title: Current induced from a magnet moving through a solenoid?

text: If we had a magnet we move it towards a solenoid a half sine wave current is induced when we make it move back, the other half is induced, inducing an AC current in the solenoid. What if the magnet moves towards the solenoid, inside it and comes the other side? Will a current be induced while the magnet is moving inside (not entering nor exiting the solenoid, given that length of solenoid is much bigger than that of the magnet). If yes, what kind of current will be induced? If no, why? If AC current is induced, how would one calculate its frequency?


title: Two-fluid description of superfluidity

text: I'm trying to teach myself about superfluidity and I'm slightly confused on the ''two-fluid'' description. From what I understand, the superfluid is considered to be a mixture of two fluids, a superfluid component and a normal fluid component. Are these two fluids a true ''mixture'' or are is it just a model to break the behavior of the one _real_ fluid into its two different aspects? Could we separate them from each other as if they were truly separate fluids? For example, if it were truly a mixture, we could make it do something that only a superfluid could do, such as pass through an extremely small opening. This would separate the fluids and leave us with just the superfluid component. Would this remaining ''pure''-superfluid still be described by the two-fluid model?


title: what is difference between these two expectation values?

text: what is difference between these two expectation values $\langle \hat A \hat B\rangle$ and $\langle \hat B \hat A\rangle$? where the $\hat B$ and $\hat A$ are two operators.


title: At what g is terminal velocity not terminal?

text: How weak would gravity need to be in order for a human to reliably survive the terminal velocity of falling through air? (Context: watching scifi on a space station with a variety of artificial gravities, it occurred to me that medium-strength gravity would have some advantages; also I note that insects already seem to have this luxury on earth, with their small masses and high air resistances...)


title: gamma-gamma collision seen from a relativistic rocket

text: This is a thought experiment. Consider two photons traveling at right angles and intersecting at some point. As seen from Earth these photons will not interact with each other because their energy is too low for particle creation. Now consider a rocket traveling at high relativistic speed towards the intersection point. People on that rocket will "see" these photons extremely blue shifted. They would appear as gamma rays. Now from the rocket point of view there is a higher probability that particle creation would occur, because these photons appear so energetic. Any produced particles like electrons or positrons would be captured by the rocket and brought back to Earth as a proof. The experiment could be repeated to get good statistics. How many particles would that rocket bring to the Earth. None? Or maybe the amount predicted by observers on the rocket? Disclaimer. I am not a physicist. Probably this "paradox" has a very simple resolution just like other relativity paradoxes. I probably made some wrong assumptions.


title: How does the Cern LHC collide particles head on if uncertainty principle limits the precision

text: I have been wondering why doesn't the uncertainty principle prevent the LHC experiment as if one want to collide two particles together one must accelerate a particle to certain speed and aim it at another particle thus requiring the scientist to know both the velocity and the position in microscopic world, violating the uncertainty principle? Thanks in advance


title: IR divergence and renormalization scale in dimensional regularization

text: * Is it possible that if a certain (loop) integral is IR divergent then that will have effect on the dimensionally regularized answer for that? (..does the epsilon expansion see the IR divergence in any way?..)    * If a certain epsilon expansion shows double poles (or higher?) then does this have an interpretation? (like is that a sign of IR divergence in the theory?)  I am not necessarily talking of QFT but dimension regularization as a method to regulate integrals in any field theory (may be classical).   * In dimension regularization is the only way to determine the value of the counterterms is solely through setting of some predefined value at a renormalizaton scale? (then one resubstitutes the counterterm thus calculated into the correlation function being calculated and hence necessarily all evidence of the counterterm is guaranteed to vanish in the final answer?)  OR after doing the dimensional regularization is it legitimate to just choose the counter-term such that it removes the 1/epsilon poles? - in this way of thinking one doesn't need to have a renormalization scale. (or after this regularization will the renormalization scale necessarily show up in trying to set the finite/physical/renormalized part of the counterterm?)


title: Ideal gas with two kinds of particles, Grand canonical partition function

text: Consider an ideal gas contained in a volume V at temperature T. If all particles are identical the Grand canonical partition function can be calculated using $$Z_g(V,T,z) := \sum_{N=0}^\infty z^N Z_c(N,V,T)$$ where $z$ is the fugacity, and $$Z_c(N,V,T):= \frac{1}{N!h^{3N}} \int_\Gamma e^{- \beta H(X)} dX$$ is the canonical partition function. For identical particles I can compute the Grand canonical partition function. Now I am asked about two different kind of particles with masses $m_2= 2m_1$ and energies of $$h_1(p) = \frac{p^2}{2m_1}$$ and $$h_2(p) = \frac{p^2}{2m_2} + \Delta$$ where $\Delta > 0$ is a constant. The fugacitys are given as $z_{1,2}=e^{\beta \mu_{1,2}}$, where $\mu_{1,2}$ are the corresponding chemical potentials. The task is to find the partition function. I am sorry to admit, but I have absolutly no clue where to start. Since no potential is present the Hamiltionian should be $$H(X) = \sum_{i=1}^{N_1} \frac{p_i^2}{2m_1} + \sum_{i=1}^{N_2} \frac{p_j^2}{2m_2} + N_2 \Delta$$ where $N_1+N_2=N$ are the numbers of particles of the differen kind. First I don't understand where that $\Delta$ comes from? And I have no idea how to continue. Simply inserting $H$ in $Z_c$ does not give any usefull results.


title: what happens to the angular velocity of star in star-black hole system?

text: What happens to the rotational and revolutionary angular velocities of star in star-black hole system as the star loses mass ?


title: How to learn the wavelet transform?

text: Is there any good literature if I want to learn the wavelet transform? Especially my project is related with marine electromagnetism?


title: Measuring tensor rank of the Higgs resonance

text: The Higgs boson is the first scalar field ever measured that is fundamental rather than composite. How do high-energy physicists determine that a given resonance in the statistics is either a scalar, vector or fermion? How is the rank of a field inferred from the input data obtained from the detectors? I'm assuming angular momentum plays some part, but I would like to have a better idea or overview about how they go about the calculation.


title: What could magnetic monopoles do that electrically charged particles can't?

text: I understand the significance to physics, but what can a magnetic monopole be used for assuming we could free them from spin ice and put them to work? What would be a magnetic version of electricity? * * * EDIT Sorry this wasn't clear. The question is mixed between the quasiparticle and the theoretical elementary particle based on some similarities between the two. I am more interested in the quasiparticle and if they have properties in some way that are similar to particle version: > There are a number of examples in condensed-matter physics where collective > behavior leads to emergent phenomena that resemble magnetic monopoles in > certain respects, including most prominently the spin ice materials. While > these should not be confused with hypothetical elementary monopoles existing > in the vacuum, they nonetheless have similar properties and can be probed > using similar techniques. > > http://www.symmetrymagazine.org/breaking/2009/01/29/making-magnetic- > monopoles-and-other-exotica-in-the-lab/ > > "The Anomalous Hall Effect and Magnetic Monopoles in Momentum Space". > Science 302 (5642) 92–95. > > "Inducing a Magnetic Monopole with Topological Surface States" > > "Artificial Magnetic Monopoles Discovered" and comments in articles about quasi-particles like this: > Many groups worldwide are currently researching the question of whether > magnetic whirls could be used in the production of computer components. led me to wonder what application might they have? Mixing these two concepts is probably a bad way to present this question. A true magnetic monopole would effect protons whereas the artificial ones don't. What I don't understand is what advantages an artificial magnetic monopole would have. And does this relate to some theoretical aspect of a true monopole?


title: Are we living in a false vacuum? Is there any way to tell?

text: I was thinking of the noted 1980 paper by Sidney Coleman and Frank de Luccia--"Gravitational effects of and on vacuum decay"-- about metastable vacuum states that could tunnel to a lower energy "true vacuum" with catastrophic results. I suspect the answer is that if our vacuum is a false one, we wouldn't "see" the true one til it hit us. http://prd.aps.org/abstract/PRD/v21/i12/p3305_1


title: Studying electrodynamics problems

text: Suppose an advanced undergraduate student has reached a moderate level of understanding on electrodynamics. Where should he **focus** on, to sharpen his **problem-solving skills**?   * Practicing integrals and/or other **mathematical tools**.   * Studying **theoretical results**.   * Working on the **physical meaning** and applications of ED equations.   * Something else. I understand this is a somewhat subjective question but please try to be as objective as possible: state what really **works** based on your experience as a teacher/TA for example


title: Basic vector addition problem

text: This is the entire problem: > _A student adds two vectors with magnitudes of 200 and 40. Taking into > account significant figures, which is the only possible choice for the > magnitude of the resultant?_ > > _a. 160_ > > _b. 240_ > > _c. 200_ > > _d. 300_ The answer is supposedly C, but it doesn't make sense to me given that 200 + 40 = 240. Could you please explain how I am wrong in my conclusion?


title: Homework Help - Physics / Calculus III / Force-Work Problem

text: I have tried to answer the problem correctly, and I have found similar (and even an identical) problem in my textbook, yet I still can't seem to yield the correct answer with this specific problem. The example problem in my textbook: A crate is hauled 8m up a ramp under a constant force of 200 N applied at an angle of 25 degrees to the ramp. Find the work done. W = F * D cos(25) = (200)(8)cos(25) = 1450J. Can anyone give me any advice on how to solve it? A woman exerts a horizontal force of 4 pounds on a box as she pushes it up a ramp that is 5 feet long and inclined at an angle of 30 degrees above the horizontal. Find the work done on the box. I would think that the work would simply be (4)(5)cos(30), but that is not the correct answer. (I've figured out how to accept answers now, so I'll accept the most helpful answer.)


title: Seeing the image as explained by Geometrical optics?

text: I was thinking about how the human visual system perceives the position of an object. I assume it essentially does this by focussing the incoming rays onto the retina and then judging from the power of the lens required to focus and the size of the image, it guesses the position. Assuming this process to be present always, the net result is that we can assume that when a set of diverging rays from a point fall on the eye, it traces the position of the object to the point from where it appears to diverge. But then in two cases I fail to see what we will "see"! 1] when using a convex lens, as in simple microscope' the image is formed at "infinity" and the object is at focal point. Then the rays are parallel. But still we can see the image and what is more, there is even a high-school formula for the angular magnification in this case! But if the rays are parallel how can they be traced back to appear to be diverging from one point where the image of the object will be perceived? 2] What if the rays are converging, in case we poke our eye in the path of the rays coming from a object placed in front of the convex lens such that our eye is pushed in between the lens and the place where real image would have been formed? Then the rays are converging so can we still see the image or the object looked at through the lens becomes invisible? And on a small side question, when we say that the object is at infinity, all we mean that the rays from each point are parallel to one another but not that the rays from different points are parallel, right?


title: Noether current for the Lagrangian without Lorentz invariance

text: I am reading an article by Watanabe & Murayama. It gives a proof on the counting of Nambu–Goldstone bosons without Lorentz invariance. I am trying to derive all the equations to get a better understanding. But I have met some trouble with Eqs. (5) and (6). How does the functions $e_i(\pi)$ are determined so that $\delta\mathcal{L}_{\mathrm{eff}}$ does not depend on $\pi^a$ explicitly? Could anyone provide a derivation of (6) from (4) and (5)? Many thanks!


title: Finding $\psi(x,t)$ for a free particle starting from a Gaussian wave profile $\psi(x)$

text: Consider a free-particle with a Gaussian wavefunction, $$\psi(x)~=~\left(\frac{a}{\pi}\right)^{1/4}e^{-\frac12a x^2},$$ find $\psi(x,t)$. The wavefunction is already normalized, so the next thing to find is coefficient expansion function ($\theta(k)$), where: $$\theta(k)=\int_{-\infty}^{\infty} \psi(x)e^{-ikx} \,dx.$$ But this equation seems to be impossible to solve without error function (as maple 16 tells me). Is there any trick to solve this?


title: About Born's rule

text: I wanted to gain a better understanding of the Born rule to make my class on quantum mechanic feel less ad hoc. To do so I attempted to show that the version (1) given in my book is equivalent to the version (2) on Wikipedia.   1. The version in my book: > $P_x = \frac{\langle\psi_x\mid\psi_x\rangle}{\langle\psi\mid\psi\rangle}$ > Where $P_x$ is the probability of getting a state $\psi_x$ when measuring.   2. Version on Wikipedia: > the probability of measuring a given eigenvalue $\lambda_i$ will equal > $\langle\psi\mid P_i\mid\psi\rangle$, where $P_i$ is the projection onto the > eigenspace of A corresponding to $P_i$. There is a short explanation but I would be glad if someone could put it into better words. Specifically I don't understand what $P_i$ is. I do know about the Hermitian operator, its eigenvalues and eigenspaces. But for example, why is $P_i = \mid\psi_i\rangle\langle\psi_i\mid$ when the eigenspaces are one- dimensional. And how do I go from there to the form in my book?


title: What is the stress-energy distribution of a string in target space?

text: If $| \psi \rangle$ is a string mode, how do you compute $\langle \psi | \hat{T}^{\mu\nu}(\vec{x}) | \psi \rangle$ where $\vec{x}$ is a point in target space? This information will tell us the energy distribution of a string. In string theory, the size of a string grows logarithmically as the worldsheet regulator scale. In the limit of zero regulator size, all strings are infinite in size, and this ought to show up in the stress-energy distribution. What about multi-string configurations? The vacuum has virtual string pairs. Only the spatial average $\int d^9x \hat{T}^{00}(\vec{x}) |0\rangle = 0$. $\hat{T}^{00}(0) | 0 \rangle \neq 0$ even though $\langle 0 | \hat{T}^{00}(0) | 0 \rangle = 0$. Here, $|0\rangle$ is the string vacuum. The stress-energy operator can create a pair of strings. It doesn't preserve the total number of strings.


title: How do stun guns not kill people?

text: Now, I've seen a lot of answers to this sort of question, but most of them provide answers that don't actually make sense from a physics perspective. As an example of such an answer, I've commonly seen it mentioned that stun guns avoid killing people because they use transformers to step up the voltage and reduce the current. Of course, because transformers reflect impedance, the current flowing between the electrodes when the human body completes the circuit will just obey Ohm's law with the stepped up voltage. Now, even with the more modest stun gun claims of, say, $150\: \mathrm{kV}$ and a generous estimate of the resistance of the human body ($100\: \mathrm{k \Omega}$), that would mean a current of $1.5\: \mathrm{A}$, more than enough to kill or cause serious damage. As, I see it, there are only a few possible explanations:   1. Manufacturers are lying, and the voltages generated by stun guns between the electrodes are much lower than reported.   2. The pulses are just too short in duration to cause serious damage.   3. The placement of the electrodes (no more than an 2-3 inches apart) means that the path taken by the current never goes much below the skin, and definitely doesn't wind through vital organs. Based solely on my intuition, I would think 3 is the most likely explanation (with a healthy dose of 1 as well, since some of the claimed voltages are obviously far too high), but I'm not too sure that's the case. Any insight on which, if any of these is the correct explanation? EDIT: The responses got me wondering what would be the limiting factor if the typical designs did not use a capacitor, which led me to another fundamental limiting factor, which I think is what was behind the point that drawing such power from a small device could not be sustained. To step up the voltage from a $9\:\mathrm{V}$ battery (the typical sort used in stun guns) to $150\: \mathrm{kV}$ would require stepping voltage up by a factor of approximately 16,667 times. Since the current in the secondary would also have to be reduced by that factor, for the current to be $1.5\: \mathrm{A}$ in the secondary, it would have to be $25\: \mathrm{kA}$ in the primary. Drawing that sort of current from a $9\:\mathrm{V}$ battery is just not possible, as the internal resistance of the battery is too high (not to mention that even if you could draw that sort of current from a $9\:\mathrm{V}$ battery the battery would probably be instantly depleted or components would melt or explode). Typical $9\:\mathrm{V}$ batteries have internal resistances around $1.5\: \mathrm{\Omega}$ (http://ww2.duracell.com/media/en- US/pdf/gtcl/Product_Data_Sheet/NA_DATASHEETS/MN1604_6LR61_US_CT.pdf, for example), so drawing a current that high is out of the question. Incidentally, if my thinking on this point is correct, that also explains why most stun guns step up the voltage a bit, charge one capacitor of higher voltage than the battery, and use that discharge to charge an even higher voltage capacitor. The batteries simply can't generate enough current to be stepped up that far without such a multi-tier approach. Does my reasoning on this point seem correct? Again, thanks for your patience and help!


title: Can we view quantum fluctuations as a type of quantum power?

text: (Note: I will let the question stand as is, it has generated a good answer, but the question is too "wishy washy" and needs to be tighter.) We know that the uncertainty principle tells us that the product of the standard deviation of energy and the standard deviation in time must always be greater than h-bar. Since there is a unit element of action, is there a unit element of power as well? In other words, if action is always present, then must power always be positive? Clarification: This is not intended to be a question related to people who think that there is "vacuum energy/zero point energy/free energy" that can be harvested. This is more along the lines of whether the universe can self- perpetuate. Clarification 2: Self perpetuate I would define as the ability for the universe, through mechanisms of quantum fluctuations related to the uncertainty principle to produce another epoch similar to our own that would appear to be later in time, from our perspective, and after an apparent thermodynamic death of our current universe. Question restatement: Is quantizing action analagous to there being a source of power at the quantum level?


title: Perturbative solution + Nonperturbative solution = Full solution?

text: I am having this silly confusion! Suppose I have a system (a Hamiltonian or an action say) and also suppose I have a perturbation parameter present (say only one in sight) in there, using which I can find out some kind of perturbative solution of that problem. Suppose now we also found out a non perturbative solution to that problem (by non-perturbative, I have in mind that the solution is not an expansion using the perturbation parameter) (well, origin of this question is that in string theory I have $\alpha^\prime$ and also $g_s$ as perturbation parameter and then we also see $D$-branes as its non perturbative d.o.f. But I am asking here as a general scenario). But is it guranteed (I thought its not) that you have ``solved'' the problem once you know that particular perturbative solution and non-perturbative solution? Let me rephrase: Unless I know the exact solution of a problem and the only way to solve that problem to us is such perturbative, non perturbative methods, how can we ever be sure that we've solved the problem and there are no more solutions? (probably it will vary from problem to problem, but can there be a problem with such ambiguity?)


title: How can I weigh liquid in a sealed container?

text: How do you measure the mass and volume of alcohol if it is in a sealed container? Are there clever ways of achieving this? The weight of the container is not known. I can't open the container, I can't lose the alcohol inside, and I can't immerse it in water (as the label will break).


title: Lead in ice block

text: This is a strange question I would like to have some explanation accompanied with: We have an ice block, with in it a lead block. This entire block is put into a glass of water, resulting in a rise of water level until the edge of the glass. The block stays floating on the surface. What will happen with the water level when the ice melts? The answer is that it will drop, but how is this even true?? My thoughts are: \- block floats on water, so rho(block) < rho(water) \- Archimedes law, the Fg of block = Fg of water pushed away. \- Since a smaller volume of water has equal mass (and so Fg) to block, and ice is less dense than water, when it melts, no mass is lost, so it will take in less space??? Please help me with this point of thinking.


title: Steps for going from Polyakov action to low-energy effective action (SUGRA?) in String Theory

text: A lot of string-theory questions have been asked here. This one is, hopefully, different in that this inquiry is specific and the expected answer would be more mathematical than philosophical in character. To give some context, I'm familiar with LQG, QFT etc. I'm a beginner in CFT and string-theory. On reading papers on AdS/CFT and String Theory the following sentence often appears: " **The low-energy effective action for $x$ dimensional ST of type $y$ reduces to that of supergravity (SUGRA) in $z$ dimensions with matter fields of types $a$, $b$ and $c$** ". The $64,000 dollar question any noob, such as me, wants answered is what are the mathematical/technical steps or arguments which lead us from the fundamental action of string-theory (the Polyakov/Nambu-Goto action) to a low- energy effective description. Of course, I might be asking a question for which the only concise answer is: " _go read GSW_ ". Here's hoping otherwise ;)


title: Moment of inertia of a sphere

text: I'm looking at sample calculations of moment of inertia of a sphere here. In the first example (disc method), it has the integral as $dI = \frac{1}{2}r^2 \,dm$, while in the second example (shell method), $dI = r^2 \,dm$. Why is this so? Where does the 1/2 come from?


title: Gravity vs. Velocity Time Dilation

text: Can some one give me the unified formula for Gravity Time Dilation and Velocity Time Dilation.


title: Do Cooper pairs act like Cheshire cats?

text: Could the pairing up of electrons be explained by their spin being in a different position? What would separating the spin from an electron in matter do in theory?


title: Reflection between two mirrors?

text: If I put my hand between two perfectly aligned mirrors and then remove it will the images continue to reflect for a few nanoseconds? If yes then will the images be of a combination of the front and back of my hand or will a clear image of both appear to occilate from one mirror to the other? In other words will the palm of my hand appear at least once on the mirror that faces the back of my hand?


title: How Safe Are Heat Ray Guns?

text: Could a little meddling with the frequencies of the Heat Ray Gun beam result in frying crowds rather than dispersing them?


title: Question about energy

text: Tom, weighted 60kg, set a world record of bunjee jumping, he jumped off at 223m height, and reached the lowest point when he's 50m above ground. If 30% of mechanical energy was lost when the cord was stretched, what was the elastic potential energy stored in the cord when he reached lowest point? My work :: 70% initial GPE = final GPE + EPE 0.7 * 60 * 9.81 * 223 = 60 * 9.81 * 50 + EPE EPE = 62450 J But the answer turn out to be 72660 J, what's wrong with my answer?


title: Why do we hear this mysterious sound?

text: Sit in a silent place and then bring your hands close to your ears and cover your ears, you will start hearing some sound but if you remove your hands you don't hear it anymore. Can anyone explain physics of this?


title: How does the full string theory potential look?

text: Is the full stringy potenetial (for which it is claimed to be 10^500 vacua) written down explicitly somewhere? Any references? Thanks, Dave


title: Why Does Dipole Moment Depend on Distance?

text: Why does the dipole moment of an electric dipole of different charges depend on the distance from the origin? Physically, I don't understand why something that measures how much of a dipole something is should depend on how far away that thing is from the origin.


title: How to Calculate temperature, given speed of sound and relative humidity?

text: I know the solution in reverse direction, but I find it very complex to go the other way around and calculate the temperature when given the speed of sound and relative humidity, any ideas?


title: Can you damage your eyes viewing Venus?

text: Whilst I know it is extremely dangerous to view the sun using a telescope (or even just looking directly at it for prolong periods) can you do permanent damage to your eyes looking at Venus through a telescope? I was out with my telescope last night looking at Jupiter and moved over to Venus as it's right next door at the moment. At the moment it seems really really bright and for a quite while afterwards I felt as if I had arc eye. I've not really used my telescope for a couple of years but I remember looking at Venus previously and whilst it was bright it seemed perfectly OK to view. Hopefully it doesn't seem like a stupid question, I was a bit surprised because it did feel like I'd just deliberately watched a welding spark. It was quite early in the evening; but the sun had fully set and the sky was dark. Is the brightness of Venus to us affected by it's position - it's probably in quite a good position to reflect sunlight as it's quite low in the west at the moment.


title: Why doesn't a close comet like Elenin look like other, famous comets (Hale Bopp, Halley...), which have nice, long trails?

text: Elenin is a comet that runs very close to the Earth (0.8 times the Earth-Moon separation at its closest point). By naked-eye observation, it doesn't look like the other famous comets (Hale Bopp, Halley, etc.)- its trail is quite invisible. Why ? Is it a dirty snow ball that does not emit any vapor? Is it really a comet by definition, or may an object lie somewhere between comet and asteroid?


title: Can the Hubble telescope bring any star into focus?

text: Lets say I am talking about a view like this supernova - 13 billions light year away. In short can Hubble bring any star into focus in the entire universe? And if so, to what definition? I also wonder, how much time would time would it need to focus on a distant star or planet, in comparison to a closer one?


title: Function for heating of a planet by its star

text: I'm looking for an algorithmic model which will gives rough estimate of the average temperature of a planet's atmosphere (good enough to say whether there will be liquid water anywhere, or if metals and plastics will melt/ignite) based on the luminosity of its star, the radius of its orbit, the constituents of its atmosphere, the mass of the planet, and the surface area of the planet. Specifically, I figure I need two mathematical models. The first would simulate the incoming thermal energy from the star, and the second would calculate how much is lost as blackbody radiation by the planet, taking into account the effects of the 'greenhouse effect'. Does anyone know how to model these things?


title: Self-consistent field approximation and uniform field approximation?

text: Can anyone give me explanation of self-consistent field approximation and uniform field approximation? I know self-consistent as when we write the Schrödinger equation as $$[ -\frac{\hbar^2}{2m} \nabla^2 +v(r) ] \psi{_k}(r)~=~ E(k)\psi{_k}(r).$$ To know the potential appearing in the above equation, we have to find its solution. If we consider a solution for determining potential and get the same value of potential by considering another solution then this is called self- consistency. I'm not actually clear on this two topics. Can anyone point me in the right direction?


title: How to find constant of motion for Hamiltonian system?

text: I have to find a constant of motion associated to this Hamiltonian but I don't know how to proceed. $$H=\frac{\mathbf{p_0}^2}{2m}+\frac{\mathbf{p_1}^2}{2m}+\frac{\mathbf{p_2}^2}{2m}-2V(\mathbf{r_1}- \mathbf{r_0})+V(\mathbf{r_2}-\mathbf{r_1})$$ where $$V(\mathbf x)=\frac {e^2}{|\mathbf x|}.$$ I don't know what $\mathbf x$ is. This Hamiltonian refers to a system of 3 particles $(0,1,2)$ with mass $m$ and charge $e$. The coordinates are $r^\alpha_i$ and conjugate momenta $p^\beta_j$ with $\alpha, \beta=0,1,2$. I have written all the information that I have.


title: What are the similarities and differences between speech and music sounds?

text: From an acoustic perspective, I guess speech sounds are produced by varying/manipulating:   1. resonance (shaping the cavity; something to do with harmony?)   2. f0 (the length of the string/pipe/whatever) However, in music, only (2) can be varied/manipulated. That is, the resonance/cavity doesn't change. As you probably guessed, I'm neither a physicist, a musician, or a speech pathologist. Can you help me understand better the similarities and differences of sound production for speech vs musical instrument?


title: Weak localization and constructive interference

text: To my knowledge weak localization is a result of constructive interference between scattering processes and their time reversed. Traveling a given path in a vector potential leads to the phase $q\int d\vec{x}\ \vec{A}(x)$. A particle traveling the time reversed path gets an additional "$-$"sign as the path is inverted. Somehow the spin flip seems to give an additional sign resulting in constructive interference. How is that? How is the phase of a particle in an vector potential connected to the spin?


title: Integrating pressure over a surface

text: Consider the 2D airfoil below. ![2D Airfoil Pressure](http://hyperphysics.phy- astr.gsu.edu/hbase/fluids/imgflu/airfoil.gif) In engineering (and maybe physics) you will often see something like the following as an expression for the pressure force acting on a surface (in this case a curve but imagine it having unit depth into the screen). $$ \mathrm{d} \mathbf{F} = p \, \mathrm{d} \mathbf{s} \\\ \text{where} \, \mathrm{d} \mathbf{s} = \mathbf{\hat{n}} \, \mathrm{d} s $$ If you attempt to integrate this over a curve C to find the force you get; $$ \int_?^? \mathrm{d} \mathbf{F} = \int_C p \, \mathrm{d} \mathbf{s} $$ where there doesn't seem to be obvious corresponding limits for integrating on the LHS. Is it fine to consider the limits as from 0 to $\mathbf{F}$ or is this some kind of engineering "shorthand" that you often see which makes no sense mathematically. I am trying to interpret it as a "change in force" but it doesn't really make sense to me.


title: Can a painting be used as a current generator via absorption of photons in the paint?

text: What happens to absorbed photons in a painting? Heat of course except for the free electrons. Could paint compounds (the dark, non-reflective) be embedded with reactive and conductive materials to generate a current?


title: Generalizing a relativistic kinematics formula for spatial-acceleration dependence

text: I'm starting from this expression $$ \alpha dt = \gamma^3 dv $$ where $\alpha$ is proper acceleration of a point particle, $dv$ and $dt$ are coordinate differentials of velocity and time, and $\gamma$ is the relativistic factor of the particle being subject to the acceleration If $\alpha$ is constant, one arrives at the usual expression: $$ t_f - t_0 = \frac{1}{\alpha} (\frac{v_f}{\sqrt{1-\frac{v_f^2}{c^2}}} - \frac{v_0}{\sqrt{1-\frac{v_0^2}{c^2}}}) $$ Now, I have some dependence of acceleration to position; $\alpha(x) = f(x)$ and I'm not sure how to integrate it in order to obtain a similar expression relating time, velocity and position. For instance, I tried the following for the left-side differential: $$ \alpha dt = \alpha(x) \frac{dt}{dx} {dx} = \int{ \frac{ \alpha(x) }{v(x)} dx } $$ and leaving the right-hand side untouched. When I do this I get a weird integral with velocity in both sides, and I'm not sure how to continue. **Update** so, after a while of staring at this, i noticed an error i was doing, and actually the 2nd derivative of acceleration should look like $$\frac{d^2 x}{d \tau^2} = \gamma^4 \frac{d^2 x}{dt^2}$$ So far so good, but the approach on the comments doesn't seems to work. I try with a simple force potential: $\alpha(x) = -k x^3$, but is not clear how to work from it $$ -k x^3 = \gamma^4 \frac{d^2 x}{dt^2}$$ $$ -k x^3 (1 - 2 \frac{1}{c^2} (\frac{dx}{dt})^2 + \frac{1}{c^4} (\frac{dx}{dt})^4 ) = \frac{d^2 x}{dt^2}$$ It looks like a (nonlinear) differential equation. I just want to be sure i'm on the right track. This is what it takes to solve this kind of problem? Or should some numerical integration/summation be enough? What throws me off in particular, is that if i would replace $-k x^3$ with just $\alpha_0$ (constant acceleration case), the $\gamma^4$ would still look pretty ugly and i wouldn't know how to solve the problem even in that case _using the above expression_ , when we already know that a closed formula exists


title: What is the physical meaning of commutators in quantum mechanics?

text: This is a question I've been asked several times by students and I tend to have a hard time phrasing it in terms they can understand. This is a natural question to ask and it is not usually well covered in textbooks, so I would like to know of various perspectives and explanations that I can use when teaching. The question comes up naturally in what is usually students' second course in quantum physics / quantum mechanics. At that stage one is fairly comfortable with the concept of wavefunctions and with the Schrödinger equation, and has had some limited exposure to operators. One common case, for example, is to explain that some operators commute and that this means the corresponding observables are 'compatible' and that there exists a mutual eigenbasis; the commutation relation is usually expressed as $[A,B]=0$ but no more is said about that object. This naturally leaves students wondering > **what is, exactly, the physical significance of the object $[A,B]$ > itself?** and this is not an easy question. I would like answers to address this directly, ideally at a variety of levels of abstraction and required background. Note also that I'm much more interested in the object $[A,B]$ itself than what the consequences and interpretations are when it is zero, as those are far easier and explored in much more depth in most resources. * * * One reason this is a hard question (and that commutators are such confusing objects for students) is that they serve a variety of purposes, with only thin connecting threads between them (at least as seen from the bottom-up perspective).   * Commutation relations are usually expressed in the form $[A,B]=0$ even though, _a priori_ , there appears to be little motivation for the introduction of such terminology.   * A lot of stock is placed behind the canonical commutation relation $[x,p]=i \hbar$, though it is not always clear what it means. (In my view, the fundamental principle that this encodes is essentially de Broglie's relation $\lambda=h/p$; this is made rigorous by the Stone-von Neumann uniqueness theorem but that's quite a bit to expect a student to grasp at a first go.)   * From this there is a natural extension to the Heisenberg Uncertainty Principle, which in its general form includes a commutator (and an anticommutator, to make things worse). Canonically-conjugate pairs of observables are often introduced, and this is often aided by observations on commutators. (On the other hand, the energy-time and angle-angular momentum conjugacy relations cannot be expressed in terms of commutators, making things even fuzzier.)   * Commutators are used very frequently, for example, when studying the angular momentum algebra of quantum mechanics. It is clear they play a big role in encoding symmetries in quantum mechanics but it is hardly made clear how and why, and particularly why the combination $AB-BA$ should be important for symmetry considerations. This becomes even more important in more rigorous treatments of quantum mechanics, where the specifics of the Hilbert space become less important and the algebra of observable operators takes centre stage. The commutator is the central operation of that algebra, but again it's not very clear why that combination should be special.   * An analogy is occasionally made to the Poisson brackets of hamiltonian mechanics, but this hardly helps - Poisson brackets are equally mysterious. This also ties the commutator in with time evolution, both on the classical side and via the Heisenberg equation of motion. I can't think of any more at the moment but they are a huge number of opposing directions which can make everything very confusing, and there is rarely a uniting thread. So: what are commutators, exactly, and why are they so important?


title: Far Field Diffraction of EM waves: what does the zero frequency signify?

text: If you have a system of independently radiating electrons/point-charges, the far field distribution of the EM waves can be approximated by the fraunhoffer diffraction integral, or simply by the Fourier Transform of the charge/electron density distribution. When taking the Fourier Transform of something, there is always a zero frequency value. What does this represent in terms of the EM wave example? The avg of the near field? The avg of the entire field summed in all space (this case it should never vary)? Thanks in advance!


title: If electric charges accelerate towards lower potential energies, why do opposite charges attract?

text: I know my logic must be wrong but I can't figure out why. I know that charges must accelerate towards lower potential energies simply because that's a general rule of nature. However, when you release a positive and negative charge close to each other, they will move towards each other, and the distance between will decrease causing the potential energy to increase (1/r^2 relationship). So aren't the two charges gaining potential energy? Does it have to do with the sign of the charges being different? I feel like that explanation doesn't fit entirely because it seems like the magnitude of the potential energy should not be increasing.


title: Entropy bounds and the size of the universe

text: The principle that the maximum amount of information or entropy a volume of space can hold is proportional to its surface area apparently applies to all space, not just black holes. Since volume grows asymptotically faster than surface area and there there is also a limit on the information per surface area this should imply limits on the possible size, shape and density of the universe. To what extent have these limits been worked out?


title: What is currently incomplete in M-theory?

text: As it is, what is currently and seemingly incomplete in M-theory? One example I can give is explaining dark matter proportions and.. how do we perceive the world as 4-dimensional. Thanks everyone.


title: Springs yet again, this time with a picture. Infinite displacement, makes no sense

text: > **Possible Duplicate:**   >  How could this damped oscillator ever go to infinity? Or negative infinity > for that matter? Consider this ![](http://img854.imageshack.us/img854/1391/78679184.jpg)! Where I purposely drew the right arrow bigger than the left one. The arrow to the right is the drag force - resistive force and the arrow to the left is the spring force. So writing out Newton's second Law I get $F = ma = bv - kx$ where I am taking the right direction as the increasing value of $a$ In differential form, I get $F = mx'' = bx' \- kx$ Before I solve and show you the solution, my intuition tells me that it with a damping force that's much greater than the restoring force, the mass won't even return to the equilibrium point. Unfortunately my intuition proves me very wrong. $ mx'' \- bx' \+ kx = 0$ Now, let b = 10, k = 2, and m = 1, x(0) = 1 and x'(0) = 0 ![](http://img36.imageshack.us/img36/4044/70539281.jpg)! Now this makes no sense to me. The mass was originally displaced to the right of the equilbrium point and was released. After being released the drag force should resist so much that the restoring force isn't enough to counter it so that it should not even reach the equilibrium point. Now I have a graph that says it will go down to negative infinity. Why? Also, what does negative infinity mean? Doesn't that mean the spring will break?


title: Functional determinant approximation

text: Let the Hamiltonian in one dimension be $H+z$, then I would like to evaluate $\det(H+z)$. I have thought that if I know the function $Z(t) = \sum_{n>0}\exp(-tE_{n})$ I can use $$\sum_{n} (z+E_{n})^{-s} \Gamma (s) = \int_{0}^{\infty}\mathrm{d}t\ Z(t)\exp(-zt)t^{s-1}$$ so i can use the zeta regularization to define the quotient $$\log\det(H+z) - \log\det(H)= -\zeta'(0,z)+ \zeta'(0,0)$$ with $\sum_{n} (z+E_{n})^{-s}= \zeta (s,z)$. The derivative is taken on the variable $s$ ... at $s=0$. However if $Z(t)$ is too hard to evaluate, can I make $Z(t)= \int \mathrm{d}x\mathrm{d}p\exp(-tp^{2}-tV(x))$ as an approximation to evaluate the functional determinant?? Thanks.


title: what's the size of carbon dioxide

text: First of all, I'm not good at English, so if you guys can use simple English it will help. My question is very simple. How big is a carbon dioxide ($\mathrm{CO}_2$) molecule, i.e. the width, length and height of it? I hope one of you guys can explain that to me.


title: Water level difference by pumping

text: Open tank is partially filled with water and is divided into two equal parts by the wall. The wall has pipe that connects both sections. Obviously, water levels are equal from both sides (Fig.1). Then, water is pumped from right section of the tank to the left one with constant flow rate (Fig.2). Q1: Is it possible to calculate the difference in water levels knowing all values (volume, flow rate, pump capacity, etc.)? Q2: How the water levels difference will change if the tank will double its size (Fig.3)? Or it will be the same? ![enter image description here](http://i.stack.imgur.com/QJcRC.png)


title: Angular momentum depends on origin?

text: The angular momentum of a particle rotating about a point is given by $\vec{L} = \vec{r} \times \vec{p}$. Imagine a particle tracing a circular path on a flat table. If I put the origin of my coordinate system at the center of the circular path such that the flat table is the $xy$-plane, the angular momentum vector points in the positive direction of the $z$-axis because $\vec{r}$ lies on the xy-plane and $\vec{p}$ is always perpendicular to $\vec{r}$. Therefore, there is no change in $\vec{L}$ throughout the motion of the particle. Now, I only move the origin vertically toward the ground. Now $\vec{r}$ makes a certain angle $\phi$ with the $z$-axis, and therefore, $\vec{L}$ also makes the same angle $\phi$ with the $z$-axis. But, now $\vec{L}$ keeps changing its direction as the particle is moving on the circular path on the flat table! To conclude, an angular momentum $\vec{L}$ depends so much on the choice of the origin pretty much unlike the linear momentum $\vec{p}$?


title: elastic potential energy of a spring when compressed

text: > A small ball with a mass of 1 kg rolls down a long frictionless inclined > ramp, which is at an angle 30 degrees above the horizon. A linear spring, > whose length is not negligible, is attached to the bottom of the ramp. The > ball is released from a distance of 1 m above the spring, measured along the > plane and the spring constant is 1N/m, then the spring is depressed by               a) 1.6m ( Answer)     b) 1.8m     c) 0.5m     d)The answer cannot be determined from the given data.     e) 2m      My attempt:   1. First, I calculated the decrease in gravitational energy $$ U_{gpe} = mgh = \frac12g $$   2. This energy is converted to kinetic energy which is then converted into elastic potential energy of the spring when the spring is compressed by the ball $$ U_{spring}=\frac12kx^2=\frac12x^2 $$   3. The change in gravitational potential energy=potential energy in spring: $$\frac12g=\frac12x^2$$ Thus, $x=\sqrt{g}=3.13$, but that answer is not even in the options. Is there anything wrong in my concept or my working?


title: Why are we blind for the era before the recombination?

text: At recombination, Universe became transparent to electromagnetic radiation after universe expanded enough to cool down to form neutral atoms. Before that, the matter plasma was effectively opaque to electromagnetic radiation due to Thomson scattering by free electrons, as the mean free path each photon could travel before encountering an electron was very short. This Thomson scattering is good to stop photons, but not Neutrinos. There are known Neutrino scatterings in theory, but it's probability is very low, AFAIK. Why can't we detect neutrinos from before recombination era which can tell something more about initial phases of universe?


title: Key difference between 3D and 4D solutions

text: In many papers in theoretical physics especially in the more advanced theories, I notice that physicists solve problems in 3 dimensions (2 spatial+ 1 temporal). In some specific papers (I can't seem to recall exactly which ones), some models solved in three dimensions gave finite solutions, whereas 4D (3+1) were either hard to solve or gave divergent results. I don't seem to understand why adding a 3rd spatial dimension should cause problems if 2 spatial dimensions gave no problems. Intuitively, it seems straightforward matter to extrapolate 4D solutions from 3D. Then what is the essential difference between the 2?


title: Melting and freezing point

text: Why is the melting and freezing point of a substance are always the same ? This was quoted in my textbook but they didn't give a reason for this being so. Thank you.


title: Nonlocal games: The CSCH game

text: The problem that I am having is that I don't see how they come up with the probabilities mentioned on Slide #7 here: https://www.ipam.ucla.edu/publications/opws1/opws1_9367.pdf first I don't understand what $a_{r}(s), b_{r}(t)$ mean. On slide #6 they are defined as $a_{r}:S \rightarrow A$ and $b_{r}:T \rightarrow B$ and am missing what the subscript $_{r}$ is supposed to dictate. Now I see that $s \in S \text{ and } t \in T$ where $S \text{ and } T$ are the set of questions for player 1 and 2 respectively. So is, for example $b_{r}:T \rightarrow B$ the probability of answering in such a way that combined with player $S$'s question and answers that they end up winning? If however this was the case, what does slide #7's saying that $a_{r}(s) = 0 \text{ and } b_{r}(s) = 0$ mean? note the $ b_{r}(s)$ instead of $b_{r}(t)$ on the slide. Is this a mistake? Either way I don't understand what they are trying to say. Lastly leading up to the question, why: $$p = max \sum_{s,t}π(s, t)V (a_{r}(s), b_{r}(t)|s, t) = \frac{3}{4} ?$$ so, how did they get $\frac{3}{4}$ in the game $s \cdotp t = a \oplus b$? Perhaps someone could show the numerical example? Thanks much, Brian


title: How far can you scatter light using a prism?

text: If I were to scatter light how far do you think it will disperse and what prism is the most effectively scatters light?


title: What would Maxwell's Equations be if we had magnetic charges and magnetic currents?

text: Mind you, we still have electric charge and electric currents. But, what would Maxwell's equations look like if we had to take magnetic charges and magnetic currents into consideration? Would there be any sign changes?


title: Does extreme cold make **everything** extremely brittle?

text: First of all, I'm genuinely sorry if this question isn't "serious" enough for this forum! A common cliche in movies and tv is that a very tough object (eg the villain) is frozen, and then hit with something, shattering into a million pieces. I've seen a demo of a flower being put into liquid nitrogen, then being crumbled, but a flower is a very delicate object to start off with. If I take a leg of lamb (for example) out of the freezer, I don't feel like it's in any danger of shattering into a million bits (unlike my foot if i were to drop it). So, is the whole "cold = brittle" thing just movie bullcrap? Or is there anything to it? Sticking with the leg of lamb example: is there a temperature to which a leg of lamb could be dropped that would make the leg of lamb prone to shattering? EDIT - i just realised that the question title could be read as "Is there anything which is rendered extremely brittle by extreme cold?". Obviously there are _some_ things, eg flowers. Hence the title change.


title: What is the relative velocity of this plane relative to the helicopter?

text: There is a plane moving 100 m/s due east relative to the ground without vertical motion. There is a helicopter facing north moving straight up at 20 m/s. From the perspective of the helicopter, is the plane moving faster or slower than 100 m/s away from the helicopter. The book I am using says the plane's relative velocity is greater than 100 m/s, but I think it is dependent on the initial position of the helicopter relative to the plane. If the helicopter is 20 meters straight below the plane in one instance, then one second later, the helicopter would be at the same altitude as the plane and the plane would be 100 meters away, so the average relative velocity would then be 80 m/s, no? That would mean that the relative velocity was, at least at some point, less than 100 m/s. Is my understanding correct, or am I missing something, and the book is correct? Here is a diagram of what I am trying to express: ![diagram](http://i.imgur.com/LSCGZFf.jpg)


title: What is entropy really?

text: On this site, change in entropy is defined as the amount of energy dispersed divided by the absolute temperature. But I want to know: What is the definition of entropy? Here, entropy is defined as average heat capacity averaged over the specific temperature. But I couldn't understand that definition of entropy: $\delta S$ = $S_{final}$ - $S_{initial}$. What is entropy initially (is there any dispersal of energy initially)? Please give the definition of _entropy_ and not its _change_. To clarify, I'm interested in the definition of entropy in terms of temperature, not in terms of microstates.


title: An electromagnetic induction problem

text: The question goes like this : > A thin non conducting horizontal disc of mass $m$ having total charge $q$ > distributed uniformly over its surface, can rotate freely about its own > axis. Initially when the disk is stationary a magnetic field $B$ directed > perpendicular to the plane is switched on at $t=0$. Find the angular > velocity $\omega$ acquired by disc as a function of time, if $B=kt$, where > $k$ is an arbitrary non-zero constant and $t$ is time The given diagram : ![Disc in time varying magnetic field](http://i.stack.imgur.com/5bxM4.jpg) What I did was to find out the induced emf in the loop due to the changing magnetic flux, using Faraday's laws. These are my results: The emf:$$ \epsilon=k\pi R^2$$ The magnetic moment of the disc when it rotates with angular velocity $\omega$ : $$\mu=\frac{q\omega R^2}{2}$$ But now I am stuck. What should be my next step? Any hints??


title: How much energy can be extracted from hydrogen?

text: http://en.wikipedia.org/wiki/B-L says that the difference between baryon number and lepton number is conserved. Ordinary hydrogen has one of each, but turning it into helium releases only the binding energy. Complete destruction of hydrogen would satisfy B-L and charge conservation, creating even more energy than fusion. Is this possible or would it violate a different conservation law?


title: vector cross products

text: Lets say you have a free particle in a rotating frame of reference with constant angular velocity $\mathbf{\omega}$. By free, I mean there are no real forces on it. Lets call the moving system "primed". The equation of the motion is $m\mathbf{a} = \mathbf{F} + \mathbf{F_{fic}}$ but $F = 0$ and the only fictitious forces acting on this particle are the centrifugal force and the coriolis force. Just to double checked, 1) there is no azimuthal (or transverse) force since there is no angular acceleration. Correct? 2) And ofcourse there is no translation force since the origin of the primed and nonprimed systems are the same? Right? Okay so lets assume I am right (which I am since I am following an example from a text :P) So we have $$m\mathbf{a} = −m \mathbf{\omega} × (\mathbf{\omega} × \mathbf{r'}) − 2m \mathbf{\omega} × \mathbf{v}'$$ Lets making the rotating axis fixed with the z axis so $\mathbf{w} = \omega \cdot \mathbf{e_z}$ $$\omega = \omega \mathbf{e_z} $$ $$\mathbf{r} = x \mathbf{e_x} + y \mathbf{e_y} + z \mathbf{e_z} $$ and similarly v is pretty much r, but x = x dot, and such. Don't know how to write x dot with latex. So my main problem is that for example lets say we are calculating the x component of everything. So then in the equation the coriolis force has a cross product between $e_z$ and $e_x$. This is $e_y$. I get that, but how do I know what sign is it? How come $e_z$ and $e_y$ gives me A NEGATIVE $e_x$? I am on page 11 of this pdf, http://www.mech.kth.se/~hanno/RelativeMotion.pdf. I am also confused about how $e_x$ crossed with $e_z$ gives you "y dot" and not "x dot" Look at the pdf (specifically the coriolis force term).


title: Why don't we use beta radiation for making electricity?

text: Why don't we use beta radiation to make our electricity? Since it is an electron being released/made, it would make sense to try and capture it. But, we obviously don't so. What am I missing in my logic?


title: How can space and time arise from nothing?

text: Lawrence Krauss said this on an Australian Q&A programme. "...when you apply quantum mechanics to gravity, space itself can arise from nothing as can time..." Can you elaborate on this please? It's hard to search for!


title: Plano convex lens with one silvered surface

text: > For the derivation of the focal length of a planoconvex lens with one > surface silvered, we consider the phenomenon taking place in three steps: > >   1. Refraction through first surface >   2. Reflection through silvered surface >   3. Refraction through first surface again > > > So the inverse of the resultant focal length can be obtained as the > summation of twice the inverse of the focal length of the planoconvex lens > and the inverse of the focal length of the silvered surface. **My question here is:** Why do we consider the focal length of the planoconvex lens (while taking into consideration the refracting surface) and why not just the focal length of that particular refracting surface?


title: Atomic transitions and density: clarification

text: A laser passes though a cell containing vapor of an alkaline element which causes atomic transitions. I was told that the formula for the atomic density in a Rubidium cell, for example, is given by: $$N=-\frac{1}{\sigma L}\ln \frac{I}{I_0}$$ where:   * $\sigma$ is the cross section and is inversely proportional to the temperature T   * $L$ is the cell length   * $I/I_0$ is the signal transmission on the peaks But I have some questions:   1. Does "atomic density" correspond to "atoms in a volume unit that participate in the stimulated emission process"?   2. Is $\sigma$ the cross section for a particular transition? (for example the Rubidium transition $D_2$)    3. $N$ depends on temperature. Is this due to the fact that if the temperature is higher, the atomic electrons can have more energy and occupy more external levels?   4. If the cell is enlightened with blue light, the peaks are deeper (Light-induced atomic desorption (LIAD) effect) and, with $T$ constant, $N$ assumes an higher value. Generally some rubidium atoms attach on cell walls and blue light give them energy for being "free". $N$ is calculated in relation to the atoms that _effectively_ make the atomic transition (thanks to peaks depth and sigma, is it right?) and the atoms that are attached to the walls aren't be counted. Is it correct? So blue light permits to (a part of) unattached atoms to give the transition and be counted in $N$. Probably this reasoning is confusing, I'm trying to respond to the question "Why the peaks are deeper and N is higher if I use blue light?"


title: How to get "massless" equation of motion from the action of Nordstrom scalar field theory?

text: There is Nordstrom theory of the particle moving in a scalar field $\varphi (x)$: $$ S = -m\int e^{\varphi (x)}\sqrt{\eta_{\alpha \beta}\frac{dx^{\alpha}}{d \lambda}\frac{dx^{\beta}}{d \lambda}}d\lambda . $$ How to get the equation of motion for massless objects? In this case I may introduce new parameter $\sqrt{\eta_{\alpha \beta}\frac{dx^{\alpha}}{d \lambda}\frac{dx^{\beta}}{d \lambda}}d\lambda = \frac{d\tau }{m}$. But when I try to derive (like this way) the equations of motion, I get $$ \partial_{\alpha } \varphi = m^{2}\frac{d }{d \tau}\left( u_{\alpha} e^{\varphi}\right)e^{-\varphi}. $$ Unfortunately, this equation doesn't predict the absense of deflection of light. But Nordstrom theory really predicts the deflection of light. So where is the mistake in my reasoning? Addition. I got the mistake. By introducing new parameter $\tau$ I must rewrite my equation in a form $$ \partial_{\alpha } \varphi = \frac{1}{m^2}\frac{d }{d \tau}\left( p_{\alpha} e^{\varphi}\right)e^{-\varphi}. $$ So it really preficts no deflection of light.


title: Molecules of a solid

text: > Question: Molecules of a solid : > > (a) are always in a state of motion > > (b) move only when heated > > (c) move because they are loosely bound > > (d) do not move at all **My attempt:** I could safely ignore (c), as molecules in a solid are tightly bound. My book says that molecules in a solid vibrate about their fixed position. But they also expand on heating. Due to this, I am totally confused whether the correct option (a) or (b) (or maybe (d)?) I know the question I am asking is really basic, and maybe does not even belong here, but I searched many sites and none revealed satisfying information. And thus, I am looking for expert help. I even doubt if my book (from where this question is) is correct or not. Thanks a lot.


title: How to derive the Aharanov-Bohm effect result?

text: In the derivations of the Aharonov-Bohm phase, it is directly mentioned that due to the introduction of the vector potential $A$, an extra phase is introduced into the wavefunction for case $A\neq0$ i.e. $$ \psi(A\neq0) = \exp(\iota\varphi)\psi(A=0),$$ where $$ \varphi = \frac{q}{\hbar} \int_P \mathbf{A} \cdot d\mathbf{x}. $$ How to derive it from the following Schordinger equation $$ \left[\frac{1}{2m}(\frac{\hbar}{i}\triangledown- eA)^{2}+V(r)\right]\psi=\epsilon\psi. $$ I tried taking the terms containing $A$ on the right and treating the equation as an inhomogeneous equation but it just becomes tedious. What is the straightforward simple way?


title: The relationship between Lorentz Lie algebra and curvature

text: Here I transfered the question from the comment The relationship between spin and spinor curvature How $\mathcal{R}_{ab} = \frac{1}{4}R_{abst}\gamma^s \gamma^t$ is from $\Psi \mapsto \Psi + \frac{1}{4} \epsilon_{\mu\nu}\gamma^\mu\gamma^\nu \Psi$ arise? When $\mathcal{R}_{ab}$ is defined by $\newcommand{\Rcal}{\mathcal{R}} \Rcal_{ab} \Psi = [D_a, D_b] \Psi$ and $R_{abst}$ is Riemann tensor constructed by metric. $D_a$ is the covariant derivative in the Dirac equation in the curved spacetime Dirac Equation in General Relativity


title: Find the minimum force that has to be applied to the block $m_1$ so that the other block just begins to slide

text: This is a problem from the book 'Problems in General Physics' by I E Irodov. Two blocks of masses $m_1$ and $m_2$ are placed on a rough horizontal surface, connected by a light spring. Find the minimum constant force that has to be applied on the block with mass $m_1$ so that the other block just begins to slide. Consider the limiting case: Suppose the spring has compression $x$ by that time. Block $m_2$ is just about to move. We can write, $$kx=\mu m_2 g$$ And, by the work energy theorem $$Fx-0.5kx^2-\mu m_1 gx- \mu m_2gx= 0.5m_1 v^2$$ For the system of $m_1$, $m_2$ and the spring. This is where I am stuck. According to the answer, the force required is minimum when $v=0$. Why is the block $m_1$ at rest in the limiting case? I cannot find any suitable justification for this, neither an intuitive explanation. Also, is something wrong with my Work Energy theorem expression?


title: Plank stopping time

text: > **Problem** : A plank of length $L$ and mass $m$ lies on a frictionless > floor, if the plank has an initial velocity $v$, what is the stopping time > of the block if it meets a floor with friction constant $\mu$. ![Problem Diagram](http://i.stack.imgur.com/z00ce.png) My attempt: I took a little piece of block with mass $\Delta m$, we know that $\Delta m = m\frac{\Delta \ell}{L}$. Now if I apply the Newton laws in this piece, I get the following equation: $$N_{piece}= \Delta mg $$ Thus my friction is $f_r=\mu\Delta mg$ If I apply Newton laws to the whole plank, I get: $$-f_r=ma$$ $$\mu\Delta mg=ma$$ $$\mu mg\frac{\Delta \ell}{L}=ma$$ $$a=\mu g\frac{\Delta \ell}{L}$$ I have this equation, but I don't know how to relate it with time.


title: What determines the speed of waves in water?

text: While I was walking my dog this morning, I passed over a canal filled with boats, barges, and kayaks all of different masses and moving at different speeds. I noticed that all of these vessels left behind wakes, and the waves of these wakes moved through the water at different rates, though my ability to make good observations was limited. This made me wonder: what determines the velocity of these waves? I know that sound travels through water at 1,480 m/s or so, but surface waves travel at different rates (tsunamis travel faster than the ripples I make in my bowl of soup when I blow on it). In particular, how is the velocity of wake waves determined?


title: Are Mathematical Physics and Occam's Razor compatible?

text: Occam's Razor and mathematical beauty appear to be compatible when reviewing Michael Atiyah's video. But are the high levels of complexity associated with mathematical _physics_ compatible with Occam's Razor?


title: Why does $x(0)=0$ for SHO in classical action

text: In leveraging this PDF to help solve the integral for $S_{cl}$ least action for a Simple harmonic oscillator, I read that I can assume $x_{cl}(0) = 0$ for the classical solution. Why is $x_{cl}(0) = 0$? Or is this the definition of "the classical solution" versus a general solution? The link is owned by a TA whose email address is no longer active. I have downloaded the PDF and attached it here as a photo![picture of PDF](http://i.stack.imgur.com/ruM3u.png) Okay, I understand I can define x(0) any way I like as long as it is on the path of the SHO so I can choose to define x(0)= 0. This simplifies the integral and I get: $S_{cl}$ = m/2 (dx/dt) (x) from ta to tb where I don't assume ti = 0 I am having a hard time getting from this equation to eqn (4) in the referenced PDF. Help on that would be great. (I can get to eqn (3) no problem) Here's where I get stuck: notation: Xb = X(tb), Xa = X (ta) $S_{cl}$ = m/2 w/sin^(w(ta-tb)) * (sin wt)*(cos wt)(Xb coswta - Xa coswtb)^2 from ta to tb I have several pages of attempts to get from this to (4). If it helps I will take a photo and upload my work...Help appreciated!


title: Classical Communication in Quantum Teleportation

text: How we could define the classical communication in the quantum teleportation protocol? I mean, classical communication means to send a classical signal. But what happens if we are in an unclear situation in which we send a signal that we cannot classify as neither classical neither non-classical, because of lack of criteria? Conceptually speaking, how critical is this **classical** in the protocol?


title: How do you assign an observable to spectral lines in Heisenberg's resolution of Rydberg-Ritz?

text: Ron's comment essentially answers the question below: Here's what I really want to know: > Suppose I have an experiment that yields 6 spectral lines corresponding to > (one-way) transitions between 4 basic frequencies in an "idealized" atom > with finitely many energy levels. According to Connes (see below) the > observable algebra of this system is a 4x4 complex matrix algebra. According > to Quantum Mechanics, one should be able to assign an observable (self- > adjoint element) of the observable algebra such that the observations of an > experiment (in this case the 6 spectral lines) index the orthogonal > eigenspaces of the observable. This, oddly, seems to say that there is a 4 > by 4 matrix with 6 eigenvalues. This is clearly nonsense. Where is the > problem with my hypothetical experiment? * * * Original question: The following is an embarrassingly basic question. Edit: The following picture is from Connes's book: _Noncommutative Geometry_. The question came out of trying to connect what is written there with the usual postulates of quantum mechanics. In classical mechanics, every observable quantity is represented by a real- valued function on the phase space X of a system. In the case of an electromagnetically radiating atom, according to Maxwell's theory each function admits a Fourier expansion which, via the Fourier transform, yields an isomorphism of the algebra of functions on X with the convolution algebra of an additive subgroup of the integers. The Rydberg-Ritz combination principle, as experimentally verified, shows that the above prediction fails and instead there is a set "I" of frequencies such that only differences of frequencies appearing in I are emitted or absorbed by the atom. The spectral lines are thus labelled by differences in frequencies in I. The convolution algebra that Heisenberg prescribes that we substitute in for the one predicted above is isomorphic to a matrix algebra. I am trying to understand the basic relation between how observables, or self- adjoint elements, of the matrix algebra above are precisely associated to the observed spectral lines, as postulated in the mathematical formulation of quantum mechanics. For example, (trying to ape this in finite dimensions...just to have a toy model) if "I" has 4 frequencies whose respective differences correspond to allowed transitions, the resulting matrix algebra will consist of 4x4 matrices. But then, since each transition labels a spectral line there ought to be 16 eigenspaces making up the self-adjoint element associated to the observed spectral lines. These are too many subspaces to be associated with any 4x4 matrix acting on $\mathbb{C}^4$. Am I right in assuming that for every distinct observed quantity (assumed discrete), we are to assign an eigenspace of a hermitian operator in the algebra describing the system? If so, how do we assign such an observable to a set of spectral lines? (Perhaps my finite-dimensionality is simply not doable.) I will appreciate any help you can offer me on this!


title: Turning an airplane - coordinated turn and inclinometer ("the ball")

text: I'm flying, turning in a stable orbit, i.e. at constant level with a constant angle of bank, at constant airspeed, with a constant radius of turn, as in the picture here (sorry it's my first post, cannot upload pictures). I am flying along the black curved line. The lift is directed as the blue axis (orthogonal to the wings). The vertical component of the lift compensate exactly the gravity, and the horizontal component of the lift is responsible for the turn (forget the component in the direction of movement). Now consider the inclinometer (aka the ball aka balance indicator aka slip indicator). The position of the ball is supposed to tell me whether my turn is "coordinated" or not. If the ball is centered the turn is coordinated, otherwise the normal practice is to use rudders ("step on the ball") in order to center the ball and obtain a "coordinated turn". Since the orbit is stable, the ball is stationary with respect to me, so the ball must be subject to the same exact acceleration I feel. Consider the frame of reference attached to the airplane, as in the picture. In this non-inertial frame of reference the forces I experience are the gravity, the lift, and the (apparent) centrifugal force. Since neither me nor the ball are moving with respect to the reference frame, all these forces must be balanced. In my case the lift is exercised on me by the seat and in the case of the ball by the wall of the cage/vial in which it is contained. In both cases the force exercised on the bodies to compensate gravity and centrifugal force must be in the same exact direction and strength of the lift. This can only happen for the ball when the ball is at the bottom of the curved vial, because only then the wall of the vial is normal to the vertical direction (blue axis). So my question is: how can the ball ever be "out of the cage"? And how can the use of rudder impact on the ball position if the airplane yaws by rotating around the blue axis (i.e. around the direction of the lift)?


title: Is $kg_f$ the same everywhere in the universe?

text: A textbook question says that a vehicle weighs $25kg_f$ on Earth, and asks us to consider certain issues related to its behaviour on the moon. My question is, does the unit kilogram-force $kg_f$ need to be converted to match the gravitational pull of the moon, or is it $25kg_f$ everywhere in the universe? I read up on $kg_f$ on Wikipedia, but regarding my question, it was inconclusive.


title: What does $\left|x\right\rangle \left|0\right\rangle$ actually mean in bra-ket notation?

text: > Initialize the registers to > > $Q^{-1/2} \sum_{x=0}^{Q-1} \left|x\right\rangle \left|0\right\rangle$ > > where ''x'' runs from 0 to ''Q'' − 1. This initial state is a superposition > of ''Q'' states. (from > http://en.wikipedia.org/wiki/Shor%27s_algorithm#Quantum_part:_Period- > finding_subroutine) I am having a hard time getting what $\left|x\right\rangle \left|0\right\rangle$ actually is. Is it outer product of ket x and ket 0? Or what is it actually meaning?


title: Superposition of waves hitting at right angle

text: I'm considering, in general, a spherical wave coming from a pin hole and landing on a wafer, both directly and off of a mirror. The mirror and wafer are at right angle. If it weren't a pin hole, and the light was coherent and moving straight at the mirror/wafer, I read that the period is $\lambda/(2\sin(\theta)$, but I have no idea why. I think the main part of this problem is just, in general, calculating the period of an interference pattern on a plane, of two light sources of the same wavelength, potentially in phase, hitting at two angles-one from the left, one from the right.


title: Determine the current and voltage for every component in the following circuit

text: Determine the current and voltage for every component in the following series circuit What is the voltage and current through R1 and what is the voltage and current through C1?   * $V_1= 20\,V$   * $R_1= 1.0\, k\Omega $   * $C_1= 1 \, \mu F$


title: Schrodinger basis kets with Time-dependent Hamiltonian

text: I was reading through the proof of the Adiabatic Theorem (in Sakurai) and I realised I'm not quite sure how Schrodinger Basis kets behave when we have a time-dependent Hamiltonian. I know that with a time-independent Hamiltonian the basis kets don't change in the Schrodinger Picture. So if $|n;t\rangle$ are the energy eigenkets of $H(t)$ at time $t$ and $|\alpha;t\rangle$ is an arbitrary state at time $t$, is the following at all true? \begin{align*} |\alpha;t\rangle = \sum_n c_n(t)|n;t\rangle = \sum_n c_n(t) e^{i\theta_n(t)}|n,t_0\rangle \end{align*} where $\theta_n(t) = -\frac{1}{\hbar}\int_{t_0}^t H(t')\,dt'$ and $e^{i\theta_n(t)}$ is a time- evolution operator Wikipedia and Sakurai both have (each in different notation): \begin{align*} |\alpha;t\rangle = \sum_n c_n(t) e^{i\theta_n(t)}|n;t\rangle \end{align*} I feel like I'm not understanding this properly at all


title: What interaction is responsible for the 21 cm Hydrogen line transition?

text: The 21 cm Hydrogen line is from the transition between the hyperfine levels of the ground state of the hydrogen atom. So, what interaction is coupling the hyperfine levels? I suspect that it is not the usual EM interaction, which causes the 1s-2p transition.


title: $k$ for magnetic monopoles?

text: Afaik, if magnetic monopoles existed, the force betweem them were very like the force between electric charges: $\bf{F}=\frac{k Q_1 Q_2}{r^2}$, where $k=9\cdot10^{11}\frac{Nm^2}{C^2}$.   * What were this constant $k$ for them?   * What were the unit of the magnetic charge?


title: Second law of thermodynamics implies a linear cosmology?

text: If one applies the second law of thermodynamics to the Universe[1] as a whole then one might expect that the entropy of the Universe always increases as time goes forward (or more accurately that the entropy never decreases). But the second law of thermodynamics is based on time-reversal invariant physics so that one would also expect that the entropy of the Universe should always increase as one goes backwards in time as well. Both these conditions are met simultaneously by a linearly expanding cosmology such that the scale factor $a(t)$ is given by: $$a(t) = \frac{t}{t_0},\tag{1}$$ where $t_0$ is the present time. According to Raphael Bousso's Covariant Entropy Conjecture, the future entropy of the Universe is bounded by the area of the event horizon at a distance $d_e(t)$ given by: $$d_e(t) = \int_{t_0}^\infty \frac{c\ dt}{a(t)}.$$ Substituting the equation (1) for a linear cosmology into the above expression we find that the distance to the event horizon, $d_e(t)$, diverges logarithmically as $t \rightarrow \infty$. Thus in a linear cosmology the future entropy of the Universe is not bounded. Similarly the past entropy of the Universe is bounded by the area of the particle horizon at a distance $d_p(t)$ given by: $$d_p(t) = \int_0^{t_0} \frac{c\ dt}{a(t)}.$$ Substituting the equation (1) for a linear cosmology into the above expression we find that the distance to the particle horizon, $d_p(t)$, diverges logarithmically as $t \rightarrow 0$. Thus in a linear cosmology the past entropy of the Universe is not bounded either. Is this reasoning correct? Does this show that such a linear cosmology is interesting even though the current evidence seems to point to an accelerating Universe with a finite future entropy? A universe with a finite future entropy leads to strange hypotheses like Boltzmann brains so it seems reasonable to consider other alternatives (see Leonard Susskind's latest talk for example). 1.Instead of "Universe" I should have stated my argument in terms of the _causal_ Universe defined by the volume inside the event/particle horizons of a particular observer.


title: Einstein Notation for Strain Energy Function

text: I encounter the following formula (for strain energy function) a lot in physics literature: $$ W(\epsilon_{kl}) = \int_0^{\epsilon_{kl}} \sigma_{ij} \textrm{d}\epsilon_{ij} $$ where all indices ranges from 1 to 3, both $\epsilon$ and $\sigma$ are 3x3 matrices. My question is, what exactly does it mean in non-Einstein notation? Is it an abuse of notation? Because when I see $W(\epsilon_{kl})=\cdots$, I feel it should interpreted as 9 separate equations (with $kl$ replaced by 11, 12, 13, 21, 22, 23, 31, 32, 33), which does not make sense in this case.


title: How did tension developed in a string when two equal and opposite forces are applied on the same body?

text: As per Newton's 3rd law > every action has an equal and opposite reaction, but both of these forces > are on different bodies under consideration Now when a single body (say rope in this case) is stretched by equal forces on the either ends of the rope, why is the tension the same as that of the applied force? (considering applied force doesn't exceed the maximum tension which that rope can hold).


title: Can resistance be directional?

text: When we think of resistance, we always think of a scalar value associated with a piece of a material. After all, resistance is but resistivity times surface geometry. But can resistance be directional meaning that it is stronger in one direction but weaker in another?


title: Why three families of multipole moments?

text: There are three families of multipole moments: The electric multipole moments, the magnetic multipole moments and the toroidal multipole moments. Is there any reason why there are this three families and not two or four or even more? Are there for example symmetry reasons or something like this?


title: Atomic Brownian Motion

text: Since atoms 'wiggle' proportionally to their energy level, I have two questions: Does it last 'forever'? _Absolute Zero question_ And so, is this 'flux' a fundamental force? Then as an extra thought, is it 'self-exciting'? If it's representative of thermal flux, and since thermal flux is almost all from the great energies from the sun, then reasonably, taking an atom infinitely far from the sun would make the atomic motion stop. So, as one more afterthought, When atomic motions stops, does the energy of the mass still exist as DC _static_ capacity?


title: How can I solve a circuit of resistances with both a voltage and a current-source in between them?

text: If I have a circuit that consists out of multiple non-parallel/serial resistances with current/voltage sources in between them like the one below ![circuit](http://i.stack.imgur.com/1KATg.png) How can I solve a system like that (in this case for I_10V and V_0)?. Would I have to do on a thevenin equivalent or is there a simpler way?


title: Does any substance possess the property color?

text: ![enter image description here](http://i.stack.imgur.com/wJzEB.jpg)For example, a leaf in normal sunlight may be seen as green, but under a red light it is seen a red or may be dark. So the color of leaf changes according to lighting conditions. So that, does any substance possess the property, Color?


title: How can such a wave exist at the surface of the sun?

text: Recently, I came across the following picture from NASA's SOHO observatory: ![enter image description here](http://i.stack.imgur.com/63hcr.jpg) It seems evident that this is a transverse wave (mind the ring which is bright and dark). But how can this be the case if the sun consists of gaseous matter?


title: String worldsheets with periodic time?

text: If we try to work out the details of string theory over a worldsheet with genus 1, does that mean we have periodic time on the worldsheet? But if we have periodic time, and in some portions of the worldsheet, the target space time $X^0$ increases with $\tau$, won't it have to decrease with $\tau$ somewhere else on the worldsheet? But wouldn't such a decrease mean we have an anti-string mode?


title: Is it possible that nuclear fission contributes to climate change?

text: This is probably a really stupid question, please forgive me. Is it possible that the use of nuclear fission on earth contributes to the increased energy in the Earth's system as according with the Laws of Thermodynamics regarding the Conservation of Energy?


title: Wavefronts and phase velocity faster than $c$

text: Lets assume we have parallel wavefronts in a glass of water: ![enter image description here](http://i.stack.imgur.com/vcwCv.jpg) and we put an inclined rod on the water surface: ![enter image description here](http://i.stack.imgur.com/aQznH.jpg) related to a very small inclining, Vy velocity is greater or much greater then Vx (Vy means, wavefronts' contact-making speed on the rod) **Now lets assume environment is space and the waves are of electro-magnetic ones: would Vx be smaller than c? Would Vy and Vx be equal? I dont think Vy would be greater than c. What do you think?** Assume inclined rod is just a metal rod and photo-electric effect is intact.


title: electron spin separation

text: > I am having doubt whether the electron's up spin moment and down spin moment > can be isolated from one another. If it got separated, will each moment acts > as magnetic monopole (stable or unstable). > > Something also came to my thought that in electrochemistry, during reactions > there is possibility of electron to come out of the reaction. For e.g.: $$ x+y \rightarrow xy+e^- $$ so whether it is possible by any laws or theories or possibilities in magnetochemistry, that during chemical reaction, individual spin moment (up or down) comes out of the reaction or the matter formed which acts as magnetic monopole (either north or south) stable in free $$x+y \rightarrow xy+\uparrow$$


title: Is the universe linear? If so, why?

text: Simple question, I'm trying to build a quantum memory system that utilizes the superposition principle to model specific phenomenon I am trying to predict, anyways, my question is this. Is the universe linear? The superposition principle would apply in ALL cases if the universe is linear, which would make sense as the quantum theory is the most accurate theory so far. If the universe is linear, why is it? Why not semi or non linear? Thanks Martin


title: Equations of fluid dynamics and differential geometry

text: Where can I look for equations of fluid motion written in terms of nifty things from differential geometry like exterior derivative, Hodge dual, musical isomorphism? Preferably both with and without assumption of Newton viscosity model. Since one can get momentum and continuity equation from the full energy equation + Galileo transform, maybe this one will suffice though continuity equation would be nice as a simple example.


title: Temperature change: boiling pasta in a lot of water or a little

text: SeriousEats gives a not-very-scientific representation of what looks like a thermodynamics calculation: > Want to hear something even more interesting? Folks will occasionally say > that "using a large volume of water will help the water come back to a boil > more quickly." > > Back up a minute there, because you know what? This is untrue. In fact, in > most real world cases, the exact opposite is the case. > > But how is this so? Doesn't adding a fixed amount of pasta to a small pot > cause the temperature in that post to drop more than it does in a large pot? > Therefore doesn't the large pot come back to a boil more quickly? Let's > examine the ideal scenario first. > > You have two pots of water. One has 1 quart of water, the other has 1 > gallon—four times as much. Both are sitting on top of identical burners and > are at a full, 212°F boil. Now add a cup of dry pasta to each one. Because > the pasta is at room temperature, it will cause the temperature of the water > in each pot to drop, and the water in the quart-sized pot will drop four > times more than the one in the gallon-sized pot. > > Ah ha!, you say. If the temperature fell four times lower in the small pot, > it must take four times longer for it to come back up to a boil! > > The problem with this line of reasoning is that it doesn't take into account > the fact that it takes four times less energy to raise a quart of water by > one degree as it does to raise a gallon of water. Since a burner puts out > energy at a constant fixed rate, the small pot, which needs to cover a > temperature gap four times as great as the large pot, serendipitously also > heats up four times faster. This means that the two pots of water return to > a boil at the exact same time!* > >   * For the record, it's also the same amount of energy and time required to > bring a cup of dry pasta from room temperature to 212°F. > > > In the real world, the "big pots boil faster" camp is even more wrong. See, > the larger a pot, the greater its surface area. And the greater the surface > area of a hot body, the more rapidly it can lose heat to the outside > environment. How does this affect heating? > > Let's say your burners put out heat energy at a very respectable 10,000 Btu. > Meanwhile, your small pot might be losing heat energy to the air in the > kitchen at, say, 1,000 Btu, giving you a net energy input of 9,000 Btu. A > large pot, on the other hand, will lose heat more rapidly due to its larger > surface area. Let's say, 2,000 Btu. Your burner is still exactly the same, > putting out 10,000 Btu, which means that with a large pot, the net energy > input is only 8,000 Btu. > > Thus, a large pot will actually return to a boil more slowly than a small > pot.** > > ** This doesn't even take into account the heat loss from evaporation, which > again compounds the case against large pots. Will the large pot really return more slowly? This isn't really intuitive, because you'd think that the more water has "more" energy in it already.


title: Difference between heat capacity and entropy?

text: Heat capacity $C$ of an object is the proportionality constant between the heat $Q$ that the object absorbs or loses & the resulting temperature change $\delta T$ of the object. Entropy change is the amount of energy dispersed reversibly at a specific temperature. But they have the same unit **joule/kelvin** like work & energy. My conscience is saying these two are different as one concerns with temperature change and other only at a specific temperature. I cannot figure out any differences. Plz explain me their differences.


title: Ricci curvature of embedded spacetime

text: 1. If I am not mistaken, there is a theorem which states that every Riemannian manifold can be embedded in the $n$-dimensional Euclidean space for some large-enough $n$.   2. Does it also hold for preudo-Riemannian manifolds and Minkowski spacetimes?   3. If so, any curved spacetime of GR can be embedded in some $n$-dimensional Minkowski spacetime. The induced metric is $g_{\mu \nu} = \partial_{\mu} \phi^A \partial_{\nu} \phi_A$, where $\phi^A$ are $n$-dimensional coordinates.   4. It seems extremely hard to calculate the Ricci curvature scalar of such embedding because the inverse metric $g^{\mu \nu}$ depends non-polynomial on partial derivatives of 'fields' $\phi^A$. But still, this Ricci curvature can be evaluated. My question is: could you provide a reference to some work where it is evaluated? Maybe it can be done in the $\det g_{\mu \nu} = 1$ gauge? After all I am interested in the Tailor expansion of the Einstein-Hilbert action with respect to derivatives of $\phi^A$ fields.


title: Help solving 2D motion problem

text: I have the following 2D motion problem: > A sprinkler mounted on the ground sends a jet of water at a 30 degree angle > to the horizontal. The water leaves the nozzle at a speed of 12 m/s. How far > does the water travel before it hits the ground? I've tried several different ways of finding ▲x, but I can't get the correct answer. My book says the answer is 13 m, but I keep getting 6.1 m. Could someone please explain this to me?


title: How to shift a laser wavelength?

text: Recently I have read something about the laser surface velocimeter, and after I have to rebuild from scratch (university project), I got a question: In the description they wrote that one of the laser beams is shifted with around 40 MHz compared to the other one. With this shift it is possible to see if the device is working and in which direction the surface is moving.   Second question: How difficult is it to create such an device, and what should I consider? For explanation as requested:   This project is done for the university racing team, the device should be build into the race car for calibration. Time: Always if I have nothing to do, Equipment: I can use the local university equipment here, and budget: I hope I do not need more than the stuff which is avaliable here (except for finalisation). Is this feasible for a one-man-project (master student, already built some interferometers)?   Ah, and btw., the distance of the moving object is between 5 to 10 cm, but the speed goes up to 30 m/s, so is this a big problem (I thought I just have to use a faster photodetector (~30 MHz clocking rate according to the formulas on wiki))?


title: A new way of looking and formulation of Observables for a new quantum theory

text: I often think of basic of QM (although it wasnt discovered that way) is that we have a physical parameter/observable and the favourite one is the displacement $x$ of a particle $P$. Its conjugate variable is its velocity $v$ (although they are vectors, i am dropping the hat for convinience and also stick to 1 dimension. Classical Mechanics says that given $x$ and $v$ at say time $t=0$, and the potential $V$, we can determine $x$ and $v$ at all other times $t>0$. It means we can determine the time evolution of the displacement and velocity of the particle. In quantum mechanics, we don't have any definite values for $x$ and $v$ of the particle. At any given time say $t=0$, all we know is that there is a state function $\Psi(x,0)$ whose magnitude gives probability density function for $x$ and its Fourier transform gives probability density function for $v$ at any time $t=0$, to begin with. We are not allowed to ask for exact values of $x$ and $v$, as they are not part of the theory. Given the potential $V$, the Schrodinger equation determines the time evolution of the state function, that is $\Psi(x,t)$ at all future times. Thats about the basic difference... This QM gives a very accurate description of our world at microscopic level. **My question :** In QM and CM we begin with the observables $x$ and $v$ at any given time, and then make a law for their time evolution. Why cant we begin with $x(t)$, the displacement as a function over $t$ as the observable, meaning we don't observe $x$ at only a given time, but over a period of time. That is our observable and thats what we intend to measure. We don't measure displacement at only a given time, but over a preiod of time. instead of single value for displacement, we measure a function over time. In this formulation, our observable is not a value but a function/signal. Here again we have classical and probabilistic formulations. In classical world given $x(t)$, we can uniquely determine $v(t)$ and potential $V$. How can we formulate this in a quantum mechanical way? Would this lead to a new useful theory?


title: Is a magnetic field affected by the presence of air?

text: Does air affect a magnetic field's ability to attract ferromagnets? If a magnetic field and a ferromagnet were placed in a vacuum would there be a better performance?


title: What is Jacobian about the "Jacobian Edge" in $E_\mathrm{T}$ distributions?

text: Particle physicists often talk of a "Jacobian Edge" in distributions, i.e. when looking at the $E_\mathrm{T}$ distribution of $W \to e \nu$ decays at rest. How is this related to the Jacobian determinant we all know about?


title: Definition of force for a scalar field

text: I have symmetry-breaking potential V that its reflection symmetry breaks in $\phi_0$. ![enter image description here](http://i.stack.imgur.com/qjjnz.jpg) The action in presence of gravity is: ![enter image description here](http://i.stack.imgur.com/X2VKk.jpg) How we calculate the scalar mediated force $F_\phi$ ? How can I obtain below relation? ![enter image description here](http://i.stack.imgur.com/7gWpX.jpg) Thanks for your Help!


title: Quartic terms as two particle interaction?

text: In some lecture notes I found the sentence: > [...] considering two toy models: the classical and quantum harmonic > oscillator with quartic perturbation. The quartic terms $\sim x^4$ and $\sim > \hat x ^4$, respectively, correspond to the two-particle interaction in > fermionic many-body models. Does the author only refer to a formal similarity (as in second quantization interaction terms are quartic) or can a perturbated harmonic oscillator really represent an interacting system? If so, how can one see this?


title: What is the linear momentum of an inelastic and elastic collision?

text: > The arrows in the figure represent the linear momentum of two balls before > they collide. If the collision is perfectly inelastic, find the linear > momentum after the collision. If the collision is perfectly elastic, find > the linear momentum after the collision. ![Figure](http://i.stack.imgur.com/YtfjL.jpg) (Ignore whatever is written in pencil) I've been trying to solve this problem, but I'm not sure what should I do, or where should I start. I asked my teacher if I had to take into account angles (which are not given), and she just said no. I honestly don't know what to do, can you help me out?


title: Statistical mechanics: What is a "microscopic realization" of a system?

text: What is a "microscopic realization" of a system? The context is statistical mechanics. The microscopic system consists of many atoms (too many to track individually) with an assigned probability density function `f(x,y,z,Vx,Vy,Vz,t)`. The macroscopic system consists of the atoms taken together, with macroscopic quantities computed as expectations of microscopic quantities.


title: Geodesic deviation on a unit sphere

text: Very little interest in the original version of this question so I've rejigged it hoping for a more positive response. I'm trying to use the geodesic deviation equation$$\frac{D^{2}\xi^{\mu}}{D\lambda^{2}}+R_{\phantom{\mu}\beta\alpha\gamma}^{\mu}\xi^{\alpha}\frac{dx^{\beta}}{d\lambda}\frac{dx^{\gamma}}{d\lambda}=0$$ to show that on the surface of a unit sphere two particles separated by initial distance $d$ , starting from the equator and travelling north (ie on lines of constant $\phi$) will have a separation $s$ given by$$s=d\sin\theta.$$ This is similar to Geodesic devation on a two sphere except that question was solved using simple spherical geometry. My plan is find $\frac{D^{2}\xi^{\mu}}{D\lambda^{2}}$ first by using the absolute derivative $$\frac{DV^{\alpha}}{d\lambda}=\frac{dV^{\alpha}}{d\lambda}+V^{\gamma}\Gamma_{\gamma\beta}^{\alpha}\frac{dx^{\beta}}{d\lambda}.$$ Then take the second derivative of this. Next find $\frac{D^{2}\xi^{\mu}}{D\lambda^{2}}$ by calculating the Riemann tensor part$$R_{\phantom{\mu}\beta\alpha\gamma}^{\mu}\xi^{\alpha}\frac{dx^{\beta}}{d\lambda}\frac{dx^{\gamma}}{d\lambda}.$$ And then try to juggle the results to show the separation $s=\xi^{\phi}$ as a function of $\theta$. The line element for spherical coordinates $$l^{2}=dr^{2}+r^{2}d\theta^{2}+r^{2}\sin^{2}\theta d\phi^{2}$$ for a great circle of constant $\phi$ on a sphere of unit radius reduces to $$dl^{2}=d\theta^{2}$$ giving $\frac{d\theta}{dl}=\frac{d\theta}{d\lambda}=1$ and $\frac{d\phi}{dl}=\frac{d\phi}{d\lambda}=0$. The absolute derivative for $\Gamma_{\theta\phi}^{\phi}=\Gamma_{\phi\theta}^{\phi}=\frac{\cos\theta}{\mathbf{\mathbf{\sin\theta}}}$ is$$\frac{D\xi^{\phi}}{d\lambda}=\frac{d\xi^{\phi}}{d\lambda}+\xi^{\theta}\Gamma_{\theta\phi}^{\phi}\frac{d\phi}{d\lambda}+\xi^{\phi}\Gamma_{\phi\phi}^{\phi}\frac{d\phi}{d\lambda}+\xi^{\theta}\Gamma_{\theta\theta}^{\phi}\frac{d\theta}{d\lambda}+\xi^{\phi}\Gamma_{\phi\theta}^{\phi}\frac{d\theta}{d\lambda}=\frac{d\xi^{\phi}}{d\lambda}+\xi^{\phi}\frac{\cos\theta}{\mathbf{\mathbf{\sin\theta}}}.$$ And for $\Gamma_{\phi\phi}^{\theta}=\sin\theta\cos\theta$ is $$\frac{D\xi^{\theta}}{d\lambda}=\frac{d\xi^{\theta}}{d\lambda}+\xi^{\phi}\Gamma_{\phi\phi}^{\theta}\frac{d\phi}{d\lambda}+\xi^{\phi}\Gamma_{\phi\theta}^{\theta}\frac{d\theta}{d\lambda}+\xi^{\theta}\Gamma_{\theta\phi}^{\theta}\frac{d\phi}{d\lambda}+\xi^{\theta}\Gamma_{\theta\theta}^{\theta}\frac{d\theta}{d\lambda}=\frac{d\xi^{\theta}}{d\lambda}.$$ However, $$\frac{D\xi^{\phi}}{d\lambda}=\frac{d\xi^{\phi}}{d\lambda}+\xi^{\phi}\frac{\cos\theta}{\mathbf{\mathbf{\sin\theta}}}$$ doesn't look right as it blows up when $\theta=0$. Any suggestions where I might be going wrong?


title: What is the sum of the electrons' magnetic moments in a wire?

text: The electron has a magnetic moment. This magnetic moment will be influenced in a electric field; the magnetic moments will be more or less aligned. During the movement of an electron in a wire under the influence of an electric potential, the electron has a chaotic movement, in addition to a drift velocity along the wire. What is the sum of the magnetic moments after such a walk? Consider only some length of the wire, not the electrons "at rest" in the source and in the sink.