title: What trajectory has the Cosmic Microwave Background radiation taken to get to earth?

text: I have a few related questions:   * Where is the CMB coming (emitted/reflected/remitted) from?    * When CMB hits the earth, is that the first thing those photons hit since they were emitted 400 thousand years after the big bang?   * Why isn't the CMB at the edges of the universe? Why is it flying around in the middle? Has the trajectory of the photons been bent by masses in the universe until it bends back inward? Or is the theory that the universe wraps around on itself?


title: Entropy and the uncertainty principle

text: According to the second law of thermodynamics, the total entropy of the Universe must always increase after any interaction (as I understand). So in the hydrogen atom, the electron has a high probability of being found in some region around the nucleus specified by its quantum number. My question is, if the hydrogen electron is probed, does this action increase the entropy of the atom's surroundings? If so, does this mean that the process of narrowing down the position of the electron (within the possible limits) increases the region over which one or more of the particles in the environment is likely to be found (i.e. does the increase in entropy imply that the 'probability cloud' of a particle expands over a larger region?)


title: What does a constant signal sound like?

text: Say I was sampling a sound incorrectly and it produced a constant signal as below: ![enter image description here](http://i.stack.imgur.com/PzXjp.png) What would this signal sound like? In Matlab, it plays nothing. Is this correct?


title: Why does a halocline form?

text: I was doing an experiment over the last couple of days to try to crystallize alum using a thermal gradient. The idea was that solute at the bottom of my container would be dissolved at a higher percentage (I used a coffee mug warmer) because of the solubility curve... http://nshs-science.net/images/alum_solubility_chart.gif Then because the local solution is warmer than the solution above it, it would become buoyant, rise to the top, cool, become super saturated, and then deposit its excess solute preferably where I want it. This setup worked fine for me with epsom salt and ordinary table salt. But in this case, using alum, a salinity gradient, or halocline formed and the solution at the bottom, though warm enough to be uncomfortable to the touch, did not become buoyant. My container was only 9 inches tall. At the top, the solution was almost room temperature. I remembered this effect when reading about solar ponds: http://en.wikipedia.org/wiki/Solar_pond But typically a solar pond is several feet deep. My question is, why does a halocline form in the first place? Is this simply the action of gravity on the solute? Is alum's greater solubility the reason I observed this effect in only 9 inches of container height?


title: Observable universe does it account for farther away galaxies measure of the observable universe?

text: From our planet we can only see so far on any direction. If a galaxy farther out than ours had a planet would its view extend beyond our measurements?


title: How is ionization explained in Quantum Mechanics?

text: I remember in my highschool chemistry classes, they taught that an atom can be ionized when it loses a valence electron and becomes positively charged. In quantum mechanics, if electrons aren't really particles orbiting the nucleus, what is it that happens during the ionization (is the wavefunction of the electron destroyed by the ionization process)?


title: Why is Mosely screening in heavy atom K-shell 1 unit?

text: The Wikipedia article on Moseley's law seems to show that the screening of heavy atoms is by 1 electron charge _exactly_ (in the limit of large Z, experimental precision, within nonrelativistic limits, and). But why is this exactly one unit? The other K-shell electron is not screening exactly one unit, and this seems to be a conspiracy of other electrons. I suspect it is because of an unappreciated hole-picture of deep holes in heavy atoms (electrons missing in deep shells), and I will describe this theory briefly. If you remove an electron from close to the nucleus, the electron-hole behaves as an object with _positive charge_ and _negative mass_ (this is why it orbits the nucleus that it is repelled by). The state is not a vacuum quite, because of the presence of other electrons, but the rigidity of the Fermi liquid near the nucleus of a heavy atom means that the hole behaves as a single particle. This single-particle behavior is in the potential background of the nucleus and the other electrons, and it is possible that the result can give an exact 1 unit screening. I developed the formalism a little bit to see what the form should be, but I did not see any reason for 1 unit screening. Perhaps there is none, but it looks to be more than a coincidence.


title: Do centripetal and reactive centrifugal forces cancel each other out?

text: In order for a body to move with uniform velocity in a circular path, there must exist some force towards the centre of curvature of the circular path. This is centripetal force. By Newton's Third Law, there must exist a reactive force that is equal in magnitude and opposite in direction. This is the reactive centrifugal force. My question is simple, and it is probably the result of lack of common sense but here it goes: In uniform circular motion, why don't these forces simply cancel each other out? If they did, how would we know they exist in that situation? When I swing a rock tied to a rope, I feel the centrifugal force, but not the centripetal force. In this situation how can the reactive force be greater than the force itself?


title: Predicting the direction of sound incorrectly?

text: When i was sitting outside my house yesterday, I heard a strange sharp sound. I could not point out it's exact direction but apparently looked like coming from left but later my friend told me that an accident had occurred (due to which the strange sound came) and it totally came from different direction? How did this happen? Why can not our ears point out the exact direction of sound sometimes?


title: Energy-Momentum Tensor of a Gravitational Wave

text: In radiation gauge ($\gamma=0$), the Einstein field equation in vacuum for a perturbation $\gamma_{\mu\nu}:=g_{\mu\nu}-\eta_{\mu\nu}$ is given by $$ \boxed{ \partial^\alpha\partial_\alpha \gamma_{\mu\nu}=0 }$$ where we use the Lorentz gauge condition $ \partial^\mu\gamma_{\mu\nu}=0$. This equation is equal to the Einstein field equation to linear order in $\gamma_{\mu\nu}$. However, it is no longer equal to the Einstein field equation to _second_ order in $\gamma_{\mu\nu}$. To fix this, we re-define our correction to the metric: $$ \eta_{\mu\nu} + \gamma_{\mu\nu} \mapsto \eta_{\mu\nu}+\gamma_{\mu\nu}+\tilde{\gamma}_{\mu\nu} $$ Where $\tilde{\gamma}_{\mu\nu}$ is a an additional dynamical field, in addition to $\gamma_{\mu\nu}$, which, in order to satisfy the Einstein field equations to second order in $\gamma_{\mu\nu}$ (we assume here that $\tilde{\gamma}_{\mu\nu}$ is of the same order of magnitude as $(\gamma_{\mu\nu})^2$ and so mixed terms proportional to $\gamma_{\mu\nu}\tilde{\gamma}_{\mu\nu}$ are proportional to third order in $\gamma_{\mu\nu}$ or higher.), $\tilde{\gamma}_{\mu\nu}$ must obey the following equation: $$ ^{(1)}\tilde{R}_{\mu\nu}-\frac{1}{2}\eta_{\mu\nu}\eta^{\alpha\beta}\,^{(1)}\tilde{R}_{\alpha\beta} =8\pi\underbrace{\left[\frac{-1}{8\pi}\left(\,^{(2)}R_{\mu\nu}-\frac{1}{2}\eta_{\mu\nu}\eta^{\alpha\beta}\,^{(2)}R_{\alpha\beta}\right)\right]}_{t_{\mu\nu}} $$ where $\tilde{R}_{\mu\nu}$ is the Ricci tensor containing only terms with the correction $\tilde{\gamma}_{\mu\nu}$ and $R_{\mu\nu}$ is the Ricci tensor containing only terms with the original perturbation $\gamma_{\mu\nu}$, and the left-hand superscript $^{(j)}R_{\mu\nu}$ means exclusively only terms portional to the $j^{th}$ power of the corresponding variable ($\gamma$ or $\tilde{\gamma}$) are contained in that term. Then we say even though we have vacuum, $t_{\mu\nu}$ represents the energy- momentum tensor of a gravitational wave travelling through that vacuum. I have a few questions:   1. Is my description so far accurate?   2. If a gravitational wave is passing throughout some part of spacetime, why do we ascribe this segment of spacetime as vacuum to begin with? My logic would say this part of spacetime should have $T_{\mu\nu}$ codify the energy-momentum density of said wave, hence, the vacuum Einstein equations should not hold, even in linear order.   3. What happens if we don't use the linear approximation for the Einstein equation? How do we see that there is some self-energy contained in the metric of the vacuum without in the exact equation?   4. What happens if we go to third order?   5. How do you prove that $t_{\mu\nu}$ is conserved, that is, that $$ \partial^{\mu}t_{\mu\nu}=0$$ I tried to just differentiate the term from here but things went south. Should I be using the radiation gauge condition? Should I use any other trick?   6. At any particular point in spacetime, should I use $\boxed{\eta_{\mu\nu} + \gamma_{\mu\nu}}$, $\boxed{\eta_{\mu\nu} + \tilde{\gamma}_{\mu\nu}}$, or $\boxed{\eta_{\mu\nu}+\gamma_{\mu\nu}+\tilde{\gamma}_{\mu\nu}}$ as the metric? It seems like it must be the second option, since $\gamma_{\mu\nu}$ has already been "used up" in the other side of the equation.   7. If I have a radiating source, for example, a spinning rod, then as far as I understand, the procedure is to first see what kind of perturbation to flat space $\gamma_{\mu\nu}$ it generates, which is proportional to the second time derivative of its mass quadrupole tensor divided by the distance from it. What happens if I then plug _that_ into $t_{\mu\nu}$? How is that different from the actual $T_{\mu\nu}$ of the spinning rod?


title: Direct and Reciprocal lattice common point

text: Does a reciprocal lattice have a common point with its direct lattice? (possibly at the origin $\vec{K}=\vec{R}=\vec{0}$)


title: Is the Lagrangian of a quantum field really a 'functional'?

text: Weinberg says, page 299, _The quantum theory of fields,_ Vol 1, that > _The Lagrangian is, in general, a functional $L[\Psi(t),\dot{\Psi}(t)$], of > a set of generic fields $\Psi[x,t]$ and their time derivatives [...]_ My hang-up concerns the use of the word _functional._ For example, the free field Lagrangian density for a quantum scalar field is: $$-\frac{1}{2} \partial_{\mu}\Phi\partial^{\mu}\Phi-\frac{m^2}{2} \Phi^2$$ Since the terms appearing here contain operators, then the Lagrangian must take values in the set of operators. Yet, I thought that the definition of a "functional", was that it took vectors in a Hilbert space to real numbers, rather than operators. Furthermore, by the above definition, what vectors does the Lagrangian act on? Should the above statement be interpreted: "The 'expectation value of the' Lagrangian is, in general, a functional...", where the expectation value is taken over some field state? I.e., the expectation value of the Lagrangian takes vectors from the Fock space of states to real numbers?


title: Covariant derivatives

text: I need correctly define covariant derivatives on the coset space $G/H$, where a group $G \equiv \\{X_i, Y_a\\}$ ($X$ and $Y$ are generators) have a subrgroup $H \equiv \\{X_i\\}$ Lie algebra of $G$ has the following structure: $$[X, X] \sim X, \quad [X, Y] \sim Y, \quad [Y, Y] \sim X + Y.$$ The coset space $G/H$ is parametrized by coordinates $\xi^a$ corresponding to the generators $Y_a$. In the exponential parametrization an element of this coset has the form $$G/H: \quad \Omega = exp\\{i\xi^a Y_a\\}$$ The ordinary derivatives $\frac{\partial}{\partial\xi^a}$ of $\psi(\xi)$ ($\psi(\xi)$ - field defined on $G/H$) do not transform covariantly. I was told once that it is quite easy to define covariant derivatives using the Cartan form $\Omega^{-1}d\Omega$. I looked through the bunch of the mathematical books about Lie groups and Lie algebras but I can't find a clear description of the procedure. Can anyone advise me a proper book (paper) or, if it's not take much time, tell it here? Thank you. Sorry for my English)


title: Systematic approach to deriving equations of collective field theory to any order

text: The collective field theory (see nLab for a list of main historical references) which came up as a generalization of the Bohm-Pines method in treating plasma oscillations are often used in the study of large N asymptotics (e.g. of matrix models). I have seen few cases where the equations for collective fields are derived; the equations are obtained using approximations and look more systematically understood by other methods than those available at dawn of collective field theory (i.e. since the works of Sakita and Jevicki). Is there now a systematic method to derive collective field theory equations to any order in 1/N ? Also if the Yaffe's methods concerning coherent state methods in large N asymptotics are ever used in collective field theory ? The subject looks interesting for doing some work (I have some background in coherent state techniques), but I do not know what the current state of advance is. EDIT: added a bounty to renew interest.


title: Does entropy measure extractable work?

text: Entropy has two definitions, which come from two different branches of science: thermodynamics and information theory. Yet, they both are thought to agree. Is it true? Entropy, as seen from information theory, measures our ignorance about a system. Entropy, as seen from thermodynamics, measures the amount of extractable work. One is epistemological, the other is ontological. Yet, they both can agree, if entropy really measures the amount of extractable work, _given one's knowledge_ about the system. But, is this true? Does Shannon's entropy computed on the probability distribution of a physical system express the amount of work we can obtain from it? How do increases in our knowledge increase the amount of extractable work? Is Landauer principle powerful enough to establish the connection?


title: The Chern-Simons/WZW correspondence

text: Can someone tell me a reference which proves this? - as to how does the bulk partition function of Chern-Simons' theory get completely determined by the WZW theory (its conformal blocks) on its boundary.. * * *   * I see that Toshitake Kohno's book seems to do a thorough study of this but its not beginner friendly at all! (...so I am kind of looking for something that will give a "simpler" proof and give me a stepping stone into that book...)   * I am often told that Witten's QFT and Jones polynomial paper proves this but I am hard pressed to find it there. If anyone can spot the proof there, it would help still..


title: Is the symmetrisation postulate unnecessary according to Landau Lifshitz?

text: The symmetrisation postulate is known for stating that, in nature, particles have either completely symmetric or completely antisymmetric wave functions. According to these postulate, these states are thought to be sufficient do describe all possible systems of identical particles. However, in Landau Lifshitz Quantum Mechanics, in the first page of Chapter IX - Identity of Particles, he comes to the same conclusion without needing to state any ad-hoc postulate. It goes like this: Let $\psi(\xi_1,\xi_2)$ be the wave function of the system, $\xi_1$ and $\xi_2$ denoting the three coordinates and spin projection for each particle. As a result of interchanging the two particles, the wave function can change only by an unimportant phase factor: $$ \psi(\xi_1,\xi_2)=e^{i\alpha}\psi(\xi_2,\xi_1) $$ By repeating the interchange, we return to the original state, while the function $\psi$ is multiplied by $e^{2i\alpha}$. Hence it follows that $e^{2i\alpha}=1$ or $e^{i\alpha}=\pm1$. Thus $$ \psi(\xi_1,\xi_2)=\pm\psi(\xi_2,\xi_1) $$ Thus there are only two possibilities: the wave function is either symmetrical or antisymmetrical. It goes on by explaining how to generalize this concept to systems with any number of identical particles, etc. Im summary, no symmetrisation postulate was ever stated in this rationale. Is "shifting by an unimportant phase factor" a too strong requirement for ensuring identity of particles?


title: Centrifugal Force

text: When a body is in circular motion, realistically, it experiences only the centripetal force, created by gravitational pull, tension etc. , which gives it acceleration towards the centre. Now, assuming we were to fill a bucket with water, and whirl it in the vertical plane. There will be a certain velocity when the water doesn't fall out even when the bucket is upside down. At that instant, if we were to write down the forces that act on the water, they would be the Normal reaction by the bucket, and it's weight. This means that the net force is downwards. Why doesn't the water fall down, since there's no force in the upward direction? Centrifugal force is a psuedo force, so it shouldn't be applicable here, as the water won't wall down even if I watch from a non inertial frame. How can one explain this phenomenon? Also, say we were to fit a ring in a rod, and rotate the rod, the ring will eventually slip and fall off. Why does this happen if there is no outward force?


title: Resistor connections

text: I can't figure out what kind of connection it would be if i put resistors like this![enter image description here](http://i.stack.imgur.com/nkaeR.png) Can someone please explain how they are connected, like is R1 and R2 in series and R4 and R5 in series and then I make a parallel of R12 and R45 and R3? And what if I replaced R3 with a capacitor would it be connected in parallel with these resistors so I could get the charge on the capacitor by doing $q = C * U$


title: Transpose Map Positive, But Not Completely Positive?

text: I am reading _Introduction to Quantum Computing_ by Kaye, Laflamme, and Mosca. Here is a question I am struggling with: > Exercise 3.5.6: Prove that the transpose map, which maps $\rho \mapsto > \rho^{T}$ is positive1, but not completely positive2. Now, positivity is defined in terms of the inner product, i.e. $\rho$ is positive iff $\forall v\in \mathcal{H}$, $\left<v, \rho v\right> \ge 0$, but "transpose" is defined in terms of an operation on matrices. So I was able to get this under the assumption that $\rho^{T} = \rho^{*}$, but not under the weaker assumption the $\rho$ is simply positive. Is this true even if I don't assume $\rho^{T} = \rho^{*}$? As for showing that the transpose map is not completely positive, I frankly do not know what I'm doing and am asking for any and all help you can give me. My attempt is given below, though it is not worth reading: Let $\rho\otimes \gamma$ be a positive map, so that $\forall u\otimes v \in \mathcal{H}_A \otimes \mathcal{H}_{B}$ $$ \left<u\otimes v, \rho u \otimes \gamma v\right> := \left<u, \rho u\right>\left< v, \gamma v \right> \ge 0. $$ Then the transpose map tensored with the identity takes $\rho \otimes \gamma$ to $\rho^{T}\otimes \gamma$, and we have (again assuming $\rho^{T}=\rho^{*}$) $$ \left<u\otimes v, \rho u \otimes \gamma v\right> := \left<u, \rho^{T} u\right>\left< v, \gamma v \right> = \overline{\left< u, \rho u\right>}\left<v, \gamma v \right> =\frac{\overline{\left< u, \rho u\right>}}{\left< u, \rho u\right>}\left<u, \rho u\right>\left< v, \gamma v \right>. $$ So $\frac{\overline{\left< u, \rho u\right>}}{\left< u, \rho u\right>}$ must be $\ge 0$. . .(at this point I am stumped.) 1 "Positive" in this context means "maps positive operators to positive operators. 2 They define completely positive as follows: A map is completely positive iff it is positive, and in addition, when tensored with the identity operation, they still map positive operators to positive operators.


title: Is nuclear fusion possible on planets in our Solar system?

text: Apart from thermo-nuclear reactions in the Sun, (and in hydrogen bombs and other methods developed by scientists) does nuclear fusion occur naturally in the planets? If not, why? Also, do the Sun and the Outer planets - Jupiter through Neptune - have rocky cores similar to the Earth (consisting of iron, nickel, silicon etc) in addition to the gases? I have read that there is not much of hydrogen or helium in the inner planets because of their low escape velocities and the solar winds from the Sun. Hence only the heavier elements remained.


title: Crystal magnetic response only skin deep?

text: The Hamiltonian for a single electron in a magnetic field reads $$H=\left(\frac{{\bf p}^{2}}{2m_{e}}+q_{e}\phi\right)+\mu_{B}\left({\bf \hat{L}}+g{\bf \hat{S}}\right)\cdot{\bf B}+\frac{e^{2}}{8m_{e}}\left({\bf B}\times{\bf r}\right)^{2}$$ where symmetric gauge ${\bf A}=\frac{1}{2}{\bf B \times r}$ is used and where the last two term are responsible for paramagnetism and diamagnetism respectively. The many-body Hamiltonian, weak electron-electron interaction considered, reads $${\cal H} = \sum_{\sigma}\int d{\bf r}\,\left[-\left(\psi_{\sigma}^{\dagger}({\bf r})\frac{\hbar^{2}}{2m^{*}}{\nabla}^{2}\psi_{\sigma}({\bf r})\right)+\boldsymbol{\mu}({\bf r})\cdot({\nabla}\times{\bf A})+\hat{n}_{e}({\bf r})\frac{e^{2}}{2m^*}{\bf A}({\bf r})^{2}\right]$$ where $\boldsymbol\mu({\bf r})=\sum_\sigma \mu_B \psi^\dagger({\bf L}+g{\bf S})\psi $ is local magnetic moment density of electron of total electron angular momentum and define $\mathbf M_\text{para} \equiv \langle \boldsymbol \mu\rangle$ Now that we vary $\bf A$, we should obtain the response current $${\bf j}={\nabla}\times{\mathbf M_\text{para}}+\frac{n_{e}e^{2}}{m^*}{\bf A}$$ It is rather strange to me that (i) the last term is essentially the same as the one in London's equation, which gives a penetration depth $\lambda$, and that (ii) the first term is non-zero only on the crystal surface if $M$ is uniform deep inside, which I think is just the magnetization current we saw in textbooks. If (i) is true, one may conclude that magnetic response for all crystals (neglect interactions) happens in a thin layer near to the surface (a few nanometer) since magnetic field can only penetrate that far. The length scale that electron density drops to zero is a few angstroms, so we are able to neglect the first term in an appropriate region.


title: Measuring the velocity of a pendulum using the width of the bob

text: I did an experiment on pendulum motion and in the method it says use the following equation to measure the velocity of the pendulum: $$v=\frac{W}{t}$$ where $W$ is the width of the pendulum bob and t is the time. Why is this? Is it correct? Edit: the time is for the bob to cross the photogate.


title: Rotating mirror - Foucault's measurement of light speed

text: Some time ago I came across a secondary web source on measurement of light speed in water made by Foucault around 1850. I append its redrawn scheme below (light is reflected from the rotating mirror to the fixed one and from the later again to the rotating one which slightly rotates in the meantime so the light ends in the direction labeled **air** rather than back in the light source). When Foucault inserted a container filled with water between the mirrors then light was reflected back at larger angle ( **water** ), because light is slower in water. Which velocity is exactly measured in this experiment? Phase velocity (light source is continuous, similar to infinite monochromatic wave), group velocity (which usually applies to a wave packet - is such a packet somehow created by rotation of mirror?) or another one? ![enter image description here](http://i.stack.imgur.com/ichYD.png) **Edit** (based on the answer by miceterminator): How would the result (the angle) change in case of _negative_ group velocity (which, as far as I know, is possible)?


title: Why does earth cool?

text: This question is boggling me for some time. We know that heat can be transferred from matter to matter and heat is nothing more than tiny atoms vibration intensity (correct me if I'm wrong). But space is a vacuum, and so arises the question: Can you heat vacuum? :)) How can it be that earth loses it's heat to space itself? Thanks


title: Neutral current: terminology

text: In particle physics, where does the term 'neutral current' originate? An example would be an electron exchanging a Z boson with another electron. I understand that the Z boson itself is neutral, but surely 'current' refers to electron and its associated amplitude in this case?


title: Did I see two Milky Ways next to each other?

text: I am from the northern hemisphere and as far as I remember, the Milky Way was "single lane", just one stripe. At least, this is what I recall the Milky Way to look like when seeing it from near Irkutsk, Siberia, in late August 1997. Yesterday I went to Blackbutt, Australia (somewhere near 27° South, 152° East) and the Milky Way seemed to have a large black stripe in the middle. I was wondering, if these were two spiral arms of the galaxy that we are part of.


title: X-ray diffraction from Metals

text: Is X-ray crystallography is possible for metals? Or not due to absorption?


title: Classical blackbody radiation 'solution'

text: I never understood how the equipartition theorem was applied electromagnetic waves inside the metallic blackbody. As hyperphysics puts it (http://hyperphysics.phy-astr.gsu.edu/%E2%80%8Chbase/mod7.html) > The classical view treats all electromagnetic modes of the cavity as equally > likely because you can add an infinitesmal amount of energy to any mode But the existence of the modes is _conditioned_ to the existence of certain movements by the particle, so I don't see it as a "degree of freedom". For example, for a high frequency mode to exist there should be electrons jiggling with a certain frequency. I expected some sort of statistic on electrons speed and acceleration for a given temperature, like from the Maxwell-Boltzmann distribution, to derive the expected electromagnetic radiation spectrum for the blackbody. 1) Why are the electromagnetic modes are considered "degrees of freedom" if their existence is conditioned to the motion of the electrons? If they actually aren't, please explain it 2) Does an alternative (non cavity + modes) approach exist? I was thinking fluctuations of charge density on the surface of a solid metal sphere due to temperature.


title: How "big" objects can WISE and NEOWISE detect?

text: I mean WISE is monitoring near-Earth objects, but cannot see the latest Russian meteor and others. Why can it not detect small objects? What is the limit of it's infrared detectors?


title: Question on the hydrostatic paradox

text: The hydrostatic paradox, states that the > height of the water in any container is independent of the shape of the > container. This does make sense since it is only the height of the fluid relative to ground which determines the pressure, and it is the pressure equilibrium that decides the shape of the fluid. But then consider the (B) container in the figure below.   ![enter image description here](http://i.stack.imgur.com/DoLKb.png) The pressure at a particular depth say, $h$ is same as in the case of the cylindrical container. But in the cylindrical case (A), any element of fluid is in equilibrium because the difference in pressure of the fluid element above it and that below it exactly balances the weight of the element. Extending this argument to (B), for any element at the edge, the pressure of the fluid above is the same as in the case of the cylindrical container, but below it is the surface of the container. This container, is in equilibrium with the atmospheric pressure outside, and therefore, would be exerting pressure equal to the atmospheric pressure onto the fluid within, which would not be enough to maintain the equilibrium of the fluid element. **How then can the equilibrium of the fluid element at the edge be explained?**


title: Generators of Poincare Groups

text: How can I determine the generators of the Poincare Group, $P(1,3)$ explicitly? Here $P(1,3)$ means a matrix Lie group.


title: $\mathcal{N}=2$ supersymmetry and $SU(2)$ holonomy

text: I was reading this Phys.SE question. I was unable to understand how an $SU(2)$ holonomy would produce $\mathcal{N}=2$ in four dimensions. Could anyone shed some light on this?


title: What is the logic behind the Fahrenheit scale?

text: The Fahrenheit scale is defined by fixed points on the scale. What interests me is the apparent arbitrary chosen numbers in these fix-points. First wikipedia wites from 32 to 212. -and later in the _History_ part we have from 0 to 32 to 96(based on a quadrupling of 7.5 to 22.5 to 60). I am stille confused. Does any of the numbers make sense in an other numeral system? Did Rømer or Fahrenheit use their lucky numbers or what is going on? **Edit:** I am not interested in what solutions or objects, that is used for the fix-points. I am asking what the logic is behind the chosen numbers?


title: Wien's fifth power Law and Stephan Boltzmann's fourth power laws of emissive power

text: Wien's fifth power law says that emissive power is proportional to the temperature raised to the fifth power. On the other hand, the Stefan–Boltzmann law says emissive power is proportional to the temperature raised to the fourth power. How can both of these be true?


title: The number of degrees of freedom of a monatomic gas

text: Suppose that I have a monatomic gas sample consisting of $N$ atoms (e.g., $N$ argon atoms); thus there are no vibrations or rotations. How many degrees of freedom does the system have?   * **Does the system have $3N$ degrees of freedom?** Each of the $N$ atoms can translate in $x$, $y$, and $z$, so $3N$ degrees of freedom seems reasonable. This seems to be supported by this website on molecular dynamics (MD), which states: "If there are $N$ atoms and $N_c$ internal constraints, then the number of degrees of freedom is $N_f = 3N - N_c$." It seems that a monatomic gas such as argon has no internal constraints, so the website seems to be suggesting $3N$ degrees of freedom.   * **Does the system have $3N - 3$ degrees of freedom?** Page 64 of Frenkel & Smit's _Understanding Molecular Simulation: From Algorithms to Applications_ (2nd edition) seems to suggest that the system has $3N - 3$ degrees of freedom: "In practice, we would measure the total kinetic energy of the system and divide this by the number of degrees of freedom $N_f$ ($ = 3N - 3$ for a system of $N$ particles with fixed total momentum)." The only thing I am unsure of is whether a monatomic gas in general has fixed total momentum: probably not, unless there is solely center-of-mass motion.


title: List of all past comet passes and area it was visible?

text: Is there a database/website which shows all comets which have passed Earth and have been viewed? Is there any country specific viewing database as well? I would like to do a search on the lines of : Show me all comets were visible from India during 1950 to 1960? Any searches on Google have not returned anything, so querying this exchange site.


title: Why does scattering depend on spin?

text: I'm reading about giant magnetoresistance (GMR), and the most important feature of this phenomenon is the spin dependance of the electron scattering inside a magnetised lattice. However, I don't quite understand why electrons with spin parallel to the magnetisation (hence, to the majority-spin direction of the electrons) scatter less than those with spin antiparallel to the magnetisation (so, parallel to the minority-spin). I'd appreciate if someone could clear this up for me.


title: Varying an action (cosmological perturbation theory)

text: I am stuck varying an action, trying to get an equation of motion. (Going from eq. 91 to eq. 92 in the image.) This is the action $$S~=~\int d^{4}x \frac{a^{2}(t)}{2}(\dot{h}^{2}-(\nabla h)^2).$$ And this is the solution, $$\ddot{h} + 2 \frac{\dot{a}}{a}\dot{h} - \nabla^{2}h~=~0. $$ This is what I get $$\partial_{0}(a^{2}\partial_{0}h)-\partial_{0}(a^{2}\nabla h)-\nabla(a^{2}\partial_{0}h)+\nabla^{2}(ha^{2})~=~0.$$ I don't really see my mistake, perhaps I am missing something. (dot represents $\partial_{0}$) It is this problem (see _Lectures on the Theory of Cosmological Perturbations,_ by Brandenburger): ![enter image description here](http://i.stack.imgur.com/hcli1.jpg)


title: Pseudo force in rotating frames

text: A bug of mass $m$ crawls out along a radial scratch of a phonographic disc rotating at $\omega$ angular velocity. It travels with constant velocity $v$ with respect to the disc. What are the forces (and their directions) which act on the bug? I have tried thinking in this way: If I look from the point of view of the bug, that is the non-inertial reference frame, forces acting on the bug are friction, centrifugal force and the Coriolis force. Since the bug crawls outward radially, the friction should act radially inward and the centrifugal force acts radially outward. I am totally confused regarding the direction of the Coriolis force. Does it act on the bug perpendicular to the radial direction? If so, then while considering the effective force on the bug (from non- inertial point of view), should I break it into horizontal and vertical components? Or can they be written in one equation?


title: Franck Condon Principle and Born Oppenheimer approximation

text: My question here is purely fundamental. I am confused with the concept in Franck Condon (FC) principle and Born Oppenheimer (BO) approximation. The FC principle is in accordance with the BO approximation or not? In FC there is a correlation between electronic states and the nuclear motion. So, BO approximation is broken. So, can we say FC is an example of breaking of BO approximation? Secondly, in the case of mega electron volt ions, is BO approximation valid? In this case, the velocity of electrons are comparable to the velocity of the nucleus!


title: Why isn't the Bekenstein-Hawking Entropy considered the quantum gravitational unification?

text: Based on the Bekenstein-Hawking Equation for Entropy, hasn't the relationship between quantum mechanics and gravity already been established.


title: Is the image formed at the retina messy?

text: The following passage has been extracted from Newton's book Opticks: > ...So if PR [in Fig.at the top] represent any Object without Doors, and AB > be a Lens placed at a hole in the Window-shut of a dark Chamber, whereby the > Rays that come from any Point Q of that Object are made to converge and meet > again in the Point q; and if a Sheet of white Paper be held at q for the > Light there to fall upon it, the Picture of that Object PR will appear upon > the Paper in its proper shape and Colours....![enter image description > here](http://i.stack.imgur.com/2TJu7.jpg) In like manner, when a Man views > any Object PQR, [in Fig. at the bottom] the Light which comes from the > several Points of the Object is so refracted by the transparent skins and > humours of the Eye, (that is, by the outward coat EFG, called the Tunica > Cornea, and by the crystalline humour AB which is beyond the Pupil mk) as to > converge and meet again in so many Points in the bottom of the Eye, and > there to paint the Picture of the Object upon that skin (called the Tunica > Retina) with which the bottom of the Eye is covered.... If we see the bluish sky with the high standing giant tall buildings, from the mountain, and analyse the validity of the above Newton's statements, it appears to be complex.   * Our eye lens is generally very small in comparison with the objects we look, and the diagram in the Newton's book doesn't seem to take concern about it. Isn't it?    * If we consider the above said situation of looking at the sky and a girl in the building, we can't expect all the rays of light of each single point of the girl or the building or the sky to come and converge in the methodical way to form the exact image of the object. The points of the objects can take a shift in the place and make messy image of the object. Isn't it?  This is what I understand with my analysis. So, is the image formed at the retina really messy?


title: What is the frequency of occurrence of stellar classifications off the HR main-sequence?

text: An alternative version of this question would be: "if was to pick a star from the $10^{11}$ or so in our galaxy at random, what are the probabilities of it being various kinds of star?" (and I do mean "in our galaxy" and not "visible in the night sky"). There is a nice table I've seen in various places (e.g wikipedia) which goes a long way towards answering this. It tells me that 0.00003% are 'O' types and 0.13% 'B' and 0.6% 'A' through to the 76% rump of 'M types. Unfortunately there is some small print associated with that table which is that it's only for stars **on the main sequence** (hence its 'M' types are all low-luminosity dwarfs - red giants aren't included - and its 'A's and 'B's are all young giants - white dwarfs aren't included). However, I've no idea what proportion of the stellar population are **off** the main sequence. One in a hundred? One in a million ? What I'd really like to find (or gain enough information to compile) is a 2D version of that table with the same axes as a HR diagram where I can look up the frequency of white dwarfs by looking in the cells in the A-B spectral and the luminosity 0.001 - 0.1 ranges, or the frequency of stars like Betelgeuse by looking in the M, $10^5$ cell. (Of course another issue with the cited table is that it claims to be frequencies in "solar neighbourhood". For example it'd be nice to have different versions specific to stellar populations of, say, globular clusters, the galactic disk and central "bulge". But I'll settle for some pan-galactic numbers initially).


title: Calculating engine starter’s energy use

text: During a discussion on start-and-stop vehicle technology some bloke began pushing the point that re-starting the car uses stored energy from the battery, which needs to be replenished by increased fuel usage once the engine is moving. Well, this is obvious, the question is that of significance of such additional usage, especially in comparison with savings from reduced idle. Common sense tells me that this is splitting hairs: if ICE efficiency outlays put all accessory usage at 2 to 3% of overall energy consumption, which includes AC, and all electricals, then it can’t be that much. More over, this dissertation explores idle-reduction technologies for long- haul commercial trucks, and one of the systems exampled is battery-powered (p. 14) that stores enough energy to power AC or heater overnight, and takes about six hours of charge while driving. Author acknowledges increased fuel consumption due to necessity of higher-amperage alternator, although there are no specific numbers provided. However, the mere existence of such commercial application available on the market leads me to believe that such added load is still better than idling. But, in the interest of science, I need some hard numbers (besides, the fellow just wouldn’t go away). I have some ideas on what to consider, but I’m not well versed in electrical and mechanical engineering, so I do not think I can account for most major factors. Energy use of the starter motor can be calculated by using the amount of current used per start (which would be 2 to 3 seconds) by the motor itself and the solenoid that engages the starter gear onto the flywheel. Both current and power demand can be found in the starter’s specifications, but I am not sure as to how reliable those numbers would be in the real-world application. Then there is fuel consumption of the start itself which is estimated at 10 to 15 seconds worth of idling (and Florida chapter of ASME even calculated six seconds for 6-cylinder engine (in a simple, non-rigorous field experiment), but the original link is broken). Now, how to calculate the increased consumption due to the charging of depleted battery from the starter motor itself, and, additionally, to account for all the accessories that were running while the engine was stopped? Is it a simple matter of using the same number calculated for energy usage of starter motor, and arriving at it by calculating the amount of fuel needed to produce that much extra energy given the losses in the engine itself and in the charging circuit? And, ultimately, how significant are those considerations in the bigger picture?


title: Advanced Heaviside-Feynman formula implies electromagnetic inertia?

text: The Heaviside-Feynman formula (see Feynman Lectures vol I Ch.28, vol II Ch. 21) gives the electric and magnetic fields measured at an observation point $P$ due to an arbitrarily moving charge $q$ $$ \mathbf{E} = -\frac{q}{4 \pi \epsilon_0} \left\\{ \left[ \frac{\mathbf{\hat r}}{r^2} \right]_{ret} + \frac{[r]_{ret}}{c} \frac{\partial}{\partial t} \left[ \frac{\mathbf{\hat r}}{r^2} \right]_{ret} + \frac{1}{c^2} \frac{\partial^2}{\partial t^2} [\mathbf{\hat r}]_{ret} \right\\} $$ $$ \mathbf{B} = -\frac{1}{c} [\mathbf{\hat r}]_{ret} \times \mathbf{E} $$ where $[\mathbf{\hat r}]_{ret}$ and $[r]_{ret}$ is the unit vector and distance _from_ the observation point $P$ at time $t$ _to_ the retarded position of charge $q$ at time $t - [r]_{ret}/c$ (hence the minus signs). This formula is remarkable in that it is completely relational. It does not refer to any external reference frame. The fields at point $P$ only depend on the vector $\mathbf{r}$ from point $P$ to the retarded position of charge $q$ and its first and second order rates of change with respect to local time $t$. Now one can imagine two ways in which the vector $\mathbf{r}$ from point $P$ to charge $q$ can change. One could move charge $q$ and keep point $P$ fixed or one could move point $P$ and keep charge $q$ fixed. The above formula is valid for the former situation giving the fields at a fixed point $P$ due to a moving charge $q$. This is the conventional retarded solution of Maxwell's equations. But what about the latter situation in which the observation point $P$ moves and the charge $q$ is fixed. The relational nature of the formula implies to me that it should still apply in this situation. Perhaps this is the situation in which the _advanced_ Heaviside-Feynman formula is valid given by $$ \mathbf{E} = -\frac{q}{4 \pi \epsilon_0} \left\\{ \left[ \frac{\mathbf{\hat r}}{r^2} \right]_{adv} - \frac{[r]_{adv}}{c} \frac{\partial}{\partial t} \left[ \frac{\mathbf{\hat r}}{r^2} \right]_{adv} + \frac{1}{c^2} \frac{\partial^2}{\partial t^2} [\mathbf{\hat r}]_{adv} \right\\} $$ $$ \mathbf{B} = -\frac{1}{c} [\mathbf{\hat r}]_{adv} \times \mathbf{E} $$ where $[\mathbf{\hat r}]_{adv}$ and $[r]_{adv}$ is the unit vector and distance from the observation point $P$ at time $t$ to the advanced position of charge $q$ at time $t + [r]_{adv}/c$. The advanced Heaviside-Feynman formula is the time-reverse of the conventional retarded formula. This interpretation of the advanced formula, if valid, implies that an accelerated observer inside a fixed charged insulating spherical shell would measure an electric field whose strength is proportional to the acceleration. This implies that an accelerated charge feels a kind of electromagnetic inertial force and thus has an electromagnetic inertia due to the presence of the charged spherical shell. For example imagine an electron with charge $-e$ inside such a fixed charged insulating spherical shell at potential $+V$ volts. Using the above advanced Heaviside-Feynman formula one can calculate that this electromagnetic inertia $m_{em}$ is given by $$m_{em} = \frac{2}{3} \frac{eV}{c^2}$$ For a shell charged to a high voltage $V=1000000$ volts this electromagnetic inertia would be of similar order to the electron's native mass and should therefore be easily observable. It would probably be important that the spherical shell is a charged insulator rather than a conductor because it is assumed that the charges inside the shell remain fixed. Finally, there is a close analogy between Maxwell's equations and Einstein's field equations in the limit of weak gravitational fields. There is a clear gravitational analogue of the advanced Heaviside-Feynman formula. Thus one would expect that a mass accelerated inside a fixed spherical shell of mass should experience a gravitational inertial force in a manner analogous to the above electrical example (just substitute mass $m$, gravitational potential $\phi$ for charge $e$, electrical potential $V$ in above expression). Perhaps this is the origin of inertia as hypothesised in Mach's Principle?


title: Fully destructive interference

text: I'm aware that this is somewhat of a frequently asked question (not only here), and i've dug through respective answer for a few hours before i decided to ask here. None of those answers helped me predict/explain my case though, so here's my problem: Suppose we have this setup: ![sketch](http://i.imgur.com/HoxRTn5.png)   A laser source on the left, two semi-transparent mirrors (grey) and two normal mirrors (black). The red lines designate the laser path. The bottom arm introduces a phase difference of half the wavelength to the respective beam (integer multiples of $\lambda/2$ could be added to the vertical distance without affecting the setup). From what i understand, the dotted lines should both exhibit destructive interference (assuming the semi-transparent mirrors both reflect and let pass exactly 50% of the light). To the question(s): I'm aware that energy conservation should still be at work, but i simply cannot figure out where the energy ends up. Assuming that full interference actually happens, it would either suggest that no work is being done in the laser source (how/why?) or that the energy is going somewhere else (where?). And if the beams don't actually cancel each other out, i'd like an explanation as to why.


title: Can something travel faster than light if it has always been travelling faster than light?

text: I know there are zillions of questions about faster than light travel, but please hear me out. According to special relativity, it is impossible to accelerate something to the speed of light. However, we can still have objects (like photons) going at speed $c$ if they never had to accelerate in the first place, i.e., if they always go at the speed of light. What I was wondering is, would it be possible for us to discover some particle that travels faster than the speed of light? Again, I don't mean something that can be accelerated beyond $c$, but rather something that always goes faster than light. Does that contradict relativity?


title: Why are particles different sizes?

text: Is it correct in saying that a particles size is it's rest energy, and that particles don't actually have size (in the way you get different size objects)? What defines what sizes a particle can be? Why do particles have discrete sizes, and there's not a continuous spectrum of particles varying in size? I ask because I was told that particles mass depends on its size, as the bigger a particle is the more it interacts with the Higgs boson and so the more mass it has. So why are there so few particles with specific masses/sizes?


title: Constants of motion vs. integrals of motion

text: Since the equation of mechanics are of second order in time, we know that for $N$ degrees of freedom we have to specify $2N$ initial conditions. One of them is the initial time $t_0$ and the rest of them, $2N-1$ are initial positions and velocity. Any function of these initial condition is a constant of motion, by definition. Also, there should be exactly $2N-1$ algebraically independent constants of motion. On the other hand, Noether's procedure gives us integrals of motion as a result of variational symmetries of the action. These integrals of motion are also conserved but they are not always $2N-1$ in number. In consequence, we classify the system by their integrability. So, what is the difference between the constant of motion and integral of motion? Why do non-integrable systems have less integrals of motion when they should always have $2N-1$ constants of motion?


title: How Uncertainty Principle, Vacumm fluctuations and Energy Conservation coexist in QFT?

text: Recently I had a debate about the uncertainty principle in QFT that made me even more confused.. Because we use Furrier transforms in QFT we should have an analogue to the usual Heisenberg uncertainty principle in QFT between the 4-vector of space- time coordinates and the conjugate momentum, but I found no reference for that, so is my guess is wrong? We know that there is no universal Hermitian operator for time in QM, even so there is uncertainty principle for time and energy, well, in QFT time is just a parameter same as spatial coordinates, so is there uncertainty principle for energy in QFT? The last question made me confused regarding energy conservation law in QFT: we use this law in QFT during calculations of propagators only (as I remember), it means we was using it with "bare" particles, while we supposing that this particles don't "interact" with vacuum fluctuations, so dose that mean energy conservation law is a statistical law? this brings to my mind vacuum expectation value, that we suppose it as zero for any observer, but it is zero statistically, in the same time we used to use Noeather theorem to deduce that energy is conserved (locally at least, and not statistically). I believe I'm missing something here, can you please advice me?


title: Is the total pressure coefficient always 1 in incompressible flow?

text: I have to do some calculations to get the drag from an experiment with a wake rake. In the equation I have to enter the total pressure coefficient $C_{pt}$, but in my calculations it seems to always be equal to +1. Is this correct?


title: Is the potential energy in a compressed spring a Lorentz invariant?

text: The total energy of an object comes from the time part of the four-momentum, and so isn't a Lorentz invariant. On the other hand, is the potential energy of a compressed spring a Lorentz invariant?


title: In condensed matter simulations, how is particle number density computed in practice?

text: I have been reading a recent paper. In it, the authors performed molecular dynamics (MD) simulations of parallel-plate supercapacitors, in which liquid resides between the parallel-plate electrodes. To simplify the situation, let us suppose that the liquid between the electrodes is argon liquid. The system has a "slab" geometry, so the authors are only interested in variations of the liquid structure along the $z$ direction. Thus, the authors compute the particle number densities averaged over $x$ and $y$: $\bar{n}_\alpha(z)$, where $\alpha$ is a solvent species. (That is, in my simplified example, $\alpha$ is argon -- an argon atom.) $\bar{n} _\alpha(z)$ has dimensions of $\frac{\text{number}}{\text{length}^3}$ or simply $\text{length}^{-3}$, I think. The $xy$-plane is given by the inequalities $-x_0 < x < x_0$ and $-y_0 < y < y_0$. The area $A_0$ of the $xy$-plane is thus given by $A_0 = 4x_0y_0$. So, the authors define the particle number density averaged over $x$ and $y$ as follows: $$\bar{n}_\alpha(z) = A_0^{-1} \int_{-x_0}^{x_0} \int_{-y_0}^{y_0} dx^\prime dy^\prime n_\alpha(x^\prime, y^\prime, z)$$ where $A_0 = 4x_0y_0$ and $n_\alpha(x, y, z)$ is the local number density of $\alpha$ at $(x, y, z)$. Thus, $\bar{n}_\alpha(z)$ is simply _proportional_ to $n_\alpha$ integrated over $x$ and $y$. But, my question is, what is $n_\alpha(x, y, z)$? How is $n_\alpha(x, y, z)$ determined in practice? As far as the computer is concerned, the argon atoms are point particles; they are modeled as having zero volume (although they interact by Lennard-Jones interactions). So how is it possible to define a number density? Do we simply "cut" the "slab" in "slices" along $z$ and then assign the particles to these slices? There might be 5 particles in the first $z$ slice, 10 in the second, 7 in the third, and so on. If I then divide 5, 10, and 7 by the volume of the respective slice, then I have a sort of number density, with units of $\frac{\text{number}}{\text{length}^3}$ or simply $\text{length}^{-3}$. But how do I now integrate this $n_\alpha(x^\prime, y^\prime, z)$ over $x$ and $y$? Do I have to additionally perform binning in the $x$ and $y$ directions?


title: Hyperbolic Cosine and Sine in Terms of 3-D Velocity

text: Intuitively, why are $\cosh(\theta) = \frac{1}{\sqrt{1-(v/c)^2}}$ and $\sinh(\theta) = \frac{v/c}{\sqrt{1-(v/c)^2}}$ true in special relativity? Is there some picture I can draw in Minkowski space (or anywhere) ![enter image description here](http://i.stack.imgur.com/q8Gbx.png) to derive these or to show that they are obvious? (Analogous to using the Pythagorean theorem to derive $\cos(\theta) = \frac{adj}{hyp}$) My derivation of the Lorentz transformations follows Landau in motivating the hyperbolic form of these equations, but the derivations of the above equalities in Landau are algebraic and I hope there is a more geometric way to see these as obvious, perhaps using the area interpretation of $\cosh$ and $\sinh$ (as giving the area of a hyperbolic sector or something, though I don't see how to relate this to velocity).


title: Matrix Representations of Galilean group

text: The general group element (in the vector representation) $$ \left [{ \begin{array} {c} \bar x^1 \\\ \bar x^2 \\\ \bar x^3 \\\ \bar t \\\ 1 \\\ \end{array} } \right] = \left[ {\begin{array}{ccccc} a_{11} & a_{12} & a_{13} & V_1 & a_1\\\ a_{21} & a_{22} & a_{23} & V_2 & a_2\\\ a_{31} & a_{32} & a_{33} & V_3 & a_3\\\ 0 & 0 & 0 & 1 & t'\\\ 0 & 0 & 0 & 0 & 1\\\ \end{array} } \right] \left [{ \begin{array} {c} x^1 \\\ x^2 \\\ x^3 \\\ t \\\ 1 \\\ \end{array} } \right] $$ This reproduces the transformations $$ \bar x^i = x^i +a^i+ a^{ij}x_j +V^it $$ and $$ \bar t = t + t' $$ where $a^{ij} \in SO(3) $ Now, if look for the generators of the group (which I don't want to enlist), but in general will have to be found using $$ S_i = I + \frac{\partial M}{\partial s_i}\Bigg|_I $$ where $S_i$ is the generator for the parameter $s_i$. But these generators don't seem to follow the commutation relations they have to, one of them for instance, $ [B_i,H] = 0 $ which is not true for the Galilean group. I am interested in knowing the faithful matrix representations for this group Galilean group, **(this is Galilean group without central extension, which will have zero commutator between $B_i$ and $P_i$).** EDIT 1 : I think the problem has to do with the fact that the parameter for for Boosts ($B_i$) is probably not velocity ($V_i$). If so, what probably is the parameter in that case ??


title: Angular momentum operator for 2 dimensions?

text: Recently I get the task to build (2 + 1)-Dirac theory. I wrote corresponding Dirac equation in a form $$ (i\sigma_{0}\partial_{0} + i\sigma_{1}\partial_{1} + i\sigma_{2}\partial_{2} - m)\Psi = 0, $$ where $$ \sigma_{0} = \sigma_{y}, \quad \sigma_{1} = i\sigma_{x}, \quad \sigma_{2} = i\sigma_{z}, \quad [\sigma^{\mu}, \sigma^{\nu}]_{+} = 2g^{\mu \nu}, \quad g^{\mu \nu} = diag (1, -1, -1). $$ Many operators, such as inversion, charge conjugation and other I can find without problems. But I can't get angular momentum operator, because I can't understand what operator I need to find. In two spatial dimensions I can get only the operator of projection of angular momentum, isn't it? Anyway, how to find it, if I haven't corresponding Lorentz group transformation law in (2 + 1)? Can I associate it with rotations on a plane?


title: Relation between different quantum excitation energy, mass energy and kinetic energy

text: When a particle enters an excited state, the energy appears in its quantum wavefunction according to $E = h \nu$. Does the $E$ in this equation also include kinetic energy, and rest mass energy? Or are they "held" somewhere else? Or are kinetic/mass energy a different framework for looking at the same phenomenon, and if so how do they relate? Also, if the particle gathers enough excitation energy to escape a potential well, and therefore escapes that well, from which 'budget' is the necessary energy taken - kinetic, frequency or mass; and why?


title: Proving a step in this field-theoretic derivation of the Bogoliubov de Gennes (BdG) equations

text: In derivation of the BdG mean field Hamiltonian as follows, I have a confusion here in the second step: $H_{MF-eff} = \int d^{3}r\psi_{\uparrow}^{\dagger}(\mathbf{r})H_{E}(\mathbf{r})\psi_{\uparrow}(\mathbf{r})+\int d^{3}r\psi_{\downarrow}^{\dagger}(\mathbf{r})H_{E}(\mathbf{r})\psi_{\downarrow}(\mathbf{r}) +\int d^{3}r\triangle^{\star}(\mathbf{r})\psi_{\downarrow}(\mathbf{r})\psi_{\uparrow}(\mathbf{r})+\int d^{3}r\psi_{\uparrow}^{\dagger}(\mathbf{r})\psi_{\downarrow}^{\dagger}(\mathbf{r})\triangle(\mathbf{r})-\int d^{3}r\frac{|\triangle(\mathbf{r})|^{2}}{U}$ $ = \int d^{3}r\psi_{\uparrow}^{\dagger}(\mathbf{r})H_{E}(\mathbf{r})\psi_{\uparrow}(\mathbf{r})-\int d^{3}r\psi_{\downarrow}(\mathbf{r})H_{E}^{\star}(\mathbf{r})\psi_{\downarrow}^{\dagger}(\mathbf{r}) +\int d^{3}r\triangle^{\star}(\mathbf{r})\psi_{\downarrow}(\mathbf{r})\psi_{\uparrow}(\mathbf{r})+\int d^{3}r\psi_{\uparrow}^{\dagger}(\mathbf{r})\psi_{\downarrow}^{\dagger}(\mathbf{r})\triangle(\mathbf{r})-\int d^{3}r\frac{|\triangle(\mathbf{r})|^{2}}{U}$ $= \int d^{3}r\left(\begin{array}{cc} \psi_{\uparrow}^{\dagger}(\mathbf{r}) & \psi_{\downarrow}(\mathbf{r})\end{array}\right)\left(\begin{array}{cc} H_{E}(\mathbf{r}) & \triangle(\mathbf{r})\\\ \triangle^{\star}(\mathbf{r}) & -H_{E}^{\star}(\mathbf{r}) \end{array}\right)\left(\begin{array}{c} \psi_{\uparrow}(\mathbf{r})\\\ \psi_{\downarrow}^{\dagger}(\mathbf{r}) \end{array}\right)+const. $ with $H_{E}(\mathbf{r})=\frac{-\hbar^{2}}{2m}\nabla^{2}$ In the second step, we have taken $\int d^{3}r\psi_{\downarrow}^{\dagger}(\mathbf{r})\nabla^{2}\psi_{\downarrow}(\mathbf{r}) = -\int d^{3}r\psi_{\downarrow}(\mathbf{r})\nabla^{2}\psi_{\downarrow}^{\dagger}(\mathbf{r})$............(1). I can prove (by integration by parts and putting the surface terms to 0) that $\int d^{3}r\psi_{\downarrow}^{\dagger}(\mathbf{r})\nabla^{2}\psi_{\downarrow}(\mathbf{r}) = \int d^{3}r \nabla^{2}\psi_{\downarrow}^{\dagger}(\mathbf{r})\psi_{\downarrow}(\mathbf{r})$ but how is it justified to now take $\int d^{3}r \nabla^{2}\psi_{\downarrow}^{\dagger}(\mathbf{r})\psi_{\downarrow}(\mathbf{r}) = - \int d^{3}r\psi_{\downarrow}(\mathbf{r})\nabla^{2}\psi_{\downarrow}^{\dagger}(\mathbf{r})$ in order to prove (1) ?


title: Types of semiconductor relaxation

text: Ok, so you've basically got tensile and compression strain in a lattice mismatched material right? So as you're epitaxially layering the material, the strained material is deforming more and more "elastically" (that is, the energy is being stored in the lattice) until "plastic relaxation" which occurs when dislocations form after a critical thickness of the material has been reached. But now I'm seeing/hearing about "elastic relaxation" where the strain is removed through "elastic relaxation". Is this just a fancy word for "no strain"? And if it is, why call it "elastic relaxation" at all?


title: Antisymmetric functions as Slater determinants

text: Can _any_ antisymmetric function, i.e., a function of $N$ spatial-plus-spin variables $\\{x_i\ | \ i= 1, \ldots, N\\}$ satisfying $$ \psi(x_1,\ldots, x_i, \ldots, x_j, \ldots, x_N) = -\psi(x_1,\ldots, x_j, \ldots, x_i, \ldots, x_N)\ ,$$ always be written as a Slater-Determinant of functions ("orbitals"), _not necessarily_ solutions to Schrödinger's Equation? (I know that there are no orbitals describing interacting electrons, but they _may_ [or may _not_ ] be solutions to a "Schrödinger-like equation, but that's beside the point.) (As context: I have been reading up on Density Functional Theory. I am teaching an optional second course on quantum mechanics and I am explaining some of DFT's basic concepts. In Parr & Yang's "Density-functional theory of atoms and molecules" (Oxford University Press, 1989) it is mentioned that there is a proof that any "reasonable" (electron) density can be derived from an antisymmetric wave function which can _always_ be written as Slater- Determinant of orbitals. The authors cite a couple of articles on some journals to which we, at the university where I work, have no subscription [Phys. Rev. A & B, Int. J. Quantum Chem.] )


title: Hydrogen Adsorption distance

text: In chemical adsorption of H2 molecule, the distance is limited. In the physical adsorption, the distance of adsorption is limited or can be any value? Is there any correspondence to adsorption distance with the van der Waals radius of the adsorbent and adsorbate?.


title: How to model a nonperiodic supercell as a periodic one?

text: Some researchers use Hydrogen atom termination, to model a non periodic finite sized 2D and 1D structures as periodic one. How far this approach is useful in predicting the properties of periodic systems? What are the other approaches to model a non-periodic supercell as periodic one?


title: Max altitude of rocket

text: > _A rocket is launched perpendicularly with an initial acceleration of 10 m/s > 2, if the fuel runs out after 1 minute, what will be the maximum height > reached by the rocket?_


title: The probability of electron-hole pair recombination as a function of physical proximity

text: When we shine line of an appropriate wavelength at a metal, e.g. gold, such that there is sufficient energy to promote an electron from the valence band to the conduction band, we'll generate with some probability a set of electron-hole pairs. We can then, to my understanding, rough approximate the electrons and holes as Brownian particles with distinct three-dimensional diffusion coefficients for movement in the metal (conditional on lattice defects / etc. being randomly or uniformly distributed). Now, when a diffusing electron comes with some critical Euclidean distance $r$ of a hole, the "particles" should recombine with some probability and annihilate one-another. My question is... how can we characterize this process in principle in some common metals or semiconductors like silicon? Is there a known distance- dependent relationship / probability distribution for recombination? Is this relationship conserved over different metals / metalloids? To be clear, I'm not talking about some overall probability distribution for recombination as a function of time - I'm strictly asking about how the physical distance between the electron and hole effects the annihilation probability. Here's a cartoon drawing from Wikipedia of an electron and a hole diffusing to the cathode and anode, respectively, of a silicon-based photovoltaic chip: http://en.wikipedia.org/wiki/File:Silicon_Solar_cell_structure_and_mechanism.svg. Of course, here there's a built-in potential to induce the appropriate directed Brownian motion of the electron and hole, but this should of no consequence for my above question.


title: What is so special about iron?

text: I remember reading something about how iron was a highly stable element. Ever since then, I have looked at iron fry pans with new-found respect. However, in a recent discussion I was unable to pinpoint this 'special' quality. So what is so special about iron?


title: What is the position and orientation of the Earth as a function of time

text: I would like to know the position and orientation of the Earth, relative to the sun, as a function of time. I.e. select any coordinate system with the Sun at the origin so that the earth lies in the x-y coordinate plane. Provide a function which takes as input the time and produces   1. The x-y position of the center of the Earth   2. The unit vector from the center of the Earth to the North Pole.   3. The orientation of the Earth around this vector This function is chaotic in general. I would be happy with a decent approximation. This problem is solved as part of the "what is the position of the sun in the sky" problem. Solutions to this problem tend not to be well separable (at least I have trouble separating my question out of the complete position-of- sun-in-sky solution.)


title: Do nearby gamma ray busts/supernova damage more than just the ozone layer?

text: So we know that many people are putting hard constraints on the galactic habitability zone based on the presence of nearby supernova/gamma ray bursts. But if they _only_ affect the ozone layer, then I doubt that it's as hard of a constraint as many people think it is. For one thing - there is practically no ozone layer around the planets of red dwarfs (and possibly even low-mass K-stars like Epsilon Eridani and Alpha Centauri B \- IMHO, K-stars offer the best prospects for life on other planets.) With this information, I am wondering wondering about the outcome, particularly in regard to life: _would a nearby supernova really do so much damage to planets around those stars?_ For instance, would a supernova really cause more damage than, say, the K/T extinction event 65 million years ago? Also, given that much marine life is shielded from UV rays by layers or ocean water, is it really going to cause significant amounts of damage to such life in that environment? As a side note, maybe this about surviving gamma ray bursts is relevant for complex life too (although this response might be imperfect for now.)


title: Could there be a sort of "Molecular Destiny"?

text: Let's say we start with the Big Bang. Every bit of matter started from this event. Therefore, given EVERY variable (every particle, movements of particles, weights, times, etc) and an INFINITE amount of time and computing power, would it be possible to predict the formation of all planets. Would it also be possible to prove that Earth would have been "created" (not religious intentions here) or that the Sun will explode in several million years? This question may be out of the scope of Astronomy.SE, but it was the most relevant SE site I could think of.


title: Relationship between the cosmic microwave background (CMB) radiation and the visible universe

text: I know that the cosmic microwave background radiation (CMBR) is the leftover radiation from the Big Bang. Have we been able to make a concrete map of the radiation and the objects in the visible universe? (Something like "this point in the CMBR map evolved into this particular region of the visible universe".)


title: Pressure at a bend in a pipe

text: This is a question from a study guide for a military officer exam: I have fluid flowing through an elbow in a pipe (the elbow is in the shape of an upside down and backwards "J") and the flow goes from south to north to east. If holes are drilled at the upper left outside part of the bend and the lower right inside part of the bend, at which point will pressure be lower? Please see attached image for a picture of the question. When I have thought about this problem, I keep coming back to pressure being transmitted equally to all parts of a fluid, so I don't understand how the pressure can be different at one of the holes. The book gives the answer as the inside bend which I can force myself to accept since I can picture the fluid attempting to go straight to the outside bend hole and thus hitting that part harder, but I don't really understand it.


title: Fermions in the same state

text: I need some clarification of what is meant when someone says "fermions cannot occupy the same quantum state". Consider two bosons: $$\psi(\vec{r_1}, s_1, \vec{r_2}, s_2) = \frac{1}{\sqrt{2}} \left( \phi_A(\vec{r_1}, s_1)\phi_B(\vec{r_2}, s_2) + \phi_A(\vec{r_2}, s_2)\phi_B(\vec{r_1}, s_1) \right)$$ This is one wavefunction of two particles. A wavefunction directly corresponds to a state, and since this is only one wavefunction, it seems there is only one state -- that two bosons occupy. But now consider two fermions: $$\psi(\vec{r_1}, s_1, \vec{r_2}, s_2) = \frac{1}{\sqrt{2}} \left( \phi_A(\vec{r_1}, s_1)\phi_B(\vec{r_2}, s_2) - \phi_A(\vec{r_2}, s_2)\phi_B(\vec{r_1}, s_1) \right)$$ Again, one wavefunction (=> one state) and two particles that occupy it. Yeah, $\psi(\vec{r_1}, s_1, \vec{r_2}, s_2) = -\psi(\vec{r_2}, s_2, \vec{r_1}, s_1)$, but it's still just one state -- occupied by two fermions. Could someone clarify?


title: what's the world record in finding the ground state of the 3D Ising model

text: Finding the ground state of the 3D ising model (with no magnetization) is known to be NP-complete. Just wondering what is the biggest size cubic lattice someone has found the ground state of for this model? dbm368


title: Boundary between supercritical fluid and gaseous phase

text: As far as I understand, going around the critical point, it is possible to transform a liquid into a gas without seeing a phase transition. Does this also mean that there is no clear separation between a gaseous phase and a supercritical fluid ? This is more or less what can be understood for the dotted line between the two that usually appears in a phase diagram such as the following: ![Phase diagram](http://i.stack.imgur.com/lbHET.png) So my question is: Is there a standard physical measure for deciding whether a given fluid is in gaseous phase or in supercritical state? Alternatively, is the boundary between the two clear-cut or somewhat fuzzy?


title: Generalisations of AdS/CFT with string theory on both sides

text: From my previous post, I found out from the comments that there are various generalisations of AdS/CFT with different things replacing the CFT on the RHS; such as AdS/CMT, AdS/QCD, and also with the AdS replaced on the LHS, like Kerr/CFT a hydrodynamic dual, etc. I am thus led to ask, " ** _Is there a generalisation of AdS/CFT with string theories on both sides?_** " I can think of at least 1 example of a/n (holographic?) equivalence between a $D$ - dimensional string theory and a $D+1$ - dimensional string theory, T-Duality. E.g. the Type I String Theory and the Type I' String Theory, etc. n


title: Dimensional transmutation in Gross-Neveu vs others

text: * Firstly I don't know how generic is dimensional transmutation and if it has any general model independent definition. Is dimensional transmutation in Gross-Neveau somehow fundamentally different than in say massless-scalar QED or QCD? I mean in the later two cases it shows up in doing perturbation theory - as in QCD, you calculate the 1-loop beta function and try to integrate it to get the flow of the coupling and in the answer you see that its possible to tradeoff the bare-coupling and the renormalization scale in terms of a fictitious mass scale.   * But doesn't it happen in a fundamentally different way in Gross-Neveau? In here because of chiral symmetry isn't it guaranteed that nothing will diverge in perturbation theory and no mass scale will appear perturbatively? So the dimensional transmutation happens only when you try to exactly try to do the path-integral (..which only works may be in the limit of large number of Fermions..) I mean - in general shouldn't one be surprised if no scale appears in doing perturbation theory?   * Also if the above is true then what is exactly so special about Gross-Neveu comapared to say QED with massless Fermions? Why is it that in the former the masslessness is perturbatively stable but so is not true for massless QED?  How well do we know or can prove that RG flow will not break any symmetries of the classical Lagrangian?


title: Senior High School work on Doppler measurements

text: Sorry this question is really vague- as a result I'm very confused. Perhaps you can help me make sense of it. "Radar measurements: Construct a space-time diagram of the radar tracking of an asteroid moving at speed $v$. By using the postulates of special relativity verify the expressions on (a book I do not have). Set $v \ll c$ and obtain the non-relativistic expressions for the relative velocity of asteroid, the one- way and two-way Doppler measurements." So postulates of SR: 1) laws of physics are the same in all frames of reference. 2) $c$ is always constant. To me Doppler shifts are to do with $f'$, $f$ and angular velocities... When it says set $v \ll c$ it makes me think that as a result $\gamma$ ($v/c$) is meant to tend to $0$ therefore it comes into the question somewhere I'm just not sure where. Perhaps $$f' ~=~( 1-\beta/1+\gamma) ^{1/2}.$$


title: Deriving the Sommerfeld expansion by contour integration (Le Bellac p. 277)

text: In Le Bellac's statistical physics book he derives the Sommerfeld expansion by a contour integral. The idea is to expand integrals of the type $I(\beta)\equiv \int_{0}^{\infty}d\epsilon\, \frac{\varphi'(\epsilon)}{e^{\beta(\epsilon-\mu)}+1}$. Integrate by parts and expand $\varphi(\epsilon)$ $$\varphi(\epsilon)=\sum_{m=0}^{\infty}\frac{(\epsilon-\mu)^{m}}{m!}\left(\frac{d ^{m}\varphi}{d \epsilon}\Bigg |_{\epsilon=\mu}\right)$$ Make a change of variables to write $I(\beta)=\sum_{m=0}^{\infty}\frac{\beta^{-m}\varphi^{(m)(\mu)}}{m!}I_{m}$ where $I_{m}=\int_{-\infty}^{\infty}dx\, \frac{x^{m}e^{x}}{(e^{x}+1)^{2}}$. The idea then to evaluate the $I_m$ is to consider the following integral $$J(p)=\int_{-\infty}^{\infty}dx\, \frac{e^{ipx}}{(e^{x}+1)(e^{-x}+1)}=\sum_{m=0}^{\infty}\frac{(ip)^{m}}{m!}\int_{-\infty}^{\infty}\, dx \frac{x^{m}}{(e^{x}+1)(e^{-x}+1)}= \sum_{m=0}^{\infty}\frac{(ip)^{m}}{m!}I_{m}$$ Integrating $J(p)$ gives $J(p)=\frac{2\pi pe^{-\pi p}}{1-e^{-2\pi p}}=1-\frac{1}{6}(\pi p)^2+\ldots$ hence, $$I(\beta)= \varphi(\mu) + \frac{\pi^{2}}{6}(k_{B}T)^{2}\varphi^{(2)}(\mu)+O(k_{B}T^{4})$$ So far so good, my problem comes when he writes the expansion of the Fermi- Dirac distribution itself as $$\frac{1}{e^{\beta(\epsilon-\mu)}+1}\simeq \Theta_{H}(\mu-\epsilon)-\frac{\pi^{2}}{6}(k_{B}T)^{2}\delta'(\epsilon-\mu)+O(k_{B}T^{4})$$ which is meant to be valid under an integral. It appears to mimic the expansion of the generating function, at least where the signs are concerned. If we consider the definition of the Sommerfeld expansion from Wikipedia, we have a formula in the form $$\int_{-\infty}^\infty \frac{H(\varepsilon)}{e^{\beta(\varepsilon - \mu)} + 1}\,\mathrm{d}\varepsilon = \int_{-\infty}^\mu H(\varepsilon)\,\mathrm{d}\varepsilon + \frac{\pi^2}{6}\left(\frac{1}{\beta}\right)^2H^\prime(\mu) + \mathrm{O} \left(\frac{1}{\beta\mu}\right)^4$$ If we give Le Bellac's form a try, $$I(\beta)\simeq\int_{0}^{\infty}d\epsilon\, \varphi'(\epsilon)\left[\Theta_{H}(\mu-\epsilon)-\frac{\pi^{2}}{6}(k_{B}T)^{2}\delta'(\epsilon-\mu)+O(k_{B}T^{4}) \right] \\\ =\varphi(\mu)-\frac{\pi^{2}}{6}(k_{B}T)^{2}\underbrace{\int_{0}^{\infty}d\epsilon\, \varphi'(\epsilon)\delta'(\epsilon-\mu)}_{K}$$ but integrating $K$ by parts gives $$K=\left[\delta(\epsilon-\mu)\varphi'(\epsilon) \right]_{0}^{\infty}- \int_{0}^{\infty}d\epsilon\, \varphi''(\epsilon)\delta(\epsilon-\mu)=-\varphi''(\mu)$$ So the signs work out, and the negative sign is probably not a typo. On the other hand, the next term in the expansion would be $+\frac{7\pi^4}{360}(k_{B}T)^{4}\delta^{(3)}(\epsilon-\mu)$ which would take a negative sign when integrated. The alternating behavior of the signs, as far as I can tell, is due to the imaginary unit that was included in the exponential for the contour integral. A very similar approach is done on p. 255 of the following notes except there is no imaginary unit in the exponential. Also, the expansion is meant to operate only on $\varphi$ instead of actually correcting the Fermi-Dirac distribution for small $T$. My questions are as follows:   1. Are the corrections to the Fermi-Dirac distribution given by Le Bellac correct to all orders? They seem to work for the 0th and 1st order terms.   2. Assuming (1) is correct, how do we "know", or what's the reasoning, that the series expansion of the generating function would give the corrections to the Fermi-Dirac distribution? Because it seemed to me that the generating function was only to determine the $I_m$ coefficients.


title: Finding the acceleration of a cart rolling on a table

text: > ![](http://i.stack.imgur.com/Cqvd9.png) > > The cart is rolls frictionless on the table. It has a mass of $1 kg$. > Attached to it are 2 strings, that go through two frictionless sheaves. The > weights have masses as in the picture. > > a) How large is the acceleration of the cart? > > b) How large is the tension in the strings? > > c) We replace the cart with a box that also has a mass of $1 kg$, it then > accelerates with $1.5m/s^2$, find the friction force between the table and > the box." I tried doing this one, and got every single answer wrong, even though I thought I knew how to do it. This is how I did it: a) The force pulling the cart to the left is equal to the force pulling the weights down since the cart is attached with a string through a sheave. The force pulling left is equal to: $F = 1 kg \times 10 m/s^2 = 10N$ and the force pulling right is $F = 2kg m/s^2 = 20N$, therefore the resulting force is $10N$ to the right. Since the cart has a mass of $1kg$ it accelerates with $10m/s^2$. b) The tension on the left string is $10N$ (from above) and on the right one is $20N$ (also from above). c) This one I just have no idea how to do. The answers given in the book are: a) $2.5m/s^2$, b) $12.5N$ and $15N$, c) $4N$


title: Stationary states with a pair of hamiltonian equations

text: I read some derivation related with probability amplitudes and hamiltonian matrix in some book, and have a few questions. Here what the book says is. We want the general solution of the pair of Hamiltonian equations, $ i\hbar \frac {d{C}_{1}}{dt } = {H}_{11}{C}_{1} + {H}_{12}{C}_{2} $ $ i\hbar \frac {d{C}_{2}}{dt } = {H}_{21}{C}_{1} + {H}_{22}{C}_{2} $ where ${C}_{x} = <x|\psi> , \psi =$ arbitrary state. Suppose that the coefficients (hamiltonians) are constant, and we can use the trial functions ${C}_{1} = {a}_{1}{e}^{-iwt} , {C}_{2} = {a}_{2}{e}^{-iwt} , w = E/\hbar $ Substituting $ {C}_{1} , {C}_{2} $ into the equations, we get $ E{a}_{1} = {H}_{11}{a}_{1} + {H}_{12}{a}_{2} \ ,\ E{a}_{2} = {H}_{21}{a}_{1} + {H}_{22}{a}_{2} $ By using a condition that ${a}_{1}$ and ${a}_{2}$ cannot be zero, and setting ${H}_{11} = {H}_{22}={E}_{0}$ and ${H}_{12} = {H}_{21}=-A$ , we can conclude that $E$ should be ${E}_{0} + A = {E}_{Ⅰ}$ or ${E}_{0} - A = {E}_{Ⅱ}$. (and by linear algebra, we can know $\frac {{a}_{1}}{{a}_{2}} = \frac {{H}_{12}}{E-{H}_{11}} $) For general case, the two solutions ${E}_{Ⅰ}$ and ${E}_{Ⅱ}$ refer to two stationary states, $|{\psi}_{Ⅰ}> = |Ⅰ>{e}^{-(i/\hbar){E}_{Ⅰ}t}$ and $|{\psi}_{Ⅱ}> = |Ⅱ>{e}^{-(i/\hbar){E}_{Ⅱ}t}$ with $|Ⅰ> = |1>{a}_{1}' + |2>{a}_{2}'$ and $|Ⅱ> = |1>{a}_{1}'' + |2>{a}_{2}''$* _................(1)_ * where $\frac {{a}_{1}'}{{a}_{2}'} = \frac {{H}_{12}}{{E}_{Ⅰ}-{H}_{11}} $ and $\frac {{a}_{1}''}{{a}_{2}''} = \frac {{H}_{12}}{{E}_{Ⅱ}-{H}_{11}} $. If the system is known to be in one of the stationary states, the sum of the probabilities that it will be found in $|1>$ or $|2>$ must equal one - ${|{C}_{1}|}^{2} +{|{C}_{2}|}^{2} = 1$* _.....................(2)_ * **Questions))**   1. Why should we struct two stationary states by setting states $|Ⅰ>$and $|Ⅱ>$ in the way of (1) ? How can we express the states $|Ⅰ>$and $|Ⅱ>$ like (1)?   2. Why should the sum of probabilities in (2) equal one?


title: Pressure at a given altitude?

text: I am trying to find out the pressure at a given altitude but have been disappointed to find the following equation does not work at all: (h = height in meters) ![enter image description here](http://i.stack.imgur.com/ujJjV.png) Where   * sea level standard atmospheric pressure p0 = 101325 Pa   * sea level standard temperature T0 = 288.15 K   * Earth-surface gravitational acceleration g = 9.80665 m/s2.   * temperature lapse rate L = 0.0065 K/m   * universal gas constant R = 8.31447 J/(mol·K)   * molar mass of dry air M = 0.0289644 kg/mol ( from Wikipedia ) Upon completion of the equation, which I think spits out Pascal values, they don't match what a pressure would realistically look like at that altitude (comparing the output value to one from Wolfram). Any idea what the issue with this equation is? EDIT: inaccuracy alone isn't the problem, it's not giving the right values for anything. I saw a value of 101800 or so for an altitude of 800ft, which is completely wrong, and then the output starts turning into negatives around 100,000ft. I do need something to work at that height, perhaps this isn't the equation for me?


title: Visible light spectrum to color space

text: I need to be able to convert an arbitrary emission spectrum in the visible spectrum range (i.e. for every wavelength between 380 and 780, I have a number between 0 and 1 that represents the "intensity" or dominance of that wavelength), and I need to be able to map any given spectrum into a particular color space (for now I need RGB or CIE-XYZ). Is it possible? For the spectrum say I have the emission spectrum of a white light, then every wavelength in the spectrum will have an intensity of 1, whereas for a green- bluish light I'd have most of the wavelengths between 500 and 550 with an intensity close to 1, with other wavelengths gradually dropping in intensity. So the first spectrum should be converted to pure white whereas the other one would be converted to a green-bluish color in any color space. Is there a way to do this?


title: How do Temperature Scales work?

text: How exactly do temperature scales work? If my understanding is correct, the Celsius scale has two fixed points: (definitions of temperature irrespective of scale) 1\. The freezing point of pure water at 1atm (labelled 0 C) 2\. The boiling point of pure water at 1atm (labelled 100 C) Note that 0 and 100 are completely random numbers. All other values of temperature are only defined using distance on a thermometer. eg. 50 deg. C is exactly half-way between the marks of 0 and 100 on a thermometer. It doesn't necessarily have to be "half as hot" as 100 deg. C (what is half as hot anyway?) Since the coefficient of expansion depends on temperature anyway, how do we justify these scales? The Kelvin Scale is simply C-273.15, to ensure that all temperatures remain positive. So, how does the RMS speed of gas molecules (which is independent of human- invented temperature scales) depend so 'cleanly' on our arbitrary temperature scales?


title: Proper distance and embedding diagrams?

text: I'm trying to understand proper distance equation in Schwarzschild spacetime. $d\sigma=\frac{dr}{\left(1-\frac{R_{S}}{r}\right)^{1/2}}$. I'm sure I'm missing something really obvious here, but how do I use this to find the coordinate distance $r$ for a particular proper distance $\sigma$ . For example, if I found the proper circumference of circle going round the Sun that roughly coincides with the Earth's orbit. Then I move radially inwards one proper mile, how would I then find the circumference of the circle I now find myself on. This example is also in the context of trying to understand the spacing of concentric circles in embedding diagrams. Thank you


title: Time to heat/cool a room

text: So, I have a basic, very basic, understanding of thermodynamics. I don't take it until next semester. I'm attempting to write a program which plots a temperature over time graph of a room being heated/cooled until a certain threshold is met, the users desired temperature, at which point the temperature will drop at the rooms natural rate of loss. I am finding it very difficult to get equations together for this. So far this is what I have:   * Room volume   * Temperature inside/outside/desired   * Heater value in kW   * Surface areas of windows/floor/ceiling   * Air density I would be immensely grateful for any help or guidance, this has been driving me crazy.


title: Proof of Canonical Commutation Relation (CCR)

text: I am not sure how $QP-PQ =i\hbar$ where $P$ represent momentum and $Q$ represent position. $Q$ and $P$ are matrices. The question would be, how can $Q$ and $P$ be formulated as a matrix? Also, what is the proof of this canonical commutation relation?


title: Why does $i ( LK-KL )$ represent a real quantity?

text: According to my textbook, it says that $i( LK-KL )$ represents a real quantity when $K$ and $L$ represent a real quantity. $K$ and $L$ are matrices. It says that this is because of basic rules. However, I was not able to recall my memories. Can anyone show the proof of this?


title: What does (001) Silicon mean?

text: If someone gives me a thin film of Si, and they tell me it's (001) Si, does that mean that the (001) planes of Si are the ones making up the surface of the film?


title: Is matrix picture of quantum mechnics further used in QFT and superstring theories?

text: Just curious: is matrix picture of quantum mechnaics further used in QFT and superstring theories? It seems like not....


title: Why do some sunsets have a green flash?

text: Some sunsets exhibit a phenomenon called "green flash," where there is a quick flash of green light right as the sun is setting. I have seen this myself several times, and sort of understand it. I am a bit frustrated at my inability to explain it better to my family. I understand how Rayleigh scattering would make the dominant color shift towards the blue just after the sun passes below the horizon. How does mirage cause the green light to refract more than the blue? Is there an easy way to use the two effects to estimate the wavelength of the flash?


title: Particle/antiparticle annihilation and entanglement

text: This is a basic question. Suppose that A and B are completely entangled particles and so are C and D. If B and C are antiparticles that annihilate each other will A and D be entangled as a result. It seems they should be in order to satisfy conservation laws. Also if the initial pairs were the results of pair creation events then the Feynman diagram of the entire can also be interpreted as a single particle starting as A emitting and absorbing photons and emerging as D. More generally if the state of the 4 particle system just before annihilation was represented as $$\alpha_{ij}\, |A_i\rangle\, |B_j\rangle\; \beta_{kl}\, |C_k\rangle\, |D_l\rangle$$ would the state of A and D just the after the B-C annihilation be $$\sum_j\alpha_{ij}\beta_{jk}\,|A_i\rangle\, |D_k\rangle$$ in a product state with the state of the emitted photons. Is this correct?


title: Can a planet form before the parent star ignites?

text: I'm unable to find an answer to my questions via my searches. This questions pertains to the timeline of a star system creation. My question: During the creation of a star system, can satellite planets form before the star has ignited and begun to burn? In more detail, I understand the basic creation of a star, with the gravitational collapse of a large cloud, leading eventually to fusion. Is this transition from protostar to maturity (or to where "ignition" occurs) long enough that planets have already started to form, even in a primitive state (just a big rock)? Or has the star long since been burning when planets finally start to form?


title: Helicity and Pseudospin in Graphene

text: The Hamiltonian for graphene at $\vec{k}$ away from the $K$ point is proportional to $$ \vec{\sigma} \cdot \vec{k} =\begin{pmatrix} 0 & k_x - i k_y \\\ k_x + i k_y & 0 \\\ \end{pmatrix} = k \begin{pmatrix} 0 & e^{-i \theta} \\\ e^{i \theta} & 0 \\\ \end{pmatrix} $$ where $\theta=\tan^{-1}(k_y/k_x)$ and $k^2=k_x^2+k_y^2$ The helicity operator is $\vec{\sigma} \cdot \vec{k}/k$, which is proportional to the Hamiltonian for a fixed k, so the eigenvectors of the Hamiltonian are also helicity eigenstates. These eigenstates are $(e^{-i \theta/2},\pm e^{i \theta/2})$. So, in graphene lingo, we say that the pseudospin is locked parallel or antiparallel to the momentum. The Hamiltonian away from the $K'$ point is proportional to $$ \vec{\sigma}^* \cdot \vec{k} =\begin{pmatrix} 0 & k_x + i k_y \\\ k_x - i k_y & 0 \\\ \end{pmatrix} = k \begin{pmatrix} 0 & e^{i \theta} \\\ e^{-i \theta} & 0 \\\ \end{pmatrix} $$ Then, it's eigenvectors are $(e^{i \theta/2},\pm e^{-i \theta/2})$. The literature claims that these are helicity eigenstates. However, these aren't eigenvectors of the matrix $\vec{\sigma} \cdot \vec{k}$ above, so they can't be helicity eigenstates. What stupid mistake am I making here? Also, does anyone have a geometric picture of what the direction of the pseudospin even means? I know the angle $\theta$ is the phase of sublattice B relative to A, but does anyone have any intuition for what it means for the pseudospin to point in the same direction as the momentum?


title: Equivalent Dose absorption from Na22 source

text: No idea what exactly the rules are or how I should ask this question, but here's the verbatim quesiton: "A Na$^{22}$ source has an activity of 100 $ \mu $C. If you handle the source, how big is the equivalent dose on your hands, if all the radiation were absorbed? Assume the volume and density to be 10 x 5 x 1 $cm^3$ and ~1 $g/cm^3$, respectively." I'm using the SI-system mostly since I live in Norway, but seem to be using $\mu$C for medical radiation. I presume the volume and density apply to the tissue of the hands, and Sodium 22 has mostly positron decay with (I think) energies of 546keV, which makes it have a quality factor of 1. And as far as I've gathered the equation I'm supposed to use is $$D_q = qD_E=q\frac{1}{\rho}\frac{W}{V}$$ where $D_q$=equivalent dose, q=quality factor, $\rho$=density, $W$=absorbed energy, and $V$=volume, but I honestly don't know how I'm supposed to find $W$. If it is as simple as to multiply the activity with the energy value of the positrons, or if I need to find something else entirely. But if it's the first, then when everything is inserted gives me a dosage of: $2.32*10^{-7} mSv / hour$, seems a bit too low dosage, I feel like it should be higher. I'm also wondering if this says that I have to consider both the positron decay, the gamma decay AND the two annihilation photons when I calculate my total energy? With the different weightfactors on the decay with the % listed there? Namely: positron decay=0.546MeV (89,9%), gamma=1.275MeV (100%), annihilation photon=0.511MeV (180%) Haven't had any classes about this so far so I'm sorta stuck trying to find things on the internet. So any help would be much appreciated!


title: Can we represent Simple Harmonic function as triangular waves?

text: Having studied the topic recently I found out that simple harmonic motion can represented well with sine and cosine functions.Take for example a pendulum swing which could look like : ![enter image description here](http://i.stack.imgur.com/iGqpV.gif) and the equations governing the motion would be ![enter image description here](http://penguinphysic.files.wordpress.com/2009/11/image0151.png) So I've been wondering why can't simple harmonic motion be represented in form of triangular waves.Although the equations above involve angular momentum so I may be contradicting myself but fundamentally the velocity time is sine function : $$-\sin(x)$$ and the gradient represents the acceleration is non-uniformly increasing and decreasing. What if instead of that you use $$-(\arcsin(\sin(x))$$ Which would represent a triangular wave whose gradient would depict that the acceleration is uniformly accelerating and decelerating.So would this represent harmonic motion or is it fundamentally incorrect.


title: Atmospheric pressure's effect on sound

text: An argument has been travelling in amongst my friends around this matter and after listening about it for **3 whole days** I decided to ask it here. The Question is: Would a high atmospheric pressure slow sound that travels through it- due to there being more matter that the waves need to travel through- or would the overall speed be un-effected? My thoughts to begin with were that sound would be slowed down, some others involved though think it would be the opposite, the denser atmosphere in-fact working to _increase_ as well as maintain the speed and distance sound waves traveled- due to there being more closely compacted matter [oxygen/gas molecules] for the waves to be passed on through- making it easier for it to move faster and further! The problem with this theory (as I thought it) is that the sound waves would loose momentum quicker- it using more energy for movement to continue hence slowing it down and reducing travel distance. The opposite view posed to this is that less atmospheric pressure would allow for sound to travel further at a faster rate, the reduced air pressure causing less energy loss as waves travelled through it. The problem with this theory (As I believe) is that if sparse atmosphere [having less pressure and gasses] made sound go further and quicker why then wouldn't it do so in an environment such as space- where there is almost no pressure or gasses at all to slow sound? (It being a near perfect vacuum.) So I now find myself on the fence asking which of the above is correct: sound travelling faster and further in sparse atmosphere or in dense atmosphere? -Harry David


title: Sum of 2 Light sources

text: When we have 2 sinusoidal waves with slightly different wavelengths and the same amplitude, the resulting wave has a changing amplitude (amplitude = intensity) which varies from 2*Amplitude to 0 ![](http://i.stack.imgur.com/9Z628.png) Suppose we have two different sources of visible light (slightly different wavelength: different color) producing an electromagnetic wave in 3D. This will result from point to point a wave similar to the sum from the picture. Now let's say we have a 2d surface (let's say a paper) where this (superposition of waves) can be projected (and finally a human can perceive it ). What will appear in this surface? ![enter image description here](http://i.stack.imgur.com/QKFyf.png)


title: Notation in a paper on quantum mechanics and gravitation (2)

text: I am currently reading the paper "Gravitation and quantum mechanics for macroscopic objects" by F. Karolyhazy (1966). The author defines formally rigorous equations in order to describe the decay of the pure state $\Phi(\vec x_i)$ at $t=0$ into a mixture of states $\Phi_1(\vec x_i), \Phi_2(\vec x_i)$ with their corresponding probabilities. He then defines a quantity $$\omega_{\chi} = average over \beta |(\Psi_\beta(\vec x_i, t), \chi(\vec x_i))|^2$$ where $\chi(\vec x_i)$ is an arbitrary, normalized function. He also defines a set $(P_k; \Phi_k(\vec x_i)$) by making $$\omega_{\chi} = P_1 |(\Phi_1, \chi)|^2 + P_2 |(\Phi_2, \chi)|^2 +...$$ for every $\chi$. And then he maximizes average over $\beta \sum_k|(\Phi_{k\beta}, \Phi_k)|^2$, in which $$\Phi_{k\beta}(\vec x_i) = \exp[-(i/\hbar)\int_0^t H_\beta dt] \exp[(i/\hbar)\int_0^tHdt]\phi_k(\vec x_i)$$ where $$H_{\beta} = \sum_i - \frac{\hbar}{2m_i} \Delta_i + \sum_{i} \gamma(\vec x_i, t) m_i c^2+\sum_{ik} V_{ik}$$ and $$H= \sum_i - \frac{\hbar}{2m_i} \Delta_i + \sum_{ik} V_{ik}$$ I could not understand the point of $\omega_\chi$ or even $\chi$ for that matter. Also, I suspect the Hamiltonians are incorrect. What is the point of the various terms and the complicated Hamiltonians? * * * Note: I can attach pictures of the paper here if needed.


title: Work to Create a Vacuum

text: I'm trying to determine the amount of work required (in Joules) to create a vacuum of a certain pressure in a tube. A pneumatic cylinder will be used to remove the air from the tube to create the vacuum. I understand the standard formula for the work to create a vacuum is: $\mathrm{pressure \; difference} \times \mathrm{volume}$. However, I've also read that when considering the use of a cylinder and piston there is actually some work that can be reclaimed since the piston and cylinder provides additional volume for the exhausted gas to expand into. Therefore, the correct formula to calculate work under this scenario is: $$W = V \left( P_{\mathrm{ambient}} - P_{\mathrm{new}} \right) + V P_{\mathrm{new}} \ln(P_{\mathrm{new}}/P_{\mathrm{ambient}})$$ Could someone validate which is the correct work formula to use for this scenario?


title: Energy in nuclear decays

text: After a nuclear decay is it a necessity that the total energy of the products is more than the energy of the original particle before decaying? (NB: by 'energy' I don't intend to include mass-energy as well, unless by definition the term 'total energy' happens to include that in the realm of particle physics) I was told that this might depend on whether the total mass of the products is less than the mass of the decaying particle. I'm not sure of the veracity of this claim. I've heard the Higgs Boson can decay to heavier particles than itself. But, I'm not sure about other decays, such as the one I described, which is not a Higgs Boson. For example, is the statement "the mass energy of the original particle is equal to the total energy of the decay products" false?


title: Applying uncertainty principle and the difference in $\Delta x$

text: These two questions seem to be very similar, but the textbook uses a bit different methods for calculating $\Delta x$ of uncertainty principle. Question A) Suppose that there is a room with the same length of its side (x-axis, y-axis, z-axis). A ball is 100g. In this case, the solution says that I should put $\Delta x$ as 15m. Question B) Find the kinetic energy of an electron bound by nucleus using uncertainty principle. Assume that the nucleus has the side of $1.0 \times 10^{-14} m$. In this case, however, the solution says that I should put $\Delta x$ as the half of $1.0 \times 10^{-14}m$. What is the difference between these two cases? (I know that the second case is only approximation.)