title: What happens to the vapor pressure when I expel the liquid phase from the vessel?

text: Suppose I put water in a closed vessel and heat it, so the vessel becomes pressurized above ambient pressure. Now, there should be liquid and gaseous water inside the vessel. Since it's under pressure, when I open the vessel at the bottom, the liquid starts to flow out. If I wait until all liquid is expelled, and close the opening before the water vapor exits too, what state do I have in the vessel? Specifically:   * Releasing pressure from a system usually means it cools down. What does that mean for the vessel's inside? If I want the vessel to stay at the temperature it was, do I have to reheat (if we ignore ordinary heat dissipation for now)?   * The pressure inside the vessel now is lower, does that mean water vapor will condense and the system moves toward a new equilibrium? If I keep the temperature constant, will it be the same pressure as before?   * Does it matter for this question to what extent the vessel is filled at the beginning? If I understood vapor pressure correctly, the contents will reach an equilibrium of liquid and gaseous such that the pressure reaches a point which is solely dependent on temperature.   * Does it matter for the purpose of this question how quickly I expel the liquid phase?   * I'm not sure what influence the ambient pressure has on this question. If you need sample values, assume 100 kPa (i. e., normal atmosphere) for the ambient pressure, 200 °C for the water and 1.555 MPa inside the vessel when I open the valve.


title: Does an electric field create a pH gradient?

text: Since pH is a measure of the effective concentration of $\mathrm{H}^+$ ions a solution, I expect that an electric field applied to a solution will create a pH gradient. The higher concentration of $\mathrm{H}^+$ ions (corresponding to lower pH) should be close to the anion, and a lower concentration of $\mathrm{H}^+$ ions (corresponding to higher pH) should be closer to the cation. This is a simple yes or no question. Does an electric field create a pH gradient as I described above? (If not, can you please provide an explanation?)


title: Euler-Lagrange Equation

text: A particle moving towards the origin has initial conditions $x(t=0) = 1$ and $\dot{x}(t=0)=0$ If the Lagrangian is L:=$\frac{m}{2}\dot{x}^2 -\frac{m}{2}ln|x|$ This should satisfy Euler Lagrange Equation $$\frac{d}{dt} (\frac{\partial L}{\partial \dot{x}}) = \frac{\partial L}{\partial x}$$ Prove the particle reaches the origin at $\Gamma(1/2) = \sqrt{\pi}$ 1) okay to begin I simply plug in and expand the D.E. $$\frac{d}{dt}[(\frac{\partial \frac{m}{2} \dot{x}^2}{\partial \dot{x}}) - \frac{\partial\frac{m}{2}ln|x|}{\partial \dot{x}}] = \frac{\partial \frac{m}{2}\dot{x}^2}{\partial x} - \frac{\partial \frac{m}{2}ln|x|}{\partial x}$$ 2) since $x(t)$ and $\dot{x}(t)$ are functions of time, the cross partials dissapear and I am left with: $$\frac{d}{dt}(\frac{\partial \frac{m}{2} \dot{x}^2}{\partial \dot{x}}) = - \frac{\partial \frac{m}{2}ln|x|}{\partial x}$$ Which reduces to: $$m \frac{d}{dt}(\dot{x}) = - \frac{m}{2x}$$ This is equivalent to: $$\ddot{x} = - \frac{1}{2x}$$ 3) Now I will separate and Integrate (Keeping in mind the particle starts from Rest): $$\dot{x} = \frac{dx}{dt} = -\frac{1}{2}ln|x|$$ $$t = -2 \int^{x}_{x_0} \frac{dx}{ln|x|}$$ All I really want to know is that Up until this point, Have I done everything correct? Because I feel like I haven't. I don't think I can even Integrate this because I put it into wolfram and I got a mess.


title: how to find the effective mass of a hole

text: how can we find out the effective mass of a hole,since a hole in the valence band is just an absence of electron?


title: Variation in measurements of $g$ on the Earth's surface due to the Moon's gravity

text: A book, Exercises for the Feynman Lectures on Physics, has recently been published. It consists of homework problems to provide practice with the techniques and concepts used in the famous Feynman lectures, which had not had published problem sets to accompany them since they came out in 1964. I bought a copy, found what I considered to be an error in the answer given for one of the problems, and, as a matter of professional courtesy, emailed one of the book's editors, M. Gottlieb, to report the erratum. Dr. Gottlieb insists that I'm wrong. I'll quote the question here and give my own answer below. > 3.12 In making laboratory measurements of $g$, how precise does one have to > be to detect diurnal variations $\Delta g$ due to the Moon's gravitation? > For simplicity, assume that your laboratory is so located that the Moon > passes through the zenith and nadir. Also, neglect earth-tide effects. The answer given in the back of the book is: > $\Delta g/g=7\times10^{-6}$


title: What is the mathematical nature of space time quantization in string theory/super string theory?

text: I don't know much about string theory, apart from it being a theory of everything which brings QM, QED and nuclear forces and gravity under one single roof. I am curious to know from a mathematical view point, how the quantization of space-time is achieved in super string theory.


title: Heat equation on ball - one-dimensional description

text: I want to solve the transient heat equation on a ball. The boundary condition is the same over the hole outer surface. So this should reduce to a one- dimensional problem in radial direction. However I cannot use the one-dim heat equation, since the surface through which the heat flows goes quadratic with the radius. Is the following approach correct: Take the heat equation, transform it into sperical coordinates and eliminate the derivatives in angular directions. I think the last part is justified, because of the rotational symmetry of the boundary condition. So I end up with this equation: $$ \frac{\partial T}{\partial t} = \alpha \frac{1}{r^2}\frac{\partial}{\partial r} \left( r^2\frac{\partial T}{\partial r}\right) $$ with thermal diffusivity $\alpha$. Is this the correct reformulation of the heat equation on a ball to a one-dimensional problem? EDIT: When I solve this with the finite element method, will there be a problem in the centre of the ball? The $1/r^2$ factor might cause a problem.


title: The time is a type of energy ?

text: You know energy is necessary for the occurrence of an event . Time is also exactly neccessary for the occurrence of an event. If there is no time , there will be no event. I mean time is a type of energy ?


title: I need help with the following Physics' questions

text: I need to know which formulas and the step-by-step answers for the following physics questions: > _1\. You throw a ball upward with an initial speed of 7.0 m/s, and it > returns to your hand o.92 seconds later, ( assuming air resistance can be > ignored)._ > >   * _(a) What is the average acceleration vector of the ball?_ >   * _(b) Find the highest up ward distance that ball can get._ > > > _2\. A ball was thrown at an angle of 45.0 degrees above the horizontal, > that it traveled a horizontal distance of 496ft, and that it was caught at > the same level from which it was thrown._ > >   * _(a) What was the ball's initial speed?_ >   * _(b) How long was the ball in the air?_ >


title: Calculating the dimensional wall-normal coordinate for a self-similar compressible boundary layer using Levy-Lees transformation

text: How can I convert my self-similar boundary layer solution that is a function of the nondimensional wall-normal coordinate $\eta$ to be a function of dimensional $y$? For instance, if I determine from my boundary layer solution that $\delta_{99}$ occurs at $\eta = 5$, for a given set of dimensional parameters how do I determine the corresponding physical coordinate? This is very straight forward for Blasius flow but is not as obvious to me for compressible flow. The definition of $\eta$ is shown below. The integral dependence on $y$ is what is throwing me off. $$\eta = \frac{U_e}{\sqrt{\int_0^x\rho_eU_e\mu_edx}}\int_0^y\rho dy$$


title: determining orbit parameters (eccentricity, length and angle of major axis)

text: I'm trying to make an orbit simulator, and I want to draw the orbit of a satellite given its position and velocity vectors in a 2-D plane. I found this other question which tells how to calculate eccentricity and semimajor axis, but how do I figure out the orientation of the major axis (if e > 0)?


title: How to argue that time travel to past would not be possible even in future?

text: I am a science fiction writer. I know how to time travel to the future (using black hole or speed). But I need to argue in my novel that time travel to the past might not be possible. Is there a way to do this without using a formula? And how basic would argument look like ?


title: Semi-conductors and induced charge

text: Can a semi-conductor ( germanium or silicon) or a semi-metal ( graphite ) carry an induced charge ?


title: Determining the length of a Torsional Pendulum

text: Currently working on this question, however I'm not sure how to solve it. > As a pendulum swings in simple harmonic motion at the surface of the Earth, > the angle the pendulum makes relative to its equilibrium position is given > by $\theta(t) = (0.140 \:\mathrm{rad})\cos(5.72t)$ where t is in seconds. > What is the length of this pendulum? > > a) 0.140 m > > b) 0.250 m > > c) 0.300 m > > d) 0.439 m > > e) 0.801 m The equation provided indicates that this is a torsional pendulum, correct? Therefore the length of the string with torsion supporting the disc at the bottom does not matter at all? If so, why?


title: How to deduce this free body diagram?

text: Can someone provide a trigonometry/geometry insight to deduce the angle of the plane is the same as the angle of the component of the weight? ![enter image description here](http://i.stack.imgur.com/W5OfC.jpg)


title: Minimize initial velocity of projectile to arbitrary point

text: I'm supposed to shoot a projectile from the origin to a point $(x_0 , y_0)$ with minimal initial velocity $v_0$. (no air resistance; regular gravity) The flight time should be $T=\frac{x_0}{v_x}$. Plugging in to the equation for y, I get $$y_0=\frac{-g}{2}\frac{x_0^2}{v_x^2}+\frac{x_0v_y}{v_x}$$ Recalling that $v_x = v_0\cos{\theta}$ and $v_y = v_0\sin{\theta}$ and some algebra yields: $$v_0=\sqrt{\frac{gx_0^2}{2\cos^2{\theta}(x_0\tan{\theta}-y_0)}}$$ To minimize initial velocity, I figure I have to take $\frac{dv}{d\theta}$, set it to 0 and hope for a minimum. This is prohibitively complicated. Am I missing some intuition about how the components are independent of each other in finding the minimum velocity (maybe minimize $v_y$ separately)? $\theta=\frac{\pi}{2}$ seems to make $v_0$ the smallest (by blowing up the denominator), but that would be shooting the object straight up, which doesn't make sense.


title: Neutrinos vs. Photons: Who wins the race across the galaxy?

text: Inspired by the wording of this answer, a thought occurred to me. If a photon and a neutrino were to race along a significant stretch of our actual galaxy, which would win the race? Now, neutrinos had better not be going faster than the speed of light _in vacuum_. However, an energetic enough neutrino can have a velocity arbitrarily close to $c$. Say we took a neutrino from a typical core-collapse supernova. It would have a speed $$ v_\nu = (1 - \epsilon_\nu) c $$ for some small $\epsilon_\nu > 0$. What is the order of magnitude for $\epsilon_\nu$? At the same time, photons can also travel slower than $c$. The interstellar medium is not completely devoid of matter, and in fact much of this matter is ionized plasma. As such, it should have a plasma frequency $\omega_\mathrm{p}$, and so it should effectively have an index of refraction depending on the ratio $\omega/\omega_\mathrm{p}$. Then the speed of a photon will be $$ v_\gamma = (1 - \epsilon_\gamma) c, $$ where $\epsilon_\gamma$ is in general frequency-dependent. What is the order of magnitude for this deviation? I know it comes into play at radio frequencies, where in fact even the variation of $v_\gamma$ with frequency is detected: Pulses from pulsars suffer dispersion as they travel over hundreds to thousands of parsecs to reach us. For simplicity, let's assume there are no obstructions like giant molecular clouds or rogue planets to get in the way of the photon. **Is it possible that some photons will be outpaced by typical neutrinos?** How big is this effect, and how does it depend on photon frequency and neutrino energy?


title: High School Physics student needs help

text: A human cannon has a spring constant of 35 000 N/m. The spring can be extended up to 4.5m. How far horizontally would a 65kg clown be fired if the cannon is pointed upward at 45 degrees to the horizontal?


title: AdS/RCFT examples?

text: RCFT's (rational conformal field theories) in two dimension are very well- studied, but it seems that there is no any gravity dual of such theories has been found. Is there example of the holographic dual of RCFT?


title: Poincare group vs Galilean group

text: * One can define the Poincare group as the group of isometries of the Minkowski space. Is its Lie algebra given either by the equations 2.4.12 to 2.4.14 (..as also given in this page - http://en.wikipedia.org/wiki/Poincaré_group..) or equations 2.4.18 to 2.4.24 of Weinberg's volume 1 of his QFT books? What confuses me is that in deriving the commutation relations between $J^{\mu \nu}$ and $P^\mu$ he did use quantum theoretic arguments about the Hilbert space operator $U$ but I guess there is nothing quantum about the Lie algebra he derives in the aforementioned equations. Is that right?   * This quantum confusion steepens when one looks at the $K_i$ (..relatvistic boost along the $i^{th}$ spatial direction..), $P_j$ (..linear momentum along the $j^{th}$ spatial direction..) commutator being non-zero. This is justified by saying that the exponential action of the boosts and the translations on the Hilbert space states do not commute and that is being reflected here. (..they pick up an extra phase proportional to the mass and the dot product of the boost velocity and the displacement vector..)  But if the afore mentioned equations are really the Lie algebra of the isometry group of the Minkowski spacetime then in the Galilean limit shouldn't they be instead reflecting the fact that Galilean boosts and translations when acting on the spacetime coordinates do infact commute. But the $K_i$ and $P_j$ commutation continues to be non-zero even when the Galilean limit is taken on page 62. This makes me strongly suspicious that the equations 2.4.12 to 2.4.14 are not the Lie algebra of the isometry group of the Minkowski spacetime but are the Lie algebra of the group whose elements are $U(\Lambda, a)$ (..using Weinberg's notation..)…right?   * So is the "low velocity" limit taken on page 62 recovering non-relativistic quantum theory ? (and not Newtonian physics)   * On page 89 of the same book he derives the topology of the inhomogeneous Lorentz group as being $R^4 \times R^3 \times S^3/Z_2$. Since this is a connected manifold, I guess that by the term ``inhomogeneous Lorentz group" he is meaning only the proper orthochronous component of the full relativistic symmetry group. right?    * I can't see how the above topology matches with the semidirect product structure for posibly the same thing as given on this Wikipedia page -http://en.wikipedia.org/wiki/Poincaré_group ?  When people talk of the Poincare group is the full symmetry group of relativity is what is being referred to or is it just its proper orthochronous component (and not the other 3 components) ?   * I am familiar with the notion of "central charge" as in the "first" term on the RHS of the TT OPE of CFT…what also has the interpretation as the zero-point energy when doing the plane-minus-point<->cylinder conformal transformation. In this light it is not clear to me as to what is meant when one says that one can add in the "mass" as a central charge to the Galilean group..an extra generator which commutes with all the rest so that with this "central extension" the free particles will lie in the unitary representations of the Galilean group rather than the projective representations before the extension. I would be grateful if someone can shed light on this issue and help reconcile the two "different" notions of central charge.   * When one takes a "low velocity" limit of the Poincare algebra to get the Galilean algebra then is one taking just a non-reltivistic limit or is one also taking a non-quantum limit ?  (..I guess this will depend on my first query about whether what Weinberg calls as the Poincare algebra in the quoted equations has any quantum effect encoded in it {as it seems to be!} or is it just the Lie algebra of the isometry group of the Minkowski spacetime..)


title: Electric field intensity of spherical shell (with cut out cap)

text: Consider a charged spherical shell of radius $R$ and surface charge density $\sigma$. Choose a point on the surface of the shell and cut a spherical cap of radius $a \ll R$. What is the electric field intensity in the middle of the hole? This was made up intended as a relatively easy problem to solve, it turns out however it is probably not, as every edition of the textbook shows a different answer and professors at our physics department can't agree upon one. What do you think? My attempt: First I calculated the intensity on the axis of a disc and integrated over the disc elements: $\vec E = \int_0^{2R-{a^2 \over 2R -1}} {\sigma \over 2\epsilon} ({ \sqrt{2Rh} - |h| })dh$. This, using the fact that $a \ll R^2$ evaluates to ${\sigma R^2 \over 3\epsilon} $, which is wrong.


title: Is there a relativistic generalization of the Maxwell-Boltzmann velocity distribution?

text: The Maxwell-Boltzmann velocity distribution in 3D space is $$ f(v)dv = 4\pi \left(\frac{m}{2\pi k_B T}\right)^{3/2} v^2 \exp\left(-\frac{m v^2}{2k_B T}\right)dv$$ It gives the probability for a single particle to have a speed in the intervall $[v,v+dv]$. But this probability is not zero for speeds $v > c $ in conflict with special relativity. Is there a generalization of the Maxwell-Boltzmann velocity distribution which is valid also in the relativistic regime so that $f(v) = 0$ for $v>c$ ? And how can it be derived? Or can a single particle distribution simply not exist for relativistic speeds, because for high energies, we always have pair- production meaning the particle number is not conserved and a single particle distribution can not be defined in a consistent way?


title: Ion Drive Propulsion Top Speed

text: I would like to know if there is some formula / graph which would provide / show the efficiency of a certain type of propeller in space. Specifically, I'm interested in the acceleration attainable at certain speeds. I'm writing a science fiction book and I'm trying to make it as correct as possible, fact wise. The propeller I'm talking about is the ion drive Now, my book takes place in a world where fusion power is finally ours. So, please, let us assume that we have unlimited energy so you could power a dozen huge ion drives non stop. OK, there's the question of argon/xenon fuel, let's assume we have 1 year of that. **So... the question is... what speed could you reach?** If a continuous acceleration of $10\frac{\mathrm{m}}{\mathrm{s}}$ is applied (I put that number because it would also constitute an advantage for my crew - living in Earth's gravity), that would mean that a ship would reach the speed of light in just 347 **DAYS** But I know that's impossible because the EFFICIENCY of the ion drive would DECREASE as the ship's speed would approach the exhaust speed of the drive's "nozzle" (well, it doesn't have a nozzle per-se, as you can see in Wiki, but anyway...) * * * Please do not fear to elaborate on top of my question. Let's suppose for example that maybe the ion drives of the future have a much higher thrust/efficiency/nozzle exhaust speed. This isn't only about currently POSSIBLE facts but also about THEORETICAL limitations which might be overcome in the future (such as fusion energy).


title: Unified field theory in layman's terms

text: I watched some videos on the unified field theory, specifically interviews with Michio Kaku and John Hagelin, and want to learn a bit more about it. I looked up the theory of everything, string theory and unified field theory but most of it goes over my head. In one of the interviews with John Hagelin he describes the unified field as being the deepest level of the universe. He refers to it as intelligence or a field that has the potential for matter. So, what does this consists of if it isn't the smallest form of matter? Also, is there a definite view in the scientific community that the unified field exists and what it is?


title: Where did the energy of the charge go?

text: Suppose there is a positron and an electron, and they both collide, and we get $E=2mc^2$ of energy from the collision. Now, the charge also got vanished. Now suppose, I create neutrinos from the collision's energy, which are neutral and that doesn't violate conservation of charge, now the mass-energy of the neutrinos is same as the mass energy of the positron and electron destroyed. But instead if I make electron and positron pair again from that energy, it doesn't violate conservation of charge either, but this implies that creating one kind of charge requires positive energy and creating the other requires negative energy in some form. Isn't it ?


title: What is the proof that a force applied on a rigid body will cause it to rotate around its center of mass?

text: Say I have a rigid body in space. I've read that if I during some short time interval apply a force on the body at some point which is not in line with the center of mass, it would start rotating about an axis which is perpendicular to the force and which goes through the center of mass. What is the proof of this?


title: Are group representations possible when the solution space is not a vector space?

text: As far as I understand, the motivation for using representation theory in high energy physics is as follows. Assume that a theory has some (internal or external) symmetry group which acts on a vector space. Then fields satisfying the theory will have to transform under some representation of that symmetry group, by construction. What happens if we have some internal or external symmetry structure that is no longer acting on a vector space? The gauge group diffeomorphisms of general relativity spring to mind. Is there some more general 'representation' type theory which comes to our aid? And are there any examples of internal symmetries where this viewpoint is needed? Apologies if this question is imprecise or flawed - I'm just starting to get my head around the foundations of the subject! Many thanks in advance!


title: Refractive index of organic solvents in near-infrared

text: I wonder if the refractive index of acetone, ethanol, methanol, heptane was measured in the range above 830 nm. The site refractiveindex.info provides it only for visible, and I couldn't find any other data so far.


title: My computer turns on when it "sees" me

text: Before asking, I swear that, for how it may seem strange, this is all true. Very briefly, when my pc is off and I am near it, it turns on. I do not touch anything. Nobody else have the same effect. I just have to still near it for 3-5 seconds and it will boot. It begun 1-2 weeks ago. The first time was at night, and I was sleeping, so I wasn't really near to it. It is not "scheduled". It doesn't happen at 3:00 every day, just random. Some days before it started I had a radiography on my head. May I have absorbed some X-rays, which are now creating some sort of magnetic field around me? I'm quite scared from this thing, even if at first it seemed funny... Thank you in advance, and excuse me for my poor english.


title: Basic question on magnetism regarding north and south pole

text: I am currently busy with some magnetism and quite shockingly (to me at least) I haven't yet read anything about the difference between the north pole and the south pole of a magnet. Before I started with magnetism I learned electricity, here there 2 opposite charges: negative charges (electrons) and positive charges (protons). Although I know for the people on this forum, that is very rudimentary, I at least find at a satisfactory explanation; different particles have different charges; a positive or a negative one. With magnetism I haven't stumbled across this yet.   * My teacher told me you can't say there is a positive and negative 'magnetic charge' (I don't know how to call it); but why?    * Also, what causes the difference between the 2?    * And, now that I'm thinking about it; are there actually magnets with only a north pole or a south pole (if not, why are the 2 so fundamentally connected that you can't have one without the other)?


Why does Lagrangian of free particle depend on the square of the velocity ? For example, $L(v^4)$ also doesn't depend on direction of $v$.


title: Why do lumps of sugar break more easily when blowing in it?

text: My question is simple, when you have a small lump of sugar, it's hard to break it with your bare hands, but when you blow in it, it appears to be more easy. (it's a piece of advice i learned for my gran, and it works !) Is it due to humidity of the breath? or the spacing between sugar cristals? or something else?


title: Temperature of a small system

text: What is wrong if I define temperature of a small system (I mean, a system which has not a large number of particles) by $$1/T = dS/dE$$ ?


title: How does sound travel in space?

text: In relation to this question: How can a black hole produce sound? Which notes that the hole "produces" sound. The top answer states that: > What you think of as the hard vacuum of outer space could just as well be > seen as a very, very, very diffuse, somewhat ionized gas. That gas can > support sound waves as long as the wavelength is considerably longer than > the mean free path of the atoms on the gas. I get that there is "stuff" in space, what I don't get is how sound travels it. I learnt in high school that sound was a wave - as an example, you could fix a shoelace at one end and vibrate the other - voila, a wave forms. And then sound kind of moves in the same way through, say, air, because of the slight molecular attraction between individual molecules is enough to create a similar waveform. And then you get to space, and there aren't any molecules nearby to irritate each other, so there's no sound. If that's the case, why does it matter _how_ long the wavelength is, if there isn't a molecule nearby for the initial molecule to affect, how can sound travel? I'd understand a _little_ better if the particle a just _hit_ particle b (like a pool ball) and b carried the sound - but that has little to do with waves and wavelengths. Or is it that the "jet" that is thrown out travels along as a contiguous blob, with a sound wave embedded within it?


title: What is the first paper to report observations of polaritons?

text: I am seeking references to the first articles regarding the observation of polaritons.


title: Light-Matter Interaction and Object's Appearance

text: I am taking a course in Computer Graphics, and the teacher said we could put materials in there main categories:   * mirror like   * glossy or specular   * diffuse He suggested that the law of reflection is immutable and applies to all types of surfaces but that glossy and diffuse surfaces appeared the way they do because of their microstructure. The roughness of these surfaces causes light rays to be reflected in random directions, thus blurring incoming light. So far, so good. He also insisted on the idea that surfaces (expected in the case of metals) reflects the incident light color. In other words, if the incident light striking a mirror is white, reflected light is white (and not yellow as it is with gold, but again metal is a different case). What I don't understand is that: if you use "roughness" to explain why some materials are shiny and others diffuse, diffuse surfaces too reflect incoming light color as mirror surfaces do. Therefore if mirror like surface do not have a color of their own (since they only reflect the incoming light color) so should diffuse surfaces. But diffuse objects have a color of their own (which mirror surfaces don not have). Thus, there's a flaw in the reasoning which I can't resolve. I understand the object's color comes from the fact that electrons making up the atoms the material is made of, selectively absorb photons of some specific wavelength, and reflect the others. However does it mean it reflects these photon always in the mirror like direction? How and why does that happen that way if this is the case? I am trying to put in one single pictures all the pieces of the puzzle together. Light can only be either reflected or refracted, however objects are coloured because they absorb some light frequencies, but I am told that "mirror reflection" does not affect the incident light color (in other words, that the reflected light color and the incident light color are the same in the case of mirror like reflections). I am totally lost and would really appreciate if someone could shed light on this mystery.


title: Why do baked goods "fall" if exposed to sudden cooler air?

text: This is a know rule when baking: > Don't open the oven until your cake is finished, otherwise it will "fall". Why/how does this happen?


title: Is there a non-technical way to understand why proving confinement is difficult?

text: As far as I understand, the basic reason why we are not able to solve QCD at low energies is because it is a strongly coupled quantum field theory, and a part from exceptional situations in simplified models (involving for instance supersymmetry), we have no tools to fully analyze such regimes. Correct me if I am wrong, but this is a rather technical obstruction. Is there thus a simple, non-technical reason why proving confinement is such a difficult problem?


title: Vibrational motion of linear diatomic molecule

text: This question concerns the following exercise from an old exam: > The vibrational motion of a linear diatomic molecule can be approximated as > simple harmonic motion. > >   * A CO molecule has a bond with force constant $k = 1900 \, Nm^{-1}$. What > frequency of radiation would excite transitions between the different > vibrational enery levels? (C and O have masses of $12 \, m_u$ and $16\, m_u$ > respecitvely, where $m_u$ is the atomic mass unit.) > >   * Explain why electromagnetic radiation can not excite vibrational > transitions in an $O_2$ or $N_2$ molecule. How is this important for the > heating of the earth's atmosphere by the sun (the greenhouse effect)? > > For the first part I know the energy levels of a harmonic oscillator (in center of mass reference frame) $$E_n = \hbar\omega(n+1/2),\quad \text{where }\omega = \sqrt{\frac{k}{\mu}}, \quad \mu = \frac{m_Om_C}{m_O + m_C}$$ and the energy of a photon of frequency $\nu$ is given by $E_{ph} = h\nu$. The photon energy must be the energy difference between two states: $E_{ph} = E_n - E_m = \hbar \omega (n-m)$. This gives me the frequency $$\nu = \frac{(n-m)}{2\pi}\sqrt{\frac{k}{\mu}}$$ Or minimum frequency $\nu_0 = \frac1{2\pi}\sqrt{\frac{k}{\mu}}$, right? However, now to my actual question: I don't know why we cannot do the exact same thing for the other two molecules? What is different for $O_2$ and $N_2$? I'd appreciate some help, thanks! =)


title: Solving Klein-Gordon equation in the Rindler coordinates - the Unruh effect

text: I am reading 't Hooft's notes on Black Holes. I want to find the solutions of the Klein-Gordon equation $(\tilde{x},\tilde{y}, \rho, \tau)$ in the Rindler coordinates which are $$x=\tilde{x}\,\,\,\,\ y= \tilde{y}\,\,\,\,\,\,z=\rho \cosh{\tau}\,\,\,\,\,t=\rho \sinh{\tau}$$. ![enter image description here](http://i.stack.imgur.com/r47yg.png) ![enter image description here](http://i.stack.imgur.com/3Tm7d.png) Firstly looking at the solution at first glance, why are there two parameters $\omega$, $\mu$, when the Klein-Gordon solution in Minkowski space has a solution with a parameter $\omega$ satisfying the mass on-shell condition. Secondly, how do you get that solution? Following 't Hooft's advice, I wrote the usual solution with $k_3=0, k_0=\mu$ in terms of the Rindler coordinates and fourier transformed with time. $$\Phi_{\mu}=\int d{\tau}\, e^{i{\tilde{k} \cdot \tilde{x}-\mu \rho \frac{e^{\tau}-e^{-\tau}}{2}}} e^{i\omega \tau}$$ Setting $s=e^{i\tau}$, I get the first expression of equation 17.7, but without the $e^{\tau}$ factors for $\alpha, \beta$. I am obviously, going wrong somewhere, I am Fourier transforming to frequency space, so why is $\tau$ still there? Shouldn't it be just $\mu$ and $\omega$ or $\mu$ and $\tau$?


title: Imposing anti-commutation relations on fermionic quasi-particles

text: In many theories of CMT, we assume the nature of quasi-particles (without giving proper justifications). For example, we assume nature of quasi- particles to be fermionic in case of a interacting fermion system we began with and impose anti-commutation relations accordingly. Like in BCS theory, while using the Bogoliubov-Valatin transformation to diagonalize the Hamiltonian, we assume that the new operators are also fermionic in nature. Please explain more on this step and how is it justified.


title: A Decrease In Momentum?

text: The specifics of a question I am working on are, "After a 0.280-kg rubber ball is dropped from a height of 1.80 m, it bounces off a concrete floor and rebounds to a height of 1.45 m." Why doesn't the ball return to the same height?


title: Thermal Penetration Depth discrepancy

text: I've been working on a project that involves thermoacoustics, and one of the commonly-used values in this field is know as the thermal penetration depth. It is calculated as follows: $$\delta_k = \sqrt{2K / (C_p 2 \pi f \rho)}$$ where $K$ is the thermal conductivity of the fluid in question (helium) $C_p$ is the isobaric specific heat of the fluid $f$ is the frequency of vibration $\rho$ is the density of the fluid For my project, I've been using this document as a reference guide In it, the given $\delta_k$ is 0.1 millimeters. The given frequency is 400 $Hz$. The fluid is helium at 10 bar. Based on this information, I found the density of helium (~1.6 $g/mL$), the thermal conductivity of helium (~0.15 $W/mK$), and its isobaric specific heat (5.193 $J/gK$). However, when I plug these values into the equation, I get delta_k = 0.00379 mm, which is considerably off. Am I doing something wrong here? Is this widely-cited paper incorrect on such a basic fact?


title: Momentum And A Car Collision

text: I am studying an example problem, concerning the very topic mentioned in the title. In this example problem, a car has a head-on collision with the wall; the initial and final velocity are known, as well as the mass. My question is, why does the car rebound off of the wall? Why doesn't the normal force of the wall reduce the car's velocity to zero, with the car remaining there, instead of reducing the car's velocity to zero, and then giving it a velocity in the opposite direction?


title: What is the meaning of negative mass (in a SUSY spectrum)?

text: I was playing around with SuSpect to generate some SUSY mass spectrums and I'm often encountering situations where I get a negative neutralino mass.   1. What is the physical meaning of this negative mass I am getting?    2. Does this mean the parameters I am putting in is giving an unphysical spectrum?


title: Calculating distance to an asteroid on a given day

text: I'd like to calculate the distance between the Earth and an asteroid at a given date. Is this possible using the data from JPL's small-body database? Is there a better way?


title: Feynman rules for SUSY

text: This might be an incredibly naive question, but I'm wondering if there are a set of rules for "translating" between Standard Model and SUSY. For instance, if I want want to go from a Standard Model feynman diagram to a SUSY diagram, can one write down a set of rule including things like: "replace quarks with the corresponding squark" with the caveat of course that R-parity is conserved (so two SUSY particles at each vertex). Given the underlying symmetry between SM particles and their superpartners, it seems like there should be some type of correspondence between SM feynman rules and the SUSY feynman rules. It seems like it would be incredibly difficult to go through and learn a completely new set of interactions and rules for SUSY. Are there things that carry over from SM Feynman rules that can be used as rules of thumb for figuring out what are the allowed SUSY diagrams for a particular process?


title: Calculations of apparent magnitude

text: I was attempting to do some calculations of apparent magnitude to help solidify my understanding of the topic, but have been running into some confusion. According to Wikipedia, the apparent magnitude can be given as: $m_x = -2.5\log_{10}(F_x/F^0_x)$ where $F_x$ is the observed flux and $F^0_x$ is a reference flux (in other words, this equation provides the _difference_ of apparent magnitude between two observed values). Also, this is assuming that the same wavelength band is used in both flux measurements. Flux, in turn, can be calculated as: $F = \frac{L}{A}$ where $L$ is the star's luminosity and $A$ is the flux density. Since stars act as point sources, this can be simplified to: $F = \frac{L}{4\pi r^2}$ where $r$ is the distance to the star. Since, historically, Vega has been used as the reference zero-point (having an apparent magnitude around 0.03), I tried doing a simple calculation to find out the apparent magnitude of Fomalhaut using the values for luminosity and distance given in Wikipedia for both of them. First, the flux of Vega: $F_{Vega} = \frac{37\,L_\odot}{4\pi (25.3\,ly)^2}$ $F_{Vega} = 4.5999\times 10^{-3}\,L_\odot/ly^2$ Next, the flux of Fomalhaut: $F_{Fomalhaut} = \frac{17.66\,L_\odot}{4\pi (25\,ly)^2}$ $F_{Fomalhaut} = 2.2485\times 10^{-3}\,L_\odot/ly^2$ Now, to calculate the apparent magnitude: $m_{Fomalhaut} = -2.5\log_{10}(\frac{2.2485\times 10^{-3}\,L_\odot/ly^2}{4.5999\times 10^{-3}\,L_\odot/ly^2})$ $m_{Fomalhaut} = 0.7777$ Huh?? Fomalhaut's apparent magnitude is supposed to be **1.16**. Even correcting for Vega's offset of 0.03, we still come up with **0.8077**. Why are the calculations failing? I don't think I've made a mistake in the mathematics. Am I using the wrong values?


title: Formation on image on screen?

text: When we first learn elementary geometrical optics, the first thing taught to us is the two broad divisions of the type of reflection.-Regular and Irregular or Diffused.   The difference in cause for both of these is that in case of regular reflection an incoming ray is perfectly reflected in a specific direction but in diffused reflection, the reflected ray are diffused in several direction probably owing to minor irregularities of the surface. Hence the diffuse reflection does not form our image as a mirror does but instead produces general luminance. But then in case of multi-media-projectors or when we use a paper or a wall as a screen for real images, an otherwise diffuse surface (wall, paper, etc) produces an image which can be seen by us. Is this not only possible when the incoming rays are REGULARLY reflected on to our eyes?


title: Interacting representation of the Poincaré group

text: In his QFT book, Weinberg claims what follows (Vol I, pag. 144-145): given a (free) representation of the Poincaré group with generators $\bf P$ (spatial translations), $H_0$ (time translations), $\bf J$ (rotations), and ${\bf K}_0$ (boosts), one obtains another (interacting) reprentation by replacing $H_0$ and ${\bf K}_0$ with $$ H = H_0 + \int \mathcal{H}({\bf x}, 0) d^3x \; \; \text{ and } \; \; {\bf K} = {\bf K}_0 - \int {\bf x} \mathcal{H}({\bf x}, 0) d^3x, $$ respectively, where $\mathcal{H}(x)= \mathcal{H}({\bf x}, t)$ is a self-adjoint operator density which is covariant and causal, that is: $$ U_0(\Lambda, a) \mathcal{H}(x) U_0(\Lambda, a)^{-1}= \mathcal{H}(\Lambda x + a) \; \; \text{ and } \; \; [\mathcal{H}(x), \mathcal{H}(y )] = 0 \text{ for } (x - y)^2 \geq 0, $$ where $U_0(\Lambda, a)$ is the free representation. Weinberg provides only a partial proof of this claim. Does somebody know if this is a rigorous result, and where one can possibly find a complete/rigorous proof?


title: How to explain spin of electron?

text: How can we explain spin of electron, or the spin of other fundamental particles? If we think the spin of electron is similar to the spin of a ball or planet we make a mistake. We say it is an intrinsic property. However, in calculating magnetic momenta and other cases we consider it as spinning entity. It's too difficult to abandon the model which we see. Without which to explain abstract one becomes challenging.


title: What exactly is the spin of a particle?

text: > **Possible Duplicate:**   >  What is spin as it relates to subatomic particles? I'm having a hard time grasping the concept of **spin** , my textbook describes it very vaguely: > _Stable matter contains just three kinds of particles: electrons, protons > and neutrons. In addition to orbital angular momentum, each of these > particles also has an intrinsic angular momentum called **spin**. The spin > angular momentum of a particle, like its mass and electric charge, is a > fundamental property of the particle that cannot be changed._ I don't really understand what is meant by 'intrinsic angular momentum'. Is there an easier way of grasping spin?


title: Is Angular Momentum truly fundamental?

text: This may seem like a slightly trite question, but it is one that has long intrigued me. Since I formally learned classical (Newtonian) mechanics, it has often struck me that angular momentum (and generally rotational dynamics) can be fully derived from normal (linear) momentum and dynamics. Simply by considering circular motion of a point mass and introducing new quantities, it seems one can describe and explain angular momentum fully without any new postulates. In this sense, I am lead to believe only ordinary momentum and dynamics are fundamental to mechanics, with rotational stuff effectively being a corollary. Then at a later point I learned quantum mechanics. Alright, so orbital angular momentum does not really disturb my picture of the origin/fundamentality, but when we consider the concept of _spin_ , this introduces a problem in this proposed (philosophical) understanding. Spin is apparently intrinsic angular momentum; that is, it applies to a point particle. Something can possess angular momentum that is not actually moving/rotating - a concept that does not exist in classical mechanics! Does this imply that angular momentum is in fact a fundamental quantity, intrinsic to the universe in some sense? It somewhat bothers me that that fundamental particles such as electrons and quarks can possess their own angular momentum (spin), when otherwise angular momentum/rotational dynamics would fall out quite naturally from normal (linear) mechanics. There are of course some fringe theories that propose that even these so-called fundamental particles are composite, but at the moment physicists widely accept the concept of intrinsic angular momentum. In any case, can this dilemma be resolved, or do we simply have to extend our framework of fundamental quantities?


title: What is spin as it relates to subatomic particles?

text: I often hear about subatomic particles having a property called "spin" but also that it doesn't actually relate to spinning about an axis like you would think. Which particles have spin? What does spin mean if not an actual spinning motion?


title: Can I usefully interpret a non-unital completely positive (CP) map as a cooling process?

text: Non-unital completely positive (CP) maps take a maximally mixed quantum state (aka a normalized identity matrix aka an infinite temperature state) and map it to something else. This necessarily decreases its von Neumann entropy and, depending on how you define it, reduces its temperature. Can a stronger connection be made between thermodynamics and CP maps? To what extent do non-unital CP maps reliably cool states that _aren't_ infinite temperature?


title: Derive $\frac{\mathrm{d}}{\mathrm{d}t}(\gamma m\mathbf{v}) = e\mathbf{E}$ from elementary principles?

text: It is experimentally known that the equation of motion for a charge $e$ moving in a static electric field $\mathbf{E}$ is given by: $$\frac{\mathrm{d}}{\mathrm{d}t} (\gamma m\mathbf{v}) = e\mathbf{E}$$ Is it possible to show this using just Newton's laws of motion for the proper frame of $e$, symmetry arguments, the Lorentz transformations and other additional principles?


title: Confusion regarding entropy, reference papers request

text: **1.a** In paragraph at Wikipedia/Entropy it is stated: > This is because energy supplied at a **high temperature (i.e. with low > entropy)** tends to be more useful than the same amount of energy available > at room temperature **1.b** But at What is Entropy? there is: > The higher the temperature of the gas, the faster the gas particles are > moving on average, so the wider the range of possible velocities for the gas > particles, and hence, the more uncertainty we have about the velocity of any > particular particle. **Thus, higher temperature** , as well as greater > volume, **mean higher entropy**. **2.a** I was trying to obtain information on relation between entropy and work. Basically, if an external process modifies a system so that its entropy is reduced, then it is expected by me that work was done on the system and energy was contributed to it. I received good answers ( Connection between entropy and energy ). However I feel that I was not properly understood. **2.b** In 1953 paper of L. Brillouin ("The Negentropy Principle of Information") wrote what I basically had in mind when I asked the question. I am not aware if the approach is correct and accepted. > An isolated system contains negentropy if it reveals a possibility for doing > mechanical or electrical work. If the system is not at a uniform temperature > T, but consists of different parts at different temperatures, it contains a > certain amount of negentropy. This negentropy can be used to obtain some > mechanical work done by the system, or it can be simply dissipated and lost > by thermal conduction. A difference in pressure between different parts of > the system is another case of negentropy. A difference of electrical > potential repre-sents another example. A tank of compressed gas in a room at > atmospheric pressure, a vacuum tank in a similar room, a charged battery, > any device that can produce high-grade energy (mechanical work) or be > degraded by some irreversible process (thermal con-duction, electrical > resistivity, friction, viscosity) is a source of negentropy. **3.a** There is often a contradict intuition about information entropy. Shannon approach sees maximum entropy as maximum uncertainty / information. **3.b** Joe Rosen in "The Symmetry Principle" states that maximum entropy is maximum symmetry – which means maximum redundancy, so minimum amount of information. **4.** Non-equilibrium thermodynamics – the branch is not mature and does not serve as a tool to solve issues like in 2.a, AFAIK. I was seeing The Physics of Maxwell's demon and information as very good guide through **equilibrium** (what about non-equilibrium?) thermodynamics, its connection with work and information. The authors did, however, when elaborating on entropy of a system in Brillouin's view, put an equal sign to it and Shannon entropy – which I think is contradict to Brillouin itself – who as I cited in 2.b stated that it is disequilibrium that allows system to work – meaning a complex, information-rich system (in other paragraph he also explicitly sates that negentropy is information). This conforms to Rosen 3.b. **Are there other good compilations on entropy, information, maybe complexity science? Such that solve the contradictions, form compatible non-equilibrium foundation?**


title: Knots and strengh of a rope

text: I read a few times that a knot can reduce the strenght of a rope, but I can't understand why this happens. Can someone explain me what happens to a rope tied with a generic knot and stretched? Is there a way to calculate the reduction of resistance from the form of the knot?


title: Possible Outcomes from Measuring a Hydrogen Atom

text: A hydrogen atom is characterized by the wavefunction $$\mid \psi \rangle =\sqrt{\frac{2}{7}}\mid 4\,2\,1\rangle +\sqrt{\frac{1}{7}}\mid 2 \,1\,\bar{1}\rangle+\sqrt{\frac{4}{7}}\mid 3\,2\,0\rangle$$ I want to know the possible outcomes and the probability for obtaining each outcome from measuring $E$, $L^2$, and $L_z$, while also calculating the expectation value of the energy expressed as some number times $E_1$ (for $E$), $\hbar^2$ (for $L^2$), and $\hbar$ (for $L_z$). I know the possible outcomes are the eigenvalues for the observable, but we didn't really go over any examples, and there are few examples of this sort in the textbook, so I don't know where to begin. For example, I know when we act $L^2$ on the system, we will get $$ L^2\mid \psi\rangle=6\hbar^2\sqrt{\frac{2}{7}}\mid 4\,2\,1\rangle +2\hbar^2\sqrt{\frac{1}{7}}\mid 2 \,1\,\bar{1}\rangle+6\hbar^2\sqrt{\frac{4}{7}}\mid 3\,2\,0\rangle$$ However, what does this tell me about the possible outcomes? Would the probabilities for each outcome be the coefficient multiplying each successive term squared divided by the Pythagorean sum of the all three terms? How do I find the expectation value? Any help would be appreciated.


title: What would happen to Earth if the Sun suddenly disappeared?

text: > **Possible Duplicate:**   >  If the earth left the solar system for interstellar space. How long would > it take for atmosphere to freeze Suppose the Sun simply vanished, what would be the consequences to Earth?


title: Is it possible create current by spinning a charged sphere?

text: If we have a sphere which has $σ$ surface charge density and rotate it in axis z will this create current ? Is it possible without any potential difference ?


title: How can we describe the electrons of multi-electron atoms (i.e. not Hydrogen) when equations/analytic solutions only exist for Hydrogen?

text: I've been digging into emission spectra of different elements and found that such things as the Rydberg equation, Bohr's model, and quantum mechanics can only fully describe the single electron in the Hydrogen atom. How did we then make the leap to s,p,d,f shells of multi-electron atoms? How accurate is our analysis of these more complicated elements? Rydberg Equation (side-note: Is this an empirical 'data-fitting' equation? What's the significance of that?) $$\frac{1}{\lambda}=R_H\left( \frac{1}{n_1^2}-\frac{1}{n_2^2}\right)$$ Hydrogen: ![enter image description here](http://i.stack.imgur.com/9yWL5.jpg) Helium: ![enter image description here](http://i.stack.imgur.com/EyQi5.jpg) Iron: ![enter image description here](http://i.stack.imgur.com/Tx0Vp.jpg) Potassium: ![enter image description here](http://i.stack.imgur.com/g23gw.jpg)


title: Fundamentals of Quantum Electrodynamics

text: In quantum electrodynamics, the classical Hamiltonian is obtained from the classical electromagnetic Lagrangian. Then the classical electric and magnetic fields are promoted to operators, as is the classical 4-vector potential $A_{\mu}$. The appropriate commutation relations are expected between the fields and their conjugate momenta. Now, my question is, do the principles of quantum electrodynamics follow as a consequence of the fact that the charged particle producing the field is a quantum particle which must follow the principles of quantum mechanics? Let me give a specific example. Consider a slow moving(for simplicity) free electron moving with a constant velocity initially. Now, classically, the magnetic field at a point $P$ would be given by a function $\vec{B} = \vec{f}(\vec{r},\vec{x},\vec{p})$, where $\vec{r}$ is the position vector of the point at which the field is being 'measured' and $x$ and $p$ are the position and momenta of the charged particle evaluated at the retarded time. Now, supposing I apply the principles of quantum mechanics to this electron and promote the above mentioned expression for the magnetic field at point $P$ to an operator by the usual quantum mechanical prescription. Would this prescription yield the correct values for the measured magnetic field at point $P$? Why? or Why not? The bottom line of my entire question is whether the quantum field theory of an electron is a direct consequence of the fact that the particle producing the field is a quantum particle (and not a classical one) or does it involve much more than that? EDIT: Thank you for your responses. I would also like to know if the above mentioned prescription for obtaining the magnetic field would yield accurate results for slow moving electrons(non-relativistic)?


title: Gravity is an intrinsic property of every atoms?

text: 1. If gravity is the force of attraction between masses, does this force is an intrinsic property of every atoms?    2. Is there any possibility of some other particles within atom which is yet to be discovered?    3. Or is it simply a result of cosmic waves which created due to big bang?   4. Does anyone have any information regarding variations in gravitational force?    5. If its not there then is it simply be "something like plasma which entire universe in within its plume and huge masses simply accelerate the attraction"?


* Does String theory say that spacetime is not fundamental but should be considered an emergent phenomenon?    * If so, can quantum mechanics describe the universe at high energies where there is no spacetime?    * Schrodinger equation assumes the presence of time dimension. So how can such equations be valid at extremely high energy?    * And is it meaningful to do physics if there is no time dimension?    * Would the concepts of Hilbert space, wavefunction collapse etc be useful in that case?


title: Why is the absolute zero -273.15ºC?

text: I can't find an answer of why the lowest temperature is -273.15ºC. Is it deduced theoretically or is it experimental? An explanation is that when any gas volume tends to zero, the temperature will be -273.15ºC (Charles law). But shouldn't this number have some kind of error? And this would only apply to ideal gases. ![enter image description here](http://i.stack.imgur.com/LOk7j.jpg)


title: Notation for anti-symmetric part of a tensor

text: I know that $A_{[a} B_{b]} = \frac{1}{2!}(A_{a}B_{b} - A_{b}B_{a})$ But how can write $E_{[a} F_{bc]}$ like the above? Can you provide a reference where this notational matter is discussed?


title: Hurricanes and bathtubs

text: Why do hurricane winds rotate? I know it´s about Coriolis force but how can I be quantitative? My problem is that I can´t imagine the direction velocity of wind and the angle with earth angular velocity rotation. Also, when can we explain other things like bathtubs, toilets.. using Coriolis? I mean when this force is relevant to explain the motion and the direction of rotation, CCW or CW(clock wise)?


title: Canonical quantum gravity

text: I am looking for an introduction (textbook or paper, or if someone wouldn't mind writing a memo on it) to canonical quantum gravity, specifically the concept of minisuperspaces. I have been reading the paper of Wheeler (in Phys Review) but he doesn't speak about minisuperspaces, maybe they were not used/invented at the time.


title: Inductively coupled DC circuit

text: The circuit under consideration has two inductively coupled loops, one with a DC battery, inductor, and resistor in series. The other loop has two inductors, one inductively coupled to the first, the other not, and a resistor, all in series. I would like to know the current in the top loop (the one without the battery) as a function of the given quantities after the battery is connected. I tried adding the three inductors and using that value to find the bottom loop's current, then substituting that into the top loop to find d(phi)/dt and then current. This gives me the standard RL current expression for the bottom loop and an exponential decay model for the current in the top loop. This is counter-intuitive for me; I believed that the current in the top loop should start at zero, peak, and decay rather than start high and decay. Thanks in advance.


title: How fast does water fall in the middle of a very very thick waterfall?

text: Let me create a very artificial experimental set up. Take a bathtub the size of Delaware and suspend it a mile above the ground. Fill it with water (though I'm not sure to what depth - and it might matter). Now pull the plug out. The effect is that you have a waterfall with an endless supply of water, high above the ground, and, in my idealization, the stream of water is an arbitrarily thick, cylinder of water. The question is, how fast does the water travel at the center of the waterfall? Obviously, I'm not looking for a numerical answer here, but I just want to understand the limits. Water at the edges of the waterfall are subject to wind resistance and will therefore quickly reach terminal velocity. But water at the center of the stream is removed from the effects of air resistence, so if the stream was thick enough and the waterfall high enough, would the center of the waterfall be unboundedly fast?


title: Will a plant grow faster or slower when affected with less/more gravitational forces

text: Will a plant grow faster when you would apply more upward gravitational force? and/or Will a plant grow slower when you would apply more downward gravitational force?


title: Fermion derivative coupling in QFT

text: I'm interested in a QFT model featuring a fermion derivative coupling like $XX^* \chi^*\gamma^\mu∂_\mu \psi$ where X is some other field operator. Has anybody seen a paper containing something like this? How would Feynman-Graphs for such an interaction look like? Are derivatives of fermions ruled out by some consideration i am missing?


title: upper critical dimension in field theory

text: Is there field theory which describe a second order phase transition without upper critical dimension ? Mermin-Wagner says something about lower critical dimension but nothing about upper dimension.


title: Probability of getting a particular spin

text: I'm a beginner in quantum mechanics, and I'm a bit confused about states and the probability to measure certain values. I would like to understand at least the following simplified situation: Suppose the operator $S_z$ is measuring the spin in the $z$ direction of a free particle. Let $$\\{e_s\\},\quad s = -L, -L+1, \dots, L-1, L$$ be a basis of eigenstates for $S_z$, with $S_z(e_s) = s e_s$. If at time $t<0$ the system is constantly in the pure state $P_{\psi}$, where $\psi$ is one basis vector, how can I compute the probability that the measuring the spin in the $z$ direction has result $-L/2$?


title: metric signature explanation

text: Can anyone explain what metric signature is? I have a basic knowledge regarding tensors, btw. Also, how is it related to fundamental understanding of general relativity? Thanks.


title: How do Stirling engines work?

text: How do Stirling engines work? I understand the heating and cooling of air, but how much faster (or more force, I'm not sure which to use) does the piston move per degree Celsius that the temperatures differ?


title: Real and imaginary parts of dielectric constant vs refractive index?

text: So for a complex dielectric constant $\epsilon = \epsilon_a + i\epsilon_b$, the wave vector and index of refraction are related to it through $k = \frac{\omega}{c}n$ and $n = \sqrt{\frac{\mu \epsilon}{\mu_0 \epsilon_0}}$. According to Jackson, the real part of the dielectric is related to polarization and anomalous dispersion, while the imaginary part is associated with dissipation of energy into the medium. If you write the wavevector as $k = \beta + i \alpha/2$ and plug it in the general wave formula (just in 1D right now) of $e^{ikr} = e^{-\alpha r/2}e^{i\beta r}$, the intensity drops as $e^{-\alpha r}$, so $\alpha$ is the attenuation constant, which tells you how quickly the wave dies out in the medium. But, if you plug that form of $k$ into the above equations to solve for $\alpha$ and $\beta$ as a function of $\epsilon_a$ and $\epsilon_b$, you find that $\alpha$ and $\beta$ are both a function of both $\epsilon_a$ and $\epsilon_b$. This is counterintuitive to me, because intuitively I'd think that the attenuation constant $\alpha$ would only be based on $\epsilon_b$, due to dissipation, and the same with $\beta$ and $\epsilon_a$. Can anyone give a good physical explanation for this "mixing"?


title: What is physics of a normal jump?

text: What is the physics behind a normal human jump ? ![a human jump](http://www.wired.com/images_blogs/wiredscience/2012/04/drawings_spring_12key_23.jpg) when a normal human wants to jump.   1. First, some energy is stored in their thighs and The elastic tendons Just like a spring. In mechanics and physics, Hooke's law of elasticity is an approximation that states that the extension of a spring is in direct proportion with the load applied to it. Many materials obey this law as long as the load does not exceed the material's elastic limit $$F=-k.x$$ where **x** is the displacement of the spring's end from its equilibrium position (a distance, in SI units: metres); **F** is the restoring force exerted by the spring on that end (in SI units: N or kg·m/s^2); and k is a constant called the rate or spring constant (in SI units: N/ m what is human rate constant **k**?


title: Do high voltage power lines attract lightning strikes?

text: I always thought that high voltage power lines would attract a lightning strike more than other structures in the same area. Turns out I was wrong. My neighbor's chimney got struck by lightning and it destroyed a lot of stuff (street lamps, wifi APs, small home electronics). We live very close to high voltage power lines - so, is my premise wrong? Do power lines not attract a lightning strike more that other objects? Also, if a chimney isn't connected to the power grid (or am I wrong?), then how is it possible, that almost all of the street lamps don't work and other electronic devices were destroyed?


title: Is there a relation between (non-) existence of magnetic monopoles and thermodynamics?

text: This question follows on previous work on connections between (other) areas of physics and thermodynamics as in here, here and even here. P. Dirac (an electrical engineer initially) was one of the first to pose the possible existence of magnetic monopoles (original paper)(for example as solenoids). This would allow more symmetric forms of certain equations (physicist fetish?). Anyway, this (a-)symmetry is also the starting point for possible connections to thermodynamical concepts. > Is there any relation to the (non-) existence of magnetic monopoles and 2nd > law? > > Would the possible existence of magnetic monopoles violate the 2nd law, or > does the 2nd law predict magnetic monopoles, or it is just irrelevant (sth > many physicists would not assume lightly)? Relevant physics areas include the intersection of materials magnetization and thermodynamics. Relevant articles include this one on spin ice and this one on Brownian motion **NOTE:** have not actually read the articles as i do not have access, however mentioned as part of existing literature on the question's subject. Relevant papers and outlines: here, here, here and an arxiv paper by P. Davies on this issue (circa 2007) UPDATE: An "unorthodox" critical view on mainstream mathematical physics (specifically Yang-Mills theory)


title: Why is the phase velocity used in the definition of the refractive index?

text: I'm aware of the so-called group index but why is the phase velocity used in the standard definition of the index of refraction? What advantage does this offer?


title: What would be measured, under current technology, of a particle moving faster than light? (Somewhat duplicate)

text: Sorry, since the question has been posed, but I insist that others posed not from this position. Can any listener deduce whether we could measure, by analyzing electromagnetic disturbance, a particle$^1$ traveling faster than light? (or current measuring device, for the sake of assuming for questioning) Studying to understand cosmic rays, inductance, and capture technologies by the artifacts of Tesla.. The name most studied by us uneducated electrical folk. \-- $^1$ For the 2011 OPERA neutrino experiment, see this Phys.SE post.


title: Nanooptics microscopy limit

text: The limit of microscopes is 200nm, but apparently STED improves this according to this coverage of the Kavil prizes 2014 : http://www.photonics.com/Article.aspx?AID=56255 What is the current limit for microscopy ?


title: Has anyone ever actually compared the weight of a heavily loaded centrifuge when it is spinning and not spinning?

text: I have searched the internet and have not found any actual test results.


title: Lagrange's equations derivations

text: While deriving lagrange's equation, for an infinitesimal displacement $\vec{dr}$, we express it using taylor series in terms of general coordinates as $\frac{\vec{dr}}{dq} \delta q$. Where $\frac{\vec{dr}}{dq}$ means derivative of $\vec{r}$. What does it mean?(this derivative term). What is the function whose derivative we are taking ?


title: include the stretch of the spring own weight in potential energy for spring pendulum?

text: we are given a problem with spring with its own mass $m$. I am confused how to set up the PE term in the Lagrangian. Assume the spring has length of $L_{0}$ when it is laying on a table horizontally. Now when we hold the spring upright and hang it vertically from a pivot in the ceiling to make a pendulum, it will stretch a little due to its own weight by $\frac{(m/3)g}{k}$ where $k$ is the spring constant. I use $m/3$ there since that is the common factor used to generate an effective mass used for a spring with its own weight. http://en.wikipedia.org/wiki/Effective_mass_%28spring-mass_system%29 Hence now the length of the spring is $L_{0}+\frac{(m/3)g}{k}$ before we even put the bob mass at the end of it. Should one use $L_{0}+\frac{(m/3)g}{k}$ as the relaxed length of the spring or use $L_{0}?$ Will the PE for the spring be $\frac{1}{2}k\left( y-L_{0}\right) ^{2}$ or $\frac{1}{2}k\left( y-(L_{0}+\frac{(m/3)g}{k})\right) ^{2}$? Assuming one is measuring $y$ here from the ceiling where the spring is hanging from                                +---------                      \     |                      /     |                      \     |                      /     | y(t)                      \     |                      /     |                      M     V      Which term should be used for PE? I know there will be additional PE term for the spring due to its own mass, which is $-mg\frac{y}{2}\cos\theta$ Hence total PE from the spring is $PE=\frac{1}{2}k\left( y-L_{0}\right) ^{2}-mg\frac{y}{2}\cos\theta -Mgy\cos\theta$ or $PE=\frac{1}{2}k\left( y-(L_{0}+\frac{(m/3)g}{k})\right) ^{2}-mg\frac{y} {2}\cos\theta-Mgy\cos\theta $ where $\theta$ is angle of rotation of the spring pendulum. not sure which to use.


title: Are the ions in a fluorescent lamps concentrated near the electrodes?

text: Are the ions in a fluorescent lamps concentrated near the electrodes , and just the electrons move to the other electrode or the positive mercury ions move too?


title: In $H_2^+$, what is the Hamiltonian of the movement of the electron?

text: An electron is orbiting two protons. With the Born-Oppenheimer approximation that the protons do not move, I'd write the Hamiltonian of the electron's movement as: $$ \mathbf{H} = -\frac{\hbar^2}{2m}\nabla^2 + E_p$$ with $$E_p = -\frac{e^2}{4\pi \epsilon_0}\left(\frac{1}{r_1}+\frac{1}{r_2}\right) $$ as the potential caused by the protons, with $r_1$ and $r_2$ denoting distances to the protons. Apparently it should be $$E_p = -\frac{e^2}{4\pi \epsilon_0}\left(\frac{1}{r_1}+\frac{1}{r_2}-\frac{1}{r_0}\right), $$ where $r_0$ denotes the distance between the protons. How can that term be explained?


title: Dimensional regularization and IR divergences and scale invariance

text: I want to know if dimensional regularization has any issues if the theory has IR divergences or is scale invariant.   * Does dimensional regularization see "all" kinds of divergences?  I mean - what does it _exactly_ mean when one says that power law divergences and IR divergences disappear in the dimensional regularization. So is more regularization needed in general over and above dimensional regularization?   * Does anything about the divergences get specially constrained if the theory is scale invariant?  I have often heard it being said that dimensional regularization "preserves" scale invariance.


title: Kinetic energy conservation in a collision

text: Kinetic energy conservation in collision. In any type of collision type of collision if we take the system to comprised of both bodies then the net external force is zero. So the work done by external force is also zero. By the relation $W=K_f-K_i$ as $W$ is zero hence kinetic energy is conserved. But this not the case. Can you tell me what is the flaw in my argument? Some people say that the argument has neglected the fact that some kinetic energy is transformed into heat. But heat is created due to the interactions between bodies during collision which is thus an internal force and thus shouldn't affect kinetic energy.


title: Is $\sigma$ or $\sigma / \sqrt{N}$ is error of a measurement?

text: I wonder whether $\sigma$ or $\sigma / \sqrt{N}$ is error of a measurement. When I measure, say $0, 1, -1, 1, -1$, I have a $\sigma = 1$. I just measure $0, 1, -1$, I also have $\sigma = 1$. But in the former case, I had more measurements, so the error should be smaller. So would the error be 1 or rather $1/\sqrt{5}$ and $1/\sqrt{3}$ respectively?


Does chaos theory occur in quantum mechanics? Or in any non-newtonian physics? Apart from perhaps thermodynamics?


title: Solving a circuit with Kirchoff/Ohms Law

text: Good evening, I haven't had physics since year 7 and now I need to use elementary things in university. Since I lack a lot of basics I am now trying my best to fill these holes. Currently I am stuck at solving circuits and I hope one of you can help me with the following circuit. I've also attached an image at the bottom of the post (large image). Given:   $R_1 = 20\Omega$   $R_2 = 100\Omega$   $R_3 = 50\Omega$   $R_6 = 100\Omega$   $U_0 = 2V$   $U_5 = 900mV$   $U_6 = 1V$   $I_5 = 10mA$   $I_7 = 80mA$ What needs to be calculated: $U_3, I_4, R_5, R_8, U_8, I_8$ My attempt on solving the circuit: 1) Since $U_5$ and $I_5$ are given we can use Ohm's law to calcualte $R_5 = 90\Omega$.   2) Knowing $U_6$ we can use the Mesh-Current Method to determine $U_7 = 1V$.   3) Again I used Ohm's law to calculate $I_7 = 80mA$ with $U_7 = 1V$ from 2)   4) Ohm's law to determine $I_6 = 10mA$ since we have $U_6, R_6$ given.   5) Now I used the Mesh-Current Method between $U_1, U_5 and U_6$. Since $U_1, U_2, U_3, U_4$ are parallel to each other I assumed that they must be equal (? not quite sure on this one). However this assumption lead to the following result for $U_1, U_2, U_3, U_4 = 100mV$.   6) Knowing $R_1, R_2, R_3$ and now $U_1, U_2, U_3$ I calculated $I_1 = 5mA, I_2 = 1mA, I_3 = 2mA$ with Ohm's law. 7) Now this is the point where I am stuck at. I'm not sure if it is allowed to reduce the circuit to only a specific area. For Instance I reduced the circuit to the upper left corner to determine $I_4 = 2mA$ with the Branch-Current Method. Having that given allows me to easily calculate $R_4 = 50\Omega$ with the help of Ohm's law. 8) Knowing that $U_0 = 2V$ I used the Mesh-Current Method for the whole outer part of the circuit to calculate $U_8 = 1V$. 9) Using the Branch-Current Method I determined the value of $I_8 = 110mV$ 10) Now I used Ohm's law to get the value of $R_8 = 9.09091\Omega$. $U_3 = 100mV, I_4 = 2mA, R_5 = 90\Omega, R_8 = 9.09091\Omega, U_8 = 1V, I_8 = 110mV$ I somehow cannot attach any images yet so I hope it is okay if I post an outisde link: https://www.dropbox.com/s/bfj64pkm0kg36ht/01.jpg (link to the circuit, large file, same as shown below) ![circuit](http://i.stack.imgur.com/04zfe.png)


title: Influence of the temperature on the ionization energies for impurities in silicon

text: Is there any dependence of the impurities ionization energy on temperature in silicon? I mean if there are any interactions between localized electron and phonons which leads to renormalization of electron binding energy.


title: Neutron scattering for a critical magnetic-ordering system : what about critical opalescence?

text: Liquid-gas transition critical point is believed to share the same universality class as the 3D Ising model. We know that the liquid-gas transition is characterized by a phenomenon called critical opalescence that occurs at the critical point and which in short happens because the system becomes correlated at all length scales and thus scatters all frequency of the incident light, thus making the liquid-gas look opaque. **Q** : What is the analogous phenomenon of critical opalescence in a spin system ? Say one takes a material that may be described by a 3D Ising model and does neutron scattering with a broad range of neutron energy : At the critical magnetic ordering temperature, will he see critical opalescence of the neutrons, i.e a diffuse scattering of the neutrons at all wave-length at the critical temperature ?


title: How to write a paper in physics

text: I really like to do research in physics and like to calculate to see what happen. However, I really find it hard to write a paper, to explain the results I obtained and to put them in order. One of the reasons is the lack of my vocabulary. How do I write physics well? I think that writing physics is more dependent of an author's taste than writing mathematics is. Are there any good reference I can consult when writing? Or could you give me advice and tips on writing a paper? What do you take into account when you start writing a paper? What are your strategies on the process such as structuring the paper, writing a draft, polishing it, etc? In addition, it is helpful to give me examples of great writing with the reason why you think it is good. Do you have specific recommendations?


title: How Exactly Does Linear Regge Trajectories Imply Stability?

text: (for a more muddled version, see physics.stackexchange: http://physics.stackexchange.com/questions/14020/whats-with-mandelstams- argument-that-only-linear-regge-trajectories-are-stable) There is a 1974 argument of Mandelstam's that linear Regge trajectories implies stability, from "Dual-Resonance Models" from 1974, sciencedirect.com/science/article/pii/0370157374900349. Expand the Regge trajectory function $\alpha(s)$ in a dispersion relation with two subtractions: $$ \alpha(s) = b + as + {1\over i\pi} \int_0^\infty {\mathrm{Im}(\alpha(s'))\over s-s'} ds'$$ The imaginary part of $\alpha(s)$ gives the decay of the string states, since where it hits an integer tells you where the poles are. So if the string resonances are exactly stable, then the imaginary part is zero, and the trajectory is linear. This argument bugged me for these reasons:   * It seems to work just as well with two subtractions, three subtractions, etc. Can you conclude that exactly quadratic or exactly cubic Regge trajectories are also stable? What is a quadratic or cubic trajectory?   * The Regge trajectory function appears in the exponent, so you have to take a log to extract it. Why is it clear that it has a representation like the above, without a cut contribution at negative s?    * In string theory, the trajectories are linear when they are long-lived, but the trajectory function doesn't look as fundamental today. Is there a more modern formulation of this, which would tell you which string limits are non-interacting just from a condition on the spectrum? Mandelstam generously emailed me a short remark, saying essentially that the trajectory function imaginary part is a lifetime, and indeed this is obvious from the fact that it gives the position of the resonances, but I am still confused regarding the questions above. Even a partial answer would be appreciated.


title: Allowed states vis-a-vis allowed dynamics in generalized probabilistic theories (GPTs)

text: In his work on information processing in GPTs http://arxiv.org/abs/quant- ph/0508211 Barrett speculates that the trade-off between allowed states and the allowed dynamics in a GPT is optimal in quantum theory, allowing for information processing capabilities not all of which occur in either the generalized no-signalling theory (GNST) or the generalized local theory (GLT). I'm wondering if this optimality is related to the symmetry of the associated polytope (of allowed states) in the theory, and in what exact manner. Any ideas?


title: Is the collapse of the wave function inherently time asymmetric?

text: Schroedinger's equation, as we all know, is time symmetric. In quantum field theory, we have to come up with a more sophisticated CPT reversal, but the essential point remains unchanged. However, the collapse of the wave function in the Copenhagen interpretation is manifestly time asymmetric. Correct me if I'm wrong, but can you uncollapse a wave function, converting it from an eigenfunction to a superposition of eigenfunctions? Is this asymmetry connected with the thermodynamic arrow of time and the second law in statistical mechanics, or are they independent? How would an uncollapse look like, and can we experimentally arrange for an uncollapse? Why are there more collapses than uncollapses? If an observer unobserves a quantum thingie, does that thingey uncollapse?


title: Similarity between spinless fermion and weakly interacting bosons

text: Why do spinless fermions behave in the same way as weakly interacting bosons.


title: What if an asteroid the size of the moon hitting earth

text: Would we all parish due to excessive heat? Or would that be only limited to the area near the impact while as the people on rest of the earth would die from other phenomenons such as mega earthquakes, volcanic activities, tsunamis etc. Does it matter where the impact is - if it landed in antarctica we would have massive floods but if it landed in the middle of eurasian continent, the effect would be different?


title: How to obtain ballistic transport in a device?

text: I would like to know the ballistic device characteristics and illustrations (if any) to understand how can we overcome the dependence of current on carrier mobility .


title: Normal modes of the 2D double pendulum

text: I'm performing an experiment with a 2D double pendulum, and in part of it I want to investigate the normal modes of the double pendulum, where the pendula are not of equal length or of equal mass. My question is - how will I actually know when I've successfully excited a normal mode? I start by setting the initial angles to be in (roughly) the correct proportion to one another in order for the initial setup to be an eigenvector, but of course once I release the pendulum there is a slight 'jolt' which means I can't be sure the initial conditions were exactly an eigenvector (of course, realistically I'm only going to be closely approximating one). Then, once data is recorded I can generate a phase portrait with the computer, and the phase portrait I see when I get quite close to an eigenvector looks kind of like a tilted cylinder (sorry, don't know how to post a Matlab graph here). Is there a way to tell from this phase portrait (with the angles $\theta_1$ and $\theta_2$) whether I've hit a normal mode or not? Would appreciate some help. (ie. if someone could post a picture of a phase portrait of the normal modes of a double pendulum, so that I know what I'm looking for, that would be very appreciated)


title: Is the wave-particle duality a real duality?

text: I often hear about the wave-particle duality, and how particles exhibit properties of both particles and waves. I most recently heard this in this video. However, I wonder; is this actually a duality? At the most fundamental level, we 'know' that everything is made up out of particles, whether those are photons, electrons, or maybe even strings. That light for example, also shows wave-like properties, why does that even matter? Don't we know that everything is made up of particles? In other words, wasn't Young wrong and Newton right, instead of them both being right?


title: How is it possible for light to be a wave and a particle?

text: I have always been interested in Physics, and lots of people say that light is a particle and a wave. How is it possible? How can a photon (a light particle) be a wave as well, when its a particle? I have tried looking this up, but I do not really understand the answers, as they are quite complicated to me. If you could answer this, please could you explain it in quite a simple way!


title: What is wave particle duality?

text: I am sort of confused about this. Wave particle duality says that sub atomic particles are waves. There is something more though. What is the actual meaning of wave particle duality?


title: Is the photon first a wave, then a particle?

text: When the 'photon' is emitted, it would reason that the result of the energy fluctuation that creates 'it' rather is created as an energy wave, which when measured by us or a surface, it 'becomes' as a particle. If my reasoning is right, the question then becomes why, or how, this wave appears as a corpuscular disturbance.. The more you know, the more complicated it will get, so I can see why knowing so little makes these questions easy for me to ask. I'm curious to see who is patient enough to educate a naiive question with a good natured 'schooling'.


title: Photons are packets of electromagnetic radiation (wave) what about other particles?

text: Photons are packets of electromagnetic radiation (wave), what about other particles? Are other particles, say electrons, really just packets of waves?


Are photons electromagnetic waves, quantum waves, or both? If I subdivide an electromagnetic field into smaller electromagnetic fields, should I eventually find an electromagnetic wave of a photon? How can individual quantum waves combine to form the macroscopic observable of an electromagnetic field?


title: Thermal Physics, Specific Question

text: Question: The melting point of copper is 1080 degrees Celsius and its heat of fusion is 200 kJ/kg. If a copper coin at this temperature is completely melted by the absorption of 2000 J of heat, what is the mass of the coin in kg? * * * I am not sure which formula(s) to use. Is the melting point even needed in this particular problem with it's given information? Thank you.


title: Equivalency of Gauge Conditions

text: How is the Lorenz gauge condition $\partial_\mu \overline{h}^{\mu \nu}=0$ equivalent to the harmonic gauge condition $\Box x^\mu=0 $?


How do the wave properties of an electron change with its motion? What about when it is stationary?


title: Unknown Mass Spring Problem

text: A student conducts an experiment to measure the motion of two unknown masses (A and B) along a smooth table. A constant force of 1 N was applied by means of a string connected by a falling weight. A motion sensor recorded the position (measured from the start line) of each mass at regular time intervals as they slid along the table. The results displayed by the data logger are tabulated: ![enter image description here](http://i.stack.imgur.com/RDHr0.png) What would be the best way of describing the motion of both masses? Do you use acceleration?


title: Faddeev-Popov Ghosts

text: When quantizing Yang-Mills theory, we introduce the ghosts as a way to gauge- fix the path integral and make sure that we "count" only one contribution from each gauge-orbit of the gauge field $A_\mu\,^a$, because physically only the orbits themselves correspond to distinct physical configurations whereas the motion within the gauge-orbit should not contribute to the path-integral. How come we don't run into this problem when we quantize the Fermions, which also have gauge transformations, and also have a gauge orbit? Shouldn't we include a gauge-fixing term for the Fermions as well, or does the term introduced for the Boson fields already pick out the gauge orbit for the Fermions as well? How does this technically come to be? So far I introduce a gauge fixing term into the Lagrangian as $$ 1 = \int d\left[\alpha\right]\det\left(\frac{\delta G\left[A_{\mu}\left[\alpha\right]\right]}{\delta\alpha}\right)\delta\left(G\left[A\left[\alpha\right]\right]\right) $$ where $\alpha(x)$ are the gauge functions, and $G[]$ is a functional which is non-zero only for a unique gauge-representative in each gauge-orbit, where we have the transformations as: $$ \begin{cases} \psi_{c_{i}} & \mapsto\left(1+i\alpha^{a}t^{a}\right)_{c_{i}c_{j}}\psi_{c_{j}}+\mathcal{O}\left(\left(\alpha^{a}\right)^{2}\right)\\\ A_{\mu}\,^{a} & \mapsto A_{\mu}\,^{a}+\frac{1}{g}D_{\mu}\,^{ab}\alpha^{b}+\mathcal{O}\left(\left(\alpha^{a}\right)^{2}\right) \end{cases} $$


title: Geometry and physics

text: I'm searching for references, articles and books, that discussed the geometrization of modern (and contemporary) physics, in a philosofical point of view. Something in the way that Michael Friedman did in "Space-time foundations theories" or Tian Yu Cao did in "The conceptual foundations and philosophical aspects of renormalization theory" or yet, what Roberto Torreti did in "Relativity and Geometry".


title: Turbine like device in space spinning with KE and converting to Watts. Acceleration is constant?

text: I have a question that I can't seem to get my head around. I'm really rusty on my physics, but I'm sitting here trying to think this out, but I'm missing a really big part of the equation. So lets say I have a turbine in space. It's purpose is to convert the energy given to it, in joules to watts. So I realize we will have friction, and we need to overturn the static friction before turning can be achieved. I want the device to turn at a relatively high speed, therefore F > Ff. However, this will create a situation where the device will constantly accelerate, because there is F and not being apposed by friction, Fn = ma. So here is my thought process for the situation. Force needed to start turning...   F = Fn - Ff   Energy needed to start turning...   W = Fd   where...   W = (KEf - KEi) where KEi = 0   Therefore,   a = F/m Since there will always be force, and constant supply of energy, the device will accelerate to infinity, supplying infinite energy in watts? I understand this to be fundamentally impossible... So where am I going wrong? I feel like this will be a facepalm moment, but I just can't seem to remember what I'm missing here. Best Regards.


title: X-ray imaging and diamond

text: Why doesn't diamond show up in X-ray imaging? Diamond is the hardest substance ever known, and as we know, X-ray radiation is produced when a cathode ray of high frequency hits a very high atomic mass target. But I am amazed: why doesn't diamond show up in X-ray imaging?


title: Boundary conditions: Particle in a ring

text: Particle in ring is a well-known example where a solution of the Schrodinger equation exists. My question is: In principle we also want that $\psi'(\theta) = \psi'(\theta + 2\pi)$. The thing is that this condition is never explicitly stated ( probably because it is fulfilled anyway, but in principle we would also need this condition, right?


title: Trouble understanding quantum mechanical picture of atom

text: This year in college we"ve had a lot of quantum stuff thrown at us, and we"ve only been doing classical mechanics till last year. We"ve been introduced to Schrodinger"s equation, electron waves, what have you. This is my understanding(rather rudimentary) so far. When you solve Schrodinger"s equation for an atom, you get a set of wave functions(what are the inputs and outputs exactly in this function?). Each wave function can describe a maximum of 2 electrons. It was stated in one place that the wave function represents the amplitude of the electron wave, and its square gives the probability of finding the electron at a place.An orbital is a surface diagram that includes the regions where the electron described by a particular wave function is most likely to be found within the surface. NOTE: I know the basic quantum formulae even though I may not understand what they mean. Is my understanding right?(I realize I had forgotten to put this sentence when I first posted the question, and it would"ve seemed rather confusing as to what it was I actually wanted to know.)


title: Effect of variable permittivity

text: If I immerse a rod vertically in a liquid with a relative permittivity gradient (the permittivity decreases with depth), will the rod stretch (will the spacing of the atoms in the rod be affected by the varying permittivity)?


title: What is it called when two particles are associated so that what happens to one happens to the other?

text: There was some experiment that I read about some time back in which two particles (or the same particle, but split into two) were sent in opposite directions, but when something happened to one, it happened to the other at the same moment, as if they were the same particle. I thought the word for this was "entrainment", but looking that up does not lead back to this experiment. What is the term for this, assuming that I am not remembering something that never happened.


title: Yet another Young Double Slit Experiment

text: If I have a laser light incident on the double slit at an angle O < 90 what kind of interference pattern will I see? Will i see the same with less intensity or different pattern?


title: Temperature in space

text: Temperature is a measure of kinetic energy transferred to particles, henceforth, space being vacuum, temperature cannot be measured. But then, there is cosmic background radiation. It is the leftover heat from the Big Bang, but then heat in space doesn't make sense, how does this radiation, heat up space time when it technically cannot be heated up? Moreover why is it referred to as heat when it is approximately just $2.5$ Kelvin, which is just above absolute zero?


title: Electric field and insulator or dielectric

text: I have a uniform electric field $E$ projected from left to right and I placed an insulator or dielectric right in the middle of it. To the left of the insulator, the $E$ is the original $E$. Inside the insulator, there is an induced field and thus the net field is less than the original. What happen to the $E$ to the right of the insulator? Will it be the same as the original? If it does, then the $E$ will have to increase back? Why?


title: Write $\epsilon_{\mu\nu\alpha\beta} F^{\mu\nu} F^{\alpha\beta}$ as a total divergence $\partial_\mu G^\mu$

text: I have the following homework problem in theoretical electrodynamics: > Show that the gauge invariant Lagrange density $\epsilon_{\mu\nu\alpha\beta} > F^{\mu\nu} F^{\alpha\beta}$ can be written as a total divergence of a four- > vector. This total divergence should be something like $\partial_\mu G^\mu$, right? With $$ \hat F^{\mu\nu} = \frac 12 \epsilon^{\mu\nu\alpha\beta} F_{\alpha\beta} $$ I figured that I write this Lagrange density (I'll refer to it as $L_2$) could be written as: $$ L_2 = 2 \hat F^{\mu\nu} F_{\mu\nu} $$ I simplified this by using the matrix representations of each $F$ and got it down to: $$ L_2 = -\frac 4c B_i E^i $$ $B$ and $E$ can be expressed in Terms of $A$ like so: $$ B_i = \epsilon_{ijk} \partial_j A^j ,\quad E_i = -\partial_i A^0 -\partial_0 A^i $$ How would I continue to find that four-vector $G$?


title: Age-ing due to Time Dilation

text: Will a person on top of hill will age faster than one at sea level due to Time Dilation?


title: Differential Realizations of certain algebras

text: I'm a first year graduate student in Mathematical Physics, and I am trying to generalise a certain method involving the so-called "Differential realizations" of certain algebras. The problem I'm having is that in papers here (Page 8) and here (Page 5-6), they express a certain type of algebra in terms of differential operators and I'm not sure how that is done. My goal is to see if such methods can be generalised to other algebras to test for a certain property called "Shape Invariance" which is mentioned in the latter paper. I'm wondering if anyone here has ever heard of differential realizations and has any idea how they can be derived for other algebras?


title: Expanding Universe now and $\sim14$Gyr ago

text: All know that Inflation may solve some of the problems of the Standard Model of Cosmology and today we suppose that the field responsible for that is the inflaton. The properties of the potential $V(\varphi)$ allow via the momentum- energy tensor to define a barotropic fluid for which $$p=-\rho$$ and when the slow-roll conditions $\epsilon\ll1$, $|\eta|\ll1$ are satisfied, this fluid with $w=-1$ has some sort of negative pressure or repulsive gravity that allow the expansion of Universe. So a scalar field may act like a cosmological constant $\Lambda$ giving some energy to the vacuum. Now seems that the Universe undergoes an epoch of "second inflation" with $$\ddot{a}>0.$$ My question is: why there is no exponential inflation like the old one if today the dominant fraction of the Universe budget is again a sort of cosmological constant $\Lambda$? Maybe the two fluids are of completely different kind, or the behavior depends on the energy and temperature conditions or i am wrong something?


title: Is current related to the length of the conductor?

text: Ohm's Law tells us that $V = IR$. This implies that $I \propto \frac{1}{R}$. But, $R \propto l$, where l is the length of the conductor. This would mean that $I \propto \frac{1}{l}$. But this does not fit with the definition of current that says that current is the amount of charge $Q$ passing through a given point per unit time. Clearly, the amount of charge passing through a given point per second does not vary with the length of the conductor. How can this contradiction exist?


title: Derive that $P = I^2R$

text: As our homework, we were asked to derive $P = I^2R$. Now, I started off with the basic relation $P = \frac{W}{T}$. I was not able to think of anything from here, so I started plugging in random formulas from a purely algebraic point of view. I tried the one for voltage $V = \frac{W}{q} \implies Vq = W$ and got $P = \frac{Vq}{T}$. And since $I = \frac{q}{T}$, $IT = q$, plugging this in we get $P = IV$. A simple application of Ohm's Law gives us the required equation. But, when I tried to get an intuitive grasp of what I'd just done, I found a few discrepancies:   1. In the definition of voltage, $W$ refers to the energy an electron loses when it goes from the negative to the positive terminal. In the definition of power, $W$, refers to the total energy consumed by the appliances. How can you relate the two?   2. In the definition of current $T$, refers to the time taken by the electron to go from one terminal to the other, whereas, in power $T$ refers to the time for which the appliances run or the energy is supplied. How can these two be the same? Is my derivation incorrect? If not, then how would you counter the arguments I mentioned above?


title: Field inside a conductor?

text: ![enter image description here](http://i.stack.imgur.com/H9uwm.jpg) If the above image is a cross section of a conductor, the field at the point shown is not zero. So the field inside a conductor is not zero at all points. You could argue that the electrons would rearrange themselves so that the field inside is zero. But if that were true, all the positive charges would not be on the surface. For example, in the diagram above, the electron at the point shown, would move, leaving a charge in the centre. What am I missing? # _**I am not looking for a proof that the field inside a conductor is zero. Please address the problem I discussed above._**


title: Fixed Angular Momentum

text: Say I'm given the following Schrodinger equation $$\frac{d^2u}{dx^2}+ \left[E - V(x)+ \frac{a}{x^2}\right]u(x) =0$$ Where $a \in \mathbb{R}$. What are the physical interpretations of this equation? I understand that it means the angular momentum is fixed, but are there any other implications associated with this?


title: Divergent issue of Madelung's constant

text: This is a question triggered by this post Madelung's constant is defined to the coefficient of electrostatic potential energy in a ionic crystal. In the example of $NaCl$, \begin{equation} M = \sum_{ijk}{}^{'}\frac{(-1)^{i+j+k}}{\sqrt{i^2+j^2+k^2}} \end{equation} is conditionally convergent. > _Since this sum is conditionally convergent it is not suitable as definition > of Madelung's constant unless the order of summation is also specified. > There are two "obvious" methods of summing this series, by expanding cubes > or expanding spheres. The latter, though devoid of a meaningful physical > interpretation (there are no spherical crystals) is rather popular because > of its simplicity. Thus, the following expansion is often found in the > literature:[2]_ $$ M = -6 +12/ \sqrt{2} -8/ \sqrt{3} +6/2 - 24/ \sqrt{5} + > \dotsb = -1.74756\dots. $$ _However, this is wrong as this series diverges > as was shown by Emersleben in 1951.[3][4] The summation over expanding cubes > converges to the correct value. An unambiguous mathematical definition is > given by Borwein, Borwein and Taylor by means of analytic continuation of an > absolutely convergent series._ I have the following questions. 1) Expanding sphere leads to a divergent series. OK, what if I make a perfect spherical sample of $NaCl$, will the experimentally measured Madelung's constant to be infinity? ( **EDIT** : I maybe have some misunderstanding of divergence: it could be the case that the series is bounded but don't have a definite limit. So is the series for expanding sphere bounded? and if there is no definite limit, what's the experimentally measured value for a perfect spherical crystal?) 2) What physical principle dictates the order of summation? or why does finite number obtained by analytic continuation should be consistent with the observed value? 3) What is the role of charge neutrality here? I ever programed to compute Madelung's constant using the fractional charge idea((assign $\frac{1}{8}$ charge to the corner, $\frac{1}{4}$ to the edge, $\frac{1}{2}$ to the face ), which makes the expanding cube charge neutral. Does that mean charge neutrality is one of conditions that _must_ be enforced? or it is just for the sake of computational efficiency?