title: Bowling ball on a rubber sheet

text: After reading a layman's guide to general relativity, I began to wonder what shape a bowling ball on a large rubber sheet would produce. For simplicity, I would like to assume that Hooke's law applies to rubber. Here's what I've done so far: The 1-dimensional case is a mass $M$ suspended in the middle of a rubber band with length $L$ and spring constant $k$. In this case, the band would create a V shape, where the angle in the middle (between the band and the horizontal) is constrained by $2Tsin(\theta) = Mg$, and the tension is related to the amount of stretching by Hooke's law, giving $T = k \Delta L = k (\sqrt{L^2 + (L\ tan\ \theta)^2 } - L)$. However, I'm not sure how to proceed in the 2-dimensional case. Is it valid to consider a rubber sheet as an infinitely fine mesh of rubber bands? Intuitively, at some distance $r$ from the point of mass, it seems that the mass must be supported equally by the material in the circle of radius $r$, so we would have $2\pi r T sin(\theta_r) = Mg$. However, this means that as $r\rightarrow 0$, we have $T\rightarrow\infty$, which is not physical (I think the tension should be constant everywhere in the sheet.) What is the correct way to model the shape of a rubber sheet with a point mass on it?


title: Could a bipolar nebula be produced by a time gradient?

text: M2-9 is an example of a bipolar nebula that resembles two back-to-back rocket nozzles. Is it possible that this shape (somewhat unusual for an explosion) is the result of a time gradient? A rotating neutron star having collapsed and preserved its angular momentum with an equatorial velocity of relativistic proportions would be at a significantly different time than the poles of the star. Frame dragging would spread this effect out from the equator to neighboring space. A nova exploding under these conditions would be constrained outward at the poles since the relativistic time gradient puts the space near the equator in the future, relative to the poles, and would not share the same "now," making that space inaccessible to the expanding explosion. Something rotating at much slower speed would explode in a more spherical manner. Is this a plausible reason for the shape of bipolar nebulae?


title: Open quantum systems and measuring devices

text: The Copenhagen interpretation by Niels Bohr insists that quantum systems do not exist independently of the measuring apparatus but only comes into being by the process of measurement itself. It is only through the apparatus that anything can be said about the system. By necessity, the apparatus has to be outside the system. An open quantum system. Can quantum mechanics be applied to closed systems where the measuring apparatus is itself part of the system? Can a measuring apparatus measure itself and bring itself into existence?


title: Infinite Redshift

text: My text says that the signals emitted at a physical size of horizon $l_{H}(t)$ come to an observer at time $t$ with an infinite redshift. How do I show this mathematically? My approach is as follows: \begin{equation}l_H(t)=a(t)\eta(t)=a(t)\int_0^t\frac{dt'}{a(t')}=\frac{1}{1+z(t)}\int_0^t(1+z(t'))dt'.\end{equation} Now I have no idea how to conclude this. Can anyone please help me?


title: How do you choose the locations of forces when calculating moments?

text: > A uniform bridge of weight 1200kN and of length 17m rests on supports at > each end which are 1m wide. A stationary lorry of weight 60kN has it’s > centre of mass 3.0m from the centre of the bridge. > > Calculate the support force on the bridge at each end. I got the answer to be 642N and 618N. Whereas, the answer in the book says 640 and 620. I took the distance between the two supports to be 15, should I take it as 17m, if so why?


title: Are these radioactive particle matter and air emmissions dangerous, 2000KM from Fukushima?

text: I have 2 questions, as a expat and new parent residing in Shanghai (2000km from Fukushima) where we are now experiencing radioactive fallout from the Fukushima Nuclear Plant (Japan), we can not get any 'real' infomation about what are safe limits for food and air. We simply do not understand the reports we are given, nor the poor explanations that accompany the reports, which are written by design to try make people feel safe and not panic. In fact, we have no idea if the numbers being released are even true. What we are being told is that yes there is fallout and some vegetables have had 1-3 Bq/Kg detected on them, i.e. spinach. And for Air in Shanghai today (mBq/m(3)) 131-I (.58), 137Cesium (.17) and 134Cesium (.16). Now I know -from what i have learned -no one can give me specifics and certainties with all this, but what are we looking at here? Can any of you geniuses in this community - and I say this respectfully-- tel me if these levels are dangerous? Would appreciate your advice...


title: Is there a good Modern English translation of Newton's "Philosophiæ Naturalis Principia Mathematica" available for free viewing or download?

text: I have found several nice versions available for purchase, but all of the free ones are either Latin, old English, or pixellated. I searched google and was only able to locate the free ones linked to from the Wikipedia page on Philosophiæ Naturalis Principia Mathematica.


title: What force does a tall building exert on air?

text: I once heard that a tall building, let's say 10 stories, actually causes wind to blow by virtue of simply existing. It also was said that is why metropolitan downtown areas tend to be windy. What are the facts relating to thins?


title: Link between Special relativity and Newtons gravitational law

text: If I make the two statements:   * General relativity is an extension of special relativity that accounts for gravity.   * Newton's law of gravitation is a special case of general relativity for when the force of gravity is weak. Firstly, are these statements valid? Secondly, if they are, does this not mean that special relativity and Newton's law of gravitation are the same thing? This would not make sence as special relativity should not account for gravity. Or does 'weak gravitational force' as in the second statement not mean the same thing as 'in the absence of gravity'?


title: Mechanistic approach to quantum mechanics

text: It seems we attribute effects to particles, but why does it seem that we don't try to answer the question: how does it have this effect? For example, in modelling the hydrogen atom with quantum mechanics, one simply posits that the Hamiltonian contains a Coulomb potential describing the force between the proton and the electron. Can we understand the origin of that force with QM?


title: mathematician or physicists

text: Mathematicians consider physicists as people who simply use mathematics as a tool but are in a way, let's say, inaccurate, as physicists tend to make assumptions a lot in their mathematics and generalize or use experimental evidence, whereas mathematicians are full of rigour, mathematical accuracy and proof where there is almost no role of experiment. Mathematicians tend to think of physicists as lower to them in terms of accuracy and rigour. My question is, who is better off in understanding how the universe works? What is the real use of all the mathematical rigour in real life? Should I in that way be a mathematician or a physicists, I have an interest in both, but I am confused.


title: What kind of physics research can a high-schooler get involved in and form new discoveries in?

text: I am a high school student with a strong background in physics, engineering, and computer science. I am very interested in research and formulating my own discoveries in order to learn more about combining these topics and how I can some day combine my passions into a longer term research project. Ideally, I would like to combine my interests by creating a computer simulation of a problem and then building a setup to gather actual results and analyze them. About what kinds of topics would I be able to form new and non-trivial discoveries through this method?


title: Circular polarization: properties and detection

text: I am thinking about the property of circular polarization of light. I have two circularly polarized beams, one is rounding in positive direction, other is negative. My task is to figure out:   1. if those two polarized beams are circular (I can use circular-polarizer detector to do so).   2. i need to verify if those two polarized beams are just opposite. I don't have any idea but I am thinking shall I add those two beams, if they are circular but opposite, then the amplitude of the superposed beam should real (since the imaginary part cancel out). However, at this step, I don't know how to use that information to determine if the beams are two opposite circular polarization?   3. To add two beams, will it work by just shining two beams onto the same photo detector? or I have to use some adder to add the beams first?


title: Confusion about P=VI and V=IR

text: If we look at $P=VI$, we see that if the current doubles then the potential difference is halved   but this doesn't seem to make sense according to $V=IR$. If we look at that equation, since the resistance remains constant, doubling the current should increase the potential difference. Can someone explain what I'm misunderstanding in simple terminology?


title: How are anyons possible?

text: If $|ψ\rangle$ is the state of a system of two indistinguishable particles, then we have an exchange operator P which switches the states of the two particles. Since the two particles are indistinguishable, the physical state cannot change under the action of the exchange operator, so we must have $P|ψ\rangle=λ|ψ\rangle$ where $|λ|=1$. Obviously switching the states of the two particles, and then switching them back, leaves the particles with their original states, so $P^2|ψ\rangle=(λ^2)|ψ\rangle=|ψ\rangle$, so $λ=±1$, and thus the state of the system must be either symmetric or anti-symmetric with respect to exchange. Now I've heard that this reasoning does not hold for two dimensions, leading to the possibility of anyons, for which you can have $λ$ be something other than 1 or -1. How in the world is that possible? Where is the flaw or oversight in the above reasoning, that makes it exclude the 2D case? Where in the above proof are we assuming that space is three-dimensional? Any help would be greatly appreciated. Thank You in Advance. EDIT: Let me present the proof in a step-by-step fashion, so the error can be more easily identified:   1. For any states $|ψ_1\rangle$ and $|ψ_2\rangle$, define $P|ψ_1\rangle|ψ_2\rangle$ to be $|ψ_2\rangle|ψ_1\rangle$   2. For identical particles, $P |\psi \rangle$ and $|\psi \rangle$ correspond to the same physical state (i.e. ray), so we must have $P |\psi \rangle = \lambda |\psi \rangle$ for some complex number $\lambda$.   3. Applying the definition of $P$ in step 1 twice, we have $P^2|ψ_1\rangle|ψ_2\rangle=PP|ψ_1\rangle|ψ_2\rangle=P|ψ_2\rangle|ψ_1\rangle = |ψ_1\rangle|ψ_2\rangle$, so for any two-particle state $|\psi\rangle$, we have $P^2|\psi \rangle = |\psi \rangle$.   4. Applying step 2 twice, we have $P^2|ψ\rangle = PP|ψ\rangle = P \lambda |ψ\rangle = \lambda P |ψ\rangle = \lambda^2 |ψ\rangle$   5. By steps 3 and 4, we have $\lambda^2 = 1$ and thus $\lambda = ±1$ I assume the problem is with step 3 somehow, but I'm not sure what the problem is, since it follows directly from the definition in step 1. Is the problem with the definition in step 1, then? But how can a definition be wrong?


title: $J_1$-$J_2$ Heisenberg antiferromagnet

text: In this paper, the authors solve for the excitation spectrum in a $J_1$-$J_2$ Heisenberg antiferromagnet using the modified spin-wave theory in the Dyson- Maleev representation. As an intermediate step, the authors computed the neighboring-spin correlators $\left<\vec{S}_i\cdot \vec{S}_j\right>$, as shown in Eq. (4). Since the quantization axes of neighboring spins are separated by an arbitrary angle $\phi_{ij}$, it seems that the spin correlators will always contain terms of the sixth power in the boson operators. However, such terms do not generate the quadratic form of Eq. (4). What am I missing here?


title: Will Earth Hour do damage to power supply system?

text: There is always a debate around Earth Hour every year, and the opposite side of Earth Hour usually claims that > The (sudden) decrease and increase of the power usage in the start and end > of Earth Hour will cause much more power loss (than the save of power), and > even do damage to the power supply system. Is this statement true? To what extent? Thank you very much.


title: Representation of quantum transformations as matrices

text: I was reading Quantum Computation explained to my mother, which makes the following claim > Postulate 2 A closed physical system in state V will evolve into a new state > W , after a certain period of time, according to W = UV where U is a n × n > unit matrix of complex numbers. Here, `V` is a column matrix with `n` rows. Can anybody justify this assumption?


title: Which one conducts heat better: water, or air?

text: If we were to assume room temperature of 20C, which of the fluids conducts heat better, and why?


title: What is the work done by friction on an arbitrary curve?

text: Here is the question, I have a particle with mass of $m$ moving on an arbitrary curve $y=y(x)$ with friction coefficient $\mu$. Force $F$ is applied to the particle with the equation, $\vec{F}= k.\vec{r}$ Where $\vec{r}$ is the location vector of the particle. ![free diagram](http://i.stack.imgur.com/jCetz.jpg) I want to find the work done by friction. And No! I do not want to find it by Energy equation! Now we know that, $W_{friction}= \int f.dr$ and on the other hand we have, $f= \mu.N$ Now I want to find $N$ using Newton's second law assuming non zero acceleration. $\sum F= m.a$ How can I find N? Thank you


title: Where can I find a complete list of metamaterials up to today?

text: Where might I find a list of all the metamaterials up-to-date?


title: Do all the electrons form Cooper pairs at absolute zero?

text: At $T=0K$, do all the electrons form Cooper pairs, or just the electrons near the Fermi Surface do?


title: Gradient is covariant or contravariant?

text: I read somewhere people write gradient in covariant form because of their proposes. I think gradient expanded in covariant basis $i$, $j$, $k$, so by invariance nature of vectors, component of gradient must be in contravariant form. However we know by transformation properties and chin rule we find it is a covariant vector. What is wrong with me? My second question is: if gradient has been written in covariant form, what is the contravariant form of gradient?


title: What kind of free energy do we use for a superconductor in a magnetic field?

text: My reasoning is as follows (using Gaussian units): Start from the second law: $$dU=TdS+dW,$$ where $dW$ is the work done by the magnetic field. To derive $dW$, we consider a solenoid with current $I$ and voltage $V$: $$dW=I\cdot V\cdot dt,$$ with $$\begin{align} V&=\frac{N}{c}\cdot\frac{d(B\cdot A)}{dt} \qquad &&\text{(Faraday's law)} \\\ I&=\frac{c}{4\pi}\cdot\frac{H}{n}. \qquad &&\text{(Ampere's law)} \end{align}$$ Now plug in $I$ and $V$ in $dW$. We get $$dW=\frac{1}{4\pi}\cdot HdB.$$ Since $B=H+4 \pi M$, if we exclude the energy of the magnetic field itself, we find $$dW=HdM.$$ Thus the second law is $$dU=TdS+HdM.$$ For the Helmholtz free energy we have $$F=U-TS \Rightarrow dF=-SdT+HdM.$$ Similarly, for the Gibbs free energy: $$G=F-HM \Rightarrow dG=-SdT-MdH.$$ In experiment, we control $T$ and $H$, therefore at any given $T$ and $H$, the system should minimize its Gibbs free energy $G$. Ergo the energy we talk about for superconductor in magnetic field should be Gibbs free energy. The questions is, why the famous superconducting textbooks (like Tinkham, Schmidt and de Gennes) use Helmholtz free energy $F$ instead of $G$? It doesn't make sense to me to minimize $F$ instead of $G$ (e.g. in deriving the GL equations using variation method).


title: Why there is a $\frac{1}{2}$ in the distance formula $d=\frac{1}{2}at^2$?

text: I'm preparing for my exam, but I have difficulties in perceiving why there is a $\frac{1}{2}$ in the distance formula $d=\frac{1}{2}at^2$?


title: Cross-Section Compendium?

text: Are there any books or references that exhaustively list cross-sections for various processes in QED or particle physics? I think such a book would make for a very useful reference, and in particular I would like to practice deriving cross-sections.


title: Add weight in front or behind the moving wheel?

text: ![enter image description here](http://i.stack.imgur.com/GKcji.jpg) Consider a wheel with bearing and axle bar rolling on level ground. If I want to help the wheel continue moving longer(or faster speed), should I add weight by moving the rider body in front or behind the wheel axle bar?


title: 1D wave equation as a function of sound speed

text: The 1D wave equation is given by $\frac{\partial ^2 p}{\partial t^2}=c^2 \frac{\partial^2 p}{\partial x^2}$ . I found in a reference that for an unsteady gas, where both the gas and sound speeds are function of $x,t$, the equation can be written as: \begin{align*} &\frac{\partial^2 a\left(x,t\right)}{\partial t^2}+2u\left(x,t\right)\frac{\partial^2 a\left(x,t\right)}{\partial x \partial t}+\left[u^2\left(x,t\right)-a^2\left(x,t\right)\right]\frac{\partial^2 a\left(x,t\right)}{\partial x^2}=0\\\ &\frac{\partial^2 u\left(x,t\right)}{\partial t^2}+2u\left(x,t\right)\frac{\partial^2 u\left(x,t\right)}{\partial x \partial t}+\left[u^2\left(x,t\right)-a^2\left(x,t\right)\right]\frac{\partial^2 u\left(x,t\right)}{\partial x^2}=0 \end{align*} where $a(x,t)$ and $u(x,t)$ are the sound speed and gas speed, respectively. I can't find any reference with the proof of this last two equations. Any help will be appreciated.


title: Is the fine-structure constant a parameter of the standard model?

text: According to the wikipedia entry on the fine-structure constant: > In fact, α is one of the about 20 empirical parameters in the Standard Model > of particle physics, whose value is not determined within the Standard > Model. but, the wikipedia list of parameters does not mention α:               me  Electron mass       511 keV     mμ  Muon mass       105.7 MeV     mτ  Tau mass        1.78 GeV     mu  Up quark mass   μMS = 2 GeV     1.9 MeV     md  Down quark mass     μMS = 2 GeV     4.4 MeV     ms  Strange quark mass  μMS = 2 GeV     87 MeV     mc  Charm quark mass    μMS = mc    1.32 GeV     mb  Bottom quark mass   μMS = mb    4.24 GeV     mt  Top quark mass  On-shell scheme     172.7 GeV     θ12     CKM 12-mixing angle         13.1°     θ23     CKM 23-mixing angle         2.4°     θ13     CKM 13-mixing angle         0.2°     δ   CKM CP-violating Phase      0.995     g1 or g'    U(1) gauge coupling     μMS = mZ    0.357     g2 or g     SU(2) gauge coupling    μMS = mZ    0.652     g3 or gs    SU(3) gauge coupling    μMS = mZ    1.221     θQCD    QCD vacuum angle        ~0     v   Higgs vacuum expectation value      246 GeV     mH  Higgs mass      ~ 125 GeV (tentative)      Is α one of the basic parameters of the Standard Model? If not, then is there a simple formula for α in terms of these other parameters? (My guess is that α can be derived from g1/g2/g3. However, I have been unable to find an explicit formula so far.)


title: Image formed in a compound light microscope

text: I am trying to understand whether the image formed in a compound light microscope is at infinity or not. I get conflicting answers everywhere I look.


title: Do electric fields generated by plane charges lose intensity over distance? If not, why?

text: Sparknotes' studyguide for the SAT II: Physics test says that for a point charge (1-dimensional, e.g. an electron), the formula for intensity of the generated electric field is given by $E=\frac{kq}{r^2}$. It also says that for a line charge (2-dimensional, e.g. a wire in a DC circuit), it's given by $E=\frac{kq}{r}$, and that for a plane charge (3-dimensional) it's given by $E=kq$. I have no example for a plane charge because I don't know what that would be, hence my question: How can a plane charge's electric field intensity NOT diminish over time? Shouldn't this violate Conservation of Energy?


title: Electric power transmission

text: If we want to transmit electic current for a long distance, we must minimize a heat that releases because of the resistanse. We cannot make a cable wide because it is expensive and it will be massive. So, we must decrease a current because $Q = I^2R\Delta t$. A power that we want to transmit is constant, $P = UI$, so we must increase the voltage. Nevertheless, $Q = \frac{U^2}{R}\Delta t$, so if we increase the voltage, the heat will increase too. And for addition, $U = IR$, so if we decrease the current, the voltage will decrease too. Could you, explaine in details where I have a mistake. P.S. sorry for my English, I am not a native speaker.


title: Best current bounds on nonconservation of momentum?

text: It's not straightforward to test conservation of momentum experimentally, and many experiments that seem like tests really aren't. For example, in a Newtonian system of identical particles that interact through collisions, conservation of momentum follows simply from Galilean invariance and the symmetry of the collisions in the center of mass frame. For reasons such as these, many freshman-physics tests of conservation of momentum may not actually test it, even approximately -- that is, they may not present even the logical possibility of falsifying the conservation law. An added problem is that both of the fundamental theories of physics, quantum mechanics and general relativity, have local conservation of the energy- momentum four-vector pretty thoroughly embedded in their structures. In general, it's hard to test a hypothesis unless you have a test theory that is consistent with the failure of the hypothesis. In the case of GR, we have the PPN formalism, which, although not really a scientific theory, does allow for nonconservation of momentum. The best test I know of within this framework is a lunar laser ranging experiment by Bartlett (1986), which verified the equality of active and passive gravitational mass to a precision of about $10^{-10}$. The validity of this test depended on the inhomogeneity of the moon -- otherwise, for reasons similar to those described in the first paragraph, an anomalous acceleration is forbidden by symmetry. More recent observations of pulsars constrain the momentum-nonconserving PPN parameter $|\alpha_3|$ to be less than $5\times10^{-16}$ (Bell 1995). What about tests at the microscopic scale, in the electromagnetic sector? Of course it's hard to imagine theoretically how conservation of momentum could fail, since it seems to follow directly from translation invariance and Noether's theorem. But this isn't the same thing as verifying it experimentally. Have there been nongravitational tests at the macroscopic scale, e.g., empirical limits on spontaneous self-accelerations of inhomogeneous kilogram- scale masses? (Seems like the kind of thing the Eot-Wash group would do.) The interpretation of this kind of thing depends on whether or not your test theory allows Lorentz violation. E.g., the PPN formalism explicitly allows for both momentum nonconservation and Lorentz violation. If Lorentz invariance is valid, then any test of conservation of energy is also a test of conservation of momentum. So there might be one bound on nonconservation of momentum if you don't assume Lorentz invariance, and some other, tighter bound if you do. [EDIT] It seems like the atomic-physics tests are conventionally described as tests of local position invariance (LPI), although by Noether's theorem that's equivalent to conservation of energy-momentum. The highest-precision experiments compare the rates of atomic clocks of different atomic species and watch for a variation in the ratio of their rates over time. One can also test for universality of gravitational redshifts, or compare atomic clocks (microscopic) with electromagnetic resonators (macroscopic). Some recent papers are Guéna 2012 and Agachev 2010. When I asked the question, I hadn't hit upon the right search term to turn up these experiments. I'd still be interested in seeing a synoptic answer, or one that touched on the strong- force sector, or one that provided an interesting review of the history of such tests. Agachev 2010, http://link.springer.com/article/10.1134%2FS0202289311010026#page-1 Bartlett and van Buren, Phys. Rev. Lett. 57 (1986) 21, summarized in Will, http://relativity.livingreviews.org/Articles/lrr-2006-3/ Bauch, http://ebookbrowse.com/2002-bauch-weyers-phys-rev-d65-pdf-d370051021 Bell, "A Tighter Constraint on post-Newtonian Gravity using Millisecond Pulsars," http://arxiv.org/abs/astro-ph/9507086 Guéna 2012, http://arxiv.org/abs/1205.4235


title: Do generators belong to the Lie group or the Lie algebra?

text: In Physics papers, would it be correct to say that when there is mention of generators, they really mean the generators of the Lie algebra rather than generators of the Lie group? For example I've seen sources that say that the $SU(N)$ _group_ has $N^2-1$ generators, but actually these are generators for the Lie algebra aren't they? Is this also true for representations? When we say a field is in the adjoint rep, does this typically mean the adjoint rep of the algebra rather than of the gauge group?


title: Help Understanding Non-conservative Fields

text: While watching a video lecture, I became uncomfortable with the results, (around 35 mins). The professor draws an electric circuit with a 1V batter, and two resistors (1 and 9 ohms). He then calculates the voltage across the 9 ohm resistor and the voltage across the battery and 1 ohm resistor so that they are the same. So far so good. Then, he "replaces" the battery with a solenoid. This is what I don't understand: How can he just replace a battery with a solenoid? He then calculates the voltage across the solenoid and 1 ohm resistor and it is different from the voltage across the 9 ohm resistor. He then adds the two different voltages to obtain the voltage supplied by the battery. He claims that this is because of non-conservative fields. I think he is right I just don't fully understand what the point of hypothetically switching out a battery for a solenoid is. What exactly is the point? We can't do this in real life without the voltage disappearing, so this seems almost like a thought experiment.


title: Experimental Verification of No Special Frames of Reference

text: Certainly, there have been numerous tests of both Special Relativity and General Relativity. Given all the phenomena and behavior stipulated by Relativity, one could perhaps divide these phenomena into two categories: Things the don't depend on "No Special Frame of Reference":   * Time Dilation   * Length Contraction   * Lorentz Covariance   * Pretty much all of General Relativity And things the do depend on "No Special Frame of Reference"   * All frames see other frames' clocks moving slower   * All frames see other frames as length contracted   * Relativity of simultaneity It seems that a lot of the particular mind bending aspects of Relativity arise from trying to maintain the "No Special Frame of Reference" rule. As far as I can tell, all of the experimental verifications of Relativity involve items contained in the first list. Is this the case? Have their been experimental results that verify any of the items in the 2nd list, that is to say that verify the "No Special Frames of Reference" rule. **Requested Clarification** Relativity says that under certain circumstances clocks will dilate. This has been verified. Relativity also says that if two frames of reference pass each other both will see the other's clock as moving slow as opposed to both frames seeing one clock being fast and the other clock being slow. I believe this assertion follows from the notion that there are no special frames of reference. Has this particular aspect of relativity been experimentally verified?


title: Equivalence Theorem of the S-Matrix

text: as far as I know the equivalence theorem states, that the S-matrix is invariant under reparametrization of the field, so to say if I have an action $S(\phi)$ the canonical change of variable $\phi \to \phi+F(\phi)$ leaves the S-matrix invariant. In Itzykson's book there is now an exercise in which you have to show, that the generating functional $$Z^\prime(j)=\int \mathcal{D}[\phi] \exp{\\{iS(\phi)+i\int d^4x\hspace{0.2cm} j(x)(\phi +F(\phi)\\}}$$ gives the same S-matrix as the ordinary generating functional with only $\phi$ coupled to the current due to the vanishing contact terms. He then writes, that this proves the equivalence theorem, which I do not fully understand. Suppose I take this canonical change of variable, then I get a new action $S^\prime(\phi)=S(\phi+F(\phi))$ and a generating functional $$Z(j)=\int \mathcal{D}[\phi] \exp\\{iS(\phi+F(\phi))+i\int d^4x\hspace{0.2cm} j(x)\phi\\} $$ If I now "substitute" $\phi+F(\phi)=\chi$ I get $$Z(j)=\int \mathcal{D}[\chi] \det\left(\frac{\partial \phi}{\partial \chi}\right) \exp\\{iS(\chi)+i\int d^4x\hspace{0.2cm} j(x)\phi(\chi)\\} $$ with $\phi(\chi)=\chi + G(\chi)$ the inverse of $\chi(\phi)$. Therefore comparing $Z(j)$ and $Z^\prime(j)$ I get an extra jacobian determinant. Where is my fallacy, or why should the determinant be 1? Thank you in advance


title: Energy dependent Maxwell-Boltzmann distribution

text: I'm having a bit of a problem figuring out the energy dependent Maxwell- Boltzmann distribution. According to my book (Ashcroft & Mermin) they write the velocity dependent distribution as: $${{f}_{MB}}\left( \mathbf{v} \right)=n{{\left( \frac{m}{2\pi {{k}_{B}}T} \right)}^{3/2}}{{e}^{-m{{v}^{2}}/2{{k}_{B}}T}},$$ where $n = N/V.$ But how do I change the variables so it will become energy ($\epsilon$) dependent? The term in the exponential, $-\frac{mv^{2}}{2k_{B}T}$, I should be able to make the switch $\epsilon = \frac{mv^{2}}{2}$ so that I will get $e^{-\frac{\epsilon}{k_{B}T}}$, but I'm pretty sure that is not the only thing I need to do to make it energy dependent $(f_{MB}(\epsilon))$, or am I wrong ?


title: How to find out whether a transformation is a canonical transformation?

text: We had a couple of examples where we were supposed to calculate the Canonical Transformation (CT), but we never actually talked about a condition that decides whether a transformation is a canonical one or not. Let me give you an example: We had the transformation: $$P=q \cdot \cot(p), \qquad Q=\ln (\frac{\sin(p)}{q}).$$ How do I see whether this transformation is a canonical one or not? You don't have to carry out the full calculation, but maybe you can give me a hint what I need to show here?


title: Help with buoyant forces and liquids (high school physics)

text: I am stuck on this problem. Here is the scenario > _Suppose you send a balloon (filled with air) from your submersible craft to > the surface of the ocean, $10.8 \text{ km}$ above, with the message “I’M > OK!” written on the balloon. Knowing that the balloon will expand as it > rises, you write very small and don’t fully inflate the balloon, leaving its > initial radius at $5.0 \text{ cm}$. The density of sea water is $1050 \text{ > kg/m}^3$ and the density of air $1.23 \text{ kg/m}^3$._ I am asked to find the acceleration and the time it will take to reach the surface (we are told we can assume the acceleration will stay constant). So far I found that the buoyant force is equal to $6.3*10^{-3}$. So the $F_{\text{net}}$ upwards is $6.3*10^{-3}$ - the weight of the balloon which I got $5.38$. I was trying to find acceleration with $F=ma=(density*volume)a$ and solving for $a$ but I'm getting a really large number. I'm still confused, do I use the density of the water or density of the air? And why?


title: Calculating the magnification of an optical microscopy system

text: I have a microscopy system that is set up as follows :   * DSLR with a crop factor of 1.6x ( equivalent to magnification, I believe )   * An adaptor optic, for attach the DSLR to the microscope, listed as a 2x magnification   * An objective on the microscope, currently set at 4x magnification I've photographed a 2mm object, and found it to be 14.24mm on the sensor ( 3618 pixels long on a 5184-pixel wide sensor with length 22.3mm ). With this, I calculate the whole system's magnification to be 7.12x. I divide by 4 to remove the objective, leaving 1.78, and then divide that by 1.6 for the camera's crop factor, to get 1.113. This is far below the 2x listed by the adaptor. In addition, the adaptor also states, "depending on your camera, you'll get between 12x and 16x magnification", which is very ambiguous, because the microscope objectives range from 4x to 100x or more. So, I'd like to know if I'm doing something wrong, why I'm not getting 2x for the adaptor, and where this "12x to 16x" comes from. Any help much appreciated !


title: A Hollow Black Hole

text: I was just reading a question about the gravity inside a hollow neutron star. It was a trivial question, obviously there is no force felt. But then it got me thinking. Suppose you had a hollow sphere that was massive enough that outside of it, it became a black hole. It's not hard to imagine a situation where the magnitude of the force of gravity at the surface (while strong) is equal to the magnitude of the force on the particles of the sphere due to some phenomenon involving one of the other 3 forces (just so I can allow this hollow sphere to continue being hollow). Obviously, I don't have to do the integral to know that any object inside the hollow would experience no net gravitational force due to the shell. However, being a few months short of General Relativity, I am curious to know whether there would be any effects from Gravity observed. Would an observer inside experience any time dilation? There may not be any net force due to gravity, but one cannot deny that there are still some fairly powerful gravitational fields present that happen to superimpose to zero. Furthermore, (assuming a relatively massless observer) would they have access to any of the information that was "lost" in the black hole? If so, then if they were long-lived enough for the black hole to be able to evaporate around them, does this mean that not only would the black hole radiate that information, but it would also store a copy of it in the hollow? I want to apologize in advance. I know this question seems a bit more like a discussion than a Q&A thing. To be clear, I am asking: In my scenario, **would there be any effects of gravity on things in the hollow?** And **would information lost in the black hole be accessible to the hollow** (which is essentially its own isolated bubble of the universe)? A good answer can address these two questions. If you want to add in your own opinions/contribute to a discussion about it, that'd be a bonus.


title: Srednicki's book chapter 8

text: Reading first page in chapter 8 of Srednicki's it reads: > To employ the $\epsilon$ trick, we multiply $H_0$ with $1-i\epsilon$. The > results are equivalent to replacing $m^2$ with $m^2-i\epsilon$. Now, our Hamiltonian is $H_0=\frac{1}{2}\Pi^2+\frac{1}{2}(\nabla\phi)^2+\frac{1}{2}m^2\phi^2$ so making the replacement he suggests ($m^2\to{}m^2-i\epsilon$) we would have $H_0(1-i\epsilon)=\frac{1}{2}\Pi^2+\frac{1}{2}(\nabla\phi)^2+\frac{1}{2}m^2\phi^2-\frac{i\epsilon}{2}\phi^2$ while on the other hand, in reality $H_0(1-i\epsilon)=\frac{1}{2}\Pi^2+\frac{1}{2}(\nabla\phi)^2+\frac{1}{2}m^2\phi^2-\frac{i\epsilon}{2}\Pi^2-\frac{i\epsilon}{2}(\nabla\phi)^2-\frac{i\epsilon}{2}m^2\phi^2$ And we see that both terms don't agree. How can both expressions be equivalent?


title: Relative velocity decomposition

text: In my fluid dynamics course notes is stated a change in relative velocity can be decomposed: $$\delta \mathbf{v} = \left\\{ \begin{pmatrix} \Delta & 0 \\\ 0 & \Delta \end{pmatrix} + \begin{pmatrix} D_1 & D_2 \\\ D_2 & -D_1 \end{pmatrix} + \begin{pmatrix} 0 & -\xi \\\ \xi & 0 \end{pmatrix}\right\\} \frac{1}{2} \begin{pmatrix} \delta x \\\ \delta y \end{pmatrix} $$ where $ \Delta = \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y}, D_1 = \frac{\partial u}{\partial x} - \frac{\partial v}{\partial y}, D_2 = \frac{\partial u}{\partial y} + \frac{\partial v}{\partial x}, \xi = \frac{\partial v}{\partial x} - \frac{\partial u}{\partial y}$ I haven't been able to find a reference to this online. Does this decomposition have a name? How is it derived?


title: When is the stress on an airplane the highest?

text: Something I´ve been wondering for a long time: At what point during a flight is the stress on an airplane the highest? Is it during take-off, landing or during (light/moderate/severe) turbulence? With "stress" I mean the load applied to the aircraft structure.


title: Does Newton's third law of motion fail anywhere?

text: Does Newton's third law of motion fail anywhere, including particle physics and other advanced topics?


title: Neutrons reflective materials in nuclear reactors

text: Nuclear reactors are driven away from criticality with the use of control rods which absorb neutrons. In case of overheating and melting of the core, the control rods may not be inserted any more leading to a disaster. A subcritical mass can be made critical by placing neutron reflective material all around it. So my question is why nuclear reactors are not operated with this kind of reflective shielding instead of control rods? The shield could be designed in order to support a maximum temperature before crashing/burning/melting driving the reactor far from the criticality automatically in a totally passive way. What am I missing?


title: Two pairs of projection operators of the Dirac equation

text: The Dirac equation may be interpreted as the action of projection operator $\frac{1 - \Delta}{2}\Psi = 0$, where $$ \Delta = \begin{pmatrix} 0 & \Delta_{b \dot {a}} \\\ \Delta^{\dot {b}a} & 0 \end{pmatrix}, \quad \Delta_{b \dot {a}} = \frac{i}{m}p_{b \dot {a}}, \Delta_{b \dot {a}}\Delta^{\dot {a} c} = \delta^{c}_{b} \Rightarrow \Delta^{2} = 1. $$ Here $\Delta_{a \dot {b}}$ changes dotted indice to undotted (transform $\left( \frac{1}{2}, 0\right)$ to $\left( 0, \frac{1}{2} \right)$). There is two projection operators $\frac{1 + \Delta }{2}, \quad \frac{1 - \Delta}{2}$, so $\frac{1 - \Delta}{2}\Psi = 0, \quad \frac{1 + \Delta }{2}\Psi = \Psi$, or vice versa. This operators leaves only two independent components of spinor $\Psi$, but formally it still refers to the direct sum of $\left( \frac{1}{2}, 0\right) + \left( 0, \frac{1}{2} \right)$. There is another two projection operators $\frac{1 + \gamma^{5}}{2}, \quad \frac{1 - \gamma_{5}}{2}$, which also projects spinor on one of the subspaces. I don't understand fully what are the difference between these pairs of operators. Maybe, the first one leaves the spinor as the direct sum representation, and the second one leaves only one, $\left( \frac{1}{2}, 0\right)$ or $\left( \frac{1}{2}, 0\right)$? Can you make understanding of it more clear?


title: Sackur-Tetrode equation - clarification required - problem with units

text: I'm a 2nd year physics undergraduate and recently I've volunteered to give a short presentation on the Sackur-Tetrode equation derivation and its use at removing the Gibbs paradox. I've looked on the Internet and in some books and found many methods for deriving this result. The one I'm going with is the derivation from the book _Thermodynamics and Statistical Mechanics_ by W. Greiner. However, there are some things I'm still not certain about (I dared to expand the argument a little) and in order to seem like I know what I'm talking about, I need some clarification. Here's more or less how I see it going: * * * In order to obtain an expression for the entropy of an ideal gas with $N$ (identical) particles of mass $m$ occupying a volume $V$ and with internal energy $E$, we use Boltzmann's entropy formula: $$S(N,V,E)=k\ln \Omega(N,V,E).$$ This reduces the problem to finding $\Omega(N,V,E)$, the number of microstates of our system which realise the macrostate $(N,V,E)$. A microstate of the system gives information about the entire system at a microscopic level, that is, it specifies the exact momentum and position of each constituent of the system. There are $N$ particles and each one has three directions of position and of momentum, therefore our system has $6N$ degrees of freedom. We imagine a $6N$ dimensional space, called the phase space, where each axis corresponds to a degree of freedom or the system. Each point in this space corresponds to a specific microstate: $(r_x^{(1)},r_y^{(1)},r_z^{(1)}, p_x^{(1)}, p_y^{(1)}, p_z^{(1)},...,r_x^{(N)},r_y^{(N)},r_z^{(N)}, p_x^{(N)}, p_y^{(N)}, p_z^{(N)})$. The microstates which correspond to our particular macrostate $(N,V,E)$ form some sort of high-dimensional hypersurface. The "volume" of this surface, $\phi(N,V,E)$, would give us a quantity related to the number of microstates. Of course in phase space, "volumes" of surfaces have dimension of action raised to some power, and since we're interested in a number, the idea is to introduce some sort of elementary bit of "volume", $\phi_0$ and then divide to produce a number $\frac{\phi(N,V,E)}{\phi_0}$ which we claim to be the number of relevant microstates. Fortunately, it physically makes sense for such an elementary "volume" to exist, since from the uncertainty principle we know that a point in phase space really cannot be specified with infinite precision, and that there is a quantum of action, which is nice because our "volumes" have dimension of action to some power. We already used the fact that we have $N$ particles. Now it's time to take our other givens into account. Let's start with $E$. Classicaly, for an isolated system of noninteracting particles, the total energy is purely kinetic and given by the sum of the kinetic energies of each particle: $$E=\sum_{k=1}^{N} \frac{ [p_{x}^{(k)} ]^2 +[p_{y}^{(k)} ]^2 +[p_{z}^{(k)} ]^2 }{2m}$$ That gives a bound for our momenta coordinates. The position coordinates can be likewise bounded using our last given: the volume $V$. We may assume that the system occupies a cube of length $l$, then for every $k=1,2,...N$ we have: $$0\le r_{x}^{(k)} \le l ; \quad 0\le r_{y}^{(k)} \le l ; \quad 0\le r_{z}^{(k)} \le l ; $$ This seems like cheating a bit (why assume a particular shape of the container), but the assumption can be later dropped because in further calculation we don't really care about the bounds on the position coordinates, thanks to the fact that the total energy is independent of them. So this works for arbitrary $V$. The two previous equations specify the desired "volume" in our phase space. The dimension of this object is $6N-1$ because we only have one constraint, namely the energy being constant. To calculate its "volume" we need to consider the wonderful looking integral: $$\phi(N,V,E)=|A|=\int_{A} dr_{x}^{(1) }dr_{y}^{(1)} dr_{z}^{(1)} dp_{x}^{(1) }dp_{y}^{(1)} dp_{z}^{(1)} ... dr_{x}^{(N) }dr_{y}^{(N)} dr_{z}^{(N)} dp_{x}^{(N) }dp_{y}^{(N)} dp_{z}^{(N)}$$ Where $A$ is the set given by the previous equations. We can simplify this notation by writing $$\phi(N,V,E)=|A|=\int_{A} d^{3N}r \ d^{3N}p$$ Notice that the position coordinates are independent of momentum coordinates, so really, the integral factors: $$\phi(N,V,E)=|A|=\int_{A_r} d^{3N}r \int_{A_p} d^{3N}p$$ Where $A_r$ is the set given by the position conditions and $A_p$ by the momentum conditions. The first factor is simply $V^N$, or $l^{3N}$. The second factor is slightly trickier to calculate, but it's just the surface of a 3N dimensional sphere with radius $\sqrt{2mE}$. This already implies that the units will be $\mbox{[momentum]}^{3N-1}$, and so the units of $|A|$ are $\mbox{[momentum]}^{3N-1} \mbox{[length]}^{3N}$. The actual derivation of the surface of an $n$-dimensional sphere of radius $r$ will be left out of this, and we just quote the result: $S_n = \frac{2 \pi ^{n/2}}{\Gamma (n/2)}r^{n-1}$ Anyway, we have obtained a solution for our "volume" $|A|$: $$\phi(N,V,E)=|A|=2 V^N \frac{\pi ^{3N/2}}{\Gamma(3N/2)}\sqrt{2mE}^{3N-1} \quad \mbox{[momentum]}^{3N-1} \mbox{[length]}^{3N}$$ * * * Here I have a big problem. As said before, the number of states is $\frac{\phi(N,V,E)}{\phi_0}$. I would _like_ for $\phi_0$ to be something like $h^{3N}$ or $h^{3N-1}$ but this gives wrong units! Ignoring this for the moment and just taking $\phi_0 = h^{3N-1}$ here's the rest of the derivation: * * * We thus obtain an expression for the number of microstates: $$\Omega(N,V,E) = 2 V^N \frac{\pi^{3N/2}}{\Gamma(3N/2)} \left( \frac{\sqrt{2mE}}{h} \right) ^{3N-1}$$ Taking the log of this and using appropriate approximations etc.(I skip this here but won't during the presentation ofc.) we arrive at: $$ S(N,V,E) = kN \left( \frac{3}{2} + \ln V \left( \frac{4 \pi m E}{h^2} \right) ^{\frac{3}{2}} \right)$$ This unfortunately leads to the Gibbs paradox. The reason for this is that we've assumed the particles to be disinguishable when in reality they are not. In order to account for this, we need to introduce the Gibbs correction factor into the total amount of states $\Omega$: $$\Omega_C(N,V,E) = \frac{1}{N!} \Omega(N,V,E) $$ Dividing by the amount of permutations accounts for indistinguishability. Going through the same derivation leads to the proper formula now. Thus the paradox is fixed. * * * I would love to hear some feedback about this presentation and particularly with the problem I had with $\phi_0$. Additionally, the introduction of the Gibbs correction factor seems kind of shady to me, I read somewhere that it's again just an approximation and that proper counting would produce a more complicated result than just diving by the factorial, but once again the error is negligible.


title: Any known relationships among quark masses?

text: Here is some numerology: Gen 1 Mass(d)/Mass(u) = 2 = 2 * 1 Gen 2 Mass(c)/Mass(s) = 12 = 4 * 3 Gen 3 Mass(t)/Mass(b) = 40 = 8 * 5 Does standard model make any predictions of relations? Do any GUT or string theories make any predictions of relations?


title: How common is compromised reference data for physical properties of materials and elements?

text: This question concerns Monte Carlo simulation of electron scattering in complex material structures, especially the reliability of publicly available source codes and reference data for physical properties of materials and elements. The source code from DTSA II contains the notice that the author noticed compromissed reference data for Rb in one publicly available source. The same source code also uses the NIST Electron Elastic-Scattering Cross-Section Database, which may be less compromised or not. The general-purpose Monte Carlo code system PENELOPE, which performs simulations of coupled electron- photon transport in complex material structures, uses the NIST ESTAR database for parts of its material data. Let me add that all this data properly links back to scientific publications describing how the data has been obtained. There are many other independent Monte Carlo codes like FLUKA, EGS or Geant, which may include reference data from other independent sources, and can be used to cross check each other. Let's assume that I have done such a cross check, and have found excellent agreement for most materials, but some additional materials/elements like Rb reported above, where something seems to be wrong. I wonder how common it is to find such compromised data especially concerning electron simulation, and whether it is possible that this is even intentional.


title: Quantum field theory quote

text: I have read this in scientific American: According to quantum field theory, all particles spend a little time as combinations of all other particles" Is this right? How long? And how can they be a combination of all other particles at once? Do they transform? Answer in laymans terms if possible


title: Gravity and Larmor effect

text: I have a Q: Does "Equivalent Principle" and "Larmor effect" imply that the charged particle should radiate electromagnetic wave if it is at rest in uniform gravitational field (like it is at rest on the surface of a massive planet)? Thanks


title: Time evolution of a quantum state

text: I have another point in QM that I would like clarified. Suppose $$\\{|n\rangle\\}$$ is a set of eigenstates of both the Hamiltonian $H$ and another operator $\hat O$ corresponding to an observable also. We have a state $$|\psi\rangle=\sum_{n=1}^3 a_n |n\rangle$$ If it is such that $$\hat O|1\rangle=1,\,\,\, \hat O|2\rangle=\hat O|3\rangle=-1$$ and a measurement of $O$ at time $t=0$ gives $-1$. How then does the system evolve In other words, what is $$|\psi(t)\rangle$$? * * * I know that $$|\psi(t)\rangle=\exp(-i\hat H t/\hbar)|\psi(0)\rangle$$ But then does it mean it is $$|\psi(t)\rangle=N(\exp(-i\hat E_1 t/\hbar)a_2|2\rangle+\exp(-i\hat E_3 t/\hbar)a_3|3\rangle)$$ with some normalization constant in front? So we can never get $|1\rangle$ back?


title: How to find distance of closest approach for a Schwarzschild geodesic?

text: What is the distance of closest approach in this Wikipedia article? I can't seem to find its definition, and this other question doesn't have an answer I can understand.


title: Time of Measurement Vs Number of Measurements

text: Let's say that an experiment has to determine the number of cosmic muon at sea level. The appropriate equipment is ready to measure the counting rate. I can think of two ways   1. Count for 10 minutes, so as to reduce the relevant error.   2. Count 10 times(1 minute/measurement) Which is the best measurement? Are they equal?


title: Relative angular momentum in two-particle problem

text: Assume the two particle problem i.e. two particles with equations of motions as follows: $\ddot{x_i} = -\frac{1}{m_i}\frac{\partial}{\partial x_i}V(|x_1 - x_2|)$ Since the angular momentum is conserved (in relative coordinates), we know that the two particles have to move in a plane and hence can write the distance between the two particles as follows: $x = r e_r$ and $\dot{x} = \dot{r} e_r + r \dot{e_r} = \dot{r}e_r + r \dot{\phi} e_{\phi}$ Now if I wanna compute the actual value of the angular momentum I seem to run into some trouble. If I do it the following way I get the right value: $|L| = |\mu x \wedge \dot{x}| = \mu |(r e_r \wedge ( \dot{r}e_r + r \dot{\phi} e_{\phi}))| = \mu | r e_r \wedge r \dot{\phi} e_{\phi} | = \mu r^2 \dot{\phi} $ But if I do it the following way it doesn't work out: $|L | = |\mu x \wedge \dot{x}| = \mu |x||\dot{x}|\sin(\theta) = \mu |x||\dot{x}| = \mu|r||\sqrt{\dot{r}^2 + r^2 \dot{\phi}^2}| $ where I used the formula for $x$ and $\dot{x}$ Why doesn't that work? I feel like I'm missing something completely trivial here, but can't find it.


title: Interpreting the equation $\vec F = R\vec v$

text: > The Aristotle's law of motion, which is incorrect, states that > > _The velocity of an object $\vec v$ is directly proportional to the force > $\vec F$ acting on it_ > > or $\vec F \propto \vec v$ > > or $\vec F = R\vec v$, > > where $R$ is the constant of proportionality which is the resistance to > motion. I want to know how can we interpret this equation $\vec F = R\vec v$ ? Does this equation imply that $\vec F$ and $\vec v$ have the same direction? If it does, then how do we know this through this equation? Does equality mean that they have the same direction? The book in which I actually saw this equation, wrote this equation in this way: $\vec v = \frac {\vec F}{R}$.


title: Distance to the galaxy which has velocity which is perpendicular to the point of view of the static observer

text: Let's have the galaxy which has velocity which is perpendicular to the point of view of the static observer in the FLRW universe. It emits the light. There is a given redshift $z_{0}$. It will be easy to find the distance from the galaxy to the observer from the relation for light which moves radially (let the universe be spatially flat): $$ dt^2 - a^2(t) dr^2 = 0 \Rightarrow r = \int \frac{dt}{a(t)}. \qquad (1) $$ But then I need to determine the relation between $a(t)$ and $z$. If the galaxy is at rest, this relation is easy: $$ \frac{a_{0}}{a(t)} = 1 + z, \quad \frac{\dot {a}}{a_{0}} = H. $$ But in this case I have to take into account the transverse Doppler redshift in a form $$ \frac{a_{0}}{a(t)} = (1 + z)\sqrt{1 - v^2(z)}. $$ So how to get $v(z)$ and how to rewrite integral $(1)$ in terms of variable $z$, $$ \int \limits_{0}^{t} \to \int \limits_{0}^{z_{0}}? $$


title: Primary direction in planet-centered equatorial reference frame

text: I am given the classical orbital elements of the orbit of a spacecraft around a planet which is not the Earth, say Venus. I assume those are referred to a reference frame whose fundamental plane is the equator of the planet, but **which is its primary direction**? I am hesitating between two options: either it is the intersection of the plane of the orbit _with the ecliptic_ , or the intersection of the plane of the orbit _with the J2000 Earth equator_. While the ecliptic options seems more reasonable to me, I've been reading lots of documentation on the Internet but either they are all Earth-minded (i.e. Wikipedia) or there is some implicit convention I am missing. The thing is that this is not important if for example I am just analyzing the motion of my spacecraft around my main attractor (Venus), but when I want to look at it from an interplanetary point of view, I think I need this information.


title: Is it possible that a matter field has a dependent on non-radial space-like coordinate in a spacetime with spherical symmetry?

text: After the work from Breitenlohner and Freedman, we know matter fields in asymptotically AdS spacetime can be stable out of the black hole under some special conditions. **My question:** In such a spherically symmetric spacetime, could the matter field has the non-radial variable of the coordinate? If it could be, what is the limit on it? Suppose the metric tensor is $$\mathrm{d}s^2 = -f(r)\mathrm{d}t^2 + \frac{\mathrm{d}r^2}{f(r)} + \frac{r^2}{L^2}\Omega_2(\theta, \phi)$$ $r \rightarrow +\infty$ is the infinite boundary, $f(r) \rightarrow \frac{r^2}{L^2}$ as $r \rightarrow +\infty$ and $L$ is the AdS radius, $\Omega_2$ is the 2-d spherical coordinate. The matter field $\Phi(r,\theta)$ in question is a complex scalar field with a charge $q$ and the Maxwell field (1-form) reads: $$A = A_t(r,\theta) \mathrm{d}t^2$$ the coordinate $r$ is the radial variable and $x$ is the non-radial. Is it possible for the set-up above? Thank you for your time. \------------------- **Update** \--------------- To avoid more confusion, rather planar AdS black hole, we consider the metric as above showed. Perhaps my question can be changed into this one: Provided the stable spacetime is of spherically symmetric, if all matter fields coupling with the black hole in this spacetime should has the $SO(2)$ symmetry, and this property is equal that all matter fields haven't the non-radical space-like variable?


title: Understanding the mean square displacement in molecular dynamics

text: In a Molecular Dynamics (MD) simulation, the mean square displacement $\text{MSD}$ is given by $$\text{MSD}(\delta t) = \left\langle\left|\vec{r}(\delta t)-\vec{r}(0)\right|^2\right\rangle,$$ where $\vec{r}$ is the position vector of an atom and $\delta t$ is some time step. Often the time step dependence of the $\text{MSD}$ is left out, which seems to relate to my question, but no problems so far. However, the practical calculation of the ensemble average $\langle\ldots\rangle$ is usually somewhat vaguely explained and seems to depend on where you look. This source states that we must average over all atoms _and_ many time steps (I would assume all time steps in a given simulation). This and this source on the other hand mention only the average over all atoms (though the latter does indicate a time (step) dependence). This last source seems to support the first one. Note that both the first and last source give the $\text{MSD}$ a dependence upon the time step. My interpretation is as follows. I would expect the $\text{MSD}$ to have a time step dependence, so I'm inclined to follow the first and last source I've quoted. So in the case of an MD simulation of $N$ atoms spanning a total time $N_{k\tau}k\tau$ one would calculate the $k\tau$ mean square displacement as $$\text{MSD}(k\tau) = \frac{1}{N}\frac{1}{N_{k\tau}}\sum_{n=1}^{N}{\sum_{i=1}^{N_{k\tau}}{\Big|\vec{r}_n\big(ik\tau\big)-\vec{r}_n\big((i-1)k\tau\big)\Big|^2}}.\qquad(*)$$ My question is really just: is this correct? And if so, what's the difference with the other two sources I've mentioned? Why do they not mention averaging over the time steps? Do they simply discuss the formula for a single time step? (i.e. $N_{k\tau} = 1$)


title: Can you make ice red hot?

text: This video clip allegedly shows an otherwise unspecified ice cube that turns red hot (and burns) due to induction heating. Can somebody explain how this works?


title: Do the same experiments yield different results due to the principle of uncertainty?

text: When thinking about small particles and their uncertainity, I've allways rather seen them being _all over the place_ rather than randomly changing location. I would think that, in the same time, you'd always have same result, have you tried to do something to them (eg. shoot another particle at them). However, in certain situations, like the evaporation of a black hole, the particle just either is here or it is not. This makes me wonder: if you try something with uncertain particles, does every experiment end up differently because, event though all conditions are the same? (The last requirement seems to be unfeasible so consider it just a thought experiment). Similarly, if there was a separated Big bang would the other universe look different than ours because particles ware randomized differently? Out of the question, I must say that I find it very confusing for particles to really have random locations. Wasn't there ever a suspicion that their location (as well as the probability of it) has a rule we can find out?


title: Air vs. Water attraction of Oxygen Molecules

text: The air we breathe is made up of nitrogen and oxygen gases. The water in a pond is made of a single hydrogen/oxygen molecule. If it wasn't for the surface tension on top of the water, oxygen molecules can enter the water and occupy space next to the hydrogen/oxygen molecules. My question is this: All things being equal, is oxygen more attracted to water or air?


title: Is there a manifestly Lorentz invariant formulation of the many worlds interpretation?

text: First, I give an absurd example. A conscious observer lives on Earth at time t. A light-year away, at a space- like separation, a nuclear bomb chain reaction goes off. A nuclear bomb chain reaction is quantum all the way and leads to a splitting into expo- exponentially many parallel worlds. In one reference frame, the explosion happens months after the observer at time t. Let's say at time t, we have N copies of the observer out there distributed across N parallel worlds. In another boosted reference frame, the explosion happens before the observer at time t. What this means is, at time t, the observer is split into N times expo-exponentially many copies, and for each version of the observer, there are expo-exponentially many exactly identical copies lying about. By anthropic reasoning, and anthropic reasoning is necessary in the many worlds interpretation, the latter reference frame would be preferred by an expo- exponential factor in complete violation of Lorentz invariance. The only way to save the many worlds interpretation from this absurdity is to find a manifestly Lorentz invariant version of splitting. How can this be done?


title: Are these IR and UV divergences equal

text: let be 2 divergent integrals $$ \int_{0}^{\infty}\frac{p^{3}dp}{(p^{2}+m^{2})^{2}}= A $$ $$ \int_{0}^{\infty}\frac{dp}{p(p+q)^{2}}=B $$ B has a divergent as $ p \to 0 $ however i can use a change of variable $ xu=1 $ so for B we have $$ \int_{0}^{\infty} \frac{udu}{(1+uq)^{2}}=B $$ so now this integral B has a logarithmic divergence, in this case is enough if we add and substract the integral $ \int_{0}^{\infty}\frac{dx}{x+1} $ to regularize it ?¿is my argument correct ??


title: Difference between momentum and kinetic energy

text: From a mathematical point of view it seems to be clear what's the difference between momentum and $mv$ and kinetic energy $\frac{1}{2} m v^2$. Now my problem is the following: Suppose you want to explain someone without mentioning the formulas what's momentum and what's kinetic energy. How to do that such that it becomes clear what's the difference between those two quantities? From physics I think one can list essentially the following differences:   1. momentum has a direction, kinetic energy not   2. momentum is conserved, kinetic energy not (but energy is)   3. momentum depends linear on velocity, kinetic energy depends quadratically on velocity I think is is relatively easy to explain the first two points using everyday language, without referring to formulas. However is there a good way to illustrate the 3. point?


title: Transmission and reflection

text: What is the transmission amplitude of a wavefunction $\phi(x)=e^{ikx}(\tanh x -ik)$? I would have thought that it is $\tanh x -ik$ since this is the factor associated with the forward travelling $e^{ikx}$ but then since the reflection coefficient is $0$, we have that the reflection probability is $0$, but $|\tanh x-ik|^2$ is dependent on $x$ so not identically $=1$? Where have I gone wrong?


title: How to build a laser in the garage?

text: So I wonder if it is any how possible to build laser at home. A powerful one to melt brick.


title: How exactly, or whats the process, rather, of energy changing into matter?

text: $E=mc^2$ this is the equation by Einstein claiming energy can change from energy to mass. this would happened at the big bang I assume, when electrons and protons were made to create hydrogen and some helium.


title: Solving by using Gauss law

text: > _Task: find the vector $ \mathbf E $ in the center of the sphere with radius > $R$, which has charge volume distribution $\rho$ , $$\rho = \mathbf a \cdot > \mathbf r ,\qquad \mathbf a = \operatorname{const}, $$ $\mathbf r$ = radius- > vector from the center of the sphere._ I solved that task by routinely method. But can it be solved with using Gauss law?   * **Addition 1**. My attempt to solve that task by using Gauss law. In first, I wrote, that $$ \oint \mathbf E \cdot d\mathbf S = \int \nabla \cdot \mathbf E\ dV = 4\pi \int \rho dV = 4\pi a \int r cos(\mathbf a , \mathbf r) dV \qquad (.1). $$ Unfortunately, I don't know how to find $\mathbf E$ using written above: $ |\mathbf E| \neq const$ into the surface of the sphere, so I can't write something like $$ \oint \mathbf E \cdot d \mathbf S = E \int dS = ...$$ and equate it to (1). Can you help me?   * **Addition 2**. My routinely solution. In the beginning   1. I can consider only surface of the sphere with surface denstity $\rho (S) = (\mathbf a \cdot \mathbf r)$.    2. I "moved" $\mathbf a$ to the center of the sphere by parallel transfer (look at the picture 1 below). ![Illustration](http://i.stack.imgur.com/5nfHT.jpg) So, I can divide the sphere into thin rings with charge $$dQ = \rho (S)dS = \rho (S) 2\pi r Rd\varphi$$ (R - length from the ring to the center of the sphere, a - length from the ring plane to the center of the sphere, $\varphi$ - an angle between $\mathbf a , \mathbf R$, and $\mathbf r = \sqrt{R^{2} - a^{2}})$. Using that, $d|\mathbf E|$ creating by this ring in the center of the sphere (using the formula for the ring E = \frac{kaQ}{R}), can be write as $$ dE = \frac{kadQ}{R^{3}} = \frac{ak\rho (S) 2\pi r R d\varphi}{R^{3}}. $$ According to the spherical symmetry, $$ d\mathbf E_{center} = -\frac{\mathbf a}{a}dE, $$ so, according, that $$ r = Rsin(\varphi ), \quad a = Rcos(\varphi ), \quad \rho (S) = (\mathbf a \cdot \mathbf r) = aRcos(\varphi ), $$ I get $$ d\mathbf E = -\frac{k R cos(\varphi) R cos(\varphi ) 2\pi R sin(\varphi) R d\varphi}{R^{3}}\mathbf a = -2\pi k R cos^{2}(\varphi )sin(\varphi)\mathbf a \Rightarrow $$ $$ \Rightarrow \mathbf E = 2\pi k \mathbf a \int \limits_{-1}^{1}cos^{2}(\varphi )d(cos(\varphi )) = \frac{4 \pi k R \mathbf a}{3}. $$ By the next step I accept, that R is a variable (takes into account the volume distribution of charge), so, with the new vector $\mathbf a$ (formally, it's changed only in a measure, $[|\mathbf a_{new}|] = \frac{[|\mathbf a_{old}|]}{meter}$), $$ d\mathbf E = \frac{4 \pi k RdR}{3}\mathbf a \Rightarrow \mathbf E = \frac{2\pi k R^{2}}{3}\mathbf a. $$


title: Spectral luminous efficiency as a function of wavelength

text: I've come across plenty of figures demonstrating the spectral luminous efficiency as a function of wavelength (meaning the humans eye's sensitivity to different wavelengths) http://www.yorku.ca/eye/photopik.htm but I've been unable to find the equation which gives this function so I can plot it myself.


title: Nonlinear polarization (second and third order)

text: Why does second order nonlinear polarization occur only in crystal materials with a non-centrosymmetric crystal structure? (Nonlinear effects at crystal surfaces are an exception). Why does third order polarization occur in basically all media? The answer, found here http://www.rp- photonics.com/nonlinear_polarization.html did not help (at least for now).


title: What actually happens to a light ray on a Schwarzschild black hole horizon?

text: I know the Schwarzschild event horizon is a null surface generated by null geodesics. But what does that actually mean in terms of the path of a light ray that reaches it? Does that mean the geodesic trajectory of light on the surface will be along the surface? Which way will the light ray go? And what equation shows this?


title: Latent heat of condensation

text: Regarding atmospheric processes, I understand that energy is required to evaporate water by moving molecules further apart i.e. a phase change from liquid to gas. The air then ascends into the atmosphere before cooling and releasing latent heat as the water molecules condense. This is often stated to be an important heat transfer from the surface to the atmosphere. I understand how the latent heat is transported by the evaporated molecules(convection)but how exactly is it released? Gas is a poor conductor of heat so it can't be from conduction to lower temperature, slower moving air molecules. Is it radiated somehow? It can't be an adiabatic process.


title: absolute defintion of the right (i. e. not left) direction

text: The other day somebody told me that there are absolute definitions of hot and cold for instance. You can always say a is warmer than b, no matter where you are. But he said, there is no absolute definition for right (i. e. not left). He told me that “they” came up with an experiment that you can set in front of you and something would go to the right. And if you turn the apparatus, it would still go to your right. I said that I cannot believe this, since it would fail with two observers and right is just arbitrary. If he meant a definition of right in a chirality kind of way, anything with a cross product in it should allow him to separate right-hand from left-hand coordinate systems. He again said that “they came up with an absolute definition of right. Just do not ask my how they did it”, which did not really allow for a substantial conversation. Is there any such thing?


title: Can a saltwater solution conduct electricity forever?

text: We know that very pure water does not conduct electricity, but salt water is a decent conductor. This is commonly explained by saying that "the ions carry the current through the solution", an explanation that does not really make sense because it is not clear what will happen when all of the ions have migrated to the electrodes. Better explanations of conduction through a salty solution (like this one or this one) explain the conduction of electricity in terms of a reduction reaction that takes place at the anode and an oxidation reaction that takes place at the cathode. In the case of salt water, chlorine gas (Cl2) is formed at the anode and hydrogen gas (H2) is formed at the cathode. This explanation makes sense to me, but it implies that the conduction of electricity through a solution is fundamentally different than the conduction of electricity through a wire. A copper wire does not change at all after we pass electricity through it. In contrast, in salt water, we are driving a two chemical reactions (one at each electrode), which fundamentally change the composition material. This implies that it is not possible for a aqueous solution to conduct electricity forever. Since we are driving a chemical reaction, we are either consuming our salt by forming gas (or plating it onto the electrodes) or we are consuming the water by forming H2 or O2 gas. This is surprising to me (for some reason). So, I'm asking if my thinking is correct: **is it possible for a salt solution to conduct electricity forever** , or it will it always eventually consume the reactants and stop as I have surmised?


title: Projectiles and escape velocity

text: > Q: _The escape velocity for a body projected vertically upwards from the > surface of earth is 11 km/s. If the body is projected at an angle of > $45^\circ$ with vertical, the escape velocity will be?_ My Approach: The new vertical velocity will be $u * \sin(45^\circ)$ = $u /\sqrt2$ **Calculating expression for escape velocity:** $$\frac{1}{2} mv^2 = \frac{GMm}{r}$$ $$\frac{1}{4} mu^2 = \frac{GMm}{r}$$ $$u = 2\sqrt\frac{GMm}{r}$$ Hence the new escape velocity = $\sqrt2 * 11$km/s. However, the **Correct Answer** is 11km/s.


title: Do particles in the solar core travel at the same speed as they would in vacuum?

text: I just read the Wikipedia article on the Sun. The section on Solar Core states > It has a density of up to 150 g/cm3[47][48] (about 150 times the density of > water) Do the particles (atomic, and sub-atomic) present in the solar core move at the same speed as they would if they were in a vacuum? EDIT: Just read the article on mean free path; i've read it before, and forgotten. Anyway, mean free path speed of the sub-atomic particles sounds like what i have in mind.


title: Can lightning be used to solve NP-complete problems?

text: I'm a MS/BS computer science guy who is wondering about why lightning can't (or can?) be used to solve NP complete problems efficiently, but I don't understand the physics behind lightning, so I'm posting here. What seems peculiar to me about lightning is that it seems to follow the "easiest" path to a given destination, as opposed to just the shortest path. This seems analogous (identical?) to the weights in the traveling salesman problem. Since the lightning process ionizes the air and seems to pick the easiest route quickly, I have 2 questions: 1) Does lightning use a method similar to Dijkstra's approach (such that a "good enough" path is found in polynomial time, but not necessarily the "best of all options")? Is it basically an NP "solver"? If the answer to the above question is "no" and it is indeed the best pick: 2) Could the selection process of lightning be used in a computer algorithm to solve a traveling salesman-type problem in polynomial time, and 3) If not, could an arbitrary series of weighted nodes be physically set up somewhere and have electricity passed through it and have the results fed back into some kind of computer mechanism for every time a user wants to know a shortest path? If you could use this approach to solve NP-hard problems quickly, you would have your name on the news and maybe in the history books. So why hasn't anyone used lightning to solve this problem? UPDATE: Another aspect I find interesting: when lightning strikes it ionizes the air to find the shortest path. If the world were totally flat (and the clouds were also) this would ionize all the air equally. Which leads me to believe the route "picked" by the lightning incorporates all points, which seems to be a requisite for finding the best of all solutions.


title: Flux compactification

text: How flux compactification solves the moduli space problem in string theory? Please provide some details and, if posible, an example. Thanks in advance.


title: why certain superpositions of quantum states are supressed?

text: it has been said that the electron is the fundamental representation of the Poincare group, with only two conmuting observables, $( \sigma , p_{\mu})$. This question regards what is usually called the _superselection rules_ what i don't understand why states that are superpositions of different $C$ like $e + \overline{e}$ (and electron and a positron superposition) are supressed and how. What about superpositions like $e + \mu$ (an electron and a muon)? and finally, why neutrino flavors are different and in this regard, can be mixed in superposition? what about superpositions of particles with different mass? what are the currently accepted mechanisms for this supression? are there other alternatives worth pursuing? are there high-decoherence impedances in some state combinations that make them decohere quicker than others or are they **fundamentally** supressed?


title: Wave equations & propagation theories

text: I'm interrested in making computer simulation but I've run into rather physics oriented problem. I have to choose how to propagate my wave. Though I've found technique called FDTD (finite-difference time-domain), I could find only explanations how to work with this method while solving electromagnetic waves but I'm more interested in mechanical waves - water, earth,... So there's first problem. EM waves are nicely discribed with Maxwells equations, but I couldn't find anything like that for mechanical ones. Only equation I could find is $y = A \sin(k x - \omega t + O)$. This is fine, one can choose a distance and a time and he gets height of that point (considering that we are working on a plane). But doesn't wave gets also damped - in time as well as distance? And what formula describes this? And the second part of a question - in school I was teached a propagation theory called Huygens principle, and it goes like: at every point where wave envelope propagate it creates elementary waves, and frontface of those again create envelope. I don't fully understand how it should be imagined, because creating waves all around a circle like on this image: http://kr.cs.ait.ac.th/~radok/physics/fig300.jpg (sorry, but I got a little problem with tags..), there is nice envelope but inside of that is mess of curves that would gave us interference. So it's just a model and we don't care what's inside as long as we have envelope, or am I missing something? And as I said, I'm creating computer simulation and propagating wave like that would be probably slow, so is there any other theory on how wave propagates or this is the best and most used one? Thanks


title: Angular and luminosity distance in general?

text: Consider a non-Friedmannian Universe in which we know the trajectories of photons, ie in which we know null geodesics $\left(\eta, x^{1}, x^{2}, x^{3}, a, z\right)$ where :   * $\eta$ is the conformal time   * $x^1$, $x^2$, $x^3$ are spatial comoving coordinates   * $a$ is the scale factor   * $z$ is the redshift In such an Universe, how do one compute the angular diameter distance and the luminosity distance and what are all the relations involving these quantities in general (non-Friedmannian) ?


title: Why is string theory a two dimensional quantum (conformal) field theory on its worldsheet?

text: In string theory, we quantize the two dimensional field theory on the string's worldsheet. I have a question about this kind of quantization of string theory: did we have similar theory for point-like particle theory, i.e., we quantize the classical theory on worldline, then we get the usual quantum mechanics? If the string theory is the two dimensional quantum field theory on its worldsheet, then what does the momentum operator measure? Does it measure the expected momentum from initial time to final time represented by the worldsheet (because usually the moment operator measure the moment of particle at some specified time)?


title: Fluid Mechanics with calculus on manifolds

text: Fluid Mechanics is a branch of physics that uses a lot of vector calculus in $\mathbb{R}^3$ to describe phenomena mathematically. Calculus on manifolds, however, is the straightforward generalization of vector calculus and has a lot of interesting and useful tools like differential forms, lie derivatives, flows of vector fields and so on. Is there some book/article that treats fluid mechanics using calculus on manifolds? I've searched a lot for some book like this but I could only find one called "A mathematical introduction to fluid mechanics" which although uses a lot of math restricts itself to vector calculus. Is there some other book that uses the tools of the calculus on manifolds to describe fluid mechanics phenomena?


title: Are there any Meson colliders?

text: There are no stable mesons as far as I know. Are there any meson-colliders analogous to for example proton-proton colliders?


title: Can the spin of a photon change during its "life"?

text: Or is the spin set in one of two possible states at its moment of creation and does not change for the rest of the duration of its "life"?


title: Optimal telescope size?

text: Consider a diffraction-limited telescope with unobstructed aperture $D$. Such a scope is capable of yielding an angular resolution $\alpha$ that scales as $\lambda/D$, with $\lambda$ denoting the wavelength of the light. However, in reality such a telescope will need to look through a turbulent atmosphere, the refractive index variations of which will cause wavefront errors with a spatial correlation length (Fried parameter) $r_0$. As a result, no stable diffraction pattern but rather a speckle pattern will form. I am interested in the angular resolution corresponding to the average size of this speckle pattern. What is this angular resolution $\alpha(D/\lambda,r_0/\lambda)$? More specifically: how does $\alpha$ scale for $D\gg r_0$, and for fixed $r_0$: is there an optimal (highest resolution) aperture $D$? Or more down-to- earth: **If I built a200 inch Hale telescope in my backyard*, would it beat a 10 inch amateur telescope in terms of optical resolution?** *You may assume my backyard to have a typical seeing characterized by a Fried parameter $r_0$ of about 4 inch.


title: Meaning of the phrase "dipole moment of the combination"

text: Here is a question I came across in a book: > Three point charges $-q$,$-q$ and $2q$ are placed on the vertices of an > equilateral triangle of side length $d$ units.What is the **dipole moment of > the combination**? I had learnt that the dipole moment is defined for 2 point charges only with equal magnitudes but opposite character.I am in a bit of a fix regarding what the author means by that phrase in bold.Thank you in advance! Edit:I just noticed theoretical physics has been merged here;I sincerely apologise if the question is unbearably stupid.


title: Supersymmetric Sigma Model

text: I was working through the Mirror Symmetry book by Clay Math Institute. It deals with supersymmetric sigma model in 10.4 section. It doesn't derive how the action is invariant under the variation. I am trying hard, but stuck at few places. Does any one have any references for such a calculations ? The Lagrangian and variation is given in the following image. ![I have given the lagrangian, if one doesn't have the soft copy of the book.](http://i.stack.imgur.com/mu6Ta.jpg)


title: QM formalism is one big confusion - lack of geometrical explaination with images

text: I have been trying to learn QM and it went well (all untill harmonic oscilator) until i had to face the formalism:   1. **Hilbert space-** As a novice to QM i am very sad that in none of the books i have read i found **the reasons** for using Hilbert space $\mathcal H$ at first place followed by a **full geometrical** explaination of this space and how we build this space out of $\mathbb R$. It goes same for for its dual space $\mathcal H_d$. Where can i get this? Every single author starts this topic by just bombard novices with bunch of rules for $\mathcal H, \mathcal H_d$ which i can't just trust and this is only **pointless learning by heart**.   2. **Dirac's notation** ... because i don't understand Hilbert space i don't understand what i am allowed to do with kets $|~ \rangle$ and bras $\langle~ |$ and for example:      * Why do we have to write an operator on the left side of kets $| \hat{A} \psi\rangle = \hat{A}|\psi\rangle$ but for bras it is vice versa (we write it on the right) $\langle \hat{A} \psi| = \langle \psi|\hat{A}$.      * Why can we factor out a constant from kets $| a\psi \rangle = a |\psi \rangle$ and we can only factor out complex conjugate from bras $\langle a \psi | = a^* \langle \psi |$.   3. **Hermitean stuff** (which i don't even know what it means as it has too many names which totally confuse me - authors should really start writing in a unified language (1 word 1 meaning) otherwise it is a big mess here. For eample here. Just check how many names there is for an _conjugate transpose_ : $A^*,A^{*T},A^{T*},A^\dagger,A^+, A^H$ $\rightarrow$ **this leads to a confusion** ). It seems to me that $A^\dagger$ is the most spread in QM by far, but $A^{*T}$ makes much more reason.    4. **And finally i need all above to just understand this one line** : $a a^\dagger = n$. I encountered this in my other topic where i tried to explain a harmonic oscilator to myself. How am i suppose to believe this? Well @Eugene B provided a proff but i don't understand it... The proff was:  * * * _By definition,_ \begin{equation} \hat{a}^\dagger \left| n \right\rangle = \sqrt{n + 1} \left| n + 1 \right\rangle , \end{equation} \begin{equation} \hat{a} \left| n \right\rangle = \sqrt{n} \left| n - 1 \right\rangle , \end{equation} _where $\left| n \right\rangle$ is the eigenstate of creation and annihilation operators, as well as of the Hamiltonian (due to the fact that they commute - homework to prove)._ _Now_ \begin{equation} \hat{a}^\dagger\hat{a} \left| n \right\rangle = \hat{a}^\dagger \sqrt{n} \left| n - 1 \right\rangle = \sqrt{n} \sqrt{n} \left| n \right\rangle = n \left| n \right\rangle , \end{equation} _so conclude that the eigenvalue of a number operator, $\hat{N}$, is just $n$..._ * * * I don't understand the meaning of definitions for $a$ and $a^\dagger$ and even if i did, how can the line above even be true? \begin{equation} \hat{a}^\dagger\hat{a} \left| n \right\rangle = \hat{a}^\dagger \sqrt{n} \left| n - 1 \right\rangle = \sqrt{n} \sqrt{n} \left| n \right\rangle = n \left| n \right\rangle , \end{equation} If i changed this equation just a bit and used a $|\psi\rangle$ in place of $|n\rangle$, i would get a different result and couldn't conclude that $a a^\dagger = n$. **Long story short i am totaly confused and i need some guidance to understand this. The best for me would be any book with a geometrical explaination of Hilbert space... Is there any?**


title: Monkey and tree - projectile motion

text: The famous scenario: A hunter is trying to shoot a Monkey hanging from a tree. However, this question doesn't mention the monkey jumping down from the tree or trying to escape. (The hunter uses a tranquilizer gun, so I'm guessing the sound goes unheard.) We only know two things: 1) The height at which the monkey is hanging. 2) Distance of the monkey from the hunter. (The hunter aims his gun directly at the monkey.) I've solved it using this approach: Solving for initial velocity required to launch a projectile to a given destination at a different height Does this fit the bill or am I missing something? Why does the answer have $d\, \text{tan}\, \theta + h$ in denominator? I'm getting $d\, \text{tan}\, \theta- h=0$ in the denominator! Does denominator = $0$ simply mean that it's not possible? In other words, does the monkey have to fall down if the bullet has to hit the monkey?


title: Before a once-warm lake starts to freeze, must its temperature be 4°C throughout at some point?

text: This is a problem I just started puzzling over, and I felt this would be a good forum to check my reasoning. So here are the relevant observations followed by my question: Water achieves its maximum density at roughly 4°C. That is, water (including ice) at all other temperatures below or above 4°C is less dense. Since matter is ordered from top to bottom by increasing density, any 4°C water in a lake will be found at the bottom. Temperature is a continuous property, and so the water in a lake must be arranged by temperature as well. That is, there can be no regions of water with temperature 4° and 6° unless there is a region between them with temperature 5°. So, my initial puzzlement arose from trying to imagine the temperature arrangement of water with temperatures from 3° to 5°. If 4° water is the most dense, it must be on the bottom of the lake, and both the 3° and the 5° water must be above it. However, if temperature is continuous, there must be 4° water _between_ any 3° and 5° water. So, there is no way for these three temperatures of water to coexist without violating either the ordering of densities or the continuous distribution of temperature. I suspect the coexistence of these temperatures in water is most likely possible, but it is short-lived. I imagine some kind of convection current where the water eventually reaches equilibrium at 4°C, but the picture of this is pretty blurry to me. So does it follow from this that for a relatively warm lake (temps from 10-20°C) to begin to freeze, all the water must reach 4° before any sub-4° temperature is reached? Otherwise, there would be a situation with 3-5° water coexisting. Right? **EDIT:**   Here's an extension of this thought that begs for some testing. Since the ordering of densities occurs because of buoyant forces, which occur in a gravitational field, I wonder if the temperature profile of cooling/freezing water is significantly different in anti-gravity. Based on the above observations, I would expect the temperature barrier at 4°C not to be observed. Whoa!


title: Spinors in more dimensions and new degeneracies?

text: As you more than probably know spinors dimensions go as $2^{\frac{D}2}$ in D spacetime dimensions. If we look at the peculiar case of D=2*4, spinors have 4 components and we usually say that's related to the 4 "degrees of freedom": particle/antiparticule & spin up/down?*. So basically linked to the degeneracy in energy and spin. But if we know go to D=6, they have 8 components, I am asking myself if it's not the appearance of a new physical degeneracy, but I don't really see what could it be. Maybe the group of rotation in 6 dimension has one more Casimir operator. Am I seeing it in a totally wrong way?


title: Where to find probability density plots for all elements?

text: Does anyone know where I can find something similar to this, but for all elements? I would love to find something with the same image quality. Also, is there any software that can produce images like these?


title: Singularities in Schwarzchild space-time

text: Can anyone explain when a co-ordinate and geometric singularity arise in Schwarzschild space-time with the element $$ ds^{2}~=~(1-\frac{2GM}{r})(dt)^{2}-(1-\frac{2GM}{r})^{-1}(dr)^{2}-r^{2}(d\theta)^{2}-(r^{2}\sin^{2}\theta)(d\theta)^{2}$$


title: Gamma ray bursts luminosity measurement

text: physicists use millisecond fluxes i.e.,fluxes calculated within a very short timescale to measure the actual 1 second flux for these grbs.But to do that they have to multiply the millisecond fluxes by a factor of 10^3. So why instead of recording and using the actual 1 second peak flux as received by the detectors they use millisecond fluxes to convert them to 1 sec fluxes from which later they can derive the luminosity? IN case of other such high energy astrophysical phenomenas which include blazars physicists use photon flux/second, but not in case of GRBs.So isn't there a chance that the luminosity derived from such millisecond fluxes might be affected with some errors?


Why does a star beyond a certain mass limit (Chandrasekhar limit) only become a black hole? A star is first made of hydrogen, it undergoes nuclear fussion reaction combining into helium and releasing a large amount of energy. This process continues till star is made up of iron core as iron has largest value of binding energy per nucleon, after this if mass of the star is above the value of Chandrasekhar limit it becomes a black star, what is the reason for this and why is certain mass limit required?


title: Relationship between stopping potential and work function

text: Suppose I have a cathode with a work function of 3eV and an anode at a potential of 2V above the cathode. If a photon having 2eV of energy hits the cathode, what happens? A. An electron is emitted from the cathode because it only takes 1eV to go from the cathode to the anode. B. An electron is not emitted because 3eV are required to escape the cathode. Sources seem unclear on this and I need to know whether it is possible to knock electrons off of a surface using light of a lower energy than the work function given a sufficient electrical potential.


title: Do electrons need specific energies to excite electrons

text: Photons need specific energy levels, equal to the difference between two energy levels to excite an electron in an atom. Is this the same case with electrons that collide with atoms?


title: The effect of high frequency AC current in a fresh corpse

text: 1. Given two AC currents of the same intensity, eg 20 mA, but different frequency, 60 Hz and 20MHz, which one would be more likely to reach the heart ?   2. I've read about the skin effect and how current tends to flow on the surface of a conductor. Does that apply to a dead human body at this particular context?    3. Would the low frequency penetrate the dead body and reach the heart and the high frequency be confined to the skin?


title: How to plot $(x,y)$ coordinate of projectile motion (with air-resistance)?

text: I am trying to plot a graph to show the difference in projectile motion when it has air-resistance and when it doesn't have air-resistance. I set the mass, $v_0$ and $\theta$ as constant I can plot the motion by using formulas. $$x=v_0t\cos \theta, $$ $$y=v_0t\sin \theta -\frac12 g t^2. $$ How do I plot the projectile motion with air resistance?


What final states can proton-proton collision generate? Where to find listing of possible final states of a particle interaction, for example proton colliding with proton?


title: Why would mode-hopping occur as the current is changed?

text: If some diode laser is given an input current, why should we see mode-hopping if the current is changed?


What is the dominant interaction in elastic collision pi- + neutron -> pi- + neutron? Is it strong nuclear interaction or electromagnetism? Edit: sorry I meant to say "elastic". There is no matter - energy transformation.


title: What are the specific requirements for a do-it-yourself quantum double-slit experiment?

text: I was shocked to recently learn that the double-slit experiment is not only possible to do with completely ordinary equipment (with photons of course), but it actually looks rather easy. This is from a MIT tech video: http://www.youtube.com/watch?v=Ippat5KPwmk&t=5m14s That time stamp starts around the description of their device. Here are some of the specifications:   * Stand 10-20m away from a light source in an otherwise mostly dark environment   * 1/4 -1/2 mm separation between the two pinholes (which act as the slits for this double-slit experiment), around 4:56 in the video    * It looks like the holes are made about 1/4 mm in diameter, but this isn't explicitly stated   * Obviously observe or resolve the pattern maybe 10 cm behind the slits with ambient light shielded out. Now, the obvious question is why these parameters work. More importantly, what is the scientific process for selection of these experimental parameters? I understand that a combination of the separation between the slits and the wavelength determines the distance between the peaks of the interference pattern, but what is the condition on the distance between the peaks? If this is a pattern that you can observe by looking into the tube, does that mean that several peaks of the pattern must fit in the diameter of a human iris? Also, what is really the limit that must be satisfied for the light to be sufficiently coherent? Will this experiment not work if the pinholes are too large? If the source is too close? What is the actual limits from the mathematics of the double-slit experiment that can be applied to a real home experiment?


title: Abstract vector form for integral of wavefunctions in subspace

text: In Landau, Lifshitz "Quantum mechanics (non-relativistic theory)" density matrix is given in the form $$\rho(x,x^\prime)=\int\Psi(q,x)\Psi^\ast(q,x^\prime)\text{d}q,$$ where $x$ is set of coordinates for the inner system and $q$ is set of coordinates for outer system. I'm trying to rewrite it in abstract (representationless) form for density operator, i.e. using the fact that $\Psi(q,x)=\left<q,x\middle|\Psi\right>$ and $\Psi^\ast(q,x^\prime)=\left<\Psi\middle|q,x^\prime\right>$. It seems to hold that $\left|q,x\right>=\left|q\right>\otimes\left|x\right>$. At the same time, as LL say, $\Psi$ doesn't generally fall into a product of functions of $x$ and $q$ alone, which I interpret that $\left|\Psi\right>$ can't be written as a tensor product of some two vectors in subspace spanned by $\left|q\right>$ and by $\left|x\right>$. So, what I try: $$\rho(x,x^\prime)=\int\Psi(q,x)\Psi^\ast(q,x^\prime)\text{d}q=\int\left<q,x\vphantom{^\prime}\middle|\Psi\right>\left<\Psi\middle|q,x^\prime\right>\text{d}q=\int\left.\left(\left<q\vphantom{^\prime}\right|\otimes\left<x\vphantom{^\prime}\right|\right)\middle|\Psi\right>\left<\Psi\middle|\left(\left|q\right>\otimes\left|x^\prime\right>\right)\right.\text{d}q$$ Clearly, the result here must be a scalar depending on $x$ and $x^\prime$, as expected, but it is only a representation of the density operator. So I want to somehow get rid of $\left|x^\prime\right>$ and $\left<x\right|$, but don't have any ideas on how to expand the tensor products in parentheses. So, my question: how can the expression I got be simplified further and rewritten in abstract representation-free form? **Edit:** I've realized that I can get an expression for _operator_ if I multiply everything by $\left|x\vphantom{^\prime}\right>$ at left side and $\left<x^\prime\right|$ at right side and then integrate over $x$ and $x^\prime$: $$\hat \rho=\int\left|x\vphantom{^\prime}\right>\left.\left(\left<q\vphantom{^\prime}\right|\otimes\left<x\vphantom{^\prime}\right|\right)\middle|\Psi\right>\left<\Psi\middle|\left(\left|q\vphantom{^\prime}\right>\otimes\left|x^\prime\right>\right)\right.\left<x^\prime\right|\text{d}q\,\text{d}x\,\text{d}x^\prime$$ Still this is not satisfactory because I have to introduce dummy basis $\left.\middle| x\right>$. So, the question which remains: how can I eliminate the need in dummy $\left.\middle| x\right>$ basis? Basically it seems I need some way to define operator (whose matrix is non-rectangular in finite-dimensional case): $$\hat q=\int\left(\left|q\right>\otimes\left|x\right>\right)\left<x\right|\text{d}x,$$ but without introducing $\left.\middle|x\right>$ basis. * * *


title: Does Noether's theorem also give rise to quantities conserved over space?

text: Noether's theorem gives rise to quantities that are conserved over time. But does it also give rise to quantities that are conserved over space?


title: Solar chamber trapped light

text: Was wondering if trapping light in a reflective chamber could heat rods sufficiently enough to heat water? Trying to think of new ways in which we can utilise solar energy. ![Ref pic A.](http://i.stack.imgur.com/WkixY.jpg)


title: Small scale water power, how does water volume and hight convert into electric energy?

text: I was playing a little bit with the basic physics behind water power production but I can't get the numbers right. Let's say that I put a windmill that pumps water into a watertank on the top of my house, then I connect some kind of pipe with generator and starts to drain the watertank. How much electric power (kWh) can I get out from a watertank with size $X\text{ m}^3$ placed $Y\text{ m}$ above the ground? How does the formulas look like? * * * Let's put some numbers on this problem, and see where we end up: Let's say the tank is $1\text{ m}^3$, and it is $10\text{ m}$ off the ground so the water will fall $10\text{ m}$ to the generator. Let's connect the generator with a standard garden hose that has a 1 inch diameter, with an area of $2.54\text{ cm}/(2\pi) \sim 5.1\text{ cm}^2$. And then I guess we would get a $10\text{ m}$ column of water pressure, that could be transformed with the area into the force the hight is putting on the system. Something like the earths gravity (9.82)*density*height = 9.82*1*10 ~ 98 Newton (???). And then maybe use that we can find that pressure=Force/Area, but how to move from pressure to energy? * * * Thanks David Zaslavsky for the example, and in theory that would mean that to store 1kWh I need like 40m3 at 10m height. That more or less mean that if one would try to build something like this in real life things need to be quite big. Also thanks Fortunato for illustrate the practical problem in extracting the energy, and that even thou it is hard to get hight numbers it can be worth the effort anyway.


title: Open problems in General Relativity

text: I would like to know if there are some open mathematical problems in General Relativity, that are important from the point of view of Physics. I mean is there something that still needs to be justified mathematically in order to have solid foundations? Please tell me. Thanks! **Edit:** The same question was asked (and answered from a mathematical perspective) on math.SE.