title: Maxwell Stress Tensor in the absence of a magnetic field

text: I'm having some trouble calculating the stress tensor in the case of a static electric field without a magnetic field. Following the derivation on Wikipedia,   1. Start with Lorentz force: $$\mathbf{F} = q(\mathbf{E} + \mathbf{v}\times\mathbf{B})$$   2. Get force density $$\mathbf{f} = \rho\mathbf{E} + \mathbf{J}\times\mathbf{B}$$   3. Substitute using Maxwell's laws $$\mathbf{f} = \epsilon_0 \left(\boldsymbol{\nabla}\cdot \mathbf{E} \right)\mathbf{E} + \frac{1}{\mu_0} \left(\boldsymbol{\nabla}\times \mathbf{B} \right) \times \mathbf{B} - \epsilon_0 \frac{\partial \mathbf{E}}{\partial t} \times \mathbf{B}$$   4. Replace some curls and combine $$\mathbf{f} = \epsilon_0\left[ (\boldsymbol{\nabla}\cdot \mathbf{E} )\mathbf{E} + (\mathbf{E}\cdot\boldsymbol{\nabla}) \mathbf{E} \right] + \frac{1}{\mu_0} \left[(\boldsymbol{\nabla}\cdot \mathbf{B} )\mathbf{B} + (\mathbf{B}\cdot\boldsymbol{\nabla}) \mathbf{B} \right] - \frac{1}{2} \boldsymbol{\nabla}\left(\epsilon_0 E^2 + \frac{1}{\mu_0} B^2 \right) \- \epsilon_0\frac{\partial}{\partial t}\left( \mathbf{E}\times \mathbf{B}\right)$$   5. Get the tensor $$\sigma_{i j} = \epsilon_0 \left(E_i E_j - \frac{1}{2} \delta_{ij} E^2\right) + \frac{1}{\mu_0} \left(B_i B_j - \frac{1}{2} \delta_{ij} B^2\right)$$   6. Assuming B=0: $$\sigma_{i j} = \epsilon_0 \left(E_i E_j - \frac{1}{2} \delta_{ij} E^2\right)$$   7. Assume flat surface with perpendicular field (z-direction) $$\sigma_{z z} = \epsilon_0 \left(E^2 - \frac{1}{2} E^2\right)=\frac{\epsilon_0}{2} E^2$$ This is the formula given in e.g. The Feynman Lectures in Physics Vol. 2 (Page 31-14), and some other text books. However, this derivation seems to assume a magnetic field until the final steps. Since most terms in eq. 4 result from the initial v x B term (even those that depend only on E, $(\mathbf{E}\cdot\boldsymbol{\nabla}) \mathbf{E}$ and $\frac{1}{2} \boldsymbol{\nabla}\epsilon_0 E^2$ ), these should not be present in my case, and in fact eq 4 should be as simple as $$ \mathbf{f} = \epsilon_0\left[ (\boldsymbol{\nabla}\cdot \mathbf{E} )\mathbf{E}\right] $$ Tensor calculus is not my strong point. To me it is not clear how to get from eq 4 to eq 5, and how modifying eq 4 alters the resulting stress tensor. Will it really still be the same as eq 6? To me it seems strange that removing terms would not affect the result, yet this seems to be what many text books claim. Or is there some reason why the initial v x B term can not be removed, even when there is no magnetic field?


title: How to produce a bigger rotational velocity from two rotational velocities?

text: If I have two electronic motors, both running at the same voltage and current, delivering the same rotational velocity, how could I produce a bigger rotational velocity from them? ![enter image description here](http://i.stack.imgur.com/sEkiT.jpg)


This question popped out of another discussion, about if the photon needs a receiver to exist. Can a photon get emitted without a receiver? A universe containing only one electron was hypothetically suggested. And a similar theory was mentioned in Feynman’s nobel lecture in 1965: > As a by-product of this same view, I received a telephone call one day at > the graduate college at Princeton from Professor Wheeler, in which he said, > "Feynman, I know why all electrons have the same charge and the same mass" > "Why?" "Because, they are all the same electron!" And, then he explained on > the telephone, "suppose that the world lines which we were ordinarily > considering before in time and space - instead of only going up in time were > a tremendous knot, and then, when we cut through the knot, by the plane > corresponding to a fixed time, we would see many, many world lines and that > would represent many electrons, except for one thing. If in one section this > is an ordinary electron world line, in the section in which it reversed > itself and is coming back from the future we have the wrong sign to the > proper time - to the proper four velocities - and that's equivalent to > changing the sign of the charge, and, therefore, that part of a path would > act like a positron." "But, Professor", I said, "there aren't as many > positrons as electrons." "Well, maybe they are hidden in the protons or > something", he said. I did not take the idea that all the electrons were the > same one from him as seriously as I took the observation that positrons > could simply be represented as electrons going from the future to the past > in a back section of their world lines. That, I stole! It was a very successful theft indeed, so let me try a theft too. Since Wheeler came up with this theory, we have come up with string theory that create electrons and other elementary particles by string vibrations. Could all strings also have equal properties or interactions, so they can create different particles with equal charge and mass? So if we merge Wheeler’s idea with string theory, we can formulate this into a hypothetical question: Could all strings be one single string which weaves the fabric of the universe? To simplify it even further we can say that a single particle drags the string along and tie the knots in the fabric of the universe together by interactions with itself. So then we get a single string or a single particle universe, which is the ultimate simplicity. The speed of light can’t be a threshold for such a particle; because the particle itself must travel with infinite speed far beyond C and probably don't even have a velocity we can put any number on, but just call infinite speed. To go past the speed of light the particle must be without mass, and then it has no inertia and is free to go everywhere to interact with its own string which is woven into time, space, particles, mass, charge, magnetism, gravity, me, you and the universe itself.


title: Wave function and Dirac bra-ket notation

text: Would anyone be able to explain the difference, technically, between wave function notation for quantum systems e.g. $\psi=\psi(x)$ and Dirac bra-ket vector notation? How do you get from one to the other formally? When you express a state in bra ket notation as a vector $(a,b)$ for instance do we have to be referring to a basis of eigenvectors for an observable? Furthermore, consider if we had a unitary function $U$ how would we express $\langle U \psi\mid$ in terms of $\langle \psi\mid$? As the bra of psi is an element of the duel Hilbert space, which is a function that takes the ket of psi to the inner product, how would we remove the unitary operator? Note, this is not a homework question, i'm just trying to improve my formal understanding of the notation I've been using, as this has been skirted over quite substantially. EDIT: Got the last question I asked $$\langle U\psi \mid \phi \rangle=\langle \psi \mid \bar{U^T}\phi \rangle$$ $$\forall \phi \in H \Rightarrow \langle U\psi\mid = \langle \psi\mid\bar{U^T}.$$


title: Are CCDs approaching the sensitivity of photomultiplier tubes?

text: I still see photomultiplier tubes (PMT) used for even benchtop spectroscopy experiments. How much more sensitive are they to CCDs? I found a post which claims about 1.5 x more sensitivity for PMTs, but any references or more complete understanding of the problem would be really appreciated. I'm also wondering if they need to be cooled or specialized chips or if commodity imaging chips like from a camera has reasonable S/N characteristics for detection?


title: Commutator with Pauli spin matrices and the momentum operator

text: How is $\left[\vec\sigma \cdot \vec p, \vec \sigma \right]$ proportional to $\vec \sigma\times \vec p$, where $\sigma$ are the Pauli spin matrices and $p$ is the momentum operator?


title: How to calibrate an Ammeter?

text: Calibrating an ammeter is tricky thing to do. The obvious way would be connecting it to another good ammeter and see if they get the same result. However, people have been telling you can use a potentiometer i.e. rheostat. How is this possible? Please explain the steps and how it needs to be done


title: Equations of graphs

text: I am doing a practical write up of an experiment I did. Here is a graph using the raw data I got. ![enter image description here](http://i.stack.imgur.com/n1JMW.png) What is the general formula of this graph, and what do I need to do in order to make that into a straight line graph? Take the natural log of the average speed and plot it against mass? Excel gives me the specific equation of the line, but I need to write down a general equation in order to get max marks. Or could it be that my empirical data is wrong with the theory? I'm having trouble understanding the maths here. Edit: Here are attempts at straight line graphs I have made. ln v vs ln M looks the straightest, is that the one I should therefore use? ![enter image description here](http://i.stack.imgur.com/eZnE8.png) Can someone walk me through how I change the general formula of the first graph into the general formula of a straight line graph?


title: Stresslet coefficient in fluid dynamics

text: I encountered a parameter called stresslet coefficient in several literatures, which seems related with viscosity and stream function (also influencing the far field displacement). But there's no Wikipedia page for this concept. Can anyone explain what it is? Thanks!


title: How do you explain spinning tops to a nine year old?

text: Why don't spinning tops fall over? (The young scientist version) My nine year old son asked me this very question when playing with his "Battle Strikers" set. Having studied Physics myself, I am very keen to encourage him to take an interest in Science and I am delighted when he asks me such questions. In this case, however, I'm stumped. How do I explain why a spinning top doesn't fall down without going into the mathematics of angular momentum? I thought the following post would help, but the answers would just bamboozle him I'm afraid. Why don't spinning tops fall over? So, how _do_ you explain spinning tops to a nine year old?


title: Why does my house seem to warm faster in summer than it cools in winter?

text: In summers when we switch off the air conditioner, the room seems to instantly get hot again. But in winter, when we switch off the heater the room seems to remain hot for some time. Why this difference?


title: Maxiumum weight a buoy can float

text: I have a buoy that can hold 2943.02 cubic millimeters of non-pressurized air, I need and equation to work out the maximum weight that the buoy can float.


title: Why is the steam from cooling towers not used?

text: If I see the steam coming out of cooling towers at an altitude of 200 meters, I cannot help but think that this energy is wasted. My question is: Why isn't this steam cooled enough to become water and used to create hydroelectric power due to the drop from 200 meters?


title: Temperature of Bose-Einstein-Condensate in space

text: Recently I heared a talk by Bill Phillips, who talked about the coldest temperatures in the universe. Among others, he sayed that the coldest temperatures created at the moment are BECs, which can reach roughly 300 pK. Furthermore he mentioned, that in the future it might be possible to create BECs in space at a satellite, where one could reach up to 1 pK. I dont understand why a BEC in space would be so much colder. The low pressure and temperature in space cannot be the reason I think, because this is not the case in a satellite (electronics etc. would not work otherwise). The only other thing that is different is that such BECs are in a free-falling environment. But I don's see how one could get a factor of 300 in temperature, given that the BEC on earth is anyway trapped, and gravity is very weak. Does anybody know an explanation or a reference for such a statement?


title: Why there is a flat band for Kagome lattice?

text: For the nearest neighbor hopping model on the Kagome lattice, there is a flat band among the three energy bands. ![](http://i.stack.imgur.com/HntsN.png) Is there some reason, such as symmetry or the special structure of the model, to ensure the existance of a flat band?


title: What is the difference between Rayleigh scattering and Thomson scattering?

text: After reading the wiki articles I know, that both Rayleigh scattering and Thomson scattering are elastic processes. But what is the essential difference between those two processes, their cross sections and energy dependence?


title: Why is an optical magnon with k=0 not an eigenenergy state?

text: I found in a paper the following explanation. Unfortunately, I can't understand it. Can anyone help me on this? > In the limit of equal spins an optical magnon with k=0 gets an acoustical > one at the boundary of the Brillouin zone where it is known to have a finite > life time.


title: Why do "mascons" perturb orbits?

text: If the moon is a rigid body, why do the mass concentrations on the moon make orbits unstable? Doesn't a satellite just orbit the center of mass of it's parent body? Satellites (e.g. GRACE) seem to be able to measure these things by perturbations in their orbit but I'm not sure how or why. What oversimplification am I making?


title: Container of liquid in free fall

text: Imagine a container of liquid in free fall. Why does the liquid fly out of the container. Is there a force exerted by the inside fluid on the surface of liquid which causes this? Also, suppose there is an object inside the fluid, what will be the force exerted by fluid on the object?


title: Google interview riddle and scaling arguments

text: I am puzzled by a riddle to which I have been told the answer and I have loads of difficulties to believe in the result. The riddle goes as follows: "imagine you are shrunk to the size of a coin (i.e. you are, say, scaled down by two orders of magnitude) but your density remains the same. You are put into a blender of hight 20 cm. The blender will start working in 60 seconds, what to you do? " One of the best answers is apparently: " I jump out of the blender to escape (yes the blender is still open luckily)" This seems ultra non intuitive to me and I have tried to find flaws in this answer but it seems to be fairly robust. There are two ways you can think of it:   * the mass scales as $\sim \: L^3$ and therefore it will be $10^6$ times smaller. If we imagine that the takeoff velocity $v_{toff}$ is the same as before being rescaled $v_{big}$. We get then the hight at which a mini us can jump by equating the takeoff kinetic energy and the potential energy i.e. $mv_{toff}^2/2=mgh $ $\Rightarrow$ $h_{mini} = v^2_{big}/(2g) = h_{big} \sim 20 \rm cm$    * The second way to see it is to look more in details on how the power produced by muscles scales with the size of muscles. Basically, the power scales with the cross section of the muscle i.e. with the number of parallel "strings" pulling on the joints to contract the muscle. This implies that $P_{mini}=P_{big}/\alpha^2$ ($\alpha$ being the factor bigger than 1 by which you have been rescaled). We know that the takeoff kinetic energy will be given by $P \Delta t$. We assume now that $\Delta t \sim L/v_{big}$ so that $\Delta t_{mini} = L/(\alpha v_{big})$. In the end, this calculation tells us that $E_{mini} \sim E_{big}/\alpha^3$. However, equating again with the potential energy to get the hight we have $h_{mini} \sim E_{mini}/(m_{mini}g) = (E_{big}/\alpha^3)/(gm_{big}/\alpha^3)=E_{big}/(m_{big}g) = h_{big} \sim 20\:\rm cm$ These two reasonings seem fair enough and yet I don't trust the result they lead to. I would like to know if I am experiencing pure denial because of my prejudices or if there is some kind of flaw in the reasonings above (e.g. the fact that it is always assumed that the speed is unchanged when changing scale). I also know that some tiny animals can jump more or less as high as human beings but it seems that most of the time these species have to use some kind of "trick" to store elastic energy in their body so as to generate enough kinetic energy at the takeoff to effectively jump super high. If anyone has any thought on this, that would be very much welcome.


title: Evolution and the arrow of time

text: I don't want to ask about potential resolutions to the arrow of time. On the contrary, I want to take it as a given that it is the case for the universe, but then ask a rather odd question about its applicability (or lack thereof) to the surface of the earth. The surface of the earth is certainly not a closed system, in that it receives tremendous energy from the sun. For that reason alone, there is no reason to believe its entropy should increase over time. Shouldn't I be able to conclude that, for processes _that are only sensitive to conditions on the surface of the earth_ , there is no arrow of time ? Shouldn't be this be the case for life as a whole ? If I were studying large scale structure formation in the universe, it would make sense to keep in mind that there will be an asymmetry in the change in entropy between one direction and another given the low entropy of the initial conditions. The evolution of life clearly also follows this asymmetry in that if I were to ask in what direction of time I would have to go to find the common ancestor of 2 animals, it would be the same as the low entropy direction of the universe, but there's no clear reason why that should be the case (in other words, if I were to reconstruct the tree of life, there would be a direction from which branches can flow out of points, but never into them). The bacteria floating around a puddle of water are hardly sensitive to the entropy conditions of the whole universe, and only care about their immediate surroundings. More generally, as I've pointed out, there is no reason to think that the surface of the earth will have greater entropy in a billion years. So why can't life appear in the future, and evolve "backwards" relative to us, leading to a separate time reversed tree of life from us ? Why isn't there a second tree of life that sees time "backwards" relative to us, with living beings that seem to multiply "backwards" in time ? More generally, if this cannot be the case for the surface of the earth if the energy from the sun simply isn't enough to accomplish it, can there exist local systems whose arrow of time will be the opposite of the global arrow of time ?


title: Is the quantization of the harmonic oscillator unique?

text: To put it a little better: > Is there more than one quantum system, which ends up in the classical > harmonic oscillator in the classial limit? I'm specifically, but not only, interested in an elaboration in terms of deformation quantization.


title: Relation of the Bloch-Siegert shift to the rotating pot lid

text: I see in Wikipedia that the Bloch-Siegert shift is analogies to the rotating pot lid, could you explain that analogy? > _The Bloch-Siegert shift is a phenomenon in quantum physics that becomes > important for driven two-level systems when the driving gets strong (e.g. > atoms driven by a strong laser drive or nuclear spins in NMR, driven by a > strong oscillating magnetic field)._ > > _When the rotating wave approximation(RWA) is invoked, the resonance between > the driving field and a pseudospin occurs when the field frequency is > identical to the spin's transition frequency. The RWA is, however, an > approximation. In 1940 Bloch and Siegert showed that the dropped parts > oscillating rapidly can give rise to a shift in the true resonance frequency > of the dipoles._ > > _The pot lid is rotating around an axis along the surface of the table that > is quickly rotating. This results in a secondary rotation which is > perpendicular to the table. This is equivalent to the Bloch-Siegert shift > and can be seen by watching the motion of the red dot._ ![The pot lid is rotating around an axis along the surface of the table that is quickly rotating. This results in a secondary rotation which is perpendicular to the table.This is equivalent to the Bloch-Siegert shift and can be seen by watching the motion of the red dot.](http://i.stack.imgur.com/fUWgk.gif)


Do salt lamps really produce negative ions? Do you know about any scientific study that could support this assertion or disprove it?


title: Hygiene thermodynamics 2

text: Two kinda related questions here:   1. Is evaporation rate and temperature difference related? There is an experiment of pouring cup of hot water out the window during winter. The water evaporates almost instantly forming a vapor cloud.   2. I was showering during winter (outside: ~15 degrees Celsius below the Celsius' zero). I opened the window and as the cold air came in the room filled with vapor. It seemed almost like the vapor was coming in through the window. What happened there? My guesses: a. It's the question 1 thing.   b. I got in a lot of dry air which allowed more water to evaporate.   c. I lowered the air temperature which caused the air humidity to condense forming this fog.   d. or something else did happen?


title: Interpreting range of conductivity diagram

text: I have found the following diagram (unfortunately in german) about the range of conductivity of conductors (Leiter), semiconductors (Halbleiter) and insulators (Isolatoren). How should one interpret the ranges of conductivity for a specific material? Is it correct that for the semiconductors the range is because of the temperature dependence of conductivity? However the conductors are also temperature dependent (positive temperature coefficient) and should therefore also occupy a range. Is it correct, that the ranges for the insulators is just because for example hard rubber is not an element and can have many different chemical implementations? Or is the conductivity of Insulators temperature dependent too? So it would be nice, if someone could clarify those points, add a bit background and give me some orders of magnitude and perhaps references for this. ![Range of conductivity](http://i.stack.imgur.com/64atr.gif) Because its in german I add some translations:   * The axis to the right is the specific electrical resistance.   * Fester Körper: solid   * Eisen: iron    * Bernstein: Amber    * Glimmer: mica    * Hartgummi: hard rubber   * Kupferoxid: Copper oxide   * Quecksilber: Mercury   * Silizium: Silicon   * Silber: Silver   * Selen: Selenium   * Germanium: Germanium   * Glas, Keramik: Glass and ceramics   * Quarz: Quartz


title: Could some design of a propeller be used in both air and water?

text: Propellers in water are smaller in diameter. They also move more slowly. On the other hand, aircraft propellers are larger in diameter, have narrower blades and operate at very high speeds. An aircraft propeller would break apart in water, while a water propeller would produce little to no thrust in air. Rotational speed can be easily adjusted while moving between environments. However, is there such a thing as a propeller shape that is halfway between the two designs? Or would it simply be extremely inefficient in both environments?


title: Formalism for BEC with short-distance sub-structure "corrections"

text: It's simple to write down a Bose-Einstein Condensate wavefunction---in the position basis, $$\Psi(r_1,\ldots,r_N)=\psi(r_1)\cdots \psi(r_N) \qquad \qquad (1)$$ But in experiments the boson is never an elementary particle, it's something like Rubidium-87 which is only a boson if you zoom out beyond the substructure of nucleus + electrons. For example, if $|r_1−r_2|=10\text{pm}$, so that the electron shells of two atoms are overlapping, then I expect that equation (1) above cannot be right. So, I expect there should be a more precise equation that reduces to equation (1) in a certain limit ... or perhaps equation (1) is a the first term in a series, and there are correction terms. How does this work?


Why Newton's law of universal gravitation is a valid law? What causes any two bodies in the universe attract each other with a force? What causes a larger body in space to have a gravitational pull on a smaller body?


title: Analytical solution of two level system driving by a sinusoidal potential beyond rotating wave approximation

text: A quantum mechanical two-level system driving by a constant sinusoidal external potential is very useful in varies areas of physics. Although the wildly used rotating-wave approximation(RWA) is very successful in treating weak coupling and near resonance cases, sometimes a analytical solution beyond RWA is desired. Are there any special cases(for example large detuning, very strong driving, etc.) where one can get the analytical solutions beyond RWA? In mathematics, this is to say that solve the following equation analytically for $C_1$ and $C_2$: $$ i\dot{C}_1(t)=\Omega~cos(\omega t)e^{-i\omega_0t}C_2(t)\\\ i\dot{C}_2(t)=\Omega~cos(\omega t)e^{i\omega_0t}C_1(t) $$ where $C_1(t)$ and $C_2(t)$ are the two level state amplitude, $\Omega$ is the coupling strength, $\omega_0$ is the two level frequency difference, and $\omega$ is the driving frequency. $\omega$, $\omega_0$, $\Omega$ are constant and $C_1$ and $C_2$ are time dependent quantities. Any suggestions or related literatures are appreciated.


title: What is the difference between gravitation and electromagnetism?

text: I am currently studying electrodynamics. And when looking at Maxwell's equations, I don't see any reason, why we cannot apply them to gravity. We know that charges generate a force field that attracts opposite charges. From this and using special relativity we can deduce that there must be a magnetic field (see this nice video from minutephysics). But masses also generate a force field that attracts other masses. Then by the same reasoning, moving masses must also generate some kind of field that interacts with other moving masses. And indeed, this happens. It's called Lense-Thirring effect. So if these fields are so similar, why aren't there some kind of Maxwell's equations for gravity? **What is the difference between electromagnetism and gravity?** **EDIT:** Thanks for the comments. So it basically boils down to the question why charge is invariant under Lorentz transformations while mass is not.


title: Can a minimum ultimate tensile strength (UTS) for an asteroid be established based on its diameter and rotation?

text: I found it fascinating that many asteroids rotate with a period of just seconds. For this fast of a rotation on this size of object, I thought that would actually cause significant acceleration on the "equator" of the rotation. This is not the case for planets where gravitation still causes a net downward force on its equator. An entity on the surface of such as asteroid will have to hold on to the surface in order to keep from flying off. Here is what I calculate for the surface acceleration (again, _off of the surface_ ) for the notable examples in the above link:               Asteroid    Period (s)  Radius (m)  v^2/r (m/s)     -----------------------------------------------     2010 JL88     24.5       15          0.987     2010 WA       31          3          0.123     2008 HJ       42.7       24          0.520     2000 DO8      78         30          0.195     2003 DW10    100         20          0.079     2003 EM1     111.6       33          0.105      I consider these accelerations to be quite considerable. If the surface was in any way sandy, for instance, the top layer would just fly off. Obviously some sort of cohesion is necessary. # My Question Are the fastest rotation periods of asteroids limited by asteroid material, or is there no astrophysical way for them to be spun fast enough to matter in the first place? Given what we know about asteroids, what is the surface acceleration at which we expect them to fall apart at, and how does that compare to what we observe?


title: How to adjust feather touch focuser for collimation?

text: I have a Starlight Instruments feather touch focuser for my Dobsonian telescope. While trying to collimate the secondary mirror with the focus all the way in, I noticed that my Howie Glatter laser collimator drifts away from the center of the primary mirror as I adjust the focus to all the way out. The collimator fits snugly into the focuser. Clearly this is not good as collimation is now dependent on the focus position. While I could work from the mid-point of focus, I'd rather resolve this issue. How can I adjust my focuser such that it is aligned with the telescope and collimation does not drift as I adjust focus? I noticed that adjusting the secondary has reduced this issue, now the drift as focus moves is more subdued, but still noticeable.


title: Simple General Relativity Relation

text: Given the identity $$\nabla_a(R^{ab}-\frac{1}{2}R g^{ab})=0,$$ how do I then show that $R_{ab}=0$ implies $$\nabla_a R^a_{\space \space \space bcd}=0$$


title: What is the Lagrangian from which the Klein-Gordon equation is derived in QFT?

text: 1. Is there a well-known Lagrangian that, writing the corresponding eq of motion, gives the Klein-Gordon Equation in QFT? If so, what is it?   2. What is the canonical conjugate momentum? I derive the same result as in two sources separately, but with opposite sign, and I am starting to suspect that the error could be in the Lagrangian I am departing from.   3. Is there any difference in the answers to that two questions if you choose (+---) or (-+++)? If so, which one?


title: For resolving different objects, are there any alternative coefficients to the one used in the Rayleigh criterion? (which is 1.22)

text: http://en.wikipedia.org/wiki/Angular_resolution#Explanation It's the point where you can't distinguish between the maxima and the first minima. But could there be cases where you have a more stringent limit? Or a more generous one?


title: Current and emf in a transformer

text: In the book Fundamentals of Physics by Halliday, Resnick and Walker(Extended)(6th edition), in a an ideal transformer it is written that the current induced in the secondary coil induces an opposing emf in the primary coil. Firstly, I want to know how we got this direction of induced emf in the primary coil using Lenz's law. According to me, the direction of magnetic field in the core due to the first coil should be downwards(assuming that the current is going down the coil). So the magnetic field in the core in contact with the secondary coil is directed upwards. So the direction of current in the secondary coil should be upwards. By similar reasoning the current induced in the primary coil should be downwards. Thus according to me the induced emf does not oppose the current in the primary coil. Secondly, the book also states that to maintain the emf in the primary circuit, the generator produces an alternating current in the circuit. How does the generator produce this extra current? If this is done by rotating the armature faster, where does the energy required to move the armature faster come from? All this is taken from Pg 793 under the topic Ideal Transformer, Chapter 33:Electromagnetic induction. Fundamentals of Physics(Extended) by Halliday, Resnick and Walker, 6th edition.(Wiley student edition)


title: What limitations are there in measuring physical properties accurately?

text: In a StackOverflow answer, I attempted to explain why a 32-bit float was perfectly adequate for representing the questioner's weight measurement: > Physical properties are inaccurately measured anyway: > >   * no measuring instrument is perfectly calibrated or completely reliable > (e.g. weighing scales cannot adjust for the exact gravitational field at the > precise time and place in which they are used, or may have undetected > mechanical/electrical faults); and > >   * no measuring instrument has infinite precision--results are actually > given across some interval, but for convenience we often adopt a "shorthand" > representation in which that information is dropped in favour of a single > number. > > > > Consequently, all that anyone can ever say about a physical property is that > we have a certain degree of confidence that its true value lies within a > certain interval: so, whereas your question gives an example weight of "5 > lbs 6.2 oz", what you will actually have is something about which you're, > say, 99.9% confident that its weight lies between 5 lbs 6.15 oz and 5 lbs > 6.25 oz. > > Seen in this context, the approximations of a 32-bit float don't become even > slightly significant until one requires _extraordinarily high accuracy_ > (relative to the scale of one's values). We're talking the sort of accuracy > demanded by astronomers and nuclear physicists. But something about this has been bugging me and I can't quite put my finger on what it is. I know that it's completely unimportant for the purposes of that StackOverflow answer, but I am curious:   1. **Is what I have said (about errors and uncertainty in measuring physical properties) completely, pedantically correct?** I acknowledge that knowing the gravitational field is only relevant if one wishes to ascertain a body's _mass_ , however at the time it struck me as a good illustration of experimental errors: systematic error from "imperfect calibration" (i.e. to the gravitational field at the scales' location of use) and random error from the instrument's "unreliability" (i.e. fluctuations in the field over time). Is there is a similarly simple and accessible illustration of error that is more relevant to _weight_? Perhaps the inability to perfectly calibrate springs, together with the randomness of their precise behaviour due to quantum effects? (If that's not complete and utter nonsense, then I'm truly amazed!)   2. **Have I omitted any further points that would help to justify my conclusion (that a 32-bit float is adequate for the OP's needs)?** Perhaps I have not fully explained the types or risks of experimental error? Perhaps I have not fully explained the limitations of physical measurements?   3. **The final sentence quoted above (re astronomers and nuclear physicists) is, of course, an exaggeration: is there a better analogy?** * * * ### _UPDATE_ I decided to remove from my original answer this rant about physical measurement, since it was pretty tangental to the purpose of _that_ question. Nevertheless, I am curious to find a good answer to this question.


title: Moment of inertia of a yo-yo

text: Considering the yo-yo like two CDs with a hollow cylinder between them, what is the moment of inertia of that object? The axis that I must choose can't pass through the CM and be parallel to rotational axis that passes through the center of mass. A force $\vec{T}$ is pulling the object. I can't use the Steiner theorem here, right? All because that cylinder cannot be ignored. Another try that I do is calculating the moment of inertia of the three (well, two) objects and sum, but I think that this isn't correct. ![enter image description here](http://i.stack.imgur.com/92mgL.png)


title: Given Angle, Initial Velocity, and Acceleration due to Gravity, plot parabolic trajectory for every "x"?

text: Given any **Angle** -> 0-90   Given any **Initial Velocity** -> 1-100   Given **Acceleration due to Gravity** -> 9.8   Plot every x,y coordinate (the parabolic trajectory) with cartesian coordinates and screen pixels (not time) This should only be one equation as far as I can tell, a "y=" type equation, telling you the height in y coordinates based on the current x coordinate which is increasing by 1 as you plot across a computer screen. I've come up with an equation that works for the angle 45, but doesn't seem to be entirely accurate for any other angle, I suspect it has something to do with the angle 45 having 1 solution and every other angle having 2 solutions (two different initial velocities land on the same spot), but I'm stumped beyond that. Here is the inaccurate equation:   it won't let me attach images


title: Showing $ m\int \frac{d\textbf{v}}{dt} \dot \normalsize \textbf{v}dt = \frac{m}{2}\int \frac{d(v^2)}{dt}{}dt$

text: Can someone please explain how this equation is valid, using intermediate steps if available? $$ m\int \frac{d\textbf{v}}{dt} \dot \normalsize \textbf{v}dt = \frac{m}{2}\int \frac{d(v^2)}{dt}{}dt$$ And what does the right side mean, if the $dt$ cancels with the $\frac {1}{dt}$?


title: Showing $ \textbf{F} \cdot d\textbf{s} = -dV$ is equivalent to $ F_s = -\frac{\partial V}{\partial s}$

text: Can someone please explain how the following $$ \textbf{F} \cdot d\textbf{s} = -dV$$ is equivalent to $$ F_s = -\frac{\partial V}{\partial s}$$ using some intermediate steps. I don't follow this in Goldstein's text. Thanks.


title: Does action really have to be Lorentz-invariant in SR?

text: From Landau & Lifshitz _The Classical Theory Of Fields_ it is said: > To determine the action integral for a free material particle (a particle > not under the influence of any external force), we note that this integral > must not depend on our choice of reference system, that is, it must be > invariant under Lorentz transformations. This seems understandable. But in comments on this answer Ján Lalinský says that "there is no good physical reason why action should be invariant". Further he suggests another Lagrangian than that given in L&L, namely, if we denote L&L Lagrangian as $L_0$, then the example could be $L_0+Cv_x$, which is clearly anisotropic. Clearly, the equations of motion must not change with this Lagrangian, because $Cv_x$ is a total time derivative (of $Cx$). On the other hand, in this answer Luboš Motl says that "the invariance of the action follows from special relativity – and special relativity is right (not only) because it is experimentally verified. [snip] If $S$ depended on the inertial system, so would the terms in the equations $\delta S=0$, and these laws of motion couldn't be Lorentz-covariant". How could I connect L&L and Luboš's arguments with Ján's example? Can both sides be simultaneously right? They seem to contradict each other.


title: How the frequency of the oscillation is affected?

text: Is the frequency of oscillation of a rotating disk affected if a mass hanging from (attached to the disk but pulled by gravity) the disk exerts a torque on the rotating disk?


title: can we apply WKB method for curved space times

text: let be the Hamiltonian of a surface $ H= g_{a,b} p^{a}p^{b} $ (Einstein summation assumed) my question is if although the space time is curved then can we use the WKB approximation to get the quantum energies from the momenta $ \oint _{C} p_{a}dq_{a}=2\pi \hbar (n_ {a}+ \mu _{a})$ and $ \oint _{C} p_{b}dq_{b}=2\pi \hbar (n_ {b}+ \mu _{b})$ for example for the hyperbolic metric $ ds^{2} = \frac{dx^{2} +dy^{2}}{y^{2}} $ with Hamiltonian $ H= -y^{2}( \partial _{x}^{2}+ \partial _{y}^{2}) $


title: Historical aspect of wave theory of light

text: Huygens thought light as a wave. Wave is a propagation of physical disturbance. We now know that light is electromagnetic field. Electric and magnetic field fluctuates here. What Huygens really thought? Light as a fluctuation of what?


title: What is the 1977 Wow signal?

text: Can someone explain about the 1977 Wow signal? I know that it is a signal received from space in 1977, in the form of a series of code "6EQUJ5". I would like to know if there is any thing or event which can generate this signal other than an intelligent life form? If yes then can you tell me which things or events can generate it??


title: In nuclear fusion reaction, what is the percentage of mass converted to energy?

text: I read somewhere that it is about one percent of the mass, but I find this too high. Also I have done some calculations, for example, the Tsar Bomba was 50 MT bomb and weighed about 27 tons. Although I don't know how much exactly fusion fuel was used, I think it is safe to assume that a large percentage of this 27 tons was fusion fuel. So, if we want to get the same amount energy from mass-energy equation, we would need something with a mass of 2.2 kg. Which proves what I mean : if fusion reaction converted one percent of the fuel mass to energy, then we would need only 100*2.2 kg = 220 kg of fusion fuel to make the Tsar Bomba, which I find much much lower than the actual number. Tell me what you think, please.


title: why does headlights dim when a starter motor is turned on

text: okay first of all, what exactly does the starter motor do? does it increase the current flow and decrease voltage? and does starter motor have its own internal resistance? i understand how there is a voltage drop in the battery terminals, so does the starter motor lower the potential difference to become less than the voltage drop? so for example if the voltage is 12V and the voltage drop is 2V. The potential difference is now 10V. what is the voltage when the starter motor is used? does it become less than 10V? but then after the engine starts the starter motor turns off and the lights return to normal. So is normal 10V?


title: Characteristic length of a triangle

text: I am reading a paper on collision detection in cloth simulation, please help me understanding following lines written in the paper : > To check if a point x4 is closer than some thickness h to a triangle x1 x2 > x3 with normal n we first check if the point is close to the plane > containing the triangle: |x43 · n| < h. If so, we project the point onto the > plane and compute the barycentric coordinates w1, w2, w3 with respect to the > triangle x1 x2 x3. If the barycentric coordinates are all within the interval [−δ , 1 + δ ] where δ is h divided by a characteristic length of the triangle, the point is close. Please help me understanding 2nd paragraph, **what does characteristic length of the triangle mean?**


title: Perturbation theory

text: I am puzzled with perturbation theory when studying quantum mechanics and solid theory. What I learn about perturbation is, from my ignorant point of view, just mathematics, or even simpler, matrix theory, regardless of being conventional perturbation theory, degenerate pertubation, or quasi-degenerate pertubation. The point is that I can only understand it as a math tool, rather than any physics-related concept. Does perturbation theory inherently build in some concepts in quantum physics? Probably its hard to be complete on this topic. Can someone make sort of "broad brush stroke"?


title: The velocity formula $\mathbf{v} = \mathbf{u} + \mathbf{a}t$ for 1D, 2D, 3D . What is the difference?

text: Could I use $\mathbf{v} = \mathbf{u} + \mathbf{a}t$ for calculating velocity in these 3 different dimensions? If not, what's the difference between these 3 dimensions? How would you calculate velocity in 3d? One more thing, is $\mathbf{v} = \mathbf{u} + \mathbf{a}t$ for 3d, 2d or both?


title: Photoionization equation

text: I really need help understanding this equation ,i am new to quantum mechanics and i cant understand the math, so i need every single symbol to be explained or given a value if it is a constant , ( lets say X is Hydrogen) ![enter image description here](http://i.stack.imgur.com/b3obQ.png)


title: Exorbitant number of counts per sec. in the Rutherford Experiment

text: May someone kindly advise on the Rutherford experiment? I have created a Gaussian of dP$_0$/$\Delta$$\theta$($\theta$), where P$_0$ is the number of counts of alpha particles through a 1.05X19.86 mm slit divided by the time elapsed and $\Delta$$\theta$($\theta$) is the angular width of the detector. At this stage the Au foil was not used, as this was merely the "background" measurement. I have integrated that Gaussian wrt $\theta$ between $-\infty$ and $+\infty$ to obtain the total P$_0$. However, this yielded a total P$_0$ equal to 24,000 counts/sec, which seems exorbitant and wrong. I'd be indebted for any help trying to figure out where the mistake is!


title: Deriving torque equation from Newton's 2nd Law

text: I'm trying to understand the derivation of the torque equation $\vec{r} \times \vec{F} = I \alpha$. My textbook derives this easily enough from Newton's 2nd Law for a _single_ point with mass $m$ and radial distance $r$, with the force applied at the same distance $r$, as (if we drop the vector notation for simplicity) $F=ma=m(r\alpha)$, so $rF=mr^2 \alpha = I\alpha$. (Note, there are both 'ay's and 'alpha's there; they look similar.) The textbook stops there and concludes that the equation holds for any rotating body. But then I tried this derivation for two points positioned along a rigid, massless rod (which points perpendicular to an axis about which it rotates). If there's a point-mass $m_1$ a distance $r_1$ from the axis of rotation and another point-mass $m_2$ at a distance $r_2$, and the force is applied at (for example) distance $r_2$, I get $F=m_1 a_1 + m_2 a_2 = m_1 r_1 \alpha + m_2 r_2 \alpha$, so $r_2 F= (m_1 r_1 r_2 + m_2 {r_2}^2 )\alpha$. But $m_1 r_1 r_2 + m_2 {r_2}^2$ isn't the correct value for $I$ here. What am I missing?


title: Using centre of mass in gravitational field calculation

text: When calculating the gravitational field of a continuous mass, like a rod, at some point outside it, why can't we just use the centre of mass of the rod and then directly plug in the distance between the point and the centre of mass in the gravitational field equation. Why do we need to integrate over the whole rod?


title: How is domain wall formation related to spontaneous symmetry breaking?

text: It is said that domain wall formation is the signature of in spontaneous symmetry breaking but not explicit symmetry breaking. Why is this so?


What is the unit of time on the Voyager Golden Record? Apparently, it has something to do with "a fundamental transition of the hydrogen atom". A high-level description was provided by NASA, as shown in this answer. How will the aliens know what this means?


title: Image formation at surface of water

text: Consider a lake full of water which has an object immersed in it. Now, if an observer was at the surface at a sufficient distance so that the light ray from the object is internally reflected, should he see the object at the surface of the water (since the light rays meet at the surface)? If not, then why?


title: What will happen when "earthing" is not an option?

text: I know that earth has a very high capacitance (of about 711$\mu$F) and it would take a lot of time for this but since we earth almost all our appliances for safety, wouldnt one day come when earth doesnt have the ability to store more charge and therefore we cannot earth our appliances? And even if it does, what would happen after that?


title: Interchange symmetry for states with identical particles

text: I was reading this web page about interchange symmetry for states with identical particles here: http://quantummechanics.ucsd.edu/ph130a/130_notes/node317.html The article states that the highest total angular momentum state will always be symmetric under interchange and the next highest state is antisymmetric. So my questions is: Why does the symmetry of the angular momentum wave function alternate? How to prove it? Thank you in advance.


What's the difference between cyclotron radiation and synchrotron radiation? The title says it. If there is a difference between a cyclotron and a synchrotron beyond the particles being accelerated in bunches, please clarify. Is bremsstrahlung a distinct mechanism from the above? Is there a gravitational equivalent, such as a relativistic particle moving just beyond an event horizon (or whatever the equivalent is since Stephen Hawking said something about black holes, the details of which I have no knowledge, maybe they were grey this whole time)?


title: Schrodinger equation from Klein-Gordon?

text: One can view QM as a 1+0 dimensional QFT, fields are only depending on time and so are only called operators, and I know a way to derive Schrodinger's equation from Klein-Gordon's one. Assuming a field $\Phi$ with a low energy $ E \approx m $ with $m$ the mass of the particle, by defining $\phi$ such as $\Phi(x,t) = e^{-imt}\phi(x,t)$ and developing the equation $$(\partial^2 + m^2)\Phi~=~0$$ neglecting the $\partial_t^2 \phi$ then one finds the familiar Schodinger equation: $$i\partial_t\phi~=~-\frac{\Delta}{2m}\phi.$$ Still, I am not fully satisfied about the transition field $\rightarrow$ wave function, even if we suppose that the number of particle is fixed, and the field know acts on a finite dimensional Hilbert Space (a subpart of the complete first Fock Space for a specific number of particles). Does someone has a reference to another proposition/argument for this derivation? Edit: for reference, the previous calculations are taken from Zee's book, _QFT in a Nutshell,_ first page in Chapter III.5.


title: Light entering in a 0° angle, is there any reflection?

text: Is there any reflection of light that enters a new medium at a 0° angle, if the electric field is such, that it is completely in the plane of the double layer?


title: Magnetic properties of matter

text: When a dielectric is placed in an electric field,it gets polarized. The electric field in a polarized material is less than the applied field. Now my query is, when a paramagnetic substance is kept in a magnetic field, the field in the substance is more than the applied fiekd. What is the reason for this opposite behaviour?


title: Round bulb and Long bulb in Series Circuit

text: If you have a series circuit with a battery, one round bulb (with thicker filament), and one long bulb (with thinner filament), the round bulb does not light up and only the long bulb lights up. Also, I understand that the voltage drop in the long bulb is greater, but I don't get how the charges distribute themselves like that. For example, if the round bulb comes first, how does the charges passing by the long bulb "know" to only give up a little bit of energy to save it up for the long bulb? Why does the round bulb not light up?


title: Why do some particles have a greater mass than others?

text: The property of mass that almost every particle possesses comes from the Higgs Field. It is this field, which permeates all of space, that particles interact with and hence obtain mass. But why do some particles have a higher mass if they are all interacting with the same Higgs Field? One would suspect that they'd all have the same mass, but there must be some property of a particular particle that makes it interact with the field in a different way, thereby giving it mass.


title: What's the relativistic energy of a moving strained spring as $k\to\infty$

text: Suppose a spring with stiffness $k$, is strained by constant forces on each end. In a frame where the strained spring moves at a constant velocity, what's the total relativistic energy of the moving strained string as $k\to\infty$?


title: How does a photon "know" that it's left one charge and that it's going to another one?

text: How does it know the same charge it left will be the same charge it will return to? My understanding is photons are neutral and have no charge. i.e. Like charges repel, unlike attract. All charged particles emit photons which are uncharged. So how does the photon "know" that it's leaving one kind of charge and "lands" on another? A neutral "particle" shouldn't be affected by charge.


title: Using photons to explain electrostatic force

text: I am trying to understand the idea of a force carrier with the following example. Let's say there are two charges $A$ and $B$ that are a fixed distance from each other. What is causing the force on $B$ by $A$? Classically charge $A$ has an associated electric field which causes a force on $B$. From the standard model, photons are the force carrier for the electromagnetic force. With this view does it mean that $A$ is constantly emitting photons but in a way that the magnetic component cancels out? If that is the case then doesn't that mean that charge $A$ is constantly losing energy?


title: Exact definition of momentarily comoving reference frame (MCRF)

text: Consider a particle $P$ in the framework of special relativity with position $r(t)=(ct,x(t),y(t),z(t))$ respect to an inertial reference frame $\Sigma=(ct,x,y,z;O)$. I need to know if the following is a correct definition of a MCRF for $P$: > A momentarily comoving reference frame (MCRF) for P is a frame reference > $\Sigma'=(ct',x',y',z';O')$ such that if $r'$ is the position of $P$ respect > to $\Sigma'$, the following two conditions are satisfied: > > 1) $r'(t'_0)=(ct'_0,0,0,0) $ for every $t'_0$ fixed. > > 2) $\frac{dr'}{dt'}(t'_0)=(c,0,0,0)$ for every $t'_0$ fixed. > > Moreover the motion of $P$ is uniformly accelerated if for every choice of a > MCRF $\frac{d^2r'}{{dt'}^2}(t'_0)$ is constant $\forall t'_0$. If the above definition is wrong I would appreciate a **formal** correct definition of a MCRF. Thanks in advance.


title: Conservation laws vs Einsteinian space-time

text: The way I understand conservation laws - which I am asking you to correct - is that if I observe any slice of the universe perpendicular to the time axis and count up all the mass/energy, momentum, charge, etc., I should obtain the same sum as I would had I observed any _other_ slice of the universe perpendicular to the time axis. Stop me right there if you have to, but here is where I start scratching my head. The general theory of relativity informs me that I _can't_ observe a slice of the universe perpendicular to the time axis. The slice I observe is skewed based on my velocity. Yet worse, the special theory of relativity informs me that I can't even observe a _flat_ slice of the universe if I'm accelerating. It's curved based on my acceleration and has all sorts of bumps and such around every massive object. Again, please re-inform me if I'm mistaken. So let's say I make my observation and sum up some conserved quantity over the entire universe. The slice of space-time that I've integrated over is S1, and my total is Q1. I change my direction of motion but not my speed such that the slice of the universe I observe S2 is an affine transformation of S1 that is not equal to S1. I then integrate the same conserved quantity over S2 and call it Q2. Does Q1 = Q2? If it doesn't, then how can we ever verify that conservation laws describe our universe, and how did we come up with them in the first place? If Q1 does = Q2, would this not imply that all conserved quantities are distributed perfectly evenly, since I can tilt my observation to exclude any symmetry-ruining lump of conserved quantity from the total? Since that doesn't _appear_ to be the case, who is the stupid: Einstein, the guy that came up the conservation laws, or me?


title: Is $v(p)\exp(ipx)$ really the positron wave function?

text: In many textbooks the negative energy solution of the Dirac equation is quoted as describing the positron. Actually I don't understand this. For me $v(p)\exp(ipx)$ is the wave function of an electron of negative energy, therefore part of the famous Dirac sea (sorry for sticking to this old model). But a positron is not an electron of negative energy, but the absence of such an electron. Furthermore, if I apply the Hamilton operator on $v(p)\exp(ipx)$ I get a negative energy as a solution. Furthermore, why should a particle which has exactly the same properties as an electron (a part from the charge which does not appear neither in $u(p)$ nor in $v(p)$) have a different wave function ? Another argument: In Feynman diagrams $u(p)\exp(-ipx)$ describes an in-going electron, but $v(p)\exp(ipx)$ is used for an out-going positron (or if you like an in time backwards in-going negative energy electron). This is my problem from the viewpoint of the Feynman-Stueckelberg interpretation: If $v(p)\exp(ipx)$ were a positron wave function should it should describe an in- going particle, but it describes an out-going particle with positive energy. Assuming that positrons should in time forward running with positive energy and only the negative energy solutions in time backward running, then $v(p)\exp(ipx)$ describes an out-going particle with positive energy, then charge conjugation transforms an in-going electron $u(p)$ in an out-going positron $v(p)$. But for me the charge conjugation should transform an in- going electron into an in-going positron. I don't know how to resolve the contradictions. At last if it were said $v(p)\exp(ipx)$ is NOT the positron function, how shall I understand the charge conjugation operation which apparently makes of an electron function $u$ a positron function $v=Cu(=Uc \bar{u})$ ? Thank you for your help.


title: Infinite potential square well solutions

text: My question is about understanding the different solutions of the potential square well. Imagine a square well defined this way: $$ V(x) = \begin{cases} ∞&\,{\rm if} x<0 \\\ 0&\,{\rm if}\,x\in\left(0,L\right) \\\ ∞&\,{\rm if} x>L. \end{cases} $$ When applying the time-independent Schrodinger Equation we get: $$ -\frac{\hbar^2}{2m} \frac{∂^2\psi}{∂x^2} =E\psi,$$ and then: $$ \frac{∂^2\psi}{∂x^2} =-\frac{2mE}{\hbar^2}\psi.$$ If we define $k_1^2=-\frac{2mE}{\hbar^2}$ we get the solution $$\psi_1=Ae^{k_1x}+Be^{-k_1x}.$$ But if we define $k_2^2=\frac{2mE}{\hbar^2}$ we get the solution $$\psi_2=Ae^{ik_2x}+Be^{-ik_2x}.$$ The way I see it one solution describes the wave-function using real-valued exponential functions, while the other describes it using complex-valued exponentials (or sines and cosines, using Euler's formula). With this "mathematical difference", can anyone help me understand if there is any "physical difference" between both solutions? Do they describe different wave-functions? Is there something simple I'm missing? PS: This question is not homework. Is about me trying to understand the solutions of the infinite potential square well.


title: Pseudo Force Other Than Centrifugal Force In A Rotating Frame

text: I read in HC Verma that if we are observing the motion of a body from a rotating frame and the body under the observation is NOT in motion with respect to our frame than centrifugal force is the sufficient pseudo force for analysis of the motion. But if the body under observation is in motion with respect to our frame than some extra pseudo forces other than centrifugal forces are required for the analysis of the motion of the object in our frame. Please explain this additional pseudo force's reason and details.


title: product solutions for PDEs, physical motivation

text: Given a boundary value problem with independent variables $x_1,x_2, \dots , x_n$ and a PDE say $U(x_i, y, \partial_j y,\partial_{ij} y, \dots )=0$ we typically begin constructing a general solution by making the ansatz $y = F_1(x_1)F_2(x_2) \cdots F_n(x_n)$ where each $F_i$ is a function of just one variable. Next, we plug this product solution into the given PDE and typically we obtain a family of ODEs for $F_1, F_2, \dots, F_n$ which are necessarily related by a characteristic constant. Often, boundary values are given which force a particular spectrum of possible constants. Each allowed value gives us a solution and the general solution is assembled by summing the possible BV solutions. (there is more for nonhomogenous problems etc... here I sketch the basic technique I learned in second semester DEqns as a physics undergraduate) Examples, the heat equation, the wave equation, Laplace's equation for the electrostatic case, Laplace's equation as seen from fluids, Schrodinger's equation. With the exception of the last, these are not quantum mechanical. My question is simply this: > what is the physical motivation for proposing a product solution to the > classical PDEs of mathematical physics? I would ask this in the MSE, but my question here is truly physical. Surely the reason "because it works" is a reason, but, I also hope there is a better reason. At the moment, I only have some fuzzy quantum mechanical reason and I have to think there must be a classical physical motivation as well since these problems are not found in quantum mechanics. Thanks in advance for any insights !


title: g as a function of Theta

text: I slide an object down a ramp of a certain length, which is inclined at a certain angle $\theta$. It reaches the bottom in time $t$. Assuming that no forces act on the object other than gravity and the normal force, derive the acceleration due to gravity $g$ as a function of $\theta$. After I did some algebra, I found that $\theta =-2h/(t\sin\theta)^2$. How do I find the same value using the slope of a graph? More precisely, which two variables would I plot to make the slope=$g$? If it was a normal graph, I would use a velocity-time graph, but that doesn't seem to work.


title: Dynamical localisation in delta-kicked rotor

text: The quantum delta-kicked rotor is a common tool for studying quantum chaos. The energy of the rotor increases ballistically when kicking at the Talbot time (resonance) and jumps between zero and some finite value when kicking at half the Talbot time (anti-resonance). Another case is **dynamical localisation** , when kicking occurs at some irrational multiple of the Talbot time. The energy grows linearly until the qwuantum break time, after which the energy levels out as you increase the number of kicks. I (more or less) understand the mathematics of the problem (e.g. from Reichl's _The Transition to Chaos_ ). The problem is just understanding this in a physical sense, because there is very little out there that gives a good description. So what is a solid (but Math-lite) description of dynamical localization?


title: Present experimental status of Moffat's Modified theory of Gravity

text: Modified theories of Gravity have been discussed before in this 2-year old question, Are modified theories of gravity credible? I was going through Moffat's modified gravity, given in http://arxiv.org/abs/gr-qc/0506021 , but before investing too much time into it, I wanted to get some clarifications: 1)What is the present experimental status of the theory? 2)I read somewhere that the theory has is free of singularities, is this true? 3)Although I don't know anything about the details of the theory, it seems a little contrived and unnatural to me, with the variable Gravitational constant and all the extra fields in the action. Am I unjustified in thinking this way?


title: Conservation of momentum when friction is present

text: Conservation of momentum applies when net force is zero. Suppose that there is a system of a canon and a canonball. Total momentum of the system is zero before canonball is fired. Now canonball is fired from the canon, and in frictionless cases, horizontal-axis momentum of the whole system would be preserved. Now friction between ground and canon is added. What happens to the momentum of the whole system in this case? I am confused because for inelastic collsion of the case when a bullet hits a block, it is often said that momentum is preserved during the collision (though not after the collision) even when friction exists between the bullet and the block. Can I say that friction is negligible at the moment of canonball being fired because $F\Delta t = \Delta p$ would be small as $\Delta t$ is very small?


title: Is special relativity axiomatic?

text: Can special relativity be somewhat described by the following statement? > _If the speed of light is a universal constant, and if light can be placed > in separate frames of motion, then time must be altered by speed._


title: Figuring out a film thickness when the wrong metal was chosen for the thickness monitor?

text: I made a silly mistake and was wondering if someone had a clever solution. I evaporated two metals, A and then B. However, when I evaporated metal B, I forgot to switch the thickness monitor to metal B, so it was reporting a thickness and evaporation rate assuming it was still metal A evaporating. Intuitively, I thought this wouldn't be too bad: I have the 2 material constants you punch in when evaporating a material (Z ratio and density) for both of the metals, so I hoped it would be a simple conversion to figure out how much was actually evaporated of metal B. However it seems to be trickier than I expected. It looks like Sauerbrey's equation is what governs it, but $f_L$ and $f_U$ (in the equation for $\Delta m$) seem to be determined at the beginning and during every run, so it's not something I can just figure out and work backwards from. Is there some way to figure it out? Thank you!


title: Why does a microphone membrane only measure pressure and not particle velocity?

text: Microphones (e.g. condenser microphone) are assumed to have a voltage output proportional to the sound pressure at the diaphragm. If the operating principle is that the voltage output is proportional to the displacement of the thin foil that is the diaphragm, then why does the particle velocity of the sound wave not affect the displacement of the diaphragm and hence the measurement? I know that in acoustics the particle velocity and the acoustic pressure do not have a fixed relationship (except for the case of plane waves), and hence in the near-field of a source, a sound intensity measurement must be taken (uses two microphones to estimate particle velocity). It is obvious that the force from the acoustic pressure at the film will cause a velocity (hence displacement) of the film, but what about the contribution from particle velocity of the fluid transferring momentum to the film?


title: Black body radiation application

text: Does it make difference if you paint a wall black or white? If yes, how does it work? Thanks for reading!


title: What are the the characteristics of a standing wave?

text: ![enter image description here](http://i.stack.imgur.com/s6Lzd.jpg) May someone explain #24 from this picture? B is apparently the correct answer. Can any of the other answers be correct here? Isn't D also correct?


title: Relative angular velocity and acceleration

text: _Background:_ (Irodov 1.55) Two bodies rotate around intersecting perpendicular axes with angular velocities $\hat\omega_1,\hat\omega_2$. Relative to one body, what is the angular-velocity and -acceleration of the other? Irodov's answer implies that $$\hat \omega=\hat\omega_1-\hat \omega_2$$ $$\hat \alpha=\hat\omega_1\times\hat \omega_2.$$ I have a hard time grokking why the above are true above (for the first) vague analogies with linear velocity. Does anyone, willing to share, have an intuitive grasp on the above equations?


title: Is it possible to calculate the force exerted by two unequal charge, separately?

text: According to the Coulomb's law, when two equal like point charges are placed 1m from each other the force between them is 9 * 109. But if two unequal like point charges are placed, the bigger charge will not be repelled or move, rather the smaller one will be repelled. This is because both the charges will exert different forces on each other and the bigger charge will exert a bigger force. But the coulomb's law tells us about a common force between the two charges, is it possible to calculate the individual forces exerted by two unequal like point charges?


title: How is gravitational force is compared to flow of water?

text: I listened to a lecture. The professor said that the gravitational field around the particle (spherical in shape) can be compared to a pond having a constant height and depth and water is constantly pumped in the center. The water flows radially outward which is similar to that of the gravitational field around a spherical object. Can anyone explain me how to compare gravitational field and this flow of water?


title: Probability amplitude in Layman's Terms

text: I am basically a Computer Programmer, but Physics has always fascinated and often baffled me. I have tried to understand probability density in Quantum Mechanics for many many years. What I understood is that Probability amplitude is the square root of the probability of finding an electron around a nucleus. But the square root of probability does not mean anything in the physical sense. Can any please explain the physical significance of probability amplitude in Quantum Mechanics? I read the Wikipedia article on probability amplitude many times over. What are those dumbbell shaped images representing?


title: What is the quantum Hall resistance R_H as a function of magnetic field?

text: For the integer quantum Hall effect, the resistance $R_H = h/(ne^2)$, where $n$ is some integer. All of the graphs of $R_H$ as a function of magnetic field, $B$, that I've seen show that at $B = 0$, $n = \infty$, and as $B$ increases, $n$ decreases in steps until it reaches 1. How do you determine the value of $n$ as a function of $B$?


title: How to create unusual sundial?

text: I am considering small "artsy" project. I would like to create sundial by placing gnomon on the window and painting hour lines on the window facing wall. Since this is to be placed in bedroom I am constrained by my geographic location, wall, window placement and orientation. The esthetic and size (I have one wall only) of the project is gating factor (otherwise known as "the wife" factor). Due to above I am perfectly OK with the fact that this sundial will "work" by limited time of the day and even limited time of the year. However whenever it will work (that is the shadow of the gnomon will be cast on the "said" wall) I would like it to be as accurate as possible. Also all of the above make the calculations for creating hour lines quite challenging (at least for me) and to be honest I do not know where to start. Could you point me the resources that would help in calculating hour lines (software, tutorials, books, math equations)?   Could you describe how would you approach the task of calculating hour lines? (Resources that are little heavy on math side are OK for me. I am also capable of wring software on my own.)


title: Does Earth have a code name?

text: Everything we discover in the sky get eventually a code name, like NGC 7293, Simeis 147, etc. Does Earth/Moon have a code name too? Or it is just Earth/Moon, etc.?


title: Resources and requirements for amateur spectroscopy

text: If I wanted to move beyond just looking through my eyepieces and taking photographs of astronomical objects, what can I do do move into astronomical spectroscopy? Are there any good resources for this? Pitfalls? And perhaps most importantly, what kind of initial outlay would I have to make?


title: How to find time taken for a spinning top to stop?

text: The angular position of a spinning top is given by $\theta = t^3 - 72t$, where $t$ is in seconds and $\theta$ in "radian".


title: Physical interpretation of Poisson bracket properties

text: In classical Hamiltonian mechanics evolution of any observable (scalar function on a manifold in hand) is given as $$\frac{dA}{dt} = \\{A,H\\}+\frac{\partial A}{\partial t}$$ So Poisson bracket is a binary, skew-symmetric operation $$\\{f,g\\} = - \\{f,g\\}$$ which is bilinear $$\\{\alpha f+ \beta g,h\\} = \alpha \\{f, g\\}+ \beta \\{g,h\\}$$ satisfies Leibniz rule: $$\\{fg,h\\} = f\\{g,h\\} + g\\{f,h\\}$$ and Jacobi identity: $$\\{f,\\{g,h\\}\\} + \\{g,\\{h,f\\}\\} + \\{h,\\{f,g\\}\\} = 0$$ How to physically interpret these properties in classical mechanics? What physical characteristic each of them is connected to? For example I suppose the anticommutativity has to do something with energy conservation since because of it $\\{H,H\\} = 0$.


title: Is there any correlation between mass-energy equivalence and Maxwell's 4th equation?

text: I wonder, how came in both equations proportionality constant is exactly $c^2$? $$c^2(\nabla \times B) = \partial E/\partial t$$ where $E$ - electric field $$c^2m = E$$ where $E$ - energy I am just curious can we relate these two equations.


title: QM interpretations

text: I don't fully appreciate what the discovery of the decoherence phenomenon adds to the Copenaghen interpretation of QM. I will be more precise: the Copenaghen interpretation, if I am not wrong, is summarized by the following concepts:   1. QM can, and has to, predict _only the probabilities_ of measurement outcomes ("has to" because descriptions of nature such as the classical one, where every quantity always assumes a certain value is in disagreement with our current experimental observations, such as the violation of Bell's Inequality.).   2. In order to do QM, we use state vectors. However, these must not be considered as a real _physical entity_ , but rather _tools_ in a similar way as probability distributions in classical physics (This means that in an EPR experiment, the change of the vector state describing Bob's qubit after Alice has measured hers in a galaxy far away is not in contrast with relativity.).   3. The probabilities are computed via the Born rule: the state after measurement is an eigenstate of the observable measured, and the outcome is the corresponding eigenvalue. Now the decoherence: if one considers the universe divided into a qubit (i.e. a two level system) system (S), apparatus (A) and the rest (E) can give a QM dynamical description of a measuring process on the qubit. Initially, the universe is in the state $|\psi\rangle = (a |0 \rangle + b |1 \rangle)\ |A_0 \rangle | \text{rest} \rangle$. If $|A_i \rangle $ are such that they can _imprint_ themselves in the environment, i.e. they do not entangle themselves with it, after the unitary evolution the SA system is described by the matrix: $\rho = |a|^2 |0\rangle\langle0||A_1\rangle\langle A_1| + |b|^2 |1\rangle\langle 1||A_2\rangle\langle A_2|$. This analysis explains why at macroscopic scales classical physics works, i.e. we don't see superposition of states and probabilities don't interfere: the environment interaction rapidly evolves the vector states into those states that can _imprint_ in the environment, or in mixture of them. This is surely interesting on its own, but is there anything else to understand from decoherence? The fact that superpositions states turn into mixtures of the eigenstates of the measured observable _seems_ to explain the "measurement problem" (that in the Copenaghen paradigm is not a problem actually, but the most important assumption, as stated in item 1 above) but it does not: after the above evolution the system is a classical mixture, but the probabilities are still there. Explaining the measurement problem means to find an evolution that deterministically bring the system into the outcome state (in the Copenaghen interpretation, this is rejected). So decoherence does not explain the Born reduction rule. Additionally, in order to interpret the mixture above as something analogue to a classical mixture, we must have already adopted the Born probability rule, so decoherence does not explain it either. To sum up: having adopted the Copenaghen interpretation of QM, decoherence explains _in its framework_ the transition from quantum world (superposition always possible, quantum interferences) to classical world (no Schroedinger's cat states, bayes rule), but does not replaces or explain any of its axioms...or does it?


title: Time Dilation Problem

text: ![Here's the formula...](http://i.stack.imgur.com/K1zGw.gif) I'm having some trouble using the time dilation formula. Say an astronaut leaves Earth for 10 years, at 0.85c. How much time has passed according to an observer on Earth? I tried using the following formula: $$t = \frac{1}{\sqrt{1-(v^2/c^2)}}$$ but couldn't seem to get an answer that made sense. Any help would be much appreciated! This concept is super confusing to me.


title: Approach to expressing $|n\rangle\langle n| $ as a polynomial when eigenvalues are degenerate?

text: If ${|n\rangle}$ are eigenvectors of an operator $A$ then $|n\rangle\langle n| $ can be expressed in terms of a finite order polynomial $$|n\rangle\langle n| =\prod_{m\ne n} \frac{A-a_m}{a_n-a_m}$$ if the eigenvalues $a_n$ of $A$ are distinct. I am looking for a way to do a similar thing but with degenerate eigenvalues. My difficulty is that the derivation of this relation starts out by considering the product $\prod_{m\ne n}(A-a_m) \mathbf{I}$ and then uses the relation $\mathbf{I}=\sum_k | k\rangle\langle k|$ to proceed to the result above. Starting with the product excluding the $n=m$ term is a bit awkward since it does not allow me to generalize to a case where two eigenvalues are the same. For the case with degenerate eigenvalues can I exclude additional terms from the product to obtain the desired result? **I am just looking for a hint on how to approach this, not a worked out solution.**


title: Young modulus and Sound Velocity in a continuus medium

text: In elasticity theory, general equations of motion are: $$\rho \partial^2_t \overline{u} = \mu \nabla^2 \overline{u} + (\mu+\lambda) \nabla(\nabla \cdot \overline{u})$$ where $\overline u$ are displacements. In a thin homogeneus rod with tractions and compressions in the x direction, stress tensor $S_{ij}$ is: $$ \pmatrix{ S_{xx} &0 &0 \\\ 0 &0 &0 \\\ 0 &0 &0 \\\\} $$ Young modulus is: $$ E=\frac{S_{xx}}{\epsilon_{11}}=\frac{\lambda+\mu}{\mu(3\lambda+2\mu)} $$ where $\epsilon_{11}=\partial_x u_x$ is the deformation on x So equation of motion for x-displacements is: $$ \rho \partial^2_t u_x=\partial_x S_{xx}= E \partial^2_x u_x= \frac{\lambda+\mu}{\mu(3\lambda+2\mu)} \partial^2_x u_x $$ Why is this different to the result of general equation of motion with only dependence to x: $$ \rho \partial^2_t u_x=(2\mu+\lambda)\partial^2_x u_x $$ ? Or: why in a elastic rod has the sound smaller speed than in a generic homogeneus indefinite medium?


title: Why do we use only nonrelativistic equations in nuclear physics?

text: What is limit between relativistic and non-relativistic equations? Which conditions do we have to use one of these?


title: Are there any liquids with zero surface tension?

text: Having read the Wikipedia page on superfluids I'm still not sure if stuff like liquid helium at the lambda point actually have surface tension or not. Is superfluidity the same thing? And are there any liquids with no surface tension at room temperature?


title: What is the relationship between Schrödinger equation and Boltzmann equation?

text: The Schrödinger equation in its variants for many particle systems gives the full time evolution of the system. Likewise, the Boltzmann equation is often the starting point in classical gas dynamics. > What is the relationship, i.e. the classical limit, which connects these two > first order in time equations of motions? How does one approach this, or is there another way in which one sees the classical time evolution? Where are these considerations relevant?


title: Really, how to decide if one can use perturbation theory or not in QM?

text: In QM, it is said that perturbation theory can be used in case the total Hamiltonian is a sum of 2 parts, one whose exact solution is known and an extra term that contains a small parameter, $\lambda$ say, so we can obtain the solution of the full Hamiltonian as a systematic expansion in terms of that small parameter. Now take as a specific example $\displaystyle H=-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}+\frac{1}{2}kx^2+\lambda x^3\equiv H^0+\lambda x^3$ Is there an intuitive way to see in what sense the $\lambda$ term can be smaller or bigger than the unperturbed Hamiltonian? Or is there someway to estimate the size of every term in the Hamiltonian, based on which we can decide the parameter space for $\lambda$ (which will consequently make our perturbation method valid)? how one can even estimate the size of a derivative!?


title: Degree of freedom paradox for a rigid body

text: Suppose we consider a rigid body, which has $N$ particles. Then the number of degrees of freedom is $3N - (\mbox{# of constraints})$. As the distance between any two points in a rigid body is fixed, we have $N\choose{2}$ constraints giving $$\mbox{d.o.f} = 3N - \frac{N(N-1)}{2}.$$ But as $N$ becomes large the second term being quadratic would dominate giving a negative number. How do we explain this negative degrees of freedom paradox?


title: What's the exact connection between bosonic Fock space and the quantum harmonic oscillator?

text: Let's suppose I have a Hilbert space $K = L^2(X)$ equipped with a Hamiltonian $H$ such that the Schrödinger equation with respect to $H$ on $K$ describes some boson I'm interested in, and I want to create and annihilate a bunch of these bosons. So I construct the bosonic Fock space $$S(K) = \bigoplus_{i \ge 0} S^i(K)$$ where $S^i$ denotes the $i^{th}$ symmetric power. (Is this "second quantization"?) Feel free to assume that $H$ has discrete spectrum. > What is the new Hamiltonian on $S(K)$ (assuming that the bosons don't > interact)? How do observables on $K$ translate to $S(K)$? I'm not entirely sure this is a meaningful question to ask, so feel free to tell me that it's not and that I have to postulate some mechanism by which creation and/or annihilation actually happens. In that case, I would love to be enlightened about how to do this. Now, various sources (Wikipedia, the Feynman lectures) inform me that $S(K)$ is somehow closely related to the Hilbert space of states of a quantum harmonic oscillator. That is, the creation and annihilation operators one defines in that context are somehow the same as the creation and annihilation operators one can define on $S(K)$, and maybe the Hamiltonians even look the same somehow. > Why is this? What's going on here? Assume that I know a teensy bit of ordinary quantum mechanics but no quantum field theory.


title: Finding moment of inertia from Lagrange equation

text: I'm getting the following information: Consider a system consisting of two rotating bars of length $\ell$ and with uniform mass density and each with total mass $m$. The bars are attached to a common axis at one end around which the can rotate. The distance between the bars on the axis of rotation is $a$. So I'm guessing it's just to bars revolving around the z-axis with the separation of $a$. Now, I'm told that the Lagrangian for the system is: $$\mathcal{L} = \frac{1}{2}I_{1}\dot\theta_{1}^{2}+\frac{1}{2}I_{2}\dot\theta_{2}^{2}-V(\theta_{1},\theta_{2}),$$ where $I_{1}=I_{2}$ is the moment of inertia, and $V$ an unspecified potential. Now I'm asked to calculate $I_{1}=I_{2}$. My idea was to just use the Euler-Lagrange equation, which I did, and ended up with $I_{1} = \frac{F_{\theta_{1}}\ell}{\alpha}=\frac{torque}{\alpha}$, where $\alpha$ is just the rotational acceleration. But if I just look up the moment of inertia for a rod with end points at the axis of rotation (As here), I find $I = \frac{m\ell^{2}}{3}$. So I'm thinking I'm a bit off here. So any idea how to do this instead, or am I just missing something ?


title: Ladder Operator killing state

text: So in my last question, @joshphysics showed me how to prove $K_\pm$ were ladder operators. Now I need to show that there is a lowest state, i.e $$\langle m_0|K_+=K_-|m_0\rangle=0$$ I am not completely sure how I should approach this. I saw that $$J_3(K_-|m\rangle)=(m-1)(K_-|m\rangle$$ But I do not see how this shows that there has to be a minimum eigenvalue. EDIT::: So I think I may have figured something out $$\langle m_0|K_+K_-|m_0\rangle = (K_-|m_0\rangle)^\dagger K_-|m_0\rangle$$ This last quantity should be greater than 0, so we have a minimum state m_0 that we act on and get 0. Does this make sense?


title: Is there such a thing as instantly stopping?

text: I'm sorry if this is a stupid question, but I've never taken a physics class and I was curious about something. But anyway, my question is, is there such a thing as instantly stopping? For example, if I'm traveling at 70MPH and I hit a wall, between the time that I am going at full speed and when I hit the wall, do I slow down until I reach zero, or do I instantly stop? Do I go every speed in between 70 and 0? I'm sorry if my question is unclear. If it is, please leave a comment and I'll try explaining it more.


title: detect movement - how does it relate to G force

text: I would like to detect if a user is moving by using an android smartphone - walking or jumping doesn't matter - want to know if the user wearing the smartphone is moving. I have found this code example               private final SensorEventListener sensorEventListener = new SensorEventListener() {             double calibration = SensorManager.STANDARD_GRAVITY;                  public void onAccuracyChanged(Sensor sensor, int accuracy) {             }                  public void onSensorChanged(SensorEvent event) {                 double x = event.values[0];                 double y = event.values[1];                 double z = event.values[2];                      double a = Math.round(Math.sqrt(Math.pow(x, 2) + Math.pow(y, 2)                         + Math.pow(z, 2)));                 currentAcceleration = Math.abs((float) (a - calibration));      basically the squarerootof the three axis, which are raised to power 2 = sqrt(x-axis^2 + y-axis^2 + z-axis^2) = G force And it gives readings between 0.01 and 0.2 G's - does anyone know how to translate these readings into "movement" - is it even possible this way - by use of G force


title: Why is electric field zero inside a hollow metal sphere ?

text: The figure below shows a hollow metal sphere with a positive point charge $Q$ sitting outside it. What is the electric field at the center of sphere ? The answer is zero (look at here at the beginning of page 4), but I do not understand why? ![enter image description here](http://i.stack.imgur.com/zYFZC.png)


title: How to find full energy of field of an arbitrary half-integer spin?

text: Let's have arbitrary half-integer spin $n + \frac{1}{2}$ representation: $$ \Psi_{\mu_{1}...\mu_{n}} = \begin{pmatrix} \psi_{a, \mu_{1}...\mu_{n}} \\\ \kappa^{\dot {a}}_{\quad \mu_{1}...\mu_{n}}\end{pmatrix}, $$ $$ (i\gamma^{\mu}\partial_{\mu} - m)\Psi = 0, \quad \gamma^{\mu}\Psi_{\mu \mu_{1}...\mu_{n - 1}} = 0, $$ $$ \Psi_{\mu_{1}\mu_{i}...\mu_{j}\mu_{n}} = \Psi_{\mu_{1}\mu_{j}...\mu_{i}\mu_{n}}, \quad \partial^{\mu}\Psi_{\mu ...\mu_{n - 1}} = 0. $$ I need to find the general expression for the full energy of corresponding particles, but I don't know how to do that. I tried to do the following.   1. The solution of equations above may be given as $$ \hat {\Psi}_{\mu_{1}...\mu_{n}} = \sum_{s}\int \left(\hat {a}^{\dagger}_{s}(\mathbf p)u^{s}_{\mu_{1}...\mu_{n}}(\mathbf p)e^{-ipx} + \hat {b}_{s}(\mathbf p)v^{s}_{\mu_{1}...\mu_{n}}(\mathbf p)e^{ipx}\right)\frac{d^{3}\mathbf p}{\sqrt{(2 \pi )^{3}2E_{\mathbf p}}}, $$ where $s$ refers to the possible polarizations, $u, v$ are "spinor-tensor" waves which have some normalization law (as example) $u^{+}u = 2E_{\mathbf p}$.   2. The Dirac equation for each $\mu_{1}...\mu_{n}$ may be rewritten in a form $$ i\partial_{0}\Psi_{\mu_{1}...\mu_{n}} = \hat {H}\Psi_{\mu_{1}...\mu_{n}} = E\Psi_{\mu_{1}...\mu_{n}}. $$ So for finding the full energy I must construct integral like $$ E = \langle \hat {H} \rangle = \int \Psi^{\dagger} \hat {H}\Psi d^{3}\mathbf r = i \int \Psi^{\dagger} \partial_{0}\Psi d^{3}\mathbf r. $$ The expression $$ \int \Psi^{\dagger}\Psi d^{3}\mathbf r $$ must be lorentz invariant. But how to make it invariant? My first idea is convolute $\Psi^{\dagger}\Psi$ as $\Psi^{\dagger}_{\mu_{1}...\mu_{n}}\Psi^{\mu_{1}...\mu_{n}}$, and I think that it is the only possible way to construct the "correct" lorentz invariant object (in vector indices space), but it is not strictly. After that I only use conservation law $\partial_{\alpha}(\bar {\Psi}\gamma^{\alpha}\Psi ) = 0$. But I'm not sure at this.  Can you help? I know that in general the expression looks like $$ E = \sum_{s}\int (\hat {a}^{\dagger}_{s}(\mathbf p)\hat {a}_{s}(\mathbf p) - \hat {b}_{s}(\mathbf p) \hat {b}^{\dagger}_{s}(\mathbf p ))E_{\mathbf p} d^{3}\mathbf p . $$


title: Measuring CO2 concentration from a respiration chamber

text: I am interested in measuring the concentration of CO2 from an open circuit respiration chamber where a sheep is living. Known variables:   * Concentration of CO2 of the air entering the chamber.    * Concentration of CO2 of the air leaving the chamber. Note. These concentrations were carried out after the CO2 analyzer removed any water vapour from the sample coming from the chamber .    * Altitude, atmospheric pressure.    * Temperature and humidity of the air leaving the chamber.    * Flow rate at which the air leaves the chamber. Not being a physicist I would appreciate you tell me the equations needed rather than pointing me in vague direction. Much appreciated. Javier


title: What is conductivity?

text: I read that if we have spin $\frac{1}{2}$-particle, where a magetic force acts on, then the force is given by a drift speed times a conductivity. This conductivity is determined to be $\frac{kT}{D}$, where $D$ is the diffusion coefficient. Now, I was wondering which conductivity is meant in that case? If anything is unclear, please let me know.


title: What is the difference between a charged rho meson and a charged pion?

text: They both seem to have the same quark content: $$\rho^{+} = u\bar{d} = \pi^{+}$$ and $$\rho^{-} = \bar{u}d = \pi^{-}$$ What is different about the two?


title: Evaluation of band gap from transmittance

text: How can I evaluate the band gap of my ZnO thin film. Thickness=d=80 nm I’m aware of alpha=-1/d*log(T) But I can’t fit a line to my data. And: How can I smoothen my data on about 346 nm? Do you have any other idea for calculation of band gap from Transmittance and Absorbtion of light by thin films?               h=4.13566733e-15;            %# eV.s      c=299792458;                 %# m/s                                   %# landa in (nm)      nu=c./(landa*1e-9);          %# 1/s      alpha=-1/(50e-7)*log(T50);   %# 1/cm      E=h*nu;                      %# eV      Y=alpha.^2;      plot(E,Y);      Data: 80.txt


title: What does this Lagrangian represents?

text: I came across this expression while doing exercises and I was wondering if it was a 'real' expression. $$\mathcal{L}=\frac{1}{2}\partial_\mu \phi \partial^\mu \phi -\frac{m^2}{2}\phi ^2 +\overline{\psi}(i\gamma^\mu\partial_\mu -M)\psi + i\lambda \phi \overline{\psi}\gamma_5\psi $$ I can recognize the first term is a kinetic term, the second as a potential term and the middle one as the Dirac Lagrangian. The last one is an interaction term, so it involves the interaction between electrons and a spin 0 particle, which one? Does this Lagrangian density represent a specific theory?


title: Neutralino Dark Matter Detection

text: Assuming supersymmetry exists and a neutralino is stable, it's often seen as a leading dark matter candidate. What would be expected from the interaction of a neutralino and its anti-particle? Has there been any unambiguous detection of such an interaction?


title: Bose-Einstein condensation in systems with a degenerate ground state

text: I understand that when a system enters the BEC phase a sizable fraction of the total number of particles enters the ground state, until at some point almost all of your particles are in the ground state. What happens in the case of degenerate ground states? Do the particles simply distribute themselves amongst these lowest energy states, or do they actually all go into the _same_ state?


title: How can something happen when time does not exist?

text: I saw this documentary hosted by Stephen Hawkins. And if I didn't get it wrong, it says that there was no time before the big bang, time was created there. So how can anything happen when there is no time (eg: the creation of time and the universe)? Is this what he calls a 'singularity'? Which I could think like this: when there is no time nothing can happen, except during this singularity. Of course I don't know of a law that says 'when there is no time nothing can happen..', I just came up with it, as something intuitive.


title: If the universe is full of dark matter, why is it only 2.73 K cold?

text: people! I am just a physics layman, but I recently watched a documentary about the universe and it was told that   1. the universe is full of dark matter and energy and   2. the universe is empty, so that the average temperature measured by the cosmic microwave background is about 2.73 Kelvin. They also said, that temperature is (also) depending on gravity or mass respectively. That's why the sun is so hot, since there is huge mass concentrated, and the empty space is cold, since there is no mass. Or is it? So now my question(s): IF there is almost only dark matter and energy in the universe with HUGE mass effects which explains interstellar movement etc., why is the universe only 2.73 K cold? Shouldn't it be warmer? Isn't the 2.73 Kelvin the temperature you get, if you calculate the temperature without dark matter and energy?


title: What is magnetic conductivity?

text: I searched google for the meaning of magnetic conductivity but couldn't figure out what is it? electric conductivity is usually means that there is the electric field parallel to the interface is continuous between to interfaces, what about magnetic conductivity material ?


title: Time it takes for Temperature Change

text: I have just been thinking about it for a while and would like to see if there is a way to do this problem. ## The Setup: We have an insulated cup with mass $m_c$ and specific heat $s_c$. The cup is at temperature $t_c$. We put a liquid with mass $m_l$ and specific heat $s_l$ in the insulated cup. The liquid is at temperature $t_l$. Assume the freezing/boiling point of the liquid is extreme enough so that it does not undergo any phase change when in the cup. Also, assume that these two objects are the only things affecting the temperatures of the liquid and the cup. (that is, ignore the atmospheric temperature) ## The Classic: We can easily find the equilibrium _temperature_ by using the familiar $Q=mc \triangle t$ and summing, setting the sum equal to zero. ## The Catch: Is there a way we can find the equilibrium _time_? Of course, since we are given definite masses, specific heats, and temperatures for all the objects in question (in the form of variables) it seems that the answer should be expressed as a function of all those (with the addition of possible some unnamed constants). Is this known? If not, how could you find it?


title: What shape is needed to contain a blade made of plasma?

text: # Although this may stray into the subject of fiction, this question requires physics expertise. If one were able to create a strong enough magnetic field to contain a blade of plasma, what shape would be needed to contain it in a loop?


title: The interference of waves and factors that affect cancellation?

text: If you had two repeated disturbances on the surface of a water, I know interference will occur. However, if I move the two sources of disturbances closer together, why would the 'gaps' between each wave increase? Thanks in advance for your help!


title: How can a glass rod become charged if it is an insulator?

text: I was reading some of the other questions, and I found this one about a glass rod and how it gains a net charge when rubbed with a silk scarf. I learned from working in a shop one summer that most solids are insulators, because their electrons are tightly bound, so it is hard to knock them off. Why would such a simple motion (like a moving scarf) knock electrons from an insulator (I looked it up and glass is an insulator)?


title: why no derivation in scientific papers?

text: Is there a reason why scientific papers rarely show derivation of an equation? I fail to see the any advantages ( besides savings latex hours for the authors) of this approach which is the standard way. Do editors get to see full derivations when one send papers to them? Why so secretive? Do these scientists live in the dark ages?


title: Do any stars exist apart from galaxies?

text: Do any stars exist apart from galaxies? Do "maverick" stars exist?


title: Representations of Lorentz Group

text: I'd be grateful if someone could check that my exposition here is correct, and then venture an answer to the question at the end! $SO(3)$ has a fundamental representation (spin-1), and tensor product representations (spin-$n$ for $n\in\mathbb{Z})$. $SO(3)$ has universal covering group $SU(2)$. The fundamental representation of $SU(2)$ and its tensor product representations descend to projective representations of $SO(3)$. We call these representations spin representations of $SO(3)$ (spin-$n/2$ for $n\in \mathbb{Z}$). The complex vector space $\mathbb{C}^2$ has elements called spinors, which transform under a rotation $R$ according to the relevant representative $D(R)$. The natural generalisation of a spinor is called a pseudotensor, and lives in the tensor product space. We can repeat the analysis for the proper orthochronous Lorentz group $L_+^\uparrow$. We find that the universal covering group is $SL(2,\mathbb{C})$ and we get two inequivalent spin-$1/2$ projective representations of $L_+^\uparrow$, namely the fundamental and conjugate representations of $SL(2,\mathbb{C})$. Now when we pass to the full Lorentz group, somehow the projective representations disappear and become genuine representations. Why, morally and mathematically, is this? If it's possible to give an answer without resorting to the Lie algebra, and just working with representations of the group I'd be delighted! Many thanks in advance.


title: Why does this state have a Schmidt rank of 1?

text: A system is entangled if and only if the Schmidt rank is greater than 1. Why does this $$\left[\frac{1}{\sqrt{2}}\left(\left|0\right\rangle+\left|1\right\rangle\right)\right]\otimes\left[\frac{1}{\sqrt{2}}\left(\left|0\right\rangle+i\,\left|1\right\rangle\right)\right] = \frac{\left|00\right\rangle+i\,\left|01\right\rangle+\left|10\right\rangle+i\,\left|11\right\rangle}{2}$$ have a Schmidt rank equal to one? or equivalently why is this not entangled? ($\otimes$ denotes the tensor product)


title: How is bound charge and free charge possible?

text: I am studying Introduction to electrodynamics by Griffiths and I came along a concept I cannot seem to understand properly. The concept of free charge AND bound charge. I do not understand how we can have both. I understand bound is created by the presence of an E field and a dielectric. So where does the free charge come from if we are inside a dielectric ? Here is a quote from the book talking about it > In Sect. 4.2 we found that the effect of polarization is to produce > accumulations of bound charge, Pb = - V . P within the dielectric and CYb = > P . n on the surface. The field due to polarization of the medium is just > the field of this bound charge. We are now rejldy to put it all together: > the field attributable to bound charge plus the field due to everything else > (which, for want of a better term, we call free charge). The free charge > might consist of electrons on a conductor or ions embedded in the dielectric > material or whatever; any charge, in other words, that is not a result of > polarization.