would be Ml- anticlockwise and ml- clockwise. Then the resultant torque would be (Ml-ml) anticlockwise. hence theoretically, the beam must rotate anticlockwise by nearly 90 degrees, which means the left pan (the pan with weight M) must move down as much as possible.(Please note the small torque due to the weight of the beam has been ignored) But what **really happens** is the beam gets tilted proportional to the weight added to each side. How can this be?


title: Electric field from conductive to dielectric media

text: I am interested in the main difference between transitions from electric fields from Conductive to Conductive/ Dielectric to Dielectric and Dielectric/Conductive media. What are the boundary conditions that an Electric field and a potential has to fulfill? I think I only know it for dielectric to dielectric media, where the potential has to be continuous and the electric field changes by the amount of surface charge.


title: Otto cycle efficiency described in pressure ratio (instead of compression ratio)

text: The Otto cycle's thermodynamic efficiency is often described using the compression ratio: $$\eta_{th} = 1 - \left( \tfrac{1}{CR} \right ) ^ {k-1},$$ (where CR is the compression ratio, and k the ratios of specific heat). Now I tried describing this using a pressure ratio (So instead of $\tfrac{V_1}{V_2}$ using $\tfrac{p_1}{p_2}$. Taking into account the compression in a cycle is (in the ideal case) an isentropic process, one could use $$p_1V_1^k = p_2V_2^k$$ $$\tfrac{P_2}{P_1} = \left ( \tfrac{V_1}{V_2} \right ) ^ {k} = \Pi$$ Combining these I would get: $$\eta_{th} = 1 - \left( \tfrac{1}{\Pi ^ {\tfrac{k-1}{k}}} \right ) $$ However this is **exactly** the same as the brayton efficiency.. While the cycles are different (otto uses isochoric combustion, while brayton uses an isobahric expansion during combustion). I wanted to compare those two, what did go wrong? EDIT: PS why is in the mathjax "\frac" not working, yet \tfrac does?


title: Are events in this experiment simultaneous if observed in platform's frame?

text: In some contexts e.g. on wiki it is defined as a matter of happening . In others(e.g. as defined by Einstein in his book "Relativity the special and general theory") it is defined as a matter of observation.   Both definitions are distinct from each other. In case of wiki's thought experiment two events $A$ and B(reaching of light at the back and front of train) are declared not simultaneous without observing them.   acc. to the Einstien's definition events can be empirically dissimultaneis only if they are observed to be dissimultaneous. So A and B are not dissimultaneous as par the criteria imposed by Einstien. > Are A and B(as explained on wiki's experiment) are dissimultaneous in > platform's frame, if yes then explain how these will be observed > dissimultaneisly.


title: How would charge distribute if electrons were balls?

text: In a conductor, any excess charge will distribute itself evenly over the surface of the conductor. Because of quantum mechanics, this is possible with small charges (i.e. 1e). But if electrons were balls, they wouldn't always be able to distribute themselves evenly over the surface of the conductor. For example, if the excess charge was 2e, there would be two points with charge 1e each and the rest would be neutral. So, what would happen in conductors with excess charge if electrons were like balls that couldn't necessarily evenly distribute themselves over the shell (on account of them being discrete)?


title: Physics problems requiring optimization?

text: I would like to know if there are simple benchmark physics problems to test a Genetic Algorithm C++ library I developed. The library supports single and multi-objective constrained optimization and I would like to publish the code showing its performance in the optimization of some physics examples.


title: Transition between spin-up and spin-down states in magnetic resonance

text: What is the classification of the transition between spin‐up and spin‐down states in magnetic resonance (such as NMR, ESR, MRI)?


title: Speed of light travel

text: I have just started special theory of relativity. The limiting speed known as speed of light fascinated me most. I asked my teacher: > _Consider two massless objects moving in the same direction at the speed of > light. What would be their relative velocity w.r.t each other?_ My teacher tells me that their relative velocity would be zero. But then the speed of light is universal constant regardless of the motion of its frame of reference, so shouldn't their relative velocity be $c$? What is their relative velocity and how? Do objects moving at the speed of light obey law of addition of velocities?


title: A curious case of Relativistic Velocity Addition

text: The relativistic velocity addition formula is $$u = \frac{v+u'}{1+ \frac{vu'}{c^2}}$$ Where $u$ = velocity of projectile seen by rest observer "A" $v$ = velocity of moving observer "B" as seen by rest observer "A" $u'$ = velocity of projectile seen by B Now the question is this: If $v=c,$ and $u'=-c $ I get an undefined answer. i.e. the relavistic velocity addition formula is undefined. Whats wrong with setting $v=c,$ and $u'=-c $?


title: Is there no radioactive decay between nuclear fusion and solid material formation?

text: I'm aware my question might be considered a duplicate of this one: Radio- dating and the age of the earth I read that one and I looked everywhere and I still can't find my piece of mind. I would really like to understand the following points:   * does one uranium nucleus start the radioactive decaying process the instant it is born inside a supernova ? (I understand that for a single nucleus the chance of decaying which is an instantaneous event, grows from 0% to 100% in the time from _the beginning_ to the half-life)   * while talking about a collection of atoms, Wikipedia states here that " _This predictability allows the relative abundances of related nuclides to be used as a clock to measure the time from the incorporation of the original nuclides into a material to the present._ "   * @JohnRennie helped me understand that chemically speaking zircon and lead do not mix whereas zircon and uranium do. So basically macroscopic contamination via accretion with flying lead is an event too rare to take into consideration right ? Also... I understand that the uranium-lead decay series has very short half-lives for every nuclide except for the beginning and the end (uranium and lead) and so, from a cosmological perspective it's quite improbable that zircon would mix with anything other than uranium, should a chemical compatibility exist with another nuclide. Please bare with me and excuse my "highly scientific" vocabulary. I have an experiment I would like to imagine in order to discover what I'm doing wrong:   1. A star goes supernova   2. We concentrate our attention on one uranium nucleus which is fused together during the violent event   3. The uranium nucleus travels for millenia towards the an accretion cloud surrounding a newborn star   4. The nucleus collides with _brand new molten zircon crystal_ and becomes part of it. Let's say that event 4. was happening after 0.99 uranium half-lives and the uranium nucleus still hadn't decayed. Isn't it imminent that the nucleus will decay pretty soon after event 4. ? I mean, I understand that if the nucleus would've become **thorium** or **protactirium** or **lead** it wouldn't've gotten stuck in the crystal for chemical reasons, but if it was still uranium what does that say about the internal clock of the nucleus which **had** been ticking for 99% of the half- life ? Is that clock reset because of external electromagnetic reasons the moment the nucleus becomes part of the crystal ? Thank you for your indulgence


title: A Variation on Laplace's equation (context: Yang-Mills N-Instantons, Rajaraman's book)

text: **Statement of the problem** I need to solve the equation \begin{align} 0 = \frac{1}{\phi} \partial_{\sigma}\partial_{\sigma} \phi \hspace{20mm} (1) \end{align} where $\phi$ is a scalar field and we're working in Euclidean 4-space. More specific questions are asked below. Basic advice on this would also be appreciated. **Context** I'm trying to work through 't Hooft's Euclidean Yang-Mills N-instanton solution. The idea is that we make the ansatz for the gauge potentials \begin{align} A_{\mu} = i \overline{\sigma}_{\mu \nu} \partial^{\nu}(\ln \phi(x)) \end{align} where $\overline{\sigma}_{\mu \nu}$ can be thought of as a constant 4x4 matrix that is unimportant to my question. We use the self-duality of the field strength tensor that arises from this ansatz to obtain the equation (1) for $\phi$ **The Difficulties** One generally good source has been Rajaraman's "Solitons and Instantons." The relevant section is displayed ![Relevant section in Rajaraman](http://i.stack.imgur.com/mdLWX.png) My difficulties are:   1. I am assuming by "singular" Rajaraman means that the function contains singularities. Why, explicitly, does $\phi$ being non-singular imply that we we need only solve $\Box \phi = 0$? Surely if $\phi$ contains any zeros then $1/\phi$ raises issues?   2. Why does $\Box \phi = 0$ only permit the constant solution and not say $\phi(x) = a_{\mu} x^{\mu} + b$ where $a_{\mu}$ and $b$ are constants? Is this because of the form of the ansatz, or is it something I'm missing?   3. In the case where $\phi$ is allowed to be singular, why does $\phi = 1/|x|^2$ solve the equation (1) for $x \neq 0$? Surely we have \begin{align} \frac{1}{\phi} \partial_{\sigma}\partial_{\sigma} \phi &= |x|^2 \partial_{\sigma}\partial_{\sigma} \frac{1}{|x|^2}\\\ &= |x|^2 \partial_{\sigma}\bigg(-\frac{2 x_{\sigma}}{|x|^4}\bigg)\\\ &=|x|^2\partial_{\sigma} (\frac{-2 x_{\sigma}}{|x|^4})\\\ &= |x|^2\partial_{\sigma} (\frac{-2 x_{\sigma}}{(\sum_{i} x_i^2)^2})\\\ &= |x|^2\frac{-2}{(\sum_i x_i^2)^2} + |x|^2\frac{8 x_{\sigma}^2}{(\sum_i x_i^2)^3}\\\ &=\frac{-2}{|x|^2} + \frac{8 x_{\sigma}^2}{|x|^4}\\\ &\neq 0 \end{align} ( _edit_ ) I see where I went wrong above, the last three lines should be \begin{align} &= \sum_{\sigma}(|x|^2\frac{-2}{(\sum_i x_i^2)^2}) + |x|^2\frac{8 x_{\sigma}^2}{(\sum_i x_i^2)^3}\\\ &=\frac{-8}{|x|^2} + \frac{8 |x|^2}{|x|^4}\\\ &= 0 \end{align}


title: Eigenstates of a Hermitian field operator

text: Consider a Hermitian field operator $\phi(x)$ with eigenstates satisfying $$ \phi(x) |\alpha\rangle = \alpha(x) | \alpha \rangle $$ I'm trying to determine the inner product between the eigenstates. To do this, I consider $$ \langle\beta|\phi(x)|\alpha\rangle = \alpha(x)\langle\beta|\alpha\rangle = \beta(x)\langle\beta|\alpha\rangle $$ which implies $$ \left[ \alpha(x) - \beta(x) \right] \langle\beta|\alpha\rangle = 0\hspace{3cm} (1) $$ **Q. What is the solution to this equation?** From the equation, I gather that $\langle\beta|\alpha\rangle = 0$ whenever $\alpha(x) \neq \beta(x)$ for any $x$ and therefore it has support only when $\alpha(x) = \beta(x)$. How can I represent this? Is it obvious that this implies $$ \langle\beta|\alpha\rangle \propto \delta \left[ \alpha(x) - \beta(x) \right] $$ This solution seems weird since it seems to imply that the norm of the eigenstate is "infinite" (naively!), but this does not follow from $(1)$. I know there are many subtleties here when dealing with infinite dimensional Hilbert spaces. The solution may lie in one of those subtleties. Any ideas?


title: Why is this the volume flow rate per unit area?

text: In fluid mechanics we consider a fluid filling a region $D\subset \mathbb{R}^3$ together with a function $\rho : D \times \mathbb{R} \to \mathbb{R}$ called the mass density such that for any $3$-dimensional submanifold $W\subset D$ the total mass of fluid in $W$ is $$m(W, t) = \int_W \rho(x , t) \ dV$$ That is fine, we also have $\mathbf{u} $ the spatial velocity field of the fluid, that is $\mathbf{u}(a, t)$ is the velocity of the particle of fluid at the point $a$ and time $t$. Suppose $\partial W$ is the boundary of $W$ and that $\mathbf{n}$ is the unit normal, then > $\mathbf{u}\cdot \mathbf{n}$ is the "volume flow rate per unit area" and > $\rho \mathbf{u}\cdot \mathbf{n}$ is the "mass flow rate per unit area" Now I simply can't grasp why. I've seem for many years people saying that for any vector field $\mathbf{V}$ and any surface with normal $\mathbf{n}$ the dot product of those is the flow of the field per unit area. This is already hard to swallow, and I never trully understood. How do we understand this in the context of fluid mechanics?


title: Intuition behind the principle of virtual work

text: To derive Lagrange's Equations we need the principle of virtual work first. This principle states that whenever a system of $K$ particles is constrained to a submanifold $\mathcal{M}\subset \mathbb{R}^{3K}$ and if the forces of constraint are perpendicular to $\mathcal{M}$ then in equilibrium holds $$\langle \mathbf{F}(c(t)), \mathbf{v}\rangle = 0 \qquad \forall \ \mathbf{v} \in \mathcal{M}_{c(t)}$$ Where $c$ is the path of evolution of the system. Now, for this to hold we need that part "the forces of constraint are perpendicular to $\mathcal{M}$" or in other words, they are workless in virtual infinitesimal displacements. Now, why is this a reasonable approach? In the books I've been looking at the justification is always "because it holds for rigid bodies so we mimic this to other cases" but this seems not natural. Is there a deeper intuition in restricting the problems being studied to those satisfying this condition?


title: why does the optical media have different refractive indices?

text: Optical density is a measure of the refracting power of a medium. In other words, the higher the optical density, the more the light will be refracted or slowed down as it moves through the medium. Optical density is related to refractive index in that materials with a high refractive index will also have a high optical density. Light will travel slower through a medium with a high refractive index and high optical density and faster through a medium which has a low refractive index and a low optical density. So to restate the question, why do optical media have different refractive indices?


title: Coarse graining on measurement vs phase space (second law)

text: I've followed Leonard Susskinds series on "the theoretical minimum". In the entropy/reversibility episode in statistical mechanics, he explains (at least one way of) how the apparent paradox of reversibility vs entropy increase is resolved. (See http://theoreticalminimum.com/courses/statistical- mechanics/2013/spring/lecture-7 and http://theoreticalminimum.com/courses/statistical- mechanics/2013/spring/lecture-8) The explanation goes something like this:   1. Assume you know that a system is in within a region of phase space $\Delta$, with a "diameter" $\delta$. The reason would be that you are unable to distinguish between points a distance $\delta$ apart. (Assuming classical mechanics, it actually is at a specific point, but we don't know which.)   2. Let the system evovle. $\Delta(t)$ will spread out sort of fractally, but keep its volume.   3. Now, since you can't tell points apart if they're within $\delta$, you basically get a "product" between a sphere with diameter $\delta$ and $\Delta(t)$, giving a new region $\Delta^{\star}(t)$, which tends to get larger.   4. Finally the entropy is just logarithm of the volume of $\Delta^{\star}(t)$. I get this, and that it will eventually decrease etc. What I don't get is why you need to do the third step. If I measure the system a second time, it still is in a specific point in phase space, and I should be able to distinguish it to the same precision again? Meaning that the entropy should decrease to the starting value immediately? EDIT: I do understand the concept of "covering with spheres". My problem with this concept is the following: Somehow we begin to know that the system is in a certain region. I assume we do this by some measurement? Now, if we can do a measurement with accuracy $\delta$, i.e. with a maximum deviation $\delta$ from the "true" point, at $t=0$, why can't we do it at again at a later time? EDIT 2: I'm going to try to explain how I think again. Let's choose to view this as a grid of size $\delta$ instead of spheres, it's easier in the following context. Assume the system is at a specific point in phase space, $x(0)$. By measuring it, we can find it resides in a certain grid element (the one that contains $x(0)$). The system evolves to a new - specific - point in phase space, $x(t)$. Regardless of our initial measurement (the grid cell) or the evolution of that initial grid cell, a new measurement would also give us a single grid element, namely the one that contains $x(t)$?


title: Is it really “time” that is dilating?

text: Consider the following variation of the twin paradox: A clock or a biological system ultimately is an electromagnetic system. First, let’s devise a new time measuring device. Imagine a tank of water with a spout on the bottom. The spout has a control mechanism that will release a single drop of water into a vacuum tube. At the bottom of the tube is a sensor that detects when the drop hits it. A wire connecting the sensor at the bottom with the spout control at the top then signals the spout to drop the next drop of water. The system is calibrated such that each drop measures precisely one second of time. Let’s refer to this time measuring device as a gravity clock. Our gravity clock almost entirely eliminates EM from the timing mechanism (at the surface of the Earth the E&M component along the signal wire makes up 1.6e-8s (4.9/c) of each second). Now empty the Universe save for 2 instances of Earth. On each of these Earths we place one normal clock (an EM clock) and one gravity clock. Both Earths are in the same inertial frame and all 4 clocks are synchronized. We now attach rockets to one of the Earth’s and accelerate it and its clocks to near the speed of light. It seems to me that the “time” dilation is caused by the longer round trip time of photons (or virtual photons) moving between charges. Since, the gravity clock (very nearly) eliminates electromagnetism from its timing mechanism, it will largely be unaffected by the shifting frame. If the wandering Earth is returned to the stationary Earth (again at near the speed of light), then the two clocks on the stationary Earth will be still be in sync, and I suspect the gravity clock on the wandering Earth will largely be in sync with the stationary clocks (aside from the effect of the signal along wire), while the wandering electromagnetic clock will vary fully as predicted by the Lorentz transformation. First, is there any reason to believe that this would not be the case; that the gravity clock would actually be affected in the same way as the EM clock? If not, is it really accurate to describe this situation as “time” dilating (as opposed to say the animation rate of EM systems dilating)? EDIT: Apologies for not noticing it earlier, but this question: (Time dilation only on electromagnetic force?) is essentially the same as mine. Although, I don't think either of the answers indicate whether any experiment has shown that a gravity clock is affected by "time" dilation.


title: Dark Matter as possible an explanation for Coronal Heating

text: It's my understanding that the mechanism behind Coronal Heating - why the Corona is so much hotter (> 1,000,000 K) than the surface of the Sun (6,000 K), and the mechanism by which the energy gets from the surface to the corona - is still very much an open question. I had an idea today that I wanted to get some information on. It is also my understanding that Rubin's work on Dark Matter explained the uncharacteristic velocity distribution of spiral galaxies by calculating what the distribution would be with Dark Matter distributed toward the outer edge of the galaxy. I was wondering, would a similar presence of Dark Matter around the sun (distributed toward the outer edge of the Sun) be a possible explanation for why the solar corona is so much hotter than the surface?


title: Interaction of ocean waves and currents

text: I was on a small island recently near high tide. As the tide started to go out, a tidal pond emptied into the ocean through a narrow channel with a strong outgoing current. There was also wind on the ocean that was driving waves onto the shore of the island and part way up the channel. I know that ocean waves are well described by a circular motion of the water and that the traveling waves are a superposition of standing waves that are 90 degrees out of phase but have no net current flow. I initially guessed that to reconcile the outgoing current and the waves the current must be somehow diving under the waves. I tested this by throwing a piece of wood into the current and watched it go out into the ocean. Unlike my expectations, it stayed on top of the water the whole time and went up and down the waves as it went out into the ocean following the current. What description of wave motion reconciles the current and the waves?


title: Linear independence of the Covariant Derivative

text: What's the easiest way to show that the covariant derivative $\nabla U^{\mu}$ is linearly independent to $U^{\mu}$, which is a vector? I mean I'm assuming they are since I'm proving the second Bianchi identity and my proof requires this to be the case; and I'm quite confident that the Bianchi identity is true! So I start with the Jacobi identity $$[\nabla_a,[\nabla_b,\nabla_c]]+\text{cyclic}=0$$ I then substitute in the definition of the Riemann tensor, giving me $$\nabla_a(R^d_{\space\space\space ebc}U^e)+\text{cyclic}=0$$ which gives $$\nabla_a(R^d_{\space\space\space ebc})U^e+R^d_{\space\space\space ebc} \nabla_aU^e+(\text{cyclic in }a,b,c)\space\space=0.$$ So we have 6 expressions, from which if the second Bianchi identity is true, the vector and the covariant derivate of the vector must be linearly independent. Or am I just OK to say that as the covariant derivative is (1,1) tensor and a vector is a (1,0) tensor, the result must follow straight away?


title: normal ordering from anti normal ordering of creation and anhilation operators

text: I am working on entanglement of systems in beam splitters. I constantly come across equations of creation and annihilation operator (ladder operators) in anti normal ordering. I want those equations in normal ordered form. Does anyone know the the relation between normal ordered and anti normal ordered. ( a^m * a' ^n)= ?


title: Deriving Gauss-Bonnet Gravity (Or just higher order corrections)

text: I have been working for some time now on deriving the equations of motion (EOM) for the Gauss-Bonnet Gravity, which is given by the action: $$\int d^D x \sqrt{|g|} (R^2-4R_{ab}R^{ab}+R_{abcd}R^{abcd}).$$ I've tried for some time to derive the first-order variation with respect to $\delta g_{ab}$. I have always been stuck with the derivation for the $\delta (R_{ab}R^{ab})$ component. Suppose I have only a Riemann tensor that contains **only constants** , \begin{align*} \delta(R_{ab}R^{ab}) & = (\delta R_{ab})R^{ab} + (\delta R^{ab})R_{ab} \\\ & = (\delta R_{ab})R^{ab}+ \delta(R_{ecfd}g^{ac}g^{bd})g^{ef}R_{ab} \\\ & = (\delta R_{ab})R^{ab}+ (\delta R_{ecfd})g^{ac}g^{bd}g^{ef}R_{ab} + (\delta g^{ac}) R_{cd}g^{bd}R_{ab}+(\delta g^{bd})R^{a}{}_{d}R_{ab}\\\&\quad+ (\delta g^{ef}) R_{e}{}^{a}{}_{f}{}^{b}R_{ab}. \end{align*} This however, gives us: \begin{align*} & = - (\delta g^{hi}) R_{ichd}R^{cd} +( \delta g^{hi}) R_{hcid}R^{cd} + 2(\delta g^{ac}) R_{cd}R_{a}{}^{d} \\\ & = 2(\delta g^{ac}) R_{cd}R_{a}{}^{d}. \end{align*} The correct answer as we know it, is $2R^{ab}R_{acbd}\delta g^{cd}$.


title: derivation of noise spectral density for Poisson distribution

text: I am trying to follow this article online to derive the noise spectral density of a Poisson distribution. I was able to follow it until equation (18). I cannot understand how he applied Wiener-Khintchine theorem to get that. It doesn't even look like the WK theorem that I know of (link to wiki). Does anyone know how he got that? The full article is from here. ![enter image description here](http://i.stack.imgur.com/YlVF5.png)


title: Quantum frequency vs classical frequency and Energy dependence

text: I recently realized what seems to be a contradiction in what is called "frequency" for light, and would enjoy some clarification of some more educated mind on the matter, thanking you in advance for your time. At the classical level, light is, in an ideal case, a plane EM wave. The electric field $\mathbf E$ for some light frequency $\nu$ is $$ \mathbf E(t,\mathbf x) = \mathbf E_\nu \sin(2 \pi\nu t-\mathbf k \mathbf x ) $$ with $\nu=c|\mathbf k|$ and $c$ is the speed of light. The energy density of the said wave is readily found to be proportional to $|E_\nu|^2$ (magnetic field is omitted for clarity). Light is described as a wave packet of such waves. The light spectrum : Radio, visible light, UV , $\gamma$ and all others in between is classified by the value of $\nu$. In quantum mechanics (or rather QED) one readily defines the energy of a light quantum, a photon, as (having usually put all constants to 1) a frequency $\nu'$. $$ E= \nu'. $$ Now to my question. For sure jumping from photons to classical wave is no light matter (!), but at a first glance, one can reasonably assume that $\nu'$ is linked to $|\mathbf E_\nu|$ due to the energy relations. One would get something like $$ |\mathbf E_\nu|^2 = \mbox{density of photons} \times \nu'. $$ However, while discussing radioactive decay for example, photons of fixed energy i.e. $\nu'$ are often emitted. At the same time the same terminology is used. That is we say some $\gamma$ ray is emitted. In what sense is that to be understood, as it refers to $\nu$ and not $\nu'$ ? It seems to me that by definition there are NO links between $\nu$ and $\nu'$, even if one introduces wave-packets to do things more rigorously. Wishing you a good day, Frédéric Edit: Thank you for your comments. The reasoning from QED, i.e. on some state clarifies the identification of $\nu$ and $\nu'$. I stupidly only recalled the classical average energy over time (which is $\nu$ independent and thus mislead me), where one should really consider the Poynting vector. Thanks again for your quick comments.


title: How is the centre of mass of a Positronium associated with that of an electron and positron?

text: How is the centre of mass of a Positronium likely associated with that of an electron and positron?


title: Fourier Transforming the Klein Gordon Equation

text: Starting with the Klein Gordon in position space, \begin{align*} \left(\frac{\partial^2}{\partial t^2} - \nabla^2+m^2\right)\phi(\mathbf{x},t) = 0 \end{align*} And using the Fourier Transform: $\displaystyle\phi(\mathbf{x},t) = \int \frac{d^3p}{(2\pi)^3}e^{i \mathbf{p} \cdot\mathbf{x}}\phi(\mathbf{p},t)$: \begin{align*} \int \frac{d^3p}{(2\pi)^3}\left(\frac{\partial^2}{\partial t^2} - \nabla^2+m^2\right)e^{i \mathbf{p} \cdot\mathbf{x}}\phi(\mathbf{p},t)&=0 \\\ \int \frac{d^3p}{(2\pi)^3}e^{i \mathbf{p} \cdot\mathbf{x}}\left(\frac{\partial^2}{\partial t^2} +|\mathbf{p}|^2+m^2\right)\phi(\mathbf{p},t)&=0 \end{align*} Now I don't understand why we are able to get rid of the integral, to be left with \begin{align*} \left(\frac{\partial^2}{\partial t^2} +|\mathbf{p}|^2+m^2\right)\phi(\mathbf{p},t)=0 \end{align*}


title: Oil Lamp Dynamics

text: Here is a photo of an oil lamp I am making. ![enter image description here](http://i.stack.imgur.com/FMfw8.jpg) I am facing a few problems.first how its made. Its a glass bulb with a custom fit plug made out of aluminium.The bulb balances by way of a neodymium magnet inside the bulb and is placed on an iron plate.A piece of cotton is rolled between oily palms to make the wick, and the oil used is sunflower oil. My problem is- the wick only glows till the base of the aluminium plug, and then extinguishes. What have i tried till now- Used a thicker oil, then thinner fuel (turpentine), a cotton wound thread, and then oiled cotton. I have also tried to lit the wick without the aluminium plug, but I get the same results. What I have read online suggests that the oil can climb on the cotton wick once it starts to get used near the flame, and this is the exact phenomenon which helps it keep lit. Can someone suggest what could be wrong, and why the lamp doesnt stay lit? Is my design faulty?


title: Could the measurement problem be solved by string theory / other ToE?

text: A quick wikipedia search will reveal the many different approaches to the measurement problem in quantum theory. What I find strange is that they even make up all these weird hypothesis like true randomness, many worlds, bohm and other interpretations when there isn't any ToE they're working it out from. I'm just a layman so maybe my assumption is wrong, but wont a fundamental theory inevitably change quantum theory? Is really a realistic model where superpositions are nothing but lack of information on our part not likely to come from the ToE? To my layman mind it seems premature to even begin to try to answer these questions if a theory of everything isn't in place. Or is there actually a reason that people are trying to solve this now? Such as: no matter what the ToE is, quantum theory wont change, superpositon will always exist etc.?


title: Given temperature, composition, column density, and radial velocity, can I find the bulk flow of a gas cloud?

text: We've got a hot star in the middle of a gas cloud. We point a spectrometer at the star, calculating the following attributes of our line of sight at the star through the cloud:   * Total number of various elements and molecules (H, H2, C, O, N, CO, lots of ions of all of the above) along the line of sight (units are number/cm^2)   * Average temperature of the atoms in our line of sight   * Average radial velocity of the atoms in our line of sight All the radial velocities are roughly the same, so we infer that the line of sight is more or less a single coherent blob of gas. Good! Now, the star itself which illuminated the line of sight is moving about 40 km/s towards us, while the gas cloud chunk we observed is moving at around 20 km/s away. Also, from comparing the luminosity of the star to the temperature of the gas, we conclude that the gas cloud is a light year or two away from the star. Given all this information, here is your task: If we take another spectrum of the _same line of sight_ ten years from now, will it be the same blob of gas? I will tell you right away that the radial velocity will only make the gas 0.002LY closer to the star, which is way less than what we need. So, in other words, given all the above, **can you conclude anything about the transverse velocity of the cloud?** Can you tell anything about the bulk flow and determine whether it is probable that the gas in our line of sight has moved with respect to the star? My physical intuition says that you'd expect yes, since the radial velocities are so different, but I'm not totally sure - the radial velocity doesn't necessarily imply anything about the transverse velocity. Or does it?


title: How to compute the speed of sound in relativistic hydrodynamic?

text: In Weinberg: Gravitation and Cosmology chapter 2.10 (Relativistic Hydrodynamics) the speed of sound is derived as $v_s^2 = \left(\frac{\partial p}{\partial \rho}\right)$ and the equation of state is given as $p=(\gamma - 1) (\rho - nm)$ with $\gamma = 4/3$ for the case of a relativistic gas and $\gamma=5/3$ for the case of a nonrelativistic monoatomic gas ($nm$ is the rest mass density and $\rho$ the energy density). Then Weinberg computes the sound speed for the relativistic case as $v_s= \sqrt{\frac{1}{3}}$ in natural units and states that this is still safely less than unity (the speed of light). However what he doesn't show is the same calculation for the nonrelativistic case. This would yield $v_s^2 = \frac{\partial ((5/3 - 1) (\rho - nm))}{\partial \rho} = 2/3$ and so the speed of sound in the non-relativistic case would actually be $v_s =\sqrt{\frac{2}{3}}$. So the speed of sound for a non-relativistic gas is actually higher than the speed of sound for an relativistic gas!? However one usually computes the non-relativistic speed of sound with another equation of state $p=K\rho^{\gamma}$ which yields $v_s^2=\frac{p}{\rho} = K \gamma \rho^{\gamma-1} = \gamma \frac{p}{\rho}$ This is a completely different result compared to the one we get from the equation of state given by Weinberg. So is the equation of state given by Weinberg wrong? If so, what is the correct equation of state for a relativistic gas and what is the real maximum speed of sound for a relativistic gas? If not, what is wrong with my calculation of the non- relativistic sound speed based on the equation of state given by Weinberg?


title: Sign convention for basic Dirac equation

text: The dirac equation;$$(i\gamma^\mu\partial_{\mu} - m)\psi=0 $$ is just; $$(i\gamma^{0}\partial_{0} - i\gamma^{i}\partial_{i} - m)\psi=0 $$ in a (+,---) metric right?


title: Bells Theorem for Dummies, How does it work?

text: I've been reading up on theoretical physics for a few years now and I feel like I am starting to get an understanding of particle physics, at least as much as you can from Wikipedia pages. One thing I have tried to understand but fails to make sense to me is Bell's Theorem. I understand that the premise of the ERP paper was that the "wave form collapse" couldn't work because it would require the two particles which made up the convoluted wave form to communicate instantly, violating the information speed limit. Instead they suggested that it was due to hidden variables (ie the values are already set, whether they have been measured or not). My question is can any one explain to me how Bell's experiment works and how it disproves this in terms that don't require an in-depth understanding of the math behind quantum mechanics? My current understanding of the experiment is you have two people who are reading a quantum value of entangle quantum particles (for my understanding lets say the spin state of a positron-electron pair produced by a pair production event). For each particle pair the two readers measure the spin at a randomly chosen angle. Here is where I need clarification: if I understand it correctly, local realism hypothesis states that when measured on the same axis the spin states should always be opposite (.5 + -.5 =0, ie conservation) when measure on opposite axis the spin states should always be the same ( .5 - .5 = 0 ) and when measured 90 degrees apart, the values are totally random. This part I get. I believe these results are predicted to be the same by both local realism and quantum mechanics. The inequalities between the two hypotheses rise when the particles are measured on axes which are between 0-90 degrees off axis from each other, correct? What I would like to have explained is the following:   1. What are the predictions made by quantum mechanics?   2. What are the predictions made by local realism?   3. How do they differ?   4. How is entanglement different from conservation?   5. Any corrections in regard to my explanation above?


title: Image formation

text: What is the real cause behind the formation of an image? It is explained as" when rays of light focus at a point image is formed." So here we have two events, one focusing of light and another formation of an image? Now my question is what is the intermediate phenomenon that converts focusing of light rays into the formation of an image?


title: Eddington-LeMaitre model

text: I'm trying to solve Friedmann equations for $k=1 $ & $p=0$ (matter). These are equations: $$3(\dot{R}^2 +1)/R^2 -λ =8πGε$$ $$(2R\ddot{R}+\dot{R}^2 +1)/R^2 -λ=0$$ If we use $(\dot{R}^2 +1) /R^2$ and put it in the second equation: $$\ddot{R}+\frac16\omega^2R=0 \\\ \omega^2=8πGε-2λ$$ If $\omega^2=0$, then we have Einstein's static model. Furthermore, for $\omega^2>0$, the solution is like $R=A\exp(\omega t)$ or $R=A\sin(\omega t)+B\cos(\omega t)$ with suitable constants. But I know Eddington-LeMaitre model (expanding universe) and the solution above, it's not that and it doesn't have coasting period or other properties of that model. I don't know what is wrong, can someone help me?


title: Electrons in Rutherford's gold foil experiment

text: I have been reading a chapter in my physics text. In it there is Rutherford's gold foil experiment. There it is written that doubly charged helium particles (alpha particles) were projected onto very thin gold foil. Most of the alpha particles passed straight through, but some of the alpha particles bounced back because positive particles (protons) in the nucleus repelled them. Positive and positive always repel each other. If charge was the reason some of the alpha particles bounced back, then why didn't the electrons (negative) of the gold atoms which are much smaller then protons get attracted to the alpha particles? Positive and negative attract. I would have thought the doubly charged alpha particles would be attracted to the negative electrons.


title: What causes ballpoint pens to write intermittently?

text: After a while, a ball point pen doesn't write very well anymore. It will write for a little distance, then leave a gap, then maybe write in little streaks, then maybe write properly again. It seems to be worse with older pens, but I have observed this with new pens right out of the box too. Experiments I have done:   * Take the cartridge out and look at the amount of ink. There is still plenty.   * Inspect the ball with a jewler's loupe, no obvious damage, everything looks smooth and clean.   * Stored new pens unused tip-down to eliminate gravity slowly pulling the ink from the ball and leaving a air pocket. Some of the pens exhibit the symptom even when used the first time with the cap never removed before and stored this way for a year.   * Stuck a wire in the open end of the ink reservoir to see if maybe the end dried to a hard plug so that new ink couldn't move down as it was removed from the reservoir by writing. I have never found anything hard, and observed the same symptoms even after "stirring" the top of the reservoir with a wire a little.   * When a pen stops writing, shake it hard, like resetting a fever thermometer. That seems to help for a brief while, but so does just waiting a few seconds, so I'm not sure the shaking is relevant.   * Stored a pen ball-down in a glass of water overnight. The thought was if the ink just above the ball had dried, maybe this would re-constitute it. Some ink clearly dissolved in the water since it was colored, but once the pen was started up again there was no apparent change to the symptoms.   * Cold seems to exaggerate the symptoms, but warming to body temperature doesn't fix them. This is not just a single pen or a single model. I have bunch of different pens of different models that do this. I'm curious, what causes this? ## Added: I have done some more experimenting, and it seems Emile Jetzer was right. The cause seems to be that the ink is so viscous that new ink doesn't flow down to replace what is removed via the ball fast enough. Two experiments support this hypothesis:   1. A pen will write again after a while by just letting it sit ball-down, but the time is significantly decreased when you shake it, like you would resetting a fever thermometer.   2. Some stick pens are sealed except for a small air hole at the top. Putting lips around the top of the pen and applying pressure as if you were trying to blow into it resets the writing action quickly. Even better, I can write with such pens much longer than they would normally go by holding my mouth over the top and applying constant air pressure. So, I think the mystery is solved. Probably ink in the reservoir dries out slowly over time by losing water vapor from the top. That makes the entire ink more visous, which explains why old but unused pens also exhibit this symptom. The next experiment is to take such a pen and add a little water at the top of the ink reservoir, then let it stand for a week and see what difference that makes.


title: Why does material resistance increase with temperature but electrolyte internal resistance decrease with increase in temperature?

text: The question is pretty much in the title itself, why does the internal resistance decrease with the increase in the temperature of electrolyte? Is it related to the electrolyte bring a fluid or something?


title: How to prove wave nature of large object?

text: In double slit experiment only small sized particles were used. however if we use large objects such as a tennis ball, the expected pattern is not observed. Why the wave nature is not visible for large objects in double slit experiment ?   How do I then prove that large objects also have wave nature?


title: Reusing the bremsstrahlung energy in circular accelerators?

text: I think the Bremsstrahlung happens as a strong radiation emitted by the accelerated particle stream. As in this question is visible, it is the main cause of the energy loss in such big particle accelerators as the Fermilab or the LHC. Are there existing plans to get this energy somehow back? The technology don't seem especially hard to this for me (at least, theoretically: taking the radiation energy with a lattice and using the induced current), especially because this very well controlled stream has probably a very stable BS.


title: Voltage in a lossless transmission line

text: I have a question where I'm deriving expressions for different variables within a lossless transmission line. Here's the question as posed: _Consider a lossless transmission line of characteristic impedance $R_0$. A time-harmonic voltage source of amplitude, $V_g$ and an internal impedance, $R_g = R_0$ is connected to the input terminals of the line, which is terminated with a load impedance $Z_L=R_L + jX_L$. Let $P_{inc}$ be the average incident power associated with the wave travelling in the $+z$ direction._ _(a) Find the expression for $P_{inc}$ in terms of $V_g$ and $R_0$_ _(b) Find the expression for the average power, $P_L$ delivered to the load in terms of $V_g$ and the reflection coefficient, $Γ$._ _(c) Express the ratio $P_L/P_{inc}$ in terms of the standing wave ratio, $S$._ Now, given the correct equations, I have no doubt I could derive this, but I feel as though I'm getting stuck on a real basic point. Consider a voltage $V_g$. Within a lossless transmission line, this time- harmonic voltage can be represented by: $V_g = V_0^{+}e^{-\gamma z} +V_0^{-}e^{\gamma z}$ And the time harmonic current, $I_g$ can be expressed in a very similar way (exchanging V's for I's). **My question is** : When we wish to consider $P_{inc}$, since it is only concerned with the wave travelling in the $+z$ direction, do we need the second term of the above expression? My thoughts are that, because the load isn't matched with the line impedance, there will be a reflection of both voltage and current within the line, and hence at any given point, the voltage will in fact be the above sum. Is this a correct assumption? Additionally, is my assumption about the general form of the current (taking the same form as the above equation) correct? And when I go to derive an expression for the power, can I simply use $P=VI$? They seem like all easy questions, but I hate to get things wrong! Help would be much appreciated :)


title: Why are the energy eigenstates realized in atomic transitions?

text: I have a question like "Why is it often assumed that particles are found in energy eigenstates?", it is a little different, though. When one solves the hydrogen atom, one can use a polynomial Ansatz and derive the energy eigenfunctions with that. The energy differences between the eigenstates are exactly the energies that are observed in spectroscopy. So it seems to me that atoms are in a pure eigenstate before and after the transition. If the state was a superposition of two eigenstates before the transition and a different one after, would the energy difference still be a difference of $E_n$? Say $$|\text{before}\rangle = \frac1{\sqrt2} \left(|1\rangle + |2\rangle \right)$$ and $$|\text{after}\rangle = \frac1{\sqrt2} \left(|0\rangle + |5\rangle \right).$$ Then before it is $\langle H \rangle = (E_1 + E_2)/\sqrt2 $ and $\langle H \rangle = (E_0 + E_5)/\sqrt2 $ afterwards. The difference would be something which is not simply $E_n - E_m$. One postulate of quantum mechanics is that each single measurement is an eigenvalue of the operator. So even though $\langle H \rangle$ could be something arbitrary, a single measurement would have to be taken out of the $E_n$? How does this relate to the energy difference (the emited photon) having energies of $E_n - E_m$?


title: Uniform Circular Motion and Centripetal Acceleration

text: In introductory physics books (or at least mine) it limits the equation $a_c=v^2/r$ to the situation where the speed around the circular path is constant. It enforces the idea that the speed is CONSTANT. But wouldn't the equation also apply to non-constant speeds? ($a_c$ would just change from being a constant to being a function of the speed) It would be very counter-intuitive to me if this equation did not apply to variable speeds (because why does this instant in time care about the speed of the next instant in time?) So my question is, can you also use this equation for variable speeds?


title: An operator on the other side of the Schrödinger equation

text: A form of the Schrödinger equation is $$ \left[-\frac{\hbar^2}{2m} \nabla^2 + V(\vec{r}, t)\right]\Psi = i\hbar \frac{\partial}{\partial t} \Psi $$ The bracketed term is of course the Hamiltonian, and the whole equation can hence be written as $$ \hat{H}\Psi = i\hbar \frac{\partial}{\partial t} \Psi $$ My question is, is there a existing/named operator represented by $i\hbar \frac{\partial}{\partial t}$? That is, can we further rewrite the equation as $$ \hat{H}\Psi = \hat{\Theta}\Psi $$ For some predefined operator $\hat{\Theta} = i\hbar \frac{\partial}{\partial t}$?


title: air flow with line fans

text: I must vent air through an 8 inch duct from one area to another using an 8 inch line fan. One goal is to vent the maximum amount of air possible through the duct. Is there any different between putting the fan at the beginning of the duct and pushing the air, or at the end of the duct and pulling the air?


title: matter anti-matter world

text: let us suppose the gedanken experiment a man isolated into a room he ask if he is made of matter oder of antimatter could he set some experiments to see if he is made of matter or if he is made of anti matter instead ?? is it possible to have an universe similar to ours but 'inverted' in the sense that this new universe would be made of anti-atoms ? is there any physical way to decide it ?


title: Color of macroscopic objects on molecular level

text: Macroscopic objects either reflect, or transmit, or absorbs electromagnetic waves. Assume now we have, say, a red apple exposed to the usual white light. The red light is apparently reflected from this apple. Next, the apple consists of molecules which have some complicated spectrum (from the quantum point of view). I was told two exactly opposite opinions about the relation between this spectrum and the red color we see: 1) The red light corresponds to the spectrum. Namely "red" photons first excite energy levels of molecules of the apple and then they are re-emitted again, and this is what we see. On the other hand all other colors cannot excite energy levels of molecules, and they are just somehow (how?) absorbed and eventually increase temperature of the apple. 2) All non-red lights corresponds to the spectrum, and the red light is the only one which does not correspond to the visible spectrum of molecules. In this approach "non-red" photons are absorbed exciting quantum levels of molecules and then this energy is somehow redistributed to increase temperature of the apple. The red photons are not absorbed and are somehow (how?) reflected. **Obviously at least one of these explanations must be completely wrong. Which one? What is the correct mechanism of visible colors on the molecular level?** A reference would be very helpful too.


title: Find acceleration along normal and tangent

text: A small block can move in a straight horizontal line a along AB. Flash lights from one side projects its shadow on a vertical wall which has horizontal cross section as a circle. Find tangential & normal acceleration of shadow of the block on the wall as a function of time if the velocity of the block is constant (v). ![enter image description here](http://i.stack.imgur.com/qaMRn.png)


title: Is there a simpler way to calculate the diminishing fringe pattern of a single edge?

text: A single edge diffraction pattern is different than a slit or multiple slit diffraction pattern. The pattern is not equally spaced and the fringe spacing's get smaller and smaller diminishing infinitely. The equations I have found for calculating these patterns are very complex and complicated. "Is there a simpler way to calculate the single edge diffraction pattern?" That is the question but what I'm really wondering is why the equations I use below are not used to work this out? I have found a way to calculate the diminishing single edge fringe pattern but explaining it can be quite involved especially when introducing a new or different way so I posted a link to my paper https://www.dropbox.com/s/lu5irtlxxe4hpis/Single%20Edge%20Certainty.docx?dl=0 If there is a better place to post and discuss this please let me know because I would love to talk about it. I can show that all slit diffraction patterns come from single edge diffraction patterns and calculating these unique diminishing fringe patterns is much simpler than the equations I have found anywhere else. Depending on the wavelength and distance from edge to screen one simple equation based on The Pythagorean Theorem will calculate the single edge fringe pattern infinitely and accurately. From there it can be shown how slit patterns can be constructed from overlapping single edge fringe patterns. Below is a few pages taken from my paper. If you could please take a look at the whole paper I would appreciate it and as I said would love to discuss it. Thanks Bill Alsept The following is based on 500nm light and a distance of one meter (1,000,000,000nm) edge to screen. The first time I calculated the distance (Y) (measured from the shadows edge) to the 1st bright fringe (B1). I tried adding ½ wavelength (250nm) to L to create the hypotenuse and then I calculated: Y=√((L+250nm)²-(L)²) or Y=√((1,000,000,000nm+250nm)²-(1,000,000,000nm)²) or Y=707,106.82nm But this was not the observed Y distance for the first bright fringe B1. I had nothing to compare my findings with because the equations I did find for single edge fringe patterns where extremely complicated. One day I came across this site about lunar occultation’s http://spiff.rit.edu/richmond/occult/bessel/bessel.html and discovered another way to look at single edge fringe patterns. This was the first time I had read about occultation. By chance I happened to recognize a sequence of numbers that were similar to the calculations I was doing for single edge fringe patterns and realized I was on the right track. Based On what I had learned about lunar occultation I could now calculate the distance of the fringes the moon would make on the earth. The creator of this web site had already calculated these distances so I finally had something to compare to. For my experiment all I had to do was convert (L) back to 1 meter instead of the 184,000 kilometers their experiment had used for the distance to the moon’s edge. Using Pythagorean Theorem and calculating backwards I converted the occultation’s measurements for Y and L and applied to them to my experiment and then calculated for the hypotenuse. What I discovered was that no matter what wavelength of light or distance you input (when calculating for the 1st bright fringe B1) the hypotenuse will always be L+3/4*1/2λ or L+3/8λ. Not quite 1/2λ This told me that these photons (maybe all photons) were emitted at the same phase of their amplitude cycle. This also told me that at the point of emission the photons were already part way into their cycles or more precisely 1/8 beyond the negative amplitude. That's why 3/8λ is added to (L) so that the first hypotenuse is the exact length needed to complete an oscillation just as it’s impacting the screen in a positive amplitude. The distance Y from the shadows edge to the first bright fringe is calculated as: Y=√((L+.75*.5*λ)²-(L)²) or Y=√((L+.75*.5*500)²-(L)²) or Y=√((1,000,000,000nm+187.5nm)²-(1,000,000,000nm)²) or Y=612372.46nm See Fig 12 below Figure 12


title: Damped Coupled Oscillations

text: I'm currently revising for a vibrations and waves module that I am taking as part of my physics degree. One of the our final questions involved finding equations for the displacements of the two masses in this system as a superposition of their normal modes: ![Two Coupled Pendulums](http://i.imgur.com/Kij0ycP.png?1?9381) I found the equations of motion for each mass to be: $$ \ddot{x_a} + \gamma\dot{x_a} + x_a(\omega_0^2+\omega_s^2) = x_b\omega_s^2\\\\\ddot{x_b} + \gamma\dot{x_b} + x_b(\omega_0^2+\omega_s^2) = x_a\omega_s^2\\\Here: \omega_0^2 = \frac{g}{l}~~\omega_s^2=\frac{k}{m}~~\gamma=\frac{b}{m}$$ Here I let $ q_1 = x_a+x_b~and~q_2 = x_a-x_b: $ $$\ddot{q_1} + \gamma\dot{q_1} + q_1\omega_0^2=0\\\\\ddot{q_2} + \gamma\dot{q_2} + q_2(\omega_0^2+2\omega_s^2)=0 $$ From here I can't see where to go. I did attempt substituting in a general solution such as $q_1 = C_1 \cos(\omega t)$ but I get a mixture of sines and cosines and I can't solve it for anything useful. Any help would be great as this is the last topic that I need to learn! Thanks, Sean.


title: Relationship between Weak Cosmic Censorship and Topological Censorship

text: The weak cosmic censorship states that any singularity cannot be in the causual past of null infinity (reference). The topological censorship hypothesis states that in a globally hyperbolic, asymptotically flat spacetime, every causal curve from $J^-$ to $J^+$ is homotopic to a topological trivial curve between the two points. (reference) I am curious as to the relationship between these two; will a found violation of weak cosmic censorship necessarily mean a violation of topological censorship? [edit: or the converse]


title: Is the liquid/solid line infinite?

text: Starting from the triple point, is the melting line between solid-phase and liquid-phase infinite? If not, why does it end? Because pressures are so high that classical inter-molecular interactions break down? Is this related in any way to the concept of degenerate matter? Can such pressure (at which the line ends) be easily computed?


title: What happens at the end of Coriolis Deflection

text: Consider we launch a cannonball due south from a point at 45 degrees latitude in the Northern Hemisphere (e.g the point defined with the co-ordinate system on this diagram). The cannonball travels for 5 minutes (through vacuum - ignore air resistance), experiences Coriolis deflection and drifts 'right/west' ending up at 40 degrees latitude. A few nanoseconds before it hits the ground, the rotational/tangential speed of the cannonball is slower than the tangential speed of the ground below. In order for the cannonball to stick/stay at the point of impact, its tangential speed must match the speed of the ground. What effect does this difference in speed have on the cannonball on landing/impact? Does it experience a sudden west-east acceleration due to friction? If yes, would this effect be even more pronounced in this case where the project is fired from North Pole?


title: Influence of choice of statistics on gas kinetics

text: In the derivation of distributions over energy states, a common assumption made is that under normal conditions (normal from a fluid dynamics standpoint, so > 300K typically) the energy states are very sparsely populated and so the vast majority of them remain empty. Because of this sparsity, the choice between **_Bose-Einstein_** and **_Fermi-Dirac_** statistics doesn't make a difference in the limiting case (the Boltzmann limit) when computing the distribution over the microstates. However, the validity of this limit comes down to the condition: $$\frac{V}{N}\times\frac{(2\pi m k T)^{3/2}}{h^3} \gg 1$$ which can be violated if the mass is small and number density is large or when T is very low. When this happens, the choice of statistics matters. I have only seen derivations when the Boltzmann limit is satisfied and the choice doesn't matter. How do the results change if Fermi-Dirac statistics are used? Do we see changes in the macro-scale results such as the specific heat coefficients or the state relations? Or are the resulting equations the same but the specific values are different? Would macro-scale features such as shocks/expansion fans or chemical non-equilibrium behave differently? Or radiation?


title: Is the spacetime interval between the Big Bang and a given event dependent only on the time component?

text: My thinking is this - at the time closest to the Big Bang when physics doesn't break down, the universe would have had a volume that was still very close to zero. If that were so, wouldn't the space component contribution be negligible, meaning that the formula becomes nearly $s^2=-c^2t^2$? I'm asking this because if it were so, and since spacetime intervals are invariant to all observers, it would make the time passed since the Big Bang also invariant to all observers, kind of like an absolute clock. But that really depends on there not being any space contribution, because the volume of the universe is nearly zero at the time of the Big Bang, and I'm not sure if that's true.


title: Mathematical physics text with plenty of applications

text: I'm looking for texts on mathematical physics. I've seen various other threads, but the texts recommended in those threads were mathematical methods of theoretical physics texts, that is to say those texts focused on the math methods useful for physicists and assumed the applications of those methods would be covered in some other text. What I'm looking for is a book which introduces some math and then looks at its applications (back and forth) rather than a text which just goes over some math. (Example of books which are NOT what I'm looking for: Mathematics for physicists by Dennery, Math methods for physicists by Weber and Arfken etc) I've taken courses in multi-variable calc, linear algebra, real analysis, and I'm currently taking a course in abstract algebra. But so far I've only seen applications of multi-variable to physics. I'm looking for books which look at the applications of the other subjects mentioned above to physics, of course books which cover subjects beyond those mentioned are also welcome. So far I'm liking the flavor of the text "A Course in Modern Mathematical Physics" by Szekeres, but I'd prefer a text which got to the physics side of things more quickly (in the Szekeres text the first eight chapters are strictly on pure math). After some amazon searching it looks like Theoretical Physics by Joos, Mathematical Physics by Henzel, and Physical Mathematics by Cahill may be good bets?


title: How much energy is carried away by neutrinos in matter-antimatter annihilation?

text: Some people say that neutrinos carry away most of the energy, some others say just a fraction. So what is the truth ? what is the percentage of energy lost due to neutrinos ?


title: Understanding the effective low-energy Lagrangian for hadrons

text: My course in Higgs Physics is discussing a two-nucleon low-energy effective theory of hadron interaction. With $\psi=(p,n)$, the pion is defined as $\vec{\pi}= i \bar{\psi}\vec{\tau} \gamma_5 \bar{\psi} $ and the sigma as $\sigma = \bar{\psi}\psi$ Then the Lagrangian is: $L=$ $\bar{\psi} i \partial_\mu \psi + \frac{1}{2}(\partial_\mu \vec{\pi} \partial^\mu \vec{\pi} + \partial_\mu \sigma \partial^\mu \sigma$) (kinetic) $+ g_\pi (i \bar{\psi} \gamma_5 \vec{\tau} \bar{\psi} ) \vec{\pi} + g_\pi \bar{\psi}\psi\sigma$ (nucleon-pion-interaction) $+ \frac{\lambda}{4}((\pi^2+\sigma^2)^2-f_\pi^2)^2$ (potential) First, I am surprised that the nucleon current in the nucleon-pion-interaction $ (i \bar{\psi} \gamma_5 \vec{\tau} \bar{\psi} )$ is almost the same as the definition of the pion $(i \bar{\psi}\vec{\tau} \gamma_5 \bar{\psi})$. In fact they seem to be equal since $[\tau_i,\gamma_5]=0$. ( $\tau_i$ acts on the SU(2)-space while $\gamma_5$ acts on the left-right space.) _Why are we reading the same term once as a pion and once as a nucleon current?_ Secondly, $f_\pi$ is referred to as the pion decay constant. It is supposed to be proportional to the coupling of a pion to the W-Boson. _Why is $\pi$ included into the potential in the way it is? How do I see that it is related to the W-Boson coupling?_


title: Angular distance Vs Angular displacement

text: Are the two terms - 'Angular distance', and, 'Angular displacement' interchangeable for a uniform circular motion?


title: Definite energy states for a single non-relativistic particle with a time dependent potential

text: Do definite energy states exist for a single particle when its potential itself changes with time? I tried solving it and the equations seem to show that they do not exist. But then i am confused as to what energies will be observed when it is measured. What does the expectation value of energy mean in this case? As a specific example, consider a 1D infinite potential well with $V(x,t)=t$. What energies are observed and with what probability when the system's energy is measured?


title: Is it possible to have a Gas heavier than a liquid?

text: Does such a pair of substances exist, that in certain physical conditions (temperature, pressure) when both are placed in the same conditions, one will be a liquid, the other - a gas, and the gas density will be higher than that of the liquid? (let us exclude special states of matter - e.g. don't count superfluid as liquid, or plasma as gas.)


title: Potential Energy in General Relativity

text: I often hear about how general relativity is very complicated because of all forms of energy are considered, including gravitation's own gravitational binding energy. I have two questions:   1. In general relativity, objects following the motion of gravitation should simply be travelling by geodesics. In such 'free fall', why would there be any 'binding energy'?   2. From Einstein's field equations, $$R_{\mu \nu} - {1 \over 2}g_{\mu \nu}\,R = {8 \pi G \over c^4} T_{\mu \nu},$$ isn't the curvature only coupled to the energy-momentum tensor? As far as I understand, potential energy is not included inside the energy-momentum tensor.


title: How to find optical toy models of entangled quantum mechanical systems?

text: I recently read Arnold Neumaier's lectures on uncovering classical aspects of quantum mechanics:   * Classical and quantum field aspects of light   * Optical models for quantum mechanics   * I can't find the third lecture, probably it doesn't exist yet. I guess it will elaborate more on the **thermal interpretation** , but it's easy enough to find published material on this. In "8. Simulating quantum mechanics", an optical model based on the second order coherence theory of the Maxwell equations is presented. Even so I can understand this model more or less, I would prefer simpler optical toy models for a start. Having explicit boundary conditions would be nice, and a restriction to (quasi-)monochromatic light could make it easier to intuitively understand such a system. For example, I guess that a complex superposition of polarized monochromatic plane waves traveling up or down in z-direction could be used to create a faithful optical model of a 2-qubit quantum system. (The periodic boundary conditions used here cannot really be reproduced in an actual physical experiment, but that doesn't worry me at the moment.) The first qubit would be the complex degree to which the plane wave is traveling up or down, and the second qubit would be the complex degree to which the plane wave is x- or y-polarized. But which measurements are allowed in such a toy model? The average intensity can of course be measured, but can the average Poynting vector be measured as well? Are destructive measurements such as putting a polarization filter into the system allowed? Is there a way to add a stochastic element (Born rule) to the measurement process in such an optical model? Of course I should be able to work out most of these questions myself, but I wonder whether people like Arnold Neumaier haven't already worked out such simple optical toy models, and put the details online somewhere...


title: what is the probability distribution for the angle of an approximate laserbeam

text: I'm trying to simulate the light distribution characteristics from a Gaussian laser beam, but having difficulty with the angular distribution. I need to generate a large number of points on an x/y plane, along with pointing vectors down the z-axis, such that their aggregate approximates the power distribution of a laser beam. The distribution on the x/y plane is a Gaussian, while the pointing vector down the z-axis approximates the laser divergence. For example, a beam with a 1 mm beam waist with a 1.5 mrad divergence has the following beam irradiance, E(r): $E(r) = exp(-r^2/b^2)/(\pi b^2)$ where $r$ is the radius from the beam center, $b$ is the beam waist (1/e radius). Therefore, I can sample from this distribution by the following equation: $r = b\sqrt(-ln(1-\mathbb{R}))$ where $\mathbb{R}$ is a random number uniformly distribution on [0,1], to get the radius from the origin on the beams starting point on the x/y plane. OK, so the question I'm having trouble with, is **how do I randomly choose the polar angle $\theta$ of divergence such that it approximates the laser divergence of 1.5 mradians**? Do I just choose the polar angle to be uniform on $[0,1.5e^{-3}]$ radians? I guess I'm getting confused by whether the distribution should be uniform over polar angle, or uniform irradiance over the solid angle, and how to sample from that. Below is an illustration to help sort things out. I'm trying to determine how to distribute the polar angle in order to approximate the laser divergence. ![enter image description here](http://i.stack.imgur.com/z4b44.png)


title: Reynolds number and inertial force

text: The Reynolds number is defined as the ratio of ´inertial´ forces to viscous forces. $$ Re = \frac{\text{Inertial Forces}}{\text{Viscous Forces}}$$ Now, viscous forces make sense to me. They are frictional shear forces that come about due to the relative motion of the different layers in a flowing fluid, resulting in different amount of friction, hence, different viscosity values. However, I am not really sure how to think about the 'inertial force'. This, to me, is somewhat of a dynamic effect since large Re numbers indicate turbulence in most cases, where there is a lot of motion, vortices and eddies. But what exactly is the inertial force and how can it be explained physically?


title: A naive question on the $U(1)$ gauge transformation of electromagnetic field?

text: For simplicity, in the following we set the electric charge $e=1$ and consider a lattice spinless free electron system in an external static magnetic field $\mathbf{B}=\nabla\times\mathbf{A}$ described by the Hamiltonian $H=\sum_{ij}t_{ij}c_i^\dagger c_j$, where $t_{ij}=\left | t_{ij} \right |e^{iA_{ij}}$ with the corresponding lattice gauge-field $A_{ij}$. As we know the transformation $\mathbf{A}\rightarrow \mathbf{A}+\nabla\theta$ does not change the physical magnetic field $\mathbf{B}$, and the induced transformation in Hamiltonian reads $$H\rightarrow H'=\sum_{ij}t_{ij}'c_i^\dagger c_j$$ with $t_{ij}'=e^{i\theta_i}t_{ij}e^{-i\theta_j}$. Now my confusion point is: Do these two Hamiltonians $H$ and $H'$ describe the same physics? Or do they describe some same _quantum states_? Or what common physical properties do they share? I just know $H$ and $H'$ have the same spectrum, thank you very much.


title: How can one build a multi-scale physics model of fluid flow phenomena?

text: I am working on a problem in Computational Fluid Dynamics, modeling multi- phase fluid flow through porous media. Though there are continuum equations to describe macroscopic flow (darcy's law, buckley leverett equations, etc), these models do not apply for heterogenous media (with transport properties). We could, however, try to use the microscopic model (lattice boltzmann, or pore network models) which would be more faithful to the dynamics of the macroscopic heterogeoneous media. But any computational simulation of this model would run too slowly to be worth it. The principals of conservation laws apply at both scales (conservation of mass, momentum, energy), but the equations that describe these laws differ at each scale. How then, can we upscale microscopic physics in a computationally efficient manner? Are there any techniques for describing microscopic phenomena at the macroscopic level without such a heavy computational cost? Is there any technique to build a continuum description at all scales of the problem?


title: What does the quantum eraser experiment tells us?

text: I am a beginner in this quantum-mechanics stuff. I understand the quantum eraser only from an experimental view. So I didn't understand the formalism that describes the quantum eraser. But what does the experiment tells us? Does the photon know that there is somebody watching it? And this is why it behaves in another way? Does the photon also see the future?


title: Starting a physics Ph.D. at 60?

text: Decades ago I was in a physics Ph.D. program, made a hash of things, and quite rightly was dropped from the program. Since then I've had a very satisfying career as a computer programmer, and picked up an M.S. in Applied Math. I'm currently working in a bioinformatics research lab, and really enjoying myself. I think I might want to retire in five or six years, and lately I've been daydreaming about returning to my first love, and entering a physics Ph.D. program. This would be entirely for my personal satisfaction. I have no intention of seeking a further career in physics, I'd just like to spend a few years working in a physics research group. I've been out of physics too long to really know what area I'd like to work in, but I'm thinking quantum computing or quantum optics. While I wouldn't require any financial support, I recognize that taking on a Ph.D. student is still a major investment of resources for a researcher. My hope is that my years of experience in software development would be enough of an asset to make it worth their while. I'm just curious if there is anybody in the audience who's done something similar, or any faculty who can give an opinion about whether this is a realistic notion. Update: Thanks to all for your answers and comments. I think nibot's comment that the thing to do is to establish relationships with some faculty I might like to work with is the most "actionable" answer. I really appreciated all the other anecdotes and advice though.


title: How to calculate the quantum expectation of frequency of a particle?

text: I know how to calculate the expectation of < $\Psi$|A|$\Psi$ > where the operator A is the eigenfunction of energy, momentum or position, but I'm not sure how to perform this for a pure frequency. In other words, what is the expectation of frequency? Indeed, is there an expectation, and is it solved using a fourier transform from position space into frequency space? **EDIT** In response to the posts by lurscher and David Zaslavsky below, I think both may be right. Frequency f is a parameter and can be considered an operator.   a) For the plane wave there exists temporal and spatial frequencies which act as parameters. $|\textbf{u}_k\rangle = e^{-i k x }$ b) there is also an operator derived from the Hamiltonian E = h f where   $\hat{H} =- \frac{\hbar }{i}\frac{\partial }{\partial t}$   Rearranging and inverting $i$   $\hat{f} = \frac{H}{\hbar } = i \frac{\partial }{\partial t}$ Intuitively this appears like an operator, it has the right units and it transforms into the frequency domain, however, it still requires that $\hat{f}$ be shown to be Hermitian and is an operator that is its own adjoint or satisfies a Poisson algebra. Also intuitively, we can observe a frequency, by taking a measurement.


title: physics- momentum ( a space question)

text: An astronaut is conducting repairs on a satellite 50.0m from a space shuttle. Her mass is 120kg (including space suit) and she is using a 0.5kg spanner. She realises that she only has 5 minutes oxygen supply remaining. With what velocity must the spanner be thrown for her to return to the shuttle safely.


title: Does the equivalence between inertial and gravitational mass imply anything about the Higgs mechanism?

text: For example: the role it might play in a theory of quantum gravity (ie causing space-time curvature)? I realize that inertial mass can result from binding energy alone. Has the equivalence principle been tested on elementary particles (like the electron) whose mass would be entirely due to the Higgs coupling? It seems like an obvious question, but I have never heard it discussed before.


title: Is there a definition of force?

text: Well, Newton's three laws talks about forces, but no definition is given. In truth, Newton's second law gives an idea of what **total** force is: the time change rate of momentum. But, if we have a force amongst others, this force alone isn't the time rate change of momentum, so that we cannot use Newton's second law to define force, but just to try defining total force. Knowing all of this, is there a definition of force or we take force as a primitive concept like that of time? Thanks very much in advance!