title: Fermat principle: which index of refraction?

text: I am somewhat puzzled by a common formulation of the Fermat principle (light travel time), because it contains index of refraction related to _phase_ velocity while light travel time through a slab of glass, for instance, should be IMO described by _group_ velocity, at least in ordinary materials like water or glass (I know that sometimes even group velocity could be greater than light velocity in vacuum). Is usage of the phase velocity correct?


title: Mixing Water at Different Temperature

text: If I have cup of water at room temperature (say, $25^\circ$C). What would be the resultant temperature if I pour another cup of same amount of water at $100^\circ$C to it? Is it simply $\frac{25+100}{2}$? If yes, what if I pour a cup of n times amount of water at $100^\circ$C to it? Is it simply the sum of all temperature (25+100) divided by the total amount of water $(n+1)$ i.e. $\frac{25+100n}{n+1}$? I am no expert in Physics. I am just thinking of a way to get a cup of water at roughly, say, $40^\circ$C without using any thermometer. I know I can obtain water at room temp and $100^\circ$C easily. But dont know how to get my target temp by mixing water at these 2 different temp.


title: Why are turbines built differently when a change of entropy or temperature is involved?

text: We are talking about turbines for extraction of energy from a fluid. Consider a typical wind turbine, or a francis turbin for water - we have one hub with on set of vanes that are driven by the medium. Now, consider a typical turbine in a steam engine - here we have alternating moving vanes and fixed counter-vanes. The fixed vanes redirect the flow after each stage, something that's apparently not neccessary in a water turbine for a high efficiency. 1 - My intuition tells me that this has something to to with the change of temperature or entropy along the turbine. Is this intuition correst? 2 - Why are the turbines built differently? Assuming it's not possible to extract a reasonable amount of energy with a one-stage turbine from steam, why is that so? 3 - Assuming an appreciable pressure drop without a big change of temperature along the turbine (maybe a pneumatically driven turbine), will I still need a multi vane design to extract most of the energy? Bonus question - can the same reasoning be applied to pumps or compressors - a multi stage design is neccessary for decent efficiency if a significant temperature change is involved?


title: 2d or 1d conduction in this scenario?

text: There is a rectangular fin attached to a heat exchanger with a base temperature of 350K. The fin has uniform properties and experiencesa uniform heat generation. It also experiences heat transfer with the surroundings through convection and radiation. Dies the fin experience 1D or 2D conduction?


title: Is ground energy of interacting fermions always higher that that of bosons?

text: Consider two systems, each made of $N$ particles. In both systems particles interact pairwise and the interaction is given by the same Hamiltonian for both systems. Any other constraints and/or requirements you'd like to add should be the same. Except for one -- the only difference between these systems is that the first particles are bosons and the second particles are fermions. I'm interested in ground states of these systems. My intuition tells me that the ground state energy of bosons should _always_ be lower than the ground state of fermions -- no matter what kind of interactions or other external properties we've chosen. But I cannot think of any reasonably general proof for that statement. Maybe, someone knows how to prove that? Or, maybe I'm wrong and there is a counterexample? * * * **Edit:** If you worry about spins of those particles, then you are free to make this difference too. I'll even give you all the following degeneracies, but the interaction shoud be the same -- spin-independent.


title: How would you explain spectrum and spectral analysis to grandma?

text: E.g. what the light or sound spectrum is, what it's useful for - in very simple terms that a grandmother or a child would understand.


title: Getting rid of double delta function in Feynman rules

text: [1] A very simple example of feynman rule for scalar fields. After computing the diagram i have got the following: $$ -i(2\pi)^4g^2\int d^4q \frac{i}{q^2 -m^2c^2}\delta^{(4)}(p_1 - p_3 -q) \delta^{(4)}(p_2 + q -p_4) $$ I'm a little confused about how the integral approached, it integrated over one delta function to get $$ -ig^2\frac{1}{(p_4 - p_2)^2 -m^2c^2}(2\pi)^4\delta^{(4)}(p_1+p_2 - p_3 - p_4) $$ Am i allowed to do that? I mean I have $q$ in both delta functions. Can I just integrate over one of it? It doesn't sound right. What I'm missing here?


title: How (why!?) does one introduce an UV cut-off in dimensional regularization?

text: This question is in reference to the confusing equation 3.7 (page 14) of this paper.   * One sees the 1-loop answers in their theory as given in their A.7 and A.8 on page 20. Each of the terms is a product of Gamma functions multiplied by $k^{2n+d}$. But how does this become their 3.7? (where an UV cut-off of $k_{NL}$ has been introduced)   * Now such a combination of Gamma functions as in 3.7 can easily produce double or higher order poles. Then in what sense does their 3.9 follow from 3.7? (..also what is the physics behind writing $(\frac{k}{k_{NL}})^{\epsilon}$ as $\sim 1 + \epsilon ln (\frac{k}{k_{NL}} )$?..) And even if the Gamma function combination in 3.7 had a simple pole and the above rewriting were to be done then still a $1/\epsilon$ term should have been there in 3.9. Why is that not there?


title: Gravity in stone

text: I want to ask is it possible for a stone to have some magnetic power different from power that we see in ordinary metal magnets (something like earth gravity that attract everything). Someone told me that he saw a stone that attract other smaller stone but did not attract metals. Is it practically possible in science?


title: Electronic, vibrational and rotational autoionization

text: Can anyone please explain me the concept of electronic, vibrational and rotational autoionization. What I understood is, autoionization usually occurs when the ion core is rotationally or vibrationally excited so that the ion core excitation energy is larger than the binding energy of the Rydberg electron. When we think more physically, the electron might get a "kick" from the ion core which can cause it to escape from the coulomb field. Well, this is the case if the ion core is rotationally or vibrationally excited. But, what is then electronic autoionization? If the electron excitation cause the electron to escape the coulomb field, it is normal ionization, not autoionization right? Please explain me.


title: Is the letter delta generally only used to express change in variable or quantity?

text: I was speaking with a friend of mine earlier and he said "Oh look, delta, the sign of uncertainty" (he doesn't study physics often so had only seen in in Heisenberg's Uncertainty Principle equations). I said "Actually, delta is used to express change in variable or quantity, and is only used for uncertainty in Heisenberg's Uncertainty Principle and other such equations". When I said "other such equations" I meant the ones that are identical to the most famous Uncertainty Principle equation, which just a change in variable, for example: you can't know both the direction of a rotating object and how fast it's spinning, as well as there being uncertainty in the relationship between the number of photons in a light beam and the phase wave associated with the beam (both are expressed with a similar equation to Heisenberg's Uncertainty Principle and use the delta symbol to represent uncertainty. Was I wrong? Are there other reasonably well-known equations that use the delta symbol to express a variable that isn't change in variable/quantity or uncertainty?


title: Quantum Mechanics and nuclear fusion

text: I've been told that, according to QM, when Hydrogen atoms are left together there is a non-zero probability that they spontaneously fuse (I accept this bit). I've been told further that, because of huge amount of Hydrogen atoms present in the Sun, a substantial proportion of the Sun's energy is released due to this phenomenon - is this true? EDIT: On second thoughts, this question may be unclear. I think I would answer myself that the conditions of the sun make fusion more probable than elsewhere, so it happens with much greater frequency. What I was meaning to ask was, with a small amount of hydrogen (neglecting the reduced gravitational effect) would we have to apply a greater pressure and heat in order to produce fusion than with a much, much greater amount - since the greater amount has the benefit of the higher probability of some of the atoms spontaneously fusing anyway? EDIT #2: http://www.youtube.com/watch?v=gS1dpowPlE8&feature=BFa&list=ECED25F943F8D6081C in this video there is a claim made regarding quantum tunnelling in the sun, can someone please explain this phenomenon to me in more detail and tell me whether the video is correct?


title: Does $y$-motion really have nothing to do with $x$-motion?

text: I am watching a Physics 1 for physical science majors on coursera.com, and in one of the concept tests there is a question that goes like this; _"A bullet is fired horizontally from a rifle on the Moon (where there is no air). The initial speed of the bullet when it leaves the gun barrel is $V_{0}$. Assume that the ground is perfectly level (and endless)."_ And then there's 3 statements, on of which is; _III) The time it takes for the bullet to hit the ground increases as $V_0$ is increased._ Apparently this statement is false. The lecturer shows a simulator here where a cannon shoots a ball in a air free environment. And when the lecturer checks the time that it took for the ball to hit the ground after several shots with different initial velocities, the time is the same. Not really understanding this, I tried it for myself because I thought; "Surely, the time it takes for the ball to hit the ground is increased as the initial velocity is increased, at least if I were to increase the initial velocity by a large amount." So I did. And sure enough the time for the ball to hit the ground did increase. How is that, and isn't it inconsistent with what the lecturer says about $x$-motion having nothing to do with $y$-motion?' The lecturer has also showed the equation; $$x = x_0+ v_{0x}t+\frac{1}{2}a_xt^2$$ $$ a = 0 \space \vec{} \space x=x_0+v_0\cos(\theta) t$$ $$a = -g \space \vec{} \space y = y_0+v_0\sin\theta-\frac{1}{2}gt^2$$ Showing that you can treat $x$- and $y$-motion independently.


title: Quantum mechanics as a Markov process

text: I am currently involved in some understanding on this matter with a colleague of mine. I know all the literature about but I do not know the state of art. Please, could you provide some relevant recent literature about? Also explanations are much appreciated. My present view is that there exists a time scale that defines the limit of validity of a Chapman-Kolmogorov like equation. It is a situation very similar to quantum chaos but I do not know if this has been currently caught.


title: Bubbles in zero gravity

text: What happens if you blow bubbles into a glass of lemonade on the international space station? Since you are weightless in orbit, there's no up, down, left nor right. We define down on the Earth surface as the direction things tend to fall, down. On Earth bubbles rise if blown into a glass of lemonade (with a straw and a cover).


title: Accelerated liquid in a U-tube

text: Consider a U-tube containing some liquid. The two ends of the tube are connected to closed volumes of air with different initial pressures, such that there is some difference in the liquid levels inside the tube. Assume that the U-tube has uniform cross section, the liquid is incompressible, and there are no losses. Now, if the U-tube is accelerated vertically, will there be any changes in the liquid level at one leg relative to that in the other, and why? I initially thought they would change. When the tube is accelerated upwards, the liquid in the tube would tend to move from the leg with higher liquid level to the other leg. The increased pressure that drives it is given as $\rho d a$, where $\rho$ is the liquid density, $d$ is the initial difference in the liquid levels of the two legs, and $a$ is the acceleration of the U-tube. But then, I thought there should be an equal and opposing pressure increase from the leg with lower liquid level, which prevents the liquid from moving. But I am not sure.


title: What is the state of the art in particle detection and localization

text: I am researching methods to detect the position of radioactive materials (emitting gamma and beta particles), and would like to know what current methods are used to do this. What type of sensors are used, and how do they process the information to detect the position of a radioactive material in 3d space? Are there any limitations/difficulties with these methods? e.g computationally expensive, inaccurate, only works for one type of element etc. I'm developing my own method to detect the position of radioactive material in 3d space, and I'd like to know what the current standard method for doing this is so I can research and compare the methods. My method involves liquid scintillation counters and a neural network. The material will be highly radioactive. e.g 100g of uranium. It is limited to (4x14x6)m area, and could be buried in a solid, surrounded by air, inside a container. However I am assuming that the matter surrounding the radioactive material is uniformly distributed with constant density.


title: Need help on Springel model for galaxy dynamics

text: I would like to have help about the response of the dark matter profile in the Springel galaxy dynamics model ( from the paper Tidal tails in CDM cosmologies(NB: Arxiv pre-print)) Here's the part of the paper that interests me : ![enter image description here](http://i.stack.imgur.com/gYUaE.png) Firstly, I have understood how to get equation (16) with the fact that angular momentum of individual dark matter orbits is conserved, i.e : $r_i M(r_i) = r_f M_f(r_f) \tag{eq 16}$ where $r_i$ and $r_f$ are the initial and final radii of some dark matter mass shell, $M(r)$ gives the initial NFW mass profile, and $M_f(r)$ is the final cumulative mass profile after the disk is formed. Now, I have still problems how to get (eq 17). Here is this equation : $$M_f (r_f ) = M_d (r_f ) + (1 - m_d )M(r_i )$$ where $M_i(r)$ is the initial and $M_f(r)$ the final cumulative mass profile and $m_d$ is defined by : > The mass $M_d$ of the disk is a given fraction $m_d$ of the halo mass Does initial and final radii correspond respectively to the initial and final radius of the dark matter halo ? So Why is it written $M_d (r_f)$ for the disk ? If this is the case, is the dark matter halo expanding from $r_i$ to $r_f$ radius during the formation of the disk ? Is the initial mass of the disk distributed proportionally to the initial mass of dark matter halo ? such that we could write : $$M_d(r_i) = m_d M(r_i) ?$$ Is the initial mass of dark matter halo the same than the final mass of dark matter halo ? such that we could write : $$M(r_i) = M(r_f) ?$$ Could you give me help to get this equation : $$M_f (r_f ) = M_d (r_f ) + (1 - m_d )M(r_i ) \tag{eq 17}$$ I tried to get it by starting from this : $$M_{total}(r) = M_f(r) = M_d(r) + M_{DM}(r) = m_d M_{DM}(r) + M_{DM}(r) = (1+m_d) M_{DM}(r)$$


title: Newton's Bucket

text: Newton's Bucket This thought experiment is originally due to Sir Isaac Newton. We have a sphere of water floating freely in an opaque box in intergalactic space, held together by surface tension and not rotating with respect to the distant galaxies. Now we set the box and water to rotate about some axis and we notice that the sphere flattens into an oblate spheroid. How does the water know it’s spinning? NOTE: Newton thought this proved the concept of absolute rotation with respect to a preferred spatial frame of reference. Perhaps these days we can do better, or different?


title: Phase on Hilbert Spaces

text: Why, in somes cases, such as spin 1/2 systems like in the Stern-Gerlach experiment, the representation of some kets has a term with an complex exponential? For example: $ \vert S_{x}; + \rangle = \dfrac{1}{\sqrt{2}} \vert + \rangle + \dfrac{1}{\sqrt{2}} \exp [i \delta] \vert - \rangle $


title: Time Delays in General Relativity

text: If gravity propagates at, or less than, the speed of light, and is therefore not instantaneous, does that introduce time delays to gravity? For example: suppose that we, the observer, sit within the future light cone of a massive star, and that the light from that star is due to reach Earth for the first time in 100 years time. Is it theoretically possible that we might we feel a sudden (possibly small but theoretically detectable) gravitational ‘tug’ in 100 years time? Further: Is it possible that, whilst we might currently sit outside of the future light cone of some a star (for example) - over time, due to expansion of the universe - the future light cone of that star could expands to encompass us – thereby enabling us to observe and be influenced by that previously unobservable and undetectable object?   (Note this is in reference to the expansion of the universe, as opposed to the light simply not having reached us yet) Apologies if I may be confusing 2 topics here. I only follow popular science at present….


title: A question from a Physics 2 exam: About electron cloud an a constant external field:

text: I am trying to prepare for my exam in Physics $2$, the following is a question from an old exam (the question also have a detailed answer to it, but I don't really understand it). > An acceptable model for an atom contains a point charge $+ze$ (nucleus) at > the center of a ball charged of charge $-ze$ with uniform density $\rho$. > The radius of the ball is $a$. > > An atom is placed under a uniform field $\vec{E}=E\hat{x}$, as a result, > there are forces on the nucleus and the electron cloud at opposite > directions. > > ![enter image description here](http://i.stack.imgur.com/ShCWk.png) > >   1. Assuming that the electron cloud moves, but not distorted: calculate > the relative shift $\delta$ between the nucleus and the center of the > electron cloud (assume $\delta<a$) > >   2. Assuming that the electrical field was increased gradually from $0$ to > its final value $\vec{E}$, calculate the work, $W$, the electrical field did > on the atom. > > > > There are another parts to the question that I omit for now, since they are > not relevant to my question. Note: I have translated the question, so the English problems origin is me, if something is not clear please ask and I'll try to answer. The answer to the first question finds $\delta$ s.t that the electric force being applied from the electron cloud equals (and is opposite in direction) from the electric force being applied from the electric field $\vec{E}$. The answer to the second question say that > the field is increased gradually so the atom is in equilibrium at each step > of the process, the force the electrical field need to overcome is the > attraction force between the nucleus and the electric cloud. For a given > distance from the center $x$, this field is given by > $E(x)=\frac{ze}{a^{3}}x$ > > The work is then calculated by $\int_{0}^{\delta}F(x)\, dx$ where $F=qE$. Note that if I calculated correctly, the calculation $E(x)$ corresponds to the field of a ball with uniform charge density $\rho$ and with total charge $-ze$. **My questions are:** 1) In the solution to the first question, the force of the attraction is calculated as if there was no external field $\vec{E}$. Shouldn't the electron cloud move somehow so that this calculation was not good anymore ? why can we still consider the ball as a ball with uniform charge density $\rho$ ? 2) In the solution to the second question, what does it mean that the atom is in equilibrium at each step of the process ? if this is the case the nucleus shouldn't be moving at all. Also, if I change the field gradually, why does the atom is in equilibrium state ? I thought that for small values of $\vec{E}$ total electrical force is $-\hat{x}$, since the force of the attraction is higher than the force from the external field, and so the nucleus would go back to the center and then pushed to the right and goes back to the center again, until $|\vec{E}|$ reaches some minimal value to overcome the attraction.


title: Official definition of astronomical units

text: As it is relatively easy to find an official value for a large number of physical constants, (thanks to CODATA), it is not so easy for some units widely used in astronomy (Wikipedia, Google and IAU give different values). So in 2012, what are the "official normalized values" (with the largest precision) for :   * Year (the one used for light-years) (in $\mathrm{s}$)?   * Solar mass (in $\mathrm{kg}$)?   * Astronomical unit (in $\mathrm{m}$)?   * Light-year (in $\mathrm{m}$)?   * Parsec (in $\mathrm{m}$)?


title: Why does a glass breaks when subject to very sharp sound like in opera?

text: I have seen in Ripleys believe it or not that an opera singer generates a very high frequency of sound and breaks a glass window.How do they do it?


title: Causality without light

text: Related to my prior question: Would a species insensitive to light have developed special relativity? Imagine a species insensate to light and other em radiation. They are very attuned to sound however. Might they conclude that any transmission of information faster than sound violated causality? For example, a bullet impacts a person, then a second later they hear the sound. Might this situation suggest to such a species that causation had been violated?


title: What's the meaning of dimensionality?

text: I got amazed with this answer and as you can see, I had a little talk with joriki. He suggested me some examples such as the 2DEG. I'm stuck with some aspects on dimensionality, see this line: > A two-dimensional electron gas (2DEG) is a gas of electrons free to move in > two dimensions, but tightly confined in the third. In mathematics, it seems that there could be $n$ dimensions, but how's it on the "real world"? I thought that such $n$-dimensional objects on the real world could exist on the absence or presence of a specific dimension, but at least in this wikipedia article, it seems that this object _have_ the third dimension - at least I conceive "confined" as something different of "abscence". Can you clarify this to me? I am open to both answers and references on this topic, I just hope that asking both is not being too broad.


title: Unitary transformations in mixed discrete-continuous representations

text: I am having trouble with the unitary transformation of a certain Hamiltonian in the paper > Zhai, H. Spin-orbit coupled quantum gases. _Int. J. Mod. Phys. B_ **26** no. > 1, 1230001 (2012). arXiv:1110.6798 [cond-mat.quant-gas]. Here they transform the hamiltonian $$ H=\begin{pmatrix} \frac{k^2_x}{2m}+\epsilon_1 & \frac{\Omega}{2}e^{i2k_0x} & 0 \\\ \frac{\Omega}{2}e^{-i2k_0x} & \frac{k^2_x}{2m} & \frac{\Omega}{2}e^{i2k_0x} \\\0 & \frac{\Omega}{2}e^{-i2k_0x} & \frac{k^2_x}{2m}-\epsilon_2 \end{pmatrix} $$ via the unitary transformation $$ U=\left(\begin{array}{ccc}e^{-i2k_0 x} & 0 & 0 \\\0 & 1 & 0 \\\0 & 0 & e^{i2k_0x}\end{array}\right) $$ to get the "effective" hamiltonian $$ H_{\text{eff}}=UHU^\dagger=\left(\begin{array}{ccc} \frac{(k_x+2k_0)^2}{2m}+\epsilon_1 & \frac{\Omega}{2} & 0 \\\ \frac{\Omega}{2} & \frac{k^2_x}{2m} & \frac{\Omega}{2} \\\0 & \frac{\Omega}{2} & \frac{(k_x-2k_0)^2}{2m}-\epsilon_2 \end{array}\right). $$ I have tried doing this transformation many times but I can't succeed.


Why is there a sharp cracking sound when we break a plastic ruler? The question appears to be simple and common but I need a apropriate answer.


title: Wave Particle duality because of discrete time?

text: If time is discrete, such as the Planck's length, would the transition from one frame of time to the next explain why it appears matter changes from a particle to a wave? During that infinitely small space between each frame we can not measure the particle and it appears as a wave?


title: What is the acceleration of a two weights connected with wire?

text: Given 2 weights: **m1 = 2 kg** **m2 = 1 kg** They are connected with wire hung over stick and considering there are no wire friction. Find the acceleration of weight **m1** going down. Find the (force) tension of the wire. * * * My try:                   F1 - F2 = ma              m1g-m2g=ma              9.8*(2-1)=ma              a=9.8/2      $$ a=4.9\; \text{m}/\text{s}^2 $$ Correct answer: $$ a=3.27\; \text{m}/\text{s}^2 $$ and **F = 13.1 H** * * * What's wrong? How to solve it?


title: Density of classical states in quantum theory

text: Let's first treat electrons as classical objects. I can evaluate the classical energy of each state in a configurational space (3N real numbers and, say, spins) using just Coulomb's law. Then I calculate the electronic wavefunction of the ground state, $\phi_0(\mathbf{r})$, in each point of the configurational space (classical state), which will give me the (square of amplitude...) probability density of each such classical state. Combining these two results, I get the "density of classical states" of the ground state electronic wavefunction - the probability as a function of the classical energy, $c(E)$. I think this $c(E)$ might have some general features which might be used for speeding up post-Hartree-Fock (HF) quantum chemistry calculations. I am not a theoretical physicists, therefore I do not know, how to ask Google / WoK, etc. Is there any study about this topic? Is there such an approach, and if so, what are its merits?


title: Why does a bullet bounce off water?

text: It is known that bullets can ricochet off a body of water. Is surface tension responsible for this or is this the same behavior we see when an asteroid ricochets off the atmosphere? I don't think surface tension has anything to do with it but I'm arguing with someone who disagrees. I think the major factor is the density of water relative to air and the density of the bullet.


How far do air particles move when a sound wave passes through them? I know that they don't actually travel, the question is how far do they oscillate or what is the physical amplitude of the oscillation? Obviously the answer is different for different mediums and volumes (and maybe frequency?), so for the purposes of the question, assume we're talking about a 100Hz sine wave, at about 75db (low bass, at listening volume), travelling through air at sea-level pressure.


title: How to find an action of $(\hat {\sigma} \cdot \hat {\mathbf L} )$ on spherical spinors?

text: Let's have the spherical spinors $\psi_{j, m, l = j \pm \frac{1}{2}}$, $$ Y_{j, m, l = j \pm \frac{1}{2}} = \frac{1}{\sqrt{2l + 1}}\begin{pmatrix} \pm \sqrt{l \pm m +\frac{1}{2}}Y_{l, m - \frac{1}{2}} \\\ \sqrt{l \mp m +\frac{1}{2}}Y_{l, m + \frac{1}{2}} \end{pmatrix}. $$ How to find an action of $(\hat {\sigma} \cdot \hat {\mathbf L}$ on it without using very long derivatives by using expressions for spin-orbital operator $(\hat {\sigma} \cdot \hat {\mathbf L} )$ (it is part of the square of total angular momentum $\hat {\mathbf J} = \hat {\mathbf L} + \hat {\mathbf S}$) in spherical coordinate system, if it is possible? P.S. I don't know that $\hat {\mathbf J}^2\psi_{j, m, l = j \pm \frac{1}{2}} = j(j + 1)\psi_{j, m, l = j \pm \frac{1}{2}}$, so I can't express $(\hat {\sigma} \cdot \hat {\mathbf L})$ from this expression.


title: How to derive the Manley-Rowe relation for the process of the second harmonics generation?

text: Derive the Manley-Rowe relation for the process of the second harmonics generation. Manley-Rowe relation: ~ The Manley-Rowe relations are mathematical expressions developed originally for electrical engineers to predict the amount of energy in a wave that has multiple frequencies. Second-harmonic generation: ~ Second harmonic generation (SHG); also called frequency doubling; is a nonlinear optical process, in which photons interacting with a nonlinear material are effectively “combined” to form new photons with twice the energy, and therefore twice the frequency and half the wavelength of the initial photons. It is example of nonlinear phenomena (I, 3). In the SHG process, the intense wave at the frequency $\omega$ propagates in the medium with second-order nonlinearity ( VI, 1).


title: How to work out bubble size and rising velocity in turbulent flow?

text: Is there a way to determine bubble size and rising velocity in turbulent flow? I know stokes law but this is only valid for laminar flow.


title: Self-adjoint functionals

text: Usually, QFT is used in operator representation, that is one can write e.g. $-i\partial_{t}\psi=H\psi$ with H being an operator. And one can ask if H is self-adjoint etc. However, there's also the possibility of the functional representation where H would be a functional. What would be the analog of self-adjointness for functonals? THX


In what environments can a dipolariton form? The article in _Science_ in which it is introduced discusses it in the context of an optical cavity in a semiconductor. Can this quasiparticle occur elsewhere -- as a free particle or on the surface or in the bulk of a metal, for example? A dipolariton is a bosonic quasiparticle that consists of two excitons, one an electron together with a hole, the other an electron together with a photon. I'm asking this question to get a sense of whether a related question that was raised not too long ago makes sense. **EDIT:** To be concrete, could a dipolariton form in the context of the system explored in this paper?


title: Can matter leave the cosmic horizon?

text: Cosmic horizon in the de Sitter space is a sphere, centered at the observer with finite radius where the red shift due to cosmic expansion becomes infinite. Given that no information can be transmitted from behind the cosmic horizon, I wonder whether any matter can ever pass behind the event horizon, because this will give rise to an information paradox, similar to the black hole information paradox. The matter approaching the cosmic horizon seems to experience time dilation, so will that time dilation prevent it from crossing the horizon? To work around the paradox, one have to postulate that the information about outgoing matter should be encoded in the de Sitter radiation.


title: Chern-Simons theory

text: In Witten's paper on _QFT and the Jones polynomial,_ he quantizes the Chern- Simons Lagrangian on $\Sigma\times \mathbb{R}^1$ for two case: (1) $\Sigma$ has no marked points (i.e., no Wilson loops) and (2) $\Sigma$ has marked points and each point has attached a representation of the gauge group. In case (1), Witten shows that the vector space should be the space of holomorphic sections of a determinant line bundle over the moduli space of flat connections. For the second case he states that the vector space should be the $G$-invariant subspace of the tensor product of all the representations associated to the marked points; by this I mean, if $\Sigma$ has $r$ marked points and each point has a rep $R_i$ then the quantum Hilbert space is $(\bigotimes_{i=1}^r R_i)^G$. Does anyone know how to interpret this second case in terms of sections of some bundle? I mean, shouldn't the second case reduce to the first when you remove the marked points? Also, Witten states, immediately following case (2), that in the presence of no marked points the quantum Hilbert space is 1-dimensional. How can one see that from the formula for the quantum Hilbert space, $(\bigotimes_{i=1}^r R_i)^G$?


title: How does my car acquire satellite signal?

text: I just got XM radio again after a brief period without it. A customer service rep said that the satellite needs to beam my signal to me in a six minute time window. How exactly does this happen? How does the satellite know where to find me, and how exactly does it "talk" to my car? Why does it have a six minute time window? I tagged this as radiometry, but I'm not quite sure what topic this is.


title: Estimate number of hairs on human head

text: A technique of vital importance at all levels in physics is estimation. This is obvious from the first chapter in any introductory physics textbook, but is also related to the working physicist. Checking orders of magnitudes during research presentations is common practice - I've seen many good questions with good followup answers that started with "If I estimated that value I would get something much different". In general, the actually result is not the interesting thing - it's what individual things will affect the result. There are even famous examples of this: Fermi's piano tuner problem, and the Drake equation. Apparently, Fermi was so good at this that he estimated the size of the Trinity nuclear bomb test to within a factor of 2 (see the wikipedia article for a discussion of that). In this spirit, I would like to see someone try and estimate the number of hairs on the human head. The answer must include the basic assumptions so we can see where the major unknowns lie, and the best answer is one which requires no specific knowledge


title: fitting free QFTs into the Haag-Kastler algebraic formulation

text: Has the free Klein-Gordon quantum field theory been fitted into the Haag- Kastler algebraic framework? (Actually, John Baez told me "yes", and he should know.) If so, can you describe the basic strategy and/or give pointers? The same question for the free Dirac field theory.


title: Why isn't this capacitor charging?

text: Let's say you have a parallel plate capacitor and you connect one plate to the positive terminal of a battery and the other plate to the negative end. So this is like a static situation, you have a large positive charge (the cathode in the battery, which keeps recharging)) connected to a metal plate, and a big negative charge (the anode) connected to another metal plate. From what I've learned about static situations, the charges should equalize so that the plate connected to the positive charge should have half the charge of the cathode, and the same for the negative side. However, my physics teacher said this would not happen and he even did an experiment to show me. However, I don't understand why my reasoning doesn't work. Could someone explain to me why I am wrong? Thanks!


title: A confusion about base states of a quantum system

text: I have been told that the eigenkets of a operator of a space form a basis for the state of the quantum system. The eigenbasis obtained from the position operator $\textbf{x}$ is the (continuously) infinite set $|x\rangle$, whereas for a general angular momentum operator the basis $|\ell,m\rangle$ is countably infinite, $|m|\leq \ell$, $\ell\in\mathbb{N}_0$. This is a contradiction as in this case the dimension of the state space does not have a well-defined cardinality. At this point one could say that these are actually different informations and the total state is some kind of a direct product of these states lying in different vector spaces. But then again I have been told that $\langle x|j,m\rangle=Y_l^m$ which means that these actually reside in the same space (because we have their inner product). So what is the state space of a quantum system with both $x$ and angular momentum and what is the correct basis?


title: Simple conservation of momentum and frame of reference problem

text: I'm making a very simple physics engine based on momentum, and I'm solving what response to use for a collision from each involved object's frame of reference. However, something about how I'm interpreting frame of reference isn't quite right... Consider the below example, with a 1g object moving 10 m/s toward a 10g object. No friction or rotation or anything complicated. ![enter image description here](http://i.stack.imgur.com/swxpU.png) That 1g object eventually hits the 10g object. Now, my engine would then look at each object from its frame of reference (so its velocity, basically) and sum the external momentums on it (in case there are multiple collisions) and use that to change its own velocity. There's only 2 objects involved in this example to keep it simple. So from the 10g unmoving object's frame of reference, the 1g object is hitting it with basically that same velocity vector (so 10 gm/s momentum). From the 1g object's frame of reference, the 10g object is moving toward it at 10 m/s and hitting it from the right with... 100 gm/s?? That doesn't make any sense! ![enter image description here](http://i.stack.imgur.com/DLCQl.png) I'm really messing up frame of reference here somehow... I need to use the frame of reference of each object, but I'm pretty sure that the momentum I'm getting in some of them is simply incorrect. That 1g object, when doing some sort of equal response to that 100 gm/s momentum, would then get launched backward at 90 m/s (10 m/s - (100 gm/s / 1g))! I'd then calculate how the 10g object moves based on the momentum it sees, resulting in it moving forward at 1 m/s. Where am I going wrong here?


title: Confused about magnetic and geographic north

text: I'm slightly confused about the "north-seeking" pole of a magnet: does it point towards magnetic north, or is it towards geographic north? I ask because I've been finding different explanations in different places.


title: audio signal arithmetic

text: BACKGROUND: I have been messing around with audio signals in Audacity and realized that by inverting a signal and adding it to the same signal I get no signal. Thus (S + invS) is really (S-S). (S+S) = 2S and S increased by 6db is also 2S (checked by subtracting an inverted S+S signal and got no signal). QUESTION: What would be S*S, 1/S, and sqrt(S)?


title: Why it happen's when I rub my finger's?

text: I have a pc with a metallic body on my CPU case. Yesterday suddenly the earthing wire of my CPU broke down and I was able to feel a little bit current on touching it's metallic body. But what I found to interesting was that when my finger rubbed on it's metal surface it was a feeling like when you rub the glass surface with a little bit wet surface a squkee sound come and your finger feel a little bit resistance on rubbing the surface and a very weird sound was coming out just like when you rub the to of a wine glass but the sound was little bit thick it was like ghnnnnnnnnnnnnnn and when I tried do that while wearing my sleepers i felt or heard nothing like that the surface felt completely smooth and there was no sound. Why was it happening?


title: Galilean invariance of the Schrodinger equation

text: I am only asking this question so that I can write an answer myself with the content found here: http://en.wikipedia.org/wiki/User:Likebox/Schrodinger#Galilean_invariance and here: http://en.wikipedia.org/wiki/User:Likebox/Schrodinger#Galilean_invariance_2 I learned about these pages in a comment to this answer by Ron Maimon. I think Ron Maimon is the original writer of this content. This is creative commons, so it's ok to copy it here. It is not on any textbook on non-relativistic quantum mechanics that I know of, and I thought it would be more accessible (if to no one else, at least for myself) and safe here. I hope this type of question is not in disagreement with site policy.


title: Magnetization energy density and Magnetic field energy density.

text: Are they the same? I mean the energy required to magnetize/demagnetize a ferromagnet is highly dependent on the hysteresis loop that is $E= \int H dB$. The energy density of the same field is equal to: $E= 0.5 \frac{B^2}{U}$ From this I assume magnetization and attraction are two different phenomenons. Are they? Is Magnetization = Attraction/Repulsion of dipole moments?


title: Why do bullets in a magazine go up instead of down?

text: I've been curious about this, and it might not belong here, but I'll ask anyways. For most modern weaponry that I've seen or used, it appears that the magazine is always placed below the barrel of it. Instead of the magazine going on top of the gun, per say, the magazine is placed below the gun barrel itself. For example, the default assault rifle used in the US military is the M16: ![enter image description here](http://i.stack.imgur.com/XMerN.jpg) As you can tell, the magazine is below the barrel of the gun, and bullets are loaded in one at a time into the chamber. What I'm curious is to why bullets in magazines are placed this way. I've had friends who say that physics plays a role, in that if you put the magazine on the top, gravity could force an extra bullet into the chamber and cause a misfire. But isn't putting the magazine in the top more efficient? Letting gravity do the work? Some guns like the Sten, used in WWII, have a magazine that go in via a slot in one side of the gun. Most sidearms have their magazines located in the grip of the gun. ![enter image description here](http://i.stack.imgur.com/fvtNr.jpg) Why was this implemented? Is there some aspect of physics that allows an optimal entry point for bullets to go in the chamber? # **Edit:** Based on a chat in Arqade's room (The Bridge), it seems there are very few guns that actually have magazines inserted from the top. The only few guns we could find were the Bren (below) and the M1 Garand, although that uses an en- bloc clip that feeds up. ![Bren Light Machine Gun](http://i.stack.imgur.com/Xnl8R.jpg)


title: Motivation for the use of Tsallis entropy

text: Every now and again I hear something about Tsallis entropy, $$ S_q(\\{p_i\\}) = \frac{1}{q-1}\left( 1- \sum_i p_i^q \right), \tag{1} $$ and I decided to finally get around to investigating it. I haven't got very deep into the literature (I've just lightly skimmed Wikipedia and a few introductory texts), but I'm completely confused about the motivation for its use in statistical physics. As an entropy-like measure applied to probability distributions, the Tsallis entropy has the property that, for two _independent_ random variables $A$ and $B$, $$ S_q(A, B) = S_q(A) + S_q(B) + (1-q)S_q(A)S_q(B).\tag{2} $$ In the limit as $q$ tends to $1$ the Tsallis entropy becomes the usual Gibbs-Shannon entropy $H$, and we recover the relation $$H(A,B) = H(A) + H(B)\tag{3}$$ for independent $A$ and $B$. As a mathematical property this is perfectly fine, but the motivation for its use in physics seems completely weird, unless I've fundamentally misunderstood it. From what I've read, the argument _seems_ to be that for strongly interacting systems such as gravitationally-bound ones, we can no longer assume the entropy is extensive (fair enough so far) and so therefore we need an entropy measure that behaves non-extensively for _independent_ sub-systems, as in Equation $2$ above, for an appropriate value of $q$. The reason this seems weird is the assumption of independence of the two sub- systems. Surely the very reason we can't assume the entropy is extensive is that the sub-systems are strongly coupled, and therefore _not_ independent. The usual Boltzmann-Gibbs statistical mechanics seems well equipped to deal with such a situation. Consider a system composed of two sub-systems, $A$ and $B$. If sub-system $A$ is in state $i$ and $B$ is in state $j$, let the energy of the system be given by $E_{ij} = E^{(A)}_i + E^{(B)}_j + E^{(\text{interaction})}_{ij}$. For a canonical ensemble we then have $$ p_{ij} = \frac{1}{Z} e^{-\beta E_{ij}} = \frac{1}{Z} e^{-\beta (E^{(A)}_i + E^{(B)}_j + E^{(\text{interaction})}_{ij})}. $$ If the values of $E^{(\text{interaction})}_{ij}$ are small compared to those of $E^{(A)}_i$ and $E^{(B)}_j$ then this approximately factorises into $p_{ij} = p_ip_j$, with $p_i$ and $p_j$ also being given by Boltzmann distributions, calculated for $A$ and $B$ independently. However, if $E^{(\text{interaction})}_{ij}$ is large then we can't factorise $p_{ij}$ in this way and we can no longer consider the joint distribution to be the product of two independent distributions. Anyone familiar with information theory will know that equation $3$ does not hold for non-independent random variables. The more general relation is $$ H(A,B) = H(A) + H(B) - I(A;B), $$ where $I(A;B)$ is the mutual information, a symmetric measure of the correlation between two variables, which is always non-negative and becomes zero only when $A$ and $B$ are independent. The thermodynamic entropy of a physical system is just the Gibbs-Shannon entropy of a Gibbs ensemble, so if $A$ and $B$ are interpreted as strongly interacting sub-systems then the usual Boltzmann-Gibbs statistical mechanics already tells us that the entropy is not extensive, and the mutual information gets a physical interpretation as the degree of non-extensivity of the thermodynamic entropy. This seems to leave no room for special "non-extensive" modifications to the entropy formula such as Equation $1$. The Tsallis entropy is non-extensive for independent sub-systems, but it seems the cases where we need a non-extensive entropy are exactly the cases where the sub-systems are not independent, and therefore the Gibbs-Shannon entropy is already non-extensive. After that long explanation, my questions are: **(i)** Is the above characterisation of the motivation for Tsallis entropy correct, or are there cases where the parts of a system can be statistically independent and yet we still need a non-extensive entropy? **(ii)** What is the current consensus on the validity of Tsallis entropy-based approaches to statistical mechanics? I know that it's been the subject of debate in the past, but Wikipedia seems to imply that this is now settled and the idea is now widely accepted. I'd like to know how true this is. Finally, **(iii)** can the argument I sketched above be found in the literature? I had a quick look at some dissenting opinions about Tsallis entropy, but surprisingly I didn't immediately see the point about mutual information and the non-extensivity of Gibbs-Shannon entropy. (I'm aware that there's also a more pragmatic justification for using the Tsallis entropy, which is that maximising it tends to lead to "long-tailed" power-law type distributions. I'm less interested in that justification for the sake of this question. Also, I'm aware there are some similar questions on the site already [1,2], but these don't cover the non-extensivity argument I'm concerned with here the answers only deal with the Rényi entropy.)


title: Why does a lump of explosive burn's with an explosion?

text: When we light a solid lump of explosive it burns slowly layer by layer, but when we light up powder form of explosive it just burns with a huge explosion and sound why does it happen. And one more question, what fraction of the energy in a blast is lost just in the form of sound?


title: Physics transformation energy

text: A 15kg child slides down a 2.3m -high playground slide. She starts from rest, and her speed at the bottom is 2.1m/s . What is the change in the thermal energy of the slide and the seat of her pants? (2.s.f.) I used the kinematics equation d=[(vi+vf)/2]∗(t) 2.3=2.1⋅t2 t=2.19 Then I used t to find acceleration then multiply acceleration with the mass which is 15kg, and then multiply that again by 2.3m to find the change in thermal energy. But I feel like I calculate t incorrectly, because it keeps telling me it's incorrect. The rest is correct, however, the first part is not correct. I keep getting 30m/s and it is incorrect.


title: Proof of Pauli group preservation by Clifford group conjugation?

text: A well know result is that Clifford group preserve the Pauli group under conjugation or, in other words:   * $C(P_{1} \otimes P_{2})C^{\dagger} = P_{3} \otimes P_{4}$, with $C \in$ Clifford group and $P_{n} \in$ Pauli group. How we can prove this? Thank's...


title: What materials are used in non thermal plasma?

text: While reading about non-thermal plasmas, I came across their ionization potentials(~1%), and other capabilities, such as their non Maxwellian energy distributions. At what temperatures, and pressures do they exhibit such properties, and what materials are used to create non thermal plasma?


title: Exoplanet detection via space-based parasol

text: I remember from watching Cosmos years ago, Carl Sagan suggested an interesting hypothetical method for directly seeing exoplanets. He proposed that in the 'future' we could launch a satellite designed to block the light from distant stars, effectively eclipsing them, perhaps by unfolding a large circular parasol, allowing a powerful telescope to directly detect the light from nearby orbiting planets. Based on this notion I have two questions:   1. Has or is this idea being seriously considered?   2. Would it be possible to position a space based telescope to take advantage of objects that are already in space to eclipse light from a star, perhaps a moon or asteroid? Related: Lack of exoplanet missions in the decadal survey Given a photo of the Moon, taken from Earth, is it possible to calculate the position of the photographer's site?


title: Why did the WMAP mission last so much longer than Planck?

text: NASA endorsed 9 years of data taken with the Wilkinson Microwave Anisotropy Probe (WMAP). The High Frequency Instrument aboard the Planck satellite ran out of coolant at the start of 2012, after about two and a half years of operation. Even if the Low Frequency instrument runs for another two years, its operating life will be much shorter than WMAP's. Why is there such a big difference between the operational life-spans of the two missions, given that they have basically the same objective: to measure the cosmic microwave background?


title: Relations between pressure and temperature

text: I have several questions concerning thermodynamics and I order them in 4 points that may be related:   1. **What's the difference between heat and work at the atomic level**? Isn't heat simply work between particles colliding with different momentum against each other?   2. Consider a chamber containing a gas. Does an increase of pressure also increases the temperature of the gas? **Is that a result of the mechanical action that puts the piston to move delivering energy to the gas through work?**     * But if one is able to displace the piston a little bit so that during this process no collision happens there will be no energy delivered, and the temperature doesn't rise. **Is that possible?**   3. Excluding water and other special materials, **why does a increase of pressure over a solid rises is melting point**?      * My teacher said that pressure would decrease the molecules motion, so in order to melt the solid we should give more energy through heat. Is this correct? If yes, then see the link referred on the $4^{th}$ point (the document says that pressure rises temperature)    4. The inner core of the Earth is solid but is also at higher temperatures than the liquid outer core. I did some research and is often stated that pressure is the reason why the inner core is solid. But returning to the $3^{rd}$ point, **if the pressure reduces the motion of the particles, how can the inner core have material with higher temperatures** (i.e. particles with higher average kinetic energy) **?**     * In this link the article states that **pressure** over the material inside earth **increases the temperature**. However if the pressure doesn't reduce the particles motion, then **how can particles that jiggle so much be in the solid state?**


title: Electrodynamics using exterior calculus

text: Can anyone suggest resource for full course of electrodynamics using exterior calculus?


title: What is the main downfall of using QPSK rather than DQPSK in fiber optics?

text: I've read that QPSK works theoretically in order to encode two bits depending on the phase of the signal. However, practically a reference for the carrier signal would be needed due to some ambiguity of the phase. DQPSK is supposed to solve this by using the previous phase as a reference to the next one. I'm not understanding how the ambiguity of phase occurs in QPSK. Wikipedia alludes to this: > there is an ambiguity of phase if the constellation is rotated by some > effect in the communications channel through which the signal passes. Can anyone explain what occurs in the communications channel that changes this phase?


title: Is there any phenomenon in physics which is sensitive to irrational numbers?

text: We can measure only rational numbers by our scale. Here is an example where irrational numbers does makes sense. If so then this question may have some theoretical importance. Is irrational numbers important for physics?


title: Boundary conditions for crystals

text: As students on solid state physics, we are all taught to use the periodic boundary condition, taking 1D as an example: $\psi(x)=\psi(x+L)$ where $L$ is the length of the 1D crystal. My question is:   1. Why this boundary condition is acceptable? Only because we think we are dealing with the bulk, and the surface are not relevant too much?   2. How about other boundary condations, such as zero boundary condtion, grain boundary condition etc? If ignoring their complexity, can they get the same answer as the periodic boundary condition?   3. How about other artificial boundary conditions, for example, if I let $\psi(x)=2 \psi(x+L)$, could I get the same energy band?


title: How was Newton able to guess that gravitational force is inversely proportional to distance squared?

text: This question is puzzling me since I learnt about the gravitation law in school. Why did Newton guess/assume that gravitational force is inversely proportional to the square of distance? Did he verify that experimentally? (I remember hearing that the first experimental verification of the law of gravitation was after Newton's death.) If the answer to the above question is no, is it for example more plausible to suppose that $F\propto1/r^2$ than to suppose that $F\propto1/r^4$? Did Newton carry out a thought experiment that makes $F\propto1/r^2$ a plausible guess? So in summary: Why did Newton choose exponent of $-2$ instead of any other exponent? Was it a guess that depended on pure luck or an educated guess?


title: Escape Velocity (Rail gun, acceleration, weight)

text: I'm not so technical and not very good with physics, but have some idea based on observation and I believe it is doable. As everyone know also this subject was repeated many times in here, Sending stuff to space is very expensive. But what if there is an easy way to do that only if we would think out side the box. So lets imagine (I do not want to go into many details where did I take it from)... An object of mass 100 kg which is on track isn't affected by gravity. I mean, it does not fall or move, unless it's touched. The only problem is air drag. So, how much energy and what would be the length of the track required to attain escape velocity? The track would be flat with a slide up at (what would be the angle of it?) the end so that the object would reach safe orbit where it would not fall back? I believe object would have to be heat resistant. Is that right? What type of engine (cheapest way) would be applied to accelerate such object? I believe people would not be able to travel inside as G force would be to big? Please describe it in simpler language, as English is not my native.


title: How would an X-ray mirror work?

text: I was wondering if light can be reflected how can someone reflect X-ray of what material does it need to be made of and is its design completely different to that of our original mirrors? Does this mean during long-space voyages in which radiation is an problem why can scientists not develop large panels of X-ray mirrors and Gamma-Mirrors and simply reflect the radiation off rather than worry about that?


title: A few questions about the concept of work

text: From Wikipedia: The work done by a constant force of magnitude F on a point that moves a displacement d in the direction of the force is the product: $$W = Fd.$$ If I lift some object from a ground, the force to be put in above equation is the gravitational force $mg$. But while I am moving the object upwards, against the gravity, I must pull the object with greater force than $mg$, right? So the net force is $F_{myforce}-mg$ which results in object being lifted. But this is not the case. Why? Does it matter how strongly I pull the object? What work will I do if I move this object in absence of gravity (in space)? Will work done be zero or not?


title: Nonequilibrium themal QFT

text: Wick rotation to thermal of QFT in Minkowski space to thermal QFT, which is after this transformation analogue to statistical mechanics, does only describe equilibrium statistical mechanics. On page 227 of this paper it is said, that for dynamical questions beyond the equilibrium properties, the time coordinate has to be analytically continued to real Minkowski time. This I do not understand at all. First of all, does analytic continuation not always generalize a real quantity to a complex quantity? This seems to go the other way round? Can somebody explain a bit why such an analytic continuation allows to describe the evolution of a system and how it works?


title: Fundamental question about the Buckingham $\pi$ theorem (dimensional analysis)

text: I have a rather fundamental question about the Buckingham $\pi$ theorem. They introduce it in my book about fluid mechanics as follows (I state the description of the theorem here, because I noticed in my search on the Internet that there are many different forms of this theorem) : if we have a physical relation between some physical quantities $X_1$, $X_2$, ..., $X_n$: $$\phi(X_1,X_2,...,X_n) = 0$$ Then we can reduce this to a relation between $m$ dimensionless quantities, where $m = n-r$, with $r$ the rank of the dimensional matrix (the matrix formed by tabulating the exponents of the fundamental dimensions of $X_1$, $X_2$,...). So we get $m$ dimensionless quantities: $\pi_1, \pi_2, ..., \pi_m$. And we have: $$\psi(\pi_1, ..., \pi_m) = 0$$ These dimensionless quantities are formed by taking a core group of $r$ physical quantities from the $n$ quantities $X_1, ..., X_n$ (for instance $X_1, ..., X_r$). Now we can form: $$\pi_1 = X_1^{a_1}*X_2^{b_1}*...*X_r^{f_1}*X_{r+1}$$ $$\pi_2 = X_1^{a_2}*X_2^{b_2}*...*X_r^{f_2}*X_{r+2}$$ and so on. These exponents are now determined so that the $\pi_1, \pi_2, ..$ are dimensionless. So this was the description of the Buckingham $\pi$ theorem they mention in my book. First of all I wonder if this description is correct and complete, because I saw many different forms on the Internet that make the understanding of the theorem rather confusing. Secondly, I tried to apply this theorem (in a way of getting familiar to it) to Newton's universal law of gravity (in a way of trying to derive the law, by the Buckingham $\pi$ theorem): $F = \frac{Gm_1m_2}{r^2}$. So while doing this, I see that there is a relation between $F, m_1, m_2$ and $r$ : $\phi(F,m_1,m_2,r) = 0$. The rank of the dimensional matrix $r$ is 3. So we have only one dimensionless number $\pi$. But if I take now as core group $m_1, m_2, r$, there is no way of making this number dimensionless (in the description of the theorem they don't mention that this can be the case). So I have to include $F$ in the core group and if I take $F,m_1, m_2$ as core group then $\pi_1 = m_1/m_2$ and I get $\phi(m_1/m_2) = 0$. This makes no sense to me, so I guess something is wrong in the way I understand this theorem, and I hope someone can explain to me what I'm missing.


title: How does one get these definitions of the energy momentum tensor?

text: I was just reading a book - Mirror Symmetry by Clay Mathematics Institute, and on Page 402 of the book, the writer says that energy momentum tensor is defined classically by $$\delta S = -\frac{1}{4 \pi} \int \sqrt{h} d^2 x \delta h^{\mu\nu}T_{\mu\nu}$$ and quantum mechanically by $$\delta_h \langle O\rangle = \left\langle O \frac{1}{4\pi} \int \sqrt{h} d^2 x \delta h^{\mu\nu}T_{\mu\nu}\right\rangle$$ **Here is my question:** I know that by Einstein-Hilbert action one can get $T_{\mu\nu} = 2 \frac{\delta \mathcal{L}}{\delta h^{\mu\nu}} $. But how can I get the exact form of $\delta S$ as above? Or $\delta_h\langle O\rangle$.


title: A thought experiment on vision and curved spacetime

text: What follows is a long self-made example to deal with my conceptual issues of visualizing curved spacetime. Imagine an observer floating somewhere in space. He feels no strain on his body, indicating he is free-falling and is thus moving along a geodesic. He is a magician and has the power to conjure up glowing sticks within his hands. Their construction is as follows (the following are valid in his vicinity of course): 1) Each stick is very thin and of 1m in length when he conjures it in his hands. 2) Each stick is rigid and mass-less 3) It glows along it's cylindrical surface uniformly. The glow varies with intensity such that it goes from complete dark to peak brightness to dark in 1 heartbeat of the magician in his local rest frame. It is thus like a clock synchronized with his heartbeat. The magician starts conjuring up billions of these sticks in his vicinity and starts assembling them into a cubic lattice in his vicinity. As he starts assembling the lattice, it grows in size and moves further out far beyond his vicinity. He effectively creates a crude cartesian co-ordinate system (without any markings). In curvature free space-time, his eyes will project the glowing grid lines as straight lines intersecting at infinity and the whole system will glow in the same way. Now if space-time had some arbitrary curvature (not too high), how would the glowing grid 'lines' be projected onto his eyes? Alternatively he could create a spherical system by making concentric glowing shells of increasing radii (where the latitudes and longitudes glow) made with uniform curvature in his vicinity. He could fix the spatial origin as wherever he wants. In flat space-time, the concentric shells would be projected onto his eyes as concentric circles. What would he see in curved space-time? Is this how experimenters construct their 'imaginary' co-ordinates systems for work in astro-physics?


title: Is mechanics needed in order to learn electromagnetism?

text: I'm a college student who wants to learn physics. I want to get started with electromagnetism as soon as possible. Is it okay to skip over mechanics, or is mechanics a big requirement before starting electromagnetism?


title: Law of attraction

text: Could you please explain that since we know Newton's law of Universal Gravitation says all masses attract each other. Thus, we humans should be attracted as well or any other daily life objects. Why is it that we don't feel that attraction to be prominent?


title: Transmission Line, TEM wave, transverse Laplacian operator (electromagnetic)

text: _Time-Harmonic Electromagnetic Fields_ (1961) by R. F. Harrington, ISBN 07-026745-6, p. 63, he discusses TEM mode in a transmission line. The wave travels in the z direction. He defines the transverse Laplacian operator as the standard Laplacian though without the z partial derivatives. He then says that the transverse Laplacian for both **E** and **H** is zero for a TEM wave, i.e., where neither **E** nor **H** components in the z direction. This is equation (2-71) on p. 63. I cannot see this for the life of me. Any thoughts would be appreciated. BTW, I am **not** a student. I did EM graduate work about 25 years ago, but then my employer, who paid for that education, assigned me to other duties, and I stopped doing EM work. I am doing this now for fun. Thanks.


title: Are $G$, $F$ and $H$ (thermodynamics potentials) extensive quantities?

text: Internal energy $E$ is an extensive quantity for most systems. But energy extensivity is not valid in systems with long-range interactions, like gravity (e.g. in astrophysical systems). For extensive systems: $$E=TS+\bf J. \bf x +\bf{\mu}.\bf N\tag{1}$$ where $\bf J$ and $\bf x$ are intensive and extensive thermodynamic variables. (like pressure and volume, respectively) Some other thermodynamic potentials like $G$, $F$, and $H$ are defined as : $$F=E-TS$$ $$G=E-TS-\bf J.\bf x$$ $$H=E-\bf J.\bf x$$ For extensive systems, physical meanings are obvious, as is written in Wikipedia: > we can say that ΔU is the energy added to the system, ΔF is the total work > done on it, ΔG is the non-mechanical work done on it, and ΔH is the sum of > non-mechanical work done on the system and the heat given to it. Apparently these interpretations are deduced from equation $(1)$, which doesn't hold in non-extensive systems. So it seems that these potentials are ,inherently, extensive quantities, and have no meaning in non-extensive systems. Do these potentials have a physical meaning (if applicable at all) in non- extensive systems like galaxies?


title: Definition and Measurment of Energy and Heat in Thermodynamics

text: The concept of work is clear from mechanics, where $dW=F\cdot dx$, leading to $dW=PdV$. Also different forms of energy are defined in mechanics, e.g., $E=\frac{1}{2}m v^2$, etc. How are the energy and heat defined and measured in thermodynamics? Because once we define them independently, we can state the first law of thermodynamics, $d E=\delta Q+\delta W$.


title: How much energy is needed to heat your body in water?

text: How can we calculate (or approximate) the energy (in kJ) needed by our body to keep it constant at $36.6^{\circ}C$ in a swimming pool at $\sim 28^{\circ}C$ for 30 minutes? Body is naked and fully submerged.


title: Where does energy in electricity come from?

text: In a simple battery-wire-lightbulb circuit the moving electrons in a wire generate electromagnetic field, and that field transports energy. And the bulb goes light. But where does the energy come from in the first place? It obviously cannot be 'generated' because of the conservation law, so it has to previously be some other form of energy? Is the energy photons radiated by moving charges? But the electrons would be going down energy levels, and then somehow going up again, only to 'discharge' all over again. And I get from the pop-sci articles all over the web that it's exactly what does _not_ happen, an electron is not an 'energy bucket'. But then again it sort of has to be, as it actually loses the energy, via heating the tungsten filament in the lightbulb? In most (all? what about moving a magnet in a coil?) circuits we'd have a potential difference, and I read somewhere there is a potential energy associated with it. Is it all this energy's 'fault'? Does it get transformed into kinetic energy that gets the charged particles moving, and also in bigger part to the E-M field energy 'bound' to these charges? Is there a mechanism through which this energy transformation is done, or is it a stupid question, it just gets transformed and that's it? What if we move a magnet inside a coil - then we get particles moving, but no potential difference (or is it an invalid conjecture, and PD is there somewhere?) It's all very confusing, sorry once again if these questions are stupid.


title: What kind of vector field is this

text: I realize that I've been asking the wrong question and perhaps has nothing to do with vector fields. The question is about fluids flowing through a homogeneous media, say ocean currents; how do you express this in math or even define just the vector field in such a situation.


title: Why does your reflection suddenly turn upside down while slowly walking away from a curved mirror?

text: Many people should have realised, when looking into a concave curved mirror (or even a rather reflective spoon in that fact) at a close distance, you will see a slightly distorted reflection. But as you move further away, the image will suddenly become upside-down. why is this?


title: Would a beam of light moving through a vacuum be considered an example of perpetual motion?

text: I have a very limited understanding of Physics but if the beam of light was moving through a vacuum like space and had saw no obstuctions during its path, would that be perpetual motion in action?


title: Vacuum energy: black hole evaporation and cosmology - a discrepancy?

text: Black hole evaporation is a result of calculating the expectation value of the stress-energy tensor of the state of the vacuum in certain spacetimes and then making a plausibility argument as to the effect backreaction on the spacetime. It is generally believed that the resulting evaporation times for black holes are in good agreement with reality (though natch there is no physical evidence to test this against). In cosmology a naive calculation of the expectation value of the stress-energy tensor of the vacuum state and it's effect on spacetime qualitatively is in agreement with reality, in as much as it predicts a positive cosmological constant, but is quantitatively absolutely hopeless in that the predicted cosmological constant is many orders larger than the observed value. Why is is expected that calculations involving the vacuum state and its effect on spacetime are quantitatively in good agreement with reality in the first situation when it is known they are not in the second situation?


title: Is rotational motion conditioned to a central force?

text: We know rotational motion as a combination (a resultant) of two effects the tangential velocity and a centripetal force. Does rotational motion turn into linear motion at the same instance this centripetal force goes absent, and that rotational motion can't be a natural motion caused by one single effect, and linear motion is the only (raw) motion present in the universe? We take an example, a spacecraft going to the moon (or anywhere into outer space). That spacecraft will leave the earth's atmosphere, and continue to move in a spiral motion outwards, increasing in radius, due to the momentum acquired by the earth's rotation. At some point that spacecraft should lose (if I'm correct) this rotational motion and shoot into a straight line, tangent to the last circle whose radius the distance from the earth's center. That's because the gravity (centripetal force) is no longer an acting force. Can this be true?? There's also a scene from the movie "Gravity" I didn't quite absorb. The astronaut keeps moving around when they separate from a rotating object and continue to move that way, (very famous scene in the trailer), is this accurate?


title: Perturbation of a Hydrogen Atom in a Quadrupole Field

text: > `Question`: A hydrogen atom is located in a quadrupole field, which gives it > a perturbation $$H_1=A(x^2-y^2)$$ where $A$ is some constant. Calculate the > quantity $\langle 211|H_1|21\bar{1}\rangle$, one of the two nonzero elements > of the degenerate perturbation matrix. `Attempt`: First, I took $R^2\equiv x^2+y^2$, thus allowing my to simplify the perturbation: $$\frac{H_1}{R^2}=A\left (\frac{x^2}{R^2}\right ) -\left (\frac{y^2}{R^2} \right)$$ $$=A\sin^2(\theta)(\cos^2(\phi)-\sin^2(\phi))=\sin^2(\theta)\cos(2\phi)$$ $$=\sin^2(\theta)\frac{1}{2}(\exp(i2\phi)+\exp(-i2\phi))=N(Y_{22}+Y_{2\bar{2}}) $$ where $Y_{\ell\,m}$ are the spherical harmonics and $N$ is a normalization constant to be determined. Now, when I calculate the matrix element, I obtain $$A\langle 211|(x^2-y^2)|21\bar{1}\rangle=AN\langle 211|Y_{22}+Y_{2\bar{2}}|21\bar{1}\rangle $$ $$=AN\int dr\,d\theta\,d\phi r^2 \sin(\theta) R_{21}^*Y_{11}^*(Y_{22}+Y_{2\bar{2}})R_{21}Y_{11} $$ Shouldn't this be zero from orthogonality conditions? It doesn't seem to make sense that it should be zero. What did I do incorrectly? Any help would be appreciated.


title: Heat transfer in a cylinder with two insulators

text: I'm used to problems which ask me to find the heat transfer when, for example I have a very long cylinder covered with an insulator, each with their respective conductivity coefficient. I'd use the formula $\frac{\partial Q}{\partial t}=\int -k\vec \nabla T \vec{ds}$ and integrate with $\rho$ varying. But now I have a situation where the cylinder is covered with two insulators, one on the left half of it, and the other one on the right. So I don't know how to use the previous formula here, as $k$ doesn't vary with $\rho$ only, but also with $\varphi$. I found this very same problem on a web but, it didn't use that formula. Instead, an electric-like circuit was built, and so on (I'm NOT supposed to solve it this way) Any advice? Thanks


title: What are the forces acting between two air bubbles in water?

text: The exact question is > Two air bubbles in water > >   1. attract each other >   2. repel each other >   3. do not exert any force on each other >   4. may attract or repel depending upon the distance between them. > The chapter is about gravitation. The given answer is > A lighter body inside a denser medium behaves like negative mass as far as > gravitational force is considered. Two air bubbles i.e. two negative masses > will attract each other. What is negative mass in this context and how can it be applied to such macroscopic objects? How would it result in attraction? My reasoning is: ![enter image description here](http://i.stack.imgur.com/PGEyt.png) Consider the bubble A in the above image. The air particles forming the bubble A would be attracted more to the left(away from B) as there are more dense particles towards that side-the air particles making up bubble B are less dense than the medium and they will attract the air from bubble A to a smaller extent than if the volume of bubble B was filled with the medium. A similar case would apply to B due to lesser density of particles forming A and the bubbles would be (indirectly) repelled. So what _is_ happening in this case?


title: Non-equilibrium Green functions

text: How do we use non-equilibrium Green's functions (NEGF) or the Keldysh formalism in the theory of quantum transport? Please take a simple example like the Hopping model with a non-equilibrium perturbation that couples two lattice sites for example, with increased hopping strength between those two particular lattice sites and explain in that context. Show with this simple example how we use the NEGF or the Keldysh formalism.


title: Adiabatic theorem and Berry phase

text: As far as I can check, the adiabatic theorem in quantum mechanics can be proven exactly when there is no crossing between (pseudo-)time-evolved energy levels. To be a little bit more explicit, one describes a system using the Hamiltonian $H\left(s\right)$ verifying $H\left(s=0\right)=H_{0}$ and $H\left(s=1\right)=H_{1}$, with $s=\left(t_{1}-t_{0}\right)/T$, $t_{0,1}$ being the initial (final) time of the interaction switching. Then, at the time $t_{0}$, one has $$H\left(0\right)=\sum_{i}\varepsilon_{i}\left(0\right)P_{i}\left(0\right)$$ with the $P_{i}$'s being the projectors to the eigenstates associated with the eigenvalue $\varepsilon_{i}\left(0\right)$, that we suppose known, _i.e._ $H_{0}$ can be exactly diagonalised. Then, the time evolution of the eigenstates is supposed to be given by $$H\left(s\right)=\sum_{i}\varepsilon_{i}\left(s\right)P_{i}\left(s\right)$$ which is fairly good because it just requires that we are able to diagonalise the Hamiltonian at any time, what we can always do by Hermiticity criterion. The adiabatic theorem (see Messiah's book for instance) $$\lim_{T\rightarrow\infty}U_{T}\left(s\right)P_{j}\left(0\right)=P_{j}\left(s\right)\lim_{T\rightarrow\infty}U_{T}\left(s\right)$$ with the operator $U_{T}\left(s\right)$ verifying the Schrödinger equation $$\mathbf{i}\hslash\dfrac{\partial U_{T}}{\partial s}=TH\left(s\right)U_{T}\left(s\right)$$ **can be proven exactly if** $\varepsilon_{i}\left(s\right)\neq\varepsilon_{j}\left(s\right)$ at any time (see _e.g._ Messiah or Kato). Now, the Berry phase is supposed to be non vanishingly small when we have a parametric curve winding close to a degeneracy, _i.e._ precisely when $\varepsilon_{i}\left(s\right) \approx \varepsilon_{j}\left(s\right)$. For more details, Berry defines the geometric phase as $$\gamma_{n}\left(C\right)=-\iint_{C}d\mathbf{S}\cdot\mathbf{V}_{n}\left(\mathbf{R}\right)$$ with (I adapted the Berry's notation to mine) $$\mathbf{V}_{n}\left(\mathbf{R}\right)=\Im\left\\{ \sum_{m\neq n}\dfrac{\left\langle n\left(\mathbf{R}\right)\right|\nabla_{\mathbf{R}}H\left(\mathbf{R}\right)\left|m\left(\mathbf{R}\right)\right\rangle \times\left\langle m\left(\mathbf{R}\right)\right|\nabla_{\mathbf{R}}H\left(\mathbf{R}\right)\left|n\left(\mathbf{R}\right)\right\rangle }{\left(\varepsilon_{m}\left(\mathbf{R}\right)-\varepsilon_{n}\left(\mathbf{R}\right)\right)^{2}}\right\\} $$ for a trajectory along the curve $C$ in the parameter space $\mathbf{R}\left(s\right)$. In particular, Berry defines the adiabatic evolution as following from the Hamiltonian $H\left(\mathbf{R}\left(s\right)\right)$, so a parametric evolution with respect to time $s$. These are eqs.(9) and (10) in the Berry's paper. Later on (section 3), Berry argues that > The energy denominators in [the equation for > $\mathbf{V}_{n}\left(\mathbf{R}\right)$ given above] show that if the > circuit $C$ lies close to a point $\mathbf{R}^{\ast}$ in parameter space at > which the state $n$ is involved in a degeneracy, then > $\mathbf{V}_{n}\left(\mathbf{R}\right)$ and hence > $\gamma_{n}\left(C\right)$, is dominated by the terms $m$ corresponding to > the other states involved. What annoys me is that the Berry phase argument uses explicitly the adiabatic theorem. So my question is desperately simple: **what's the hell is going on there?** Can we reconcile the adiabatic theorem with the Berry phase elaboration? Is the Berry phase a kind of _correction_ (in a perturbative expansion sense) of the adiabatic theorem? Is there some criterion of _proximity to the degeneracy_ that must be required in order to find the Berry phase? REFERENCES:   * Berry, M. V. _Quantal Phase Factors Accompanying Adiabatic Changes._ Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, **392** 45–57 (1984).   * Kato, T. _On the Adiabatic Theorem of Quantum Mechanics._ Journal of the Physical Society of Japan, **5** 435–439 (1950).    * Messiah, A. _Mécanique quantique_ Dunod (French reprint of the 1958 edition) (1995).


title: Riemann invariants....Any physical interpretation?

text: I am really new to the CFD simulation, and started some simple algorithms recently. I then got introduced to the Riemann Invariants.   1. Can any one provide some physical interpretation?   2. Also, why is it the case, that when we have an open tube, and the flow is entering with a subsonic speed, then at this point, only one characteristic exist dx/dt=u+a ?


title: What is *uplift* in respect to extra dimensions and their stability?

text: What is _uplift_ in respect to extra dimensions and their stability? It's notoriously hard to find something on this, as all possible keyword combinations pull up plethora of unrelated Google hits.


title: Electromagnetic 4-potential and basic index contraction

text: I'm trying to learn about relativistic electrodynamics on my own, and I am struggling with derivatives of the 4-potential and index (Einstein) notation. I think I understand expressions such as $\partial_\mu A^\mu$. The index is repeated and is once up and once down, so I would expand the sum as: $\partial_0 A^0 + \partial_1 A^1 + \partial_2 A^2 + \partial_3 A^3$, which gives me a scalar.   1. How am I to interpret something like this: $(\partial_\mu A_\nu)(\partial^\mu A^\nu)$ ? We are summing over the two indices this time, which is fine. What confuses me is that we are taking a covariant derivative of a covariant vector. Does one need to "convert" $A_\nu$ in the first term to contravariant, like so: $(\partial_\mu A^\rho \eta_{\nu\rho})(\partial^\mu A_\sigma\eta^{\nu\sigma})$? I guess my doubts arise from the fact that I see a covariant vector as being an entirely different object from a contravariant one. The covariant derivative $\partial_\mu = \frac{\partial}{\partial x^\mu}$ differentiates with respect to the components of the contravariant vector $x$. So I don't understand how such an operation can be applied to a vector that isn't also contravariant.   2. How should I interpret terms such as $(\partial_\mu A^\mu)^2$ ? Is it just $\left(\partial_0 A^0 + \partial_1 A^1 + \partial_2 A^2 + \partial_3 A^3\right)^2$ or is there something else going on?    3. According to some textbook, $(\partial_\mu \phi)^2 = \eta^{\mu\nu}\partial_\mu \phi\partial_\nu\phi$, but I don't understand why. For me $\partial_\mu \phi$ is just the derivative of a scalar $\phi$ with respect to some (unspecified) component $\mu$ of a contravariant 4-vector $x$. Instead, judging from the right-hand side, it is to be interpreted as a vector $(\frac{\partial}{\partial x^0},\boldsymbol{\nabla})\phi$ which is then squared. Is it just sloppy notation, or am I being stupid? Thanks. EDIT:   1. Are the following then true? $$\frac{\partial}{\partial(\partial_\mu A_\nu)} \left(\partial_\mu A_\nu\right) = 1$$ $$\frac{\partial}{\partial(\partial_\mu A_\nu)} \left(\partial^\mu A^\nu\right) = 0$$   2. Can I also raise and lower the indices of a partial derivative?


title: Do communicating vessels work when water is moving?

text: I want to know if the water level equalize in communicating vessels, when in one of the vessels there is a vortex, or any other water movement, and if the level does equalize, what will be the level? I made a simple drawing: ![enter image description here](http://i.stack.imgur.com/8aKU6.png) Here's a link for full size: http://i49.tinypic.com/2gsn85h.jpg Thanks!


title: Derivation of Green's Function for Wave Equation

text: In the textbook _Modern Methods in Analytical Acoustics_ (Crighton-1992) the following relates the 3D Green's function in the time-domain to the frequency domain g(x-y): ![enter image description here](http://i.stack.imgur.com/f24jO.png) I cannot see how the integration has eliminated the variable c. To me the integration should leave $e^{i k_0 r}$ where $k_0=\omega/c$ and then the answer $$ g(x-y) = -\frac{c}{4 \pi r} e^{i k_0 r} $$ which I know to be incorrect by a factor of c. Is the text wrong? and if so, then how do I derive the expression for g(x-y)? Note: the $-c^2$ factor is used to relate the Green's function $G(x,y,t) = \frac{\delta(r-ct)}{4 \pi c r}$ for the equation $(\frac{\partial^2}{\partial t^2} - c^2 \nabla^2) G(x,y,t) = \delta |x-y|$ to the reduced wave equation: $$ ( \nabla^2 + k_0^2) G(x,y,\omega) = -\frac{1}{c^2} \delta |x-y| $$


title: Why are free electrons always moving?

text: Electrons move when a field acts on them. If the electrons move towards the field, they cancel out the field when enough electrons build up. Shouldn't the free electrons stop moving eventually and reach a stable system where there is no field acting on any electron? I would expect them to keep moving until they build up to cancel out the field acting on other electrons resulting in the system being static. Please don't make your answers too technical, I don't know a lot about physics. EDIT: I have no knowledge of quantum mechanics but I believe the random motion I'm talking about is called the Fermi velocity.


title: Are Born-Oppenheimer energies analytic functions of nuclear positions?

text: I am looking for references to bibliography that explores the smoothness and analyticity of eigenvalues and eigenfunctions (and matrix elements in general) of a hamiltonian that depends on some parameter. Consider, for example, the original setting of the Born-Oppenheimer approximation, to molecular dynamics, where the nuclear wavefunction is momentarily ignored and the hamiltonian becomes parametrized by the positions $\mathbf{R}_m$ of the nuclei, $$ \hat{H}(\mathbf{R}_m)=-\sum_{i=1}^N \frac{\hbar^2}{2m}\nabla^2_i+\sum_{i>j}\frac{e^2}{|\mathbf{r}_i-\mathbf{r}_j|}-\sum_{i,m}\frac{Z_m e^2}{|\mathbf{r}_i-\mathbf{R}_m|}. $$ The energies $E_n(\mathbf{R}_m)$ then become functions of all the nuclear coordinates and therefore make up the energy landscape which governs the nuclear wavefunctions' evolution. Since the original appearance of the $\mathbf{R}_m$ is in the analytic (well, meromorphic) functions $\frac{1}{|\mathbf{r}_i-\mathbf{R}_m|}$, I would expect further dependence on the $\mathbf{R}_m$ to be meromorphic (and would definitely expect physical meaning from poles and branch cuts). What I am looking for is references to bibliography that will establish or disprove results of this type in as general a setting as possible. In particular, given a hamiltonian that depends on a set of parameters $z_1,\ldots,z_m$ in a suitably defined analytic way, I would like to see results establishing the analyticity of matrix elements (and thus, for example, of eigenvalues) involving the eigenvectors of the hamiltonian. I would also be interested in knowing what quantities can be extended analytically to the complex plane. Any and all pointers will be deeply appreciated.


title: Entangled electron-positron pair

text: Usually when we talk about entanglement, we mean entangled spin states (of electrons) or polarizations (of photons). My questions are: Does pair production guarantee the product electron and positron entangled? If there's no observer measuring either particle, can we say the types, or charge, of the particles are also entangled, with a wavefunction like: $\frac{1}{\sqrt{2}}( |+e\rangle \pm |-e\rangle)$?


title: energy production in the nuclear fission

text: ![enter image description here](http://i.stack.imgur.com/FEj0S.jpg) according to this image of nuclear fission, before fisson reaction, the total mass of uranium and the slow neutron is 236 amu. and after adding the masses of Barium, krypton and the 3 neutrons after the fission reaction, you will again get 236 amu. Question: if we get back all the mass, where did energy come from? is it the binding energy that is released?


title: Kerr throat solution derivative

text: I'm going through this article, since I'll need a part of it for my thesis. And I am trying to derive the Kerr throat solution, from which I should be able, with the change of coordinates get to near-horizon extreme Kerr metric. I am following the article, and so far I got the $dt^2$, $dr^2$ and $d\theta^2$ part, exactly like in the article, but when I try to get the $\left(d\phi+\frac{r^2}{r_0^2}dt\right)^2$ part, I get stuck. First I tried by putting already extremal expression for $\omega$, but then I get the correct result without(!) $dt$ part :\ All the mixed terms ($dtd\phi$) and $dt^2$ parts cancel each other out. So I figured, ok, maybe I need to put everything in, before I let $\lambda\to 0$, and see if things will cancel each other out, but in that case my expression becomes infinite! (I either have extra $1/\lambda$ or $1/\lambda^2$ term that don't cancel). What am I doing wrong here? Is there something I'm missing?


title: Does the expression of the orbital magnetic dipole moment have $c$?

text: The orbital magnetic dipole moment of a particle with mass $m$ and charge $q$ can be shown to be related to the orbital angular momentum through the equation $$\displaystyle \boldsymbol\mu_L=\frac{q}{2m}\bf{L}.$$ One of the quantum mechanics textbooks has instead the same equation but with the speed of light in the denominator, $$\displaystyle \boldsymbol\mu_L=\frac{q}{2mc}\bf{L}.$$ The same book also defined the spin magnetic dipole moment for the electron with a $c$, $$\displaystyle \boldsymbol\mu_S=-g\frac{e}{2m_ec}\bf{S},$$ where $g$ is the usual Lande factor (~2). Is the speed of light appearing in those equations just a typo? (but it is not in the book errata)


title: How does time dilation work without a privileged reference frame?

text: As I understand special relativity, light travels at the same speed in all reference frames. What I fail to understand is why time dilation would occur in one reference frame, but not by an equal amount in all other reference frames, at the same time. To put it another way, if I am sitting on earth, observing a space ship that is flying at a speed near $c$, I should observe him moving through time slower than me. However, looking out his window, he should observe me traveling relative to him at a speed near $c$ and should likewise see me moving through time slower than himself. What am I missing from my understanding of special relativity?


title: Are there still Big Bang Gravitational effects

text: If all matter was concentrated at one point before the big bang, I assume that the gravity was at a huge scale. For a long time many believed that the universe would eventually collapse back upon itself, until it was discovered that the speed at which the universe is expanding is actually increasing. So my question is.... Is there still a strong gravitational force from the big bang which is acting on the universe to try and contract it again to a single point? If not where is the gravity coming from that would cause the universe to collapse again? Is it just all the collective matter including dark matter and the gravitational pull from all matter to itself? And the only thing stopping it is the increasing speed of expansion?


title: Book recommendations for fluid dynamics self study

text: I have my bachelors degree in engineering science: computer science, option electrotechnic. I have a good understanding of basic mechanics and dynamics, calculus (differential equations). The books I'm now having a look at are:   * _Fox and McDonald's Introduction to Fluid Mechanics, 8th Edition_   * _Fundamentals of Aerodynamics, John D. Anderson_ What book do you recommend for studying fluid dynamics on my own and why?


title: Is Mach's Principle Wrong?

text: This question was prompted by another question about a paper by Woodward (not mine). IMO Mach's principle is very problematic (?wrong) thinking. Mach was obviously influenced by Leibniz. Empty space solutions in GR would result in a Minkowski metric and would suggest no inertia. Mach's principle seems incompatible with GR. Gravitational waves could also be a problem. I had thought that papers like one by Wolfgang Rindler had more or less marginalised the Mach Principle, but I see lots of Internet discussion of it. Is it correct? Wrong? Is there evidence? (frame dragging experiments)? Let's use this definition from ScienceWorld.Wolfram.com: > _In his book The Science of Mechanics (1893), Ernst Mach put forth the idea > that it did not make sense to speak of the acceleration of a mass relative > to absolute space. Rather, one would do better to speak of acceleration > relative to the distant stars. What this implies is that the inertia of a > body here is influenced by matter far distant._


title: Propagators and Probabilities in the Heisenberg Picture

text: I'm trying to understand why $$\Bigl|\langle0|\phi(x)\phi(y)|0\rangle\Bigr|^2$$ is the probability for a particle created at $y$ to propagate to $x$ where $\phi$ is the Klein-Gordon field. What's wrong with my reasoning below? We can write $$\phi(x) = \int \frac{\textrm{d}^3\mathbf{p}}{(2\pi)^3\sqrt{2E_{\mathbf{p}}}}(a_{\mathbf{p}}e^{-ip.x}+a_{\mathbf{p}}^{\dagger}e^{ip.x})$$ Then acting with the position operator (which presumably we can write in terms of the $a$ somehow) we find that $$\mathbf{\hat{x}}\phi(x)|0\rangle=\mathbf{x}\phi(x)|0\rangle$$ This tells us that $\phi(x)|0\rangle$ can be interpreted as a particle at $x$. Now define a state $|\psi(t)\rangle$ with $$|\psi(0)\rangle=\phi(y)|0\rangle$$ We have time evolution defined by $$|\psi(t)\rangle=e^{iHt}|\psi(0)\rangle=e^{iHt}\phi(y)|0\rangle$$ Pick some point $x=(T,X,Y,Z)$ (a four vector). The probability that the particle propagates from $y$ to $x$ is precisely the probability that we find $|\psi(T)\rangle$ in the state $\phi(x)|0\rangle$. But by the measurement postulate this is just the square of $$\langle0|\phi(x)e^{iHT}\phi(y)|0\rangle$$ Now I don't know how or why to get rid of the $e^{iHT}$. Since $H$ doesn't commute with $\phi$ we seem to be a bit stuck! Am I missing something simpler than this? Many thanks in advance!


title: What came first, Rice Crispy or "Snap," "Crackle," and "Pop"?

text: The fourth, fifth, and sixth derivatives of position are called "Snap" "Crackle" and "Pop". What came first, the rice crispy characters, or the physics units?


title: Alpha Centauri Bb: Comparing astrometric precision vs doppler precision

text: How do you compare astrometric precision vs doppler spectroscopic precision? In particular will the Gaia spacecraft to be launched next year be able to confirm the existence of Alpha Centauri Bb? I.e. is Gaia's astrometric precision at least as great as the precision of the doppler spectroscopy that was used to discover the planet?


title: Do all vacua in the string theory landscape have a different cosmological constant?

text: Or can two vacua with the same energy differ in other ways?


title: How can the electromagnetic stress energy tensor be restricted to flat space-time

text: The Wikipedia article describing the electromagnetic stress energy tensor seems to suggest that this tensor can only be defined in flat space-time. How is it possible to define an electromagnetic stress energy tensor this way since any available electromagnetic energy/momentum must render the space-time curvature nonzero? How in practice would someone extract useful information with this stress energy tensor?


title: Determining fluid flow velocity from experimental volumetric flow rate data

text: I would be very grateful if someone helps me with my issue. I have a pump pumping water into a tube. My goal is to find **velocity** of the fluid flow in another tube that will be connected to the first one. To determine the **flow rate** in the first tube, I used a ~$1000~ml$ container and a stopwatch. I obtained the **mean volumetric flow rate** , let it be about $100~ml/s$. I use the following relation for **laminar flow** as a fisrt approximation: $$ V_1=\frac{Q_1}{S_1}, $$ where $V_1$ is **flow velocity** ($m/s$), $Q_1$ is **volumetric flow rate** ($\text{m}^3/s$), and $S_1$ is inner area of the first tube ($\text{m}^2$). Let the inner radius of the first tube be $4~mm$, so $S_1=\pi \cdot 0.004^2 \approx 5 \cdot 10^{-5}~\text{m}^2$. Thus, we have $$ V_1=\frac{100~ml/s}{5 \cdot 10^{-5}~\text{m}^2}=\frac{1 \cdot 10^{-4}~\text{m}^3/s}{5 \cdot 10^{-5}~\text{m}^2}=2~m/s. $$ Then I connect the second tube, which has $6~mm$ inner diameter, to the first one ($8~mm$ inner diameter). Outer diameter of the second tube is equal to the inner diameter of the first tube. I use a pump, so I assume flow rates $Q_1$ and $Q_2$ to be equal. From $Q_1=Q_2$ I get $V_1S_1=V_2S_2$, and $$ V_2=\frac{V_1S_1}{S_2}=\frac{2\cdot0.004^2}{0.003^2}\approx3.6~m/s. $$ The result looks close to what I have seen, but I want to know the Reynolds number to determine if the flow is laminar or turbulent: $$ \textrm{Re}=\frac{\rho V_2 D_H}{\eta}, $$ where $\rho$ is density of the fluid ($kg/\text{m}^3$), $D_H$ is hydraulic diameter of the second tube ($m$; we let $D_H$ be equal to geometric diameter of the tube $D_2$), and $\eta$ is dynamic viscosity of the fluid ($N\cdot s/\text{m}^2$). In our case, $\rho\approx974~kg/\text{m}^3$ and $\eta\approx 422\cdot10^{-6}~N\cdot s/\text{m}^2$, so $$ \textrm{Re}\approx\frac{974 \cdot 3.6 \cdot 0.008}{422\cdot10^{-6}}\approx66500, $$ which is way more than $2000$, so the flow is **turbulent** , and the first approximation of the flow velocity **should be corrected**. If I have experimental data of the **volumetric flow rate** in the first tube, then how can I find the flow velocity in the second tube in the case of turbulent flow? In the literature, they usually use volumetric flow rate in the case of laminar flow only. In some sources, equation $Q_1=Q_2$ ($V_1S_1=V_2S_2$) is used for pumped flows without indication of the flow type (laminar or turbulent). I am a bit confused.


title: Energy density in Magnetic fields?

text: I don't understand what does magnetic field energy density stand for? The possible work that the magnetic field can do? So we can calculate the amount of "work" it can do just from the energy density? Example: A magnetic field is created by a solenoid, there is a energy density of 10J/m3. From that can I derive the magnitude of magnetic force using work = F*d Thus, it would be F = W/d?


title: Synchronization phenomenon: A simple explanation?

text: Being from a mathematical background, physicists' intuitive arguments always seemed challenging for me to follow. I am currently reading a book called "Synchronization: A Universal Concept in Nonlinear Sciences". The book is well-written but from time to time, I have hard time following the intuitive arguments. In its most simple case the authors discuss the effect of a periodic external force on a self-sustained quasilinear oscillator. I have to emphasize that these are arguments in chapter 3 of the book. Things get more formal in chapter 7. The only problem is that in chapter 7 there is no argument about potentials anymore. In chapter 3 the argument mainly relies on a few parameters, $\overline\phi_e$ being the initial phase of the forcing oscillator which has the following phase relation $\phi_e=\omega t+\overline\phi_e$ and the driven oscillator is based on $x(t)=A\sin(\omega_0 t+\phi_0)$ and the external force varies based on $\epsilon \cos(\omega t+\overline\phi_e)$. The following points are the main things that I don't understand:   1. What does potential mean?(I know that potential function defines an invariance most of the times, but how it is defnied for oscillators case?    2. What are big parallel vectors and the small ones in figure 3.3(a) representing?   3. Why they phase locking take place in $\phi_0$   4. Why point 1 is stable and point 2 is not? I imagine stability means the point with minimum level of potential. I hope the answer to this question becomes clear if I get an answer for the first question. The following figures might be helpful for this clarification: ![enter image description here](http://i.stack.imgur.com/pHyPj.png) ![enter image description here](http://i.stack.imgur.com/AH8ju.png)


title: Wave vector $\vec{k}$ vs position vector $\vec{x}$

text: My question is about the $k$-vectors in first Brillouin zone. If I am not misunderstood, the relation k = 2π/(Na) tells that when k goes to zero, we are very very far away from the reference atom and when k = 2π/a, we are one lattice constant (a) away in real space from reference atom. Now, for nearest neighbour tight binding approximation, the energy at k = 0 is (E-2t) and that at k = π/a is (E+2t). With respect to k = 0 eigenvalue, how do i make the connection in real space picture. Can we say that when we are infinitely far away from our reference atom, the energy is E-2t? If we cannot, then how should we interpret it? Or if we can then what is the justification with respect to real space?