title: Deriving equations of motion of polymer chain with Hamilton's equations

text: This is related to a question about a simple model of a polymer chain that I have asked yesterday. I have a Hamiltonian that is given as: $H = \sum\limits_{i=1}^N \frac{p_{\alpha_i}^2}{2m} + \frac{1}{2}\sum\limits_{i=1}^{N-1} m \omega^2(\alpha_i - \alpha_{i+1})^2 $ where $\alpha_i$ are generalized coordinates and the $p_{\alpha_i}$ are the corresponding conjugate momenta. I want to find the equations of motion. From Hamilton's equations I get $\frac{\partial H}{\partial p_{\alpha_i}} = \dot{\alpha_i} = \frac{p_{\alpha_i}}{m} \tag{1}$ $- \frac{\partial H}{\partial {\alpha_i}} = \dot{p_{\alpha_i}} = -m \omega^2 (\alpha_i - \alpha_{i+1} ) \tag{2}$ , for $i = 2,...,N-1$. Comparing this to my book, (1) is correct, but (2) is wrong. (2) should really be $- \frac{\partial H}{\partial {\alpha_i}} = \dot{p_{\alpha_i}} = -m \omega^2 (2\alpha_i - \alpha_{i+1} - \alpha_{i-1}) \tag{$2_{correct}$}$ Clearly, I am doing something wrong. I suspect that I'm not chain-ruling correctly. But I also don't get, where the $\alpha_{i-1}$ is coming from. Can anybody clarify?


title: Acoustic Metamaterials: Negative Bulk Modulus?

text: In acoustic metamaterials we have simultaneously negative bulk modulus, $\beta$, and effective mass density, $\rho$. I understand how one can obtain a -ve $\rho$ by constructing a solid-solid system with vastly different speeds of sound, as this can be considered as a mass-in-mass system connected by springs see here. But how to get a negative bulk modulus eludes me, I am aware that in doubly negative acoustic metamaterials the negative bulk modulus is achieved by having a sphere of water containing a gas i.e. bubble-contained-water spheres. But I can't see why this would result in a negative composite bulk modulus. I mean obviously water has an extremely high bulk modulus and is practically incompressible, whilst air is highly compressible so external pressure on the system would fail to compress the water however the pressure would be transmitted to the gas which would compress, thus creating an extremely low- pressure region in between the gas and water (the water wouldn't expand due to it's large bulk modulus right?) and thus the gas would then re-expand and possibly exert pressure on the water - I suppose if this pressure exceeded the external pressure then the sphere might expand resulting in an expansion of the system upon application of external pressure and thus a negative bulk modulus??? I believe the answer may lie in this paper and I will attempt to get my University vpn to work to see if I can access it. Any information about acoustic metamaterials would be greatly appreciated. P.S: How is metamaterials not an existing tag?!


title: Use of Monte-Carlo simulation in High-energy Physics

text: I've been doing some research into the analysis used in particle physics when determining the significance of a finding (e.g. the recent Higgs candidate was announced as a boson in the 125-126 GeV/$c^{2}$ mass range with a $5\sigma$ significance). I believe this confidence level is determined by estimating the Standard Model background cross-section they should observe if all known processes except for Higgs production occurred and then taking the ratio of the observed cross- section with that predicted. I am interested in how they determine the background cross-section. I believe that they use a Monte-Carlo simulation normalized to fit with well-known processes such as $Z^{0}Z^{0}$-production, but how exactly does this work? I am aware that the tool primarily used in High-energy physics for this kind of modelling is Geant, and I would like to know more about how this works. I have looked through the source code, but it is very fragmented and thus quite hard to understand, especially since I am not 100% certain what it is that should be occuring in the code.


title: Difference between nautical and terrestrial miles

text: Does someone know the historical reason behind the difference in physical units between nautical and terrestrial miles?


title: Spin state of electron after measurement

text: I have a system of two spin 1/2 particles in a superposition of spin states in the z-direction given by: $\psi = \frac{1}{2} |+ +\rangle + \frac{1}{2} |+ -\rangle + \frac{1}{\sqrt{2}} |- -\rangle$ where $+$ designates spin up, $-$ designates spin down and the first particle's state is the first term in each ket and the second particles' state is the second term in each ket. If I measure the spin on the first particle and get a value of $-\hbar / 2$ (corresponding to a spin down state) is the new state of the particles simply $\psi = | - - \rangle$ meaning that the first particle is now "set" to being spin down? And if I determine the spin on the first particle to be spin up, would the subsequent state be $\psi = \frac{1}{\sqrt{2}} |+ +\rangle + \frac{1}{\sqrt{2}} |+ - \rangle$ ? Basically, my question is once I make a measurement of a spin of a particle, does the wavefunction stay collapsed on the spin determined? And does having a second particle affect this in any way?


title: Hypotheses a hindrance in physics?

text: In a recent episode of the _Big Picture Science_ podcast, there was an interview with Stuart Firestein (chair of the Columbia University Biology Department) in which he discussed his book _Ignorance: How It Drives Science_. In particular, he mentioned that for the most part he dislikes hypotheses--- they tend to pigeonhole people's idea of what "good data" is, so that if when unexpected happens, it's often thrown out as "bad data". Is this a problem in physics? If so, how much of a problem is it, and what, if anything, do physicists do to attempt to mitigate it? My instinct tells me that this is a problem, albeit possibly not as much as in so-called "soft" fields such as psychology and sociology where eliminating variables and bias is much more difficult (at least I assume it's much more difficult).


title: Renormalization: Why is only a finite number of counter-terms allowed?

text: I have a question please about renormalization in QFT. Why a renormalizable theory requires only a finite number of counter-terms?


title: What is the force between two perpendicular wire?

text: What is the force between two perpendicular wire carrying current, one to the north and one to the east?


title: Pressure loss in a syringe

text: I'm currently working on a problem which is really giving me some issues. The problem concerns the force required to expel water from a syringe. We have a 20 ml syringe (which is $2\times10^{-5}$ meters cubed) with a diameter of 1 cm, full of water. The needle of the syringe is 40 mm in length and has a diameter of 0.2 mm. All of the water must be expelled from the syringe in 20 s. How much force must be applied to the syringe head to achieve this? Ordinarily this is fine, but we have to include the pressure loss as a result of the friction in the needle. I'm using the Darcy–Weisbach equation to determine this. I calculated the speed the fluid needs to flow at by dividing the flow rate by the cross-sectional area of the needle. I've used a Moody chart to get $f_D$ as 0.046, and I'm using $\rho = 998.21$. I'm guessing the pressure loss in the needle is therefore $$0.046\times\frac{0.04}{0.002}\times\frac{998.21\times31.8^2}{2} = 4.64\,\mathrm{MPa}$$ Is that correct? In which case, how do I now get to the force from here?


title: Does special relativity unify the two phenomena at the base of Faraday's flux law (was Feynman wrong in this case)?

text: Consider Faraday's flux law for the EMF generated in a conductor loop: $$ \varepsilon = - \frac{d \phi}{dt},$$ where $\varepsilon$ is the EMF, and $\phi$ is the magnetic flux through the loop. There are two possible causes for the flux to variate over time: variations in the magnetic field ("transformer EMF") and variations in the area enclosed by the loop ("motional EMF").   Feynman has noted that this is a unique case where a single rule is explained by two different phenomena: > We know of no other place in physics where such a simple and accurate > general principle requires for its real understanding an analysis in terms > of _two different phenomena_. Usually such a beautiful generalization is > found to stem from a single deep underlying principle. Nevertheless, in this > case there does not appear to be any such profound implication.   >  —Richard P. Feynman, _The Feynman Lectures on Physics_ (Volume II, 17-2). **Is it really true that there is no way to view these two phenomena as one?**   For example, it says in this Wikipedia article that this apparent dichotomy was part of what led Einstein to develop special relativity.   **Does special relativity give us a unifying principle to derive Faraday's law from?**


title: Pulsar beam radiant intensity distribution

text: I'm curious about the radiant intensity distribution of pulsars: what's the general dependence of intensity on angle, and what are typical angular beam widths? How much does the beam width vary between pulsars? (Presumably this is tied to magnetic field strength.) Even something as simple as a very sketchy plot of intensity vs. angle would be great. It's easy enough to find plots of observed intensity vs. time for individual pulsars, but it takes a bit to get from those to the distribution at the source.


title: Where are the Voyagers going?

text: Given enough time, where are the Voyager spacecrafts heading? (Assuming some alien civilization doesn't pick them up.) Will they pass by any interesting stars on the way to the black hole at the center of our galaxy or will it perhaps leave the galaxy? What are the highlights on their journey that we can reasonably predict?


title: Which direction before dawn to look for Comet Lovejoy (C/2011 W3)?

text: Various websites today are reporting with photos and videos of Comet Lovejoy. However, I can't seem to find a definition of which direction to look for it tomorrow morning. I'm in Christchurch, New Zealand (roughly 43°S 173°E).


title: Seeking chiral anomaly EFT example

text: If an effective field theory has a chiral anomaly it means that chiral symmetry isn't a symmetry of the underlying theory which has been cut off to make the EFT. My question is whether there's a good example where this can seen explicitly. The kind of thing I'm picturing would be an EFT at cutoff A which has no chiral symmetry, while the same EFT at cutoff B << A has chiral symmetry which is anomalous.


title: Time Dilation at zero velocity (and zero gravity)

text: From what I've learned, the more an object travels closer and closer to the speed of light, the more time will slow down for that object.. at least from an outside perspective.. It was shown that atomic clocks run slower in high speed orbit than clocks on earth.. I assume that the rate of radioactive decay (for example) is also slowed down at high speeds (correct me at any time, please). We are moving through space right now at 760 miles per second (0.40771% the speed of light), which I can only assume is _our_ current "cosmic clock", which also regulates how fast radioactive decay happens on earth (if we continue with that example). When an astronaut is traveling at high velocity, his/her velocity is being added to the overall velocity of our galaxy moving through space, right? So my question is this: What will happen if an object were to stay completely _stationary_ in space- time? Far away from any galaxy.. Will time go infinitely fast for that object? Will it instantly decay? Since space is expanding, I realize you can't really stay "stationary".. but I mean: not having velocity of moving _through_ space. Thanks :)


title: Derivation of angular momentum commutator relations

text: I'm trying to understand the derivation of the angular momentum commutator relations. How is $$[zp_y, zp_x] ~=~ 0?$$ How is $$[yp_z, zp_x] ~=~ y[p_z, z]p_x?$$


title: General integral to find resistance

text: My question is: **is there a simple and truly general equation for the resistance between two electrical equipotential surfaces?**. Obviously, if so, what is it, and if not, why? It would be very difficult to solve, granted, but I just want to see a calculus equation that is fully descriptive. I have two frameworks under which this could be entertained, I'll write those out and then explain the motivation. To start with, we need propose that the volume separating the two surfaces has a volumetric resistivity, $\rho$ in units of $(\Omega m)$. # Single Volume Framework We can limit the discussion to a defined volume, then the surfaces reside in that volume or on the surface of it. This volume may have a constant resistivity $\rho$ while everywhere outside the volume is completely electrically insulating. # Infinite Volume Framework An alternative to the above approach that might make the task more or less difficult would be to replace a constant resistivity with a spatial dependence $\rho(\vec{r})$ and no longer require a boundary condition. In that case we only have 3 mathematical inputs to the problem, which is the resistivity defined for all $\vec{r}$ and a definition of the two surfaces, $S_1$ and $S_2$. # Known Algebraic Analogs The basic algebraic formulation that I find insufficient is: $$R = \rho \frac{\ell}{A}$$ Where $l$ is the length of the restive material that is any shape which has translation symmetry over that length, and $A$ is the cross-sectional area. Obviously, this is a rather simple equation that won't apply to more complicated geometry. Even more sophisticated academic sources seem to give equations that fall short of what I'm asking. For example: $$R = \rho \int_0^l \frac{1}{A(x)} dx$$ I think it's obvious that an equation such as this is built upon a myriad of assumptions. For a thought experiment, imagine that the area starts out as very small and then pans out to very large quickly. Well, accounting for the larger area in the above sense underestimates the resistance, because the charge has to diffuse out perpendicular to the average direction of flow as well as parallel to it. I have some reasons to suspect this might actually be rather difficult. A big reason is that all the approaches I'm familiar with require the flow paths to be established beforehand, which can't be done for what I'm asking. So maybe this will result in two interconnected calculus equations. # Motivation I had an interest in Squishy Circuits, and it occurred to me that I can't quickly and simply write down the equation for resistance between two points. The unique thing about Squishy Circuits is that it calls for two types of dough, one that conducts and one that is mostly insulating. However, the recipes aren't perfect and because of that, the young children who play with these circuits regularly encounter the limits of conductor and insulator definitions. If you make your conductor dough too long and/or too thin, you will encounter dimming of the light you connect with it. Similarly, a thin insulator layer will lead to a lot of leakage current which also dims the light.


title: Gravitational potential energy of mass between two planets

text: Suppose I want to launch a rocket from earth to some point $O$ between the center of earth and the center of moon (on a straight line connecting their centers), where the gravitational force of the moon 'cancels out' the gravitational force of the earth (this point is located at $\approx 54 R_E$ from the center of earth where $R_E$ is the radius of earth). I want to know how much energy I should spend in order for the rocket to get there (neglecting the atmosphere and the rotation of the earth around its axis). So, I know that this is basically the difference between the potential energy at the start point and at the end point of the destination. However, $O$ is located not only in the gravitational field of the earth, but also in the gravitational field of the moon. And it seems that I cannot neglect the gravitational potential energy of the body at the moon's gravitational field. So my question is - how can I combine these two? How can I calculate the total GPE of the body in two (or even more) intersecting gravitational fields?


title: What are the frameworks of physics?

text: Are there physical theories in use, which don't fit into the frameworks of either _Thermodynamics_ , _Classical Mechanics_ (including General Relativity and the notion of classical fields) or _Quantum Mechanics_ (including Quantum Field Theory and friends)?


title: Does the Moon's orbital inclination follow the wobble in the Earth's rotation?

text: As the Earth wobbles during rotation, does the higher gravity at the equator tend to pull the moon toward an equatorial orbit even as the earth does that thousands of years wobble cycle? It would seem to me that the higher gravity, due to the larger diameter at the equator, would keep the Moon's orbit close to the equator. Or does the orbit of the moon with respect to the earth stay relatively stable with respect to the solar plane around the sun?


title: Moment of inertia of a sector about its center point?

text: What is the moment of inertia of a pizza slice that has a radius r, an angle (radians) of theta, and a height of h about the center point perpendicular to the cheese plane?


title: A simple action inequality

text: How do I prove that $$\int \dot{x}^2 dt\geq \int \langle \dot{x}\rangle^2 dt $$ (i.e a free particle not in any external potential field movies with uniform veloctiy)


title: Cosmic radiation cutoff at LOW energies?

text: The energy spectrum of the cosmic radiation (not CMB) is limited to both sides. I know about the GZK-cutoff at high energies. Basically the interaction probability for photons of energies above 10^20 eV becomes so high, that all have interacted before they can reach us. But why is there a limit at lower energies? Earth magnetic field and/or athmosphere, radiation belt? Perhaps someone can explain that to me.


title: Can the expectation value of the square of momentum be negative?

text: I've been solving a problem in quantum mechanics, and I was deriving the standard deviation of $P$, knowing that $\langle P\rangle=0$. Because $\Delta P=\sqrt{\langle P^2 \rangle - \langle P \rangle ^2} = \sqrt{\langle P^2 \rangle}$, I was trying to calculate the expectation value of the square of the momentum. The wave function was given by $\psi(x)=\sqrt{\alpha}e^{-\alpha|x|}$ where $\alpha>0$. Here is what I've done. $$\langle P^2\rangle = \int_{-\infty}^{\infty}\psi^*(x) \left(-\hbar^2 \frac{d^2}{dx^2}\psi(x)\right)dx = -\hbar^2\alpha^2$$ Now, we have negative expectation value of the square of the momentum, which I think is wierd, and we have to take square root of that value. That's impossible. I couldn't find out what's wrong with my idea. Can somebody help me with this?


title: Why do the stars in the galaxy core move so fast?

text: There are a bunch of stars orbiting the black hole in the center of our galaxy. These stars move at huge speed. Why do we see this? Why do the black hole not impose any noticeable time dilation on these stars?


title: Is the frequency/wavelength of light modified when multiple light sources are combined?

text: Let's say I light a wall with two spotlights: One red and one green one. Where they overlap, I'll see a yellow area at the wall. My question is, whether this is caused by an modification of the frequency/wavelength or simply by my eye combining the two incoming lights. Light is "added", wavelength is modified: ![enter image description here](http://i.stack.imgur.com/6DOCu.png) The eye combines two separate lights: ![enter image description here](http://i.stack.imgur.com/bdPjA.png)


title: Why did population III stars lack planets?

text: Jay Wacker1 (professor of physics at the SLAC National Accelerator Laboratory) stated: > The first stars (known as Pop III) were made out of hydrogen and helium. > They had no planets. Why couldn't they have had gas planets? [1] Requires login at LinkedIn


title: Recommended first accessories for starblast 4.5

text: My son got an orion starblast 4.5 for Christmas. It comes with orion explorer II 17mm and 6mm eyepieces. We are looking at some additional accessories and wondering what you would recommend as "first accessories" to get the most out of the telescope. Our initial inclination is towards:   1. A barlow lens. Probably the orion shorty 2x barlow. Is the shorty-plus twice as good? (It is twice the price.)   2. A 25mm eye piece. From what we've read, a reflector with these specs is best for wide field views.   3. A solar filter. We think it would be cool to look at the sun.


title: How can I determine whether the mass of an object is evenly distributed?

text: How can I determine whether the mass of an object is evenly distributed without doing any permanent damage? Suppose I got all the typical lab equipment. I guess I can calculate its center of mass and compare with experiment result or measure its moment of inertia among other things, but is there a way to be 99.9% sure?


title: Generator of local symmetries

text: Let us only consider classical field theories in this discussion. Noether's theorem states that for every global symmetry, there exists a conserved current and a conserved charge. The charge is the generator of the symmetry transformation. Concretely, if $\phi \to \phi + \varepsilon^a\delta_a \phi$ is a symmetry of the action, then there exists a set of conserved charges $Q_a$, such that $$ \\{Q_a, \phi\\} = \delta_a \phi $$ If the symmetry is a gauge symmetry, then no such conserved current or charge exists (although certain constraints can be obtained via Noether's second theorem). However, I have the following question > Is there a quantity that, like $Q_a$ generates the gauge transformation? This quantity need not be conserved. To be precise, if an action is invariant under local transformations $\phi \to \phi + [\varepsilon^a(x) \delta_a] \phi$. Here, heuristically, the gauge transformation might look like $[\varepsilon^a \delta_a] \phi \sim d \varepsilon + \varepsilon \phi + \cdots$ where $d \varepsilon$ is heuristically some derivative on $\varepsilon$. Does there exist a quantity such that $$ \\{ Q_{\varepsilon}, \phi \\} = [ \varepsilon^a \delta_a] \phi $$ I know there exist such charges in some theories, and I have explicitly computed them as well. I was wondering if there is a general formalism to construct such quantities? PS - I have some ideas about this, but nothing quite concrete.


title: In Which of the following situations would force be exerted on on object and no work be done on the object?

text: Consider the following question: > In which of the following situations would a force be exerted on an object > and no work be done on the object? > > I. a centripetal force is exerted on a moving object > > II. a force in the opposite direction as the object is moving > > III. a force is exerted on an object that remains at rest > > A.) I only > > B.) I and II > > C.) I and III > > D.) II and III > > E.) I, II, and III My response to this question is only II, but this is not a choice. I do not understand how I or III could have work. My logic is that in I, the force is centripetal and therefore is not parallel to the object's path, in any case it would be perpendicular because centripetal is towards the center. Work must be parallel. In III, if an object remains at rest, $d = 0$. And $W = fd$, so $W = 0$. I can't find the flaw in my logic. Any help is appreciated! Thanks!


title: Change of basis in non-linear Schrodinger equation

text: At the mean-field level, the dynamics of a polariton condensate can be described by a type of nonlinear Schrodinger equation (Gross-Pitaevskii-type), for a classical (complex-number) wavefunction $\psi_{LP}$. Its form in momentum space reads: \begin{multline} i \frac{d}{dt}\psi_{LP}(k) =\left[\epsilon(k) -i\frac{\gamma(k)}{2}\right] \psi_{LP}(k) +F_{p}(k)\,\, e^{-i\omega_{p}t} \\\ \+ \sum_{q_1,q_2} g_{k,q_1,q_2}\, \psi^{\star}_{LP}(q_1+q_2-k) \, \psi_{LP}(q_1)\, \psi_{LP}(q_2). \end{multline} The function $\epsilon(k)$ is the dispersion of the particles (polaritons). The polaritons are a non-equilibrium system, due to their finite lifetime (damping rate $\gamma$). Therefore, they need continuous pumping with amplitude $F_p$ at energy $\omega_p$. Finally, there exists a momentum- dependent nonlinear interaction $g_{k,q_1,q_2}$ that depends of the so-called Hopfield coefficients $X$ (simple functions of momentum) as: \begin{equation} g_{k,q_1,q_2}=g\, X^{\star}(k)\, X^{\star}(q_1+q_2-k)\, X(q_1)\, X(q_2) \end{equation} How can one transform the equation for $\psi$ to real-space?


title: Contact electricity and photoelectric effect

text: Most universities provide an experiment about the photoelectric effect to determine $h$ by measuring the stop voltage against the light frequency and calculating the slope $h/e$. But mostly they also talk about the contact voltage, which just changes the offset, but not the slope. I have several related questions regarding this contact voltage in combination with the photoelectric effect:   * Why do they talk about contact voltages at all if it's irrelevant for the slope?   * Why are the photo cells build with two different materials at all? The same material would give no contact voltage and everything would be fine.   * Books and Wikipedia tell me contact voltage is the voltage between two materials that are connected _directly_. But the photo cell's electrodes are not contacted directly, but normaly through copper cables and with several devices inbetween. Doesn't this change the contact voltage inpredictable?   * Are there papers or books or something that explains contact electricity from ground up?


title: Bosonic-Fermionic interactions in supersymmetry

text: There are a lot of supersymmetric theories, and, sometimes,in the Lagrangian, there are interacting terms between bosonic and fermionic degrees of freedom, and sometimes not. Why ? For instance, for the basic chiral superfield without superpotential, there is no interaction, while for 11-D supergravity, there is an interaction term between the (bilinear expression of) gravitinos and the strength 4-form.


title: Eigenvalue problem for differential equations in QM

text: I have a very simple question with regard to numerical methods in physics. I want to solve the eigenvalue problem for a particle moving in an arbitrary potential. Let's take 1D to be concrete. I.e. I want to find $(E,\psi(x))$ satisfying \begin{align} \left[-\frac{1}{2}\partial_x^2 + V(x) \right]\psi(x)=E \psi(x). \end{align} Now how do I do it exactly? Naively I would implement the following algorithm: 1) Pick some $E$. 2) I want to find $\psi(x)$ which is normalizable. So I could pick a large $L > 0 $, set $\psi(-L) = \epsilon > 0$ and $\psi'(-L) = \epsilon'>0$ and numerically integrate from there using the Schrodinger equation. 3) If I encounter a solution which is exponentially small far to the right of the origin, then I say the solution is normalizable (since it is decaying at $|x|\to\infty$), and I accept the pair $(E,\psi(x))$. 4) I increment $E \to E + dE$ and I repeat the process. In doing so, I should get the spectrum around my starting value of $E$. Does this algorithm actually work?? It also seems to me like a very uncontrolled way of doing it; I have no idea how accurate the spectrum is going to be. For example, would changing $L, \epsilon, \epsilon'$ make a difference? The thing is, I know from Sturm-Liouville theory that the spectrum $E$ is going to be discrete (given $V(x)$ satisfying some nice properties). So the spectrum is going to be a set of measure 0 amongst the entire real line that $E$ lives in. This means that I'm almost surely (i.e. with probability 1) never going to get a solution that is normalizable, and whatever solution I try to numerically integrate from my starting point is always going to blow up having integrated far enough to the right. So, what algorithm do people use to numerically obtain the spectrum and the eigenvalues? How do I also control accuracy of the spectrum generated?


title: Does the tension of a rope/cable change if acceleration/velocity changes?

text: For example, imagine an object of a certain mass which is attached to a rope and is dangling in the air. The rope needs to have a certain tension to be able to hold the object up, that is, negate the effects of gravity pulling it down. I'm wondering, if you took the other end of the rope and started pulling it upwards, would the tension need to be higher to still keep the object from falling? For example if a helicopter is carrying an object attached to a rope under it, would the tension of the rope need to be different for when the helicopter is hovering in the air and when it's flying upwards? I'm wondering this because it feels like that should be the case, especially when I think about how swinging the rope rapidly in circles might make the object detach and fly off, even if it would've stayed on had the rope not been in motion.


title: How would one experimentally prove AdS/CFT correspondence?

text: What would be an experimental test of AdS/CFT correspondence? Or it's extensions? I've heard that people are studying AdS/CMT (condensed matter) correspondence, but I don't know the details of it? But in general, how would you test it?


title: Help with incline plane intuition?

text: I am doing a simple project for a self-learn class with a friend to understand physics better over the summer. We slide down a hill on a tarp on an ice-block and then half way down, we hit grass and we want to compare the friction between the grass and the tarp. We calculated the angle of the hill as 9.7 degrees, the mass of myself and the ice-block is 63.5 kilograms. How can I calculate the force of friction at this point since we do not have the coefficient of friction?


title: Diffusion and Drift currents in a PN junction

text: In a forward-biased PN junction, the potential barrier decreases, allowing more majority carriers from one side to diffuse to the other side where they are minority carriers. After they cross the potential barrier, they form a diffusion current, the drift current of minority carriers is insignificant, then they recombine with majority carriers and form a drift current under the effect of the applied electric field. Why do minority carriers form a diffusion current not a drift current after they cross the potential barrier? It is counter-intuitive that the main current is diffusion when there is an applied electric field. This is according to all the microelectronics book I'm currently reading. There is one which says this can be proved but without providing anything. Can someone please provide a proof for this.


title: Electric field of a negative charge

text: How was it discovered that the electric field of a negative charge points towards the charge itself? Is it true? ![field of a negative point charge](http://i.stack.imgur.com/bh9uO.png) (Courtesy of wikipedia)


title: Is there a limit to acceleration?

text: As we all know the speed of light is the limit at which energy/matter can travel through our universe. My question being: is there a similar limit for acceleration? Is there a limit to how quickly we can speed something up and if so why is this the case? Short question, not sure what else I can elaborate on! Thanks in advance!


title: Unstable particles and quantum field theory

text: I am searching for not too old literature on the quantum description of unstable particles. I am referring to something beyond the ad-hoc S-matrix description based on the optical theorem common to textbooks such as those given by Peskin and Schröder or Weinberg etc. The book "Open Quantum Systems and Feynman Integrals" by Exner seems to go in this direction. But I find the formulation there very mathematical. Certainly it is possible to understand it but I am worried if I will get the connection to physics.


title: Frequency Of Light

text: I am confused on few topics... What is meant by "Frequency of Light"? Does the Photon(s) vibrate, that is known as its frequency? If the Photons vibrate, then they have a specific frequency, then What is meant by "Higher frequency light" as used in Photo- electric Effect? In which directions/axis do they vibrate to have a specific frequency. Why is it that nothing can go faster than the Speed of light. Does these two things have any relation to each other? -Thanks.


title: origin of magnetic field in a permanent magnet

text: I have a simple question. I hope I don't get a stupid answer. Where does the magnetic field of a permanent magnet comes from AND why is it permanent (are we dealing with perpetual motion)? This is what wiki is saying: "The spin of the electrons in atoms is the main source of ferromagnetism, although there is also a contribution from the orbital angular momentum of the electron about the nucleus. When these tiny magnetic dipoles are aligned in the same direction, their individual magnetic fields add together to create a measurable macroscopic field." What the wiki is saying is correct, but we have to think further or go one step further... So put your head out of the box: THIS IS NOT THE ANSWER I WANT TO HEAR. Because then I can ask why is the electron spinning (or what is the origin of spin) and that for an infinit time (CRAZY)? This is of course a physics question......We are missing an important energy source, unless you know where this form of energy is coming from and why it is permanent. Thank you very much! Edit: Wikipedia says: **"Truly isolated systems cannot exist in nature..."** Just be aware of this, before you are answering. (So before you are answering or commenting ==>THINK!)


title: Doubt in Kinematics

text: I know that this isn't the place for such basic questions, but I didn't find the answer to this anywhere else. It's pretty simple: some particle moves in straight line under constant acceleration from one point $x_0$ to another point $x_1$ during the time interval $\Delta t_1$. When the particle reaches the point $x_1$ it reverses it's movement and goes to another point $x_2$ during another time interval $\Delta t_2$. I want to determine $x_2$, however I don't understand how to do it. My try was: Let $\Delta x_1 =x_1 -x_0$ be the first displacement and let $\Delta x_2 = x_2 - x_1$ be the second displacement. Then I can calculate two velocities: $$v_1 = \frac{\Delta x_1}{\Delta t_1}$$ $$v_2 = \frac{\Delta x_2}{\Delta t_2}$$ My thought is then to find the acceleration as: $$a = \frac{v_2-v_1}{\Delta t_2 + \Delta t_1}$$ But I'm not sure it'll work, since the movement reverses at $x_1$ and since I'm assuming the velocities constant on the intervals. Can someone help me how to think with this problem, and how to solve it?


title: Smaller mass in gravity well?

text: When sitting in a gravity well, as we do on earth, does our effective mass become smaller than our rest mass due to having negative potential energy? Correspondingly, does a free falling mass (from infinity, radial path) have an effective mass equal to its rest mass, as here potential and kinetic energies are ballanced? The question is based on the concept of equivalence of energy and mass.


title: Earthquake as a transverse wave

text: I have a very simple mental picture that earthquake waves travel like shear (transverse) waves through the earth. a. Does the speed of this wave give any valuable information about the mechanical properties of the geological medium? b. Can this data be used to characterize mechanical properties of the earth? c. Can the speed of this wave be estimated by recording the time taken for the earthquake wave to reach sensors at different locations?


title: wave-particle duality

text: I have been trying to understand "wave-particle duality" and other cases related to it. I am currently a college level student. I have few question which I am not getting answers clearly. In double slit experiment, A particle behave like a wave, then how is "wave- particle duality" explained? I mean, If the particle behave like wave, then is it generating a wave or behaving as a wave? Is that wave going horizontally through slits(Double Slit Experiment) or vertically up and down or in which direction/axis the particle is vibrating to have a specific frequency? How does light behave as waves in it? and how does observer modify the experiment? I may be thankful to you, If you clear my problems. I have read the theories on Wikipedia and other informational sites and tried to understand it.


title: Decay/Counts/Number of Nuclei

text: I'm writing up a lab report and have a question about the following formula $$N = N_0e^{-\lambda t}$$ $N$ indicates the number of nuclei left after a time $t$ and $N_0$ indicates how much there was to begin with. In the experiment, we used a scintillation counter (consists of a crystal and a photomultiplier tube) to detect the gamma rays beginning emitted by a decaying metastable Ba-137 to Ba-137. After recording the "counts" $C$, we corrected for the background radiation counts $BG$ by calculating $C -BG$. Then we plotted $\ln(C-BG)$ vs $t$. We did this because from the above formula $$C-BG = (C-BG)_0e^{-\lambda t}$$ Then, $$\ln(C-BG) = \ln(C-BG)_0 - \lambda t$$ Thus a plot of $\ln(C-BG)$ vs $t$ yields a straight line with slope $-\lambda$. My question is, how does counts relate to $N$? I thought first formula above indicates the number of nuclei left. However, the detector is detecting gamma rays from decaying nuclei, i.e. counting how many nuclei are decay? So why can I replace $N$ with counts?


title: How does electric current generate thermal energy?

text: How does free electrons moving through a wire cause random vibrational motion of the positive ions?


title: Can a burning laser be directed through fiber optics?

text: Can a laser powerful enough to cut(100w for example) be redirected using fiber optics ? What issues need to be considered? To be clear it should retain the ability to cut after exiting the fiber.


title: Alice and Bob moving in a circular ring of radius $R$

text: Alice and Bob are moving in opposite direction around a circular ring of Radius $R$, which is at rest in an inertial frame. Both move with constant speed $V$ as measured in that frame. Each carries a clock, which they synchronize to zero time at a moment when they are at the same position on the ring. Bob predicts that when they next meet, Alice's clock will read less than his because of the time dilation arising because she is moving relative to him. Alice predicts that Bob's clock will read less with the same reason. They both can't be right. What's wrong with their arguments? What will the clocks really read? I have try to answer it that their all wrong, Since the they are all moving at the same speed, and they will all cover the same distance ($\pi R$), so they will be at same place with the same time? but I am not sure about my reasoning, Then the clock reading will be $\Delta t=\sqrt{1-\frac{V^2}{c^2}}\Delta t_B.$ Can any one give me a clear reasons on What's wrong with their arguments?


title: Can temperature be objectively related to energy?

text: Temperature is disordered kinetic energy, with 0K being 0 Joules disordered kinetic energy, if I"m not mistaken. So, given required parameters(temperature, number of particles, mass of each particle, whatever is required) can you find the disordered kinetic energy of each particle?


title: Any undergrad exercises about nuclear bomb design?

text: Are there any handy exercises about nuclear weapon design that are suitable for advanced undergrads in a nuclear physics or similar level physics course? I'm most curious about questions that actually appeared in textbooks or course homeworks. It doesn't matter if it's about gun weapon design or implosion weapon design, but it should be suitably idealized so as to be a reasonable exercise for an individual homework assignment in some undergraduate course.


title: What happens when we bring an electron and a proton together?

text: I have a couple of conceptual questions that I have always been asking myself. Suppose we have an electron and a proton at very large distance apart, with nothing in their way. They would feel each the other particle's field - however weak - and start accelerating towards each other. Now: 1) Do they collide and bounce off? (conserving momentum) 2) Does the electron get through the proton, i.e. between its quarks? 3) Do both charges give off Brehmsstrahlung radiation while moving towards each other? Different scenario: Suppose I can control the two particles, and I bring them very close to each other (but they are not moving so quickly as before, so they have almost no momentum). Then I let them go: 1) Would an atom be spontaneously formed? 2) If anything else happens: what kind of assumptions do we make before solving the TISE for an Hydrogen atom? Does the fact that the electron is bound enter in it? This is to say: is quantum mechanics (thus solving the Schrödinger equation) the answer to all my questions here?


title: If the universe is infinite, shouldn't I already have been contacted by a time and space travelling doppelgänger?

text: If the universe is infinite, by virtue of chance it means that every possible configuration of matter must exist somewhere (according to this documentary). Therefore, if we accept that the universe is infinite and it's possible to travel through time and space near-instantaneously, by sheer chance there must be a version of me out there that can do so and that wants to contact me. Since this has not happened yet, can I conclude that either near-instantaneous travel through time and space is impossible or that the universe isn't infinite? On a broader scale: given the infinity of the universe, should certain occurences that occur everywhere at the same time not occur always throughout the universe? Or do such occurences simply not exist?


title: How does QED deal with wavelength of quanta

text: Since QED treats photons as individual units (quanta) how does it treat the concept of the "wavelength" associated with the photon?


title: What are examples of classical physical systems having polynomial observables of degree greater than 2?

text: Specifically: What are empirically well-understood examples of (integrable) Hamiltonian systems whose Hamiltonians include polynomial expressions, in the canonical coordinates $\\{q^i,p_i\mid i=1,\ldots,n\\}$, having degree greater than 2? Below are follow up questions/replies in response to the comments/questions of Ron Maimon: With respect to integrable motion in one dimension, what are physical examples of one-dimensional potentials containing polynomial expressions of degree greater than 2? Beyond the integrable motion of a single particle in one dimension, what are empirically well-understood examples of many-body one-dimensional (integrable) Hamiltonian systems whose Hamiltonians include polynomial expressions, in the canonical coordinates $\\{q^i,p_i\mid i=1,\ldots,n\\}$, having degree greater than 2? What are well-understood examples of single particle (or many-body) greater than one-dimensional (integrable) Hamiltonian systems whose Hamiltonians include polynomial expressions, in the canonical coordinates $\\{q^i,p_i\mid i=1,\ldots,n\\}$, having degree greater than 2? As to the naturalness of these questions, the restriction is to the at least cubic polynomials in the Poisson algebra of classical polynomial observables in the $q^i$ and $p_i$ on phase space (presumably ${\mathbb R}^{2n}$ with $n>1$ in the single particle case). That said, could you expand on your observation concerning the naturalness of this restriction in the context of QFTs?


title: Transition of electric charge to "magnetic charge" when $\alpha$ gets >> 1 in QED?

text: I`ve just learned that electrically charged particles and magnetically charged monopoles in QED are S-dual to each other such that it depends on the value of the fine structure constant which of the two is the ligher (and more fundamental) one and which is heavier (and more complicated). Whereas I think I understand reasonably well how and why the roles of the light and the heavy particle concerning their mass get interchanged as the coupling sonstant increases from $\alpha << 1$ to $\alpha >> 1$, I fail to grasp how the electric charge is turned into a magnetic charge in the course of this (hypothetical) process. Should an electrically charged particle, such as an electron for example, not be fundamental different to some kind of a magnetically charged (hypothetical) monopole (I mean two different fundamental particles)? Non of the different "pictures" I`ve seen so far to "visualize" S-duality explain what happens to the kind of charge ...


title: Can the field generated by a magnet domain extend to infinity?

text: As a thought experiment let us assume that we have isolated a magnetic domain. This domain is of finite size and we know its dimensions. Assuming that we can measure an infinitesimal field, will there be a certain region beyond which the field won't be applicable? The instinctive answer to this question is no, but if you think about it we see the magnet's influence on the space around it as the result of equipotential regions then the contention is that only so many discrete equipotential regions are possible (the fact that something is not countable doesn't automatically mean it's infinite). Hence, that line of thought goes, there should be a limit theoretically and practically until which a field can exist. Can you please clarify this sticking point for me? Am I pushing the analogies we use to understand fields too far? What conceptual mistake am I making over here?


title: How long does a permanent magnet remain a magnet?

text: I have a bunch of magnets (one of those game-board thingies) given to me when I was a school-going lad over 20 years ago, and the magnets feel just as strong as it was the day it was given. As a corollary to this question Do magnets lose their magnetism?, is there a way to determine how _long_ a permanent magnet will remain a magnet? Addendum: Would two magnets remain a magnet for a shorter duration if they were glued in close proximity with like poles facing each other?


title: If we had three eyes, would our visual perspective be fourth dimensional?

text: If one covers up one eye, then he loses depth perception (two dimensional perspective). When we uncover that eye, we can now see depth (three dimensional perspective). My question is if we had four eyes, would we be able to see from a four dimensional perspective?


title: How fast is wave propagation in interference?

text: When someone performs Young's Double Slit experiment, the person sees an interference pattern on the screen. What is the time taken to for the pattern to appear on the screen? Is it distance between slit and the screen divided by speed of light? Another way to put the question is when photons are converted to waves is wave propagation speed = speed of light ?


title: Why does Matter occupies space? Matter and space are two things... If Space does not has mass, why should matter have space-ful nature?

text: This thought has completely changed my perspective towards matter. If the matter in a star can collapse to a point to form a Black hole, surely the true nature of matter should be able explain this behavior. I find this collapse easy to imagine if I visualize Matter as something which lies in space, but which _itself_ is not space, or _space-like_... It does not occupies space... What I mean is this... Why should matter occupy space? _Space-fulness is the nature of space, why associate it with matter?_ I use this modified picture of matter to explain the collapse of a star to a point. Suppose matter in a star has _0 volume_ , and it is the _space_ in between matter that does the job of occupying the volume, then we can easily eliminate this _space_ to explain a point sized infinitely dense black hole...! A particle of matter with no volume is hardly like a particle I had imagined earlier. It is more surprising... _I think that matter lies in space, but does not occupies it._ Is the general association of space-fulness with matter a misunderstanding? Is it correct to view our matter as a space-less entity? Or matter is actually woven to the fiber with space, such that it itself behaves a bit like space, by occupying it? Does Matter REALLY occupies space? Am I am under a misconception?


title: Calculating the Reflection Coefficient of a Potential Step Explicitly

text: So I'm using the following definition for the Reflection coefficient : $$\frac{\vec{j}_{reflected}}{\vec{j}_{incident}}$$ Hence, since : $$\psi_{reflected}=Be^{-ikx}$$ and $$\psi^{*}_{reflected}=B^{*}e^{ikx}$$ We can perform the usual to obtain the incident probability density current as : $$\vec{j}_{reflected} =2ik\lvert{B}\rvert^{2}$$ Now for the incident we use : $$\psi_{incident}=e^{ikx}$$ and $$\psi^{*}_{incident}=e^{-ikx}$$ Again we obtain : $$\vec{j}_{incident} =-2ik$$ This would imply a coefficient of $-\lvert{B}\rvert^{2}$ which is incorrect. What have I done incorrectly ?


title: Young double slit experiment without first slit

text: What if I take a laser pen and direct it towards the middle of two wide slits? Will I get interference pattern or just two lines? Do I need to scatter photons before they hit the slits to observe interference pattern?


title: Why are holomorphic boundary CFT2 primary operators massless in the AdS3 bulk?

text: I saw a claim in this paper that holomorphic boundary CFT$_2$ primary operators correspond to massless states in the AdS$_3$ bulk. Specifically, > As always, we simplify the situation by assuming the absence of holomorphic > primary operators. (These would have a little group diﬀerent from that of a > massive particle in the bulk of AdS; therefore for small $\Lambda = −L^{-2}$ > they can only correspond to massless states, which do not have a rest frame, > or else to states which do not propagate into the bulk of AdS at all.) My question is: how did he arrive at this conclusion/where can I find an explanation? I can't figure it out, and nowhere near the claim does he give any relevant sources. It's certainly conceivable: holomorphic primary operators will include gauge fields, for example.


title: How can we test if something is a wave?

text: More specifically, I want to understand why a wave is a wave but a wave packet is not considered a wave (as discussed in this question). I would think that if something have these characteristics: 1. Wavelength; 2. Frequency; 3. Period; 4. Amplitude; and 5. Wave velocity; then it must be a wave. Does a wave packet have these characteristics? Do you have a better set of rules to apply to test if something is a wave?


title: Breakdown voltage of a dielectric

text: I know that a capacitor with a dielectric can operate normally up till a certain voltage (AFAIK called breakdown voltage) which depends on the strength of the dielectric placed between the plates. After this voltage, the circuit becomes short and current flows between the plates and thus the capacitor breaks down. But i want to know what is exactly happening when we say a dielectric "breaks down" ? What I know about a dielectric is that due to the electric field (because of the plates of the capacitor) the molecules of the dielectric align themselves accordingly and set up an electric field in the opposite direction, thus decreasing the net electric field. So, please can anyone tell me what happens during breakdown?


title: Does a concave or flat bottom pan use heat more efficiently?

text: This may be anecdotal. Playing in the kitchen I realized the frying pan comes with both a flat, and a concave bottom. So here's the question - Given two pans made of brass, one has a concave base & the other a flat base, which of the two would use heat more efficiently? I would believe the latter because the flat base means any heat must necessarily travel through the cooking medium, and the recipe before reaching open air.


title: If I'm running away from chasing gamma photons, I can see them, right?

text: Suppose Bruce Banner goes back in time, convinced that the Hulk is a stupid menace. Let's say he also borrows Quicksilver's speed abilities (maybe the Flash is more appropriate, but based on the recent X-Men movie Quicksilver is plenty fast enough). Anyway he decides to run away from the burst of gamma rays that turns him into the Hulk (assuming this would actually do the trick). Now obviously, they will still travel at the speed of light. But, due to frequency shifts, shouldn't he eventually look behind and "see" the burst of gamma rays as really just color, starting with violet? If he kept running, would he not see a rainbow cycle of colors? Would this mean that the flash could see the whole spectrum in terms of the visible spectrum (that is, he wouldn't see new colors, but new sources of colors), depending of course on his speed? I guess this is all just a red-shift blue-shift (one shift two shift) problem, so I think it is a valid hypothesis. Am I right, or is Banner doomed to become invincible and mean?


title: Is this description of particle's spin strict?

text: I'm trying to understand a concept spin in QM. I've read some explainations, and it seems that the idea is _very_ easy and straighforward, however I didn't find these exact wordings in my books.   My question: is the following explaination **fully correct and strict** , or I'm missing something? The explaination: under an inifinitesimal rotation $\vec{\delta\varphi}$ the position vector $\vec{r}$ transforms to $$\vec{r}' = \vec{r} + \delta\vec{r} = \vec{r}+\vec{\delta\varphi}\times\vec{r} = \left(1+\vec{\delta\varphi}\cdot\hat{\textbf{R}}\right)\vec{r}$$ $\hat{\textbf{R}}$ are three $3\times 3$ matrices of rotation. Thus for any function $f(\vec{r})$ $$f(\vec{r}') = f( \vec{r} ) + \delta \vec{r}\cdot\nabla f(\vec{r}) = \left(1+\vec{\delta\varphi}\cdot[\vec{r}\times\nabla]\right) f(\vec{r}) \equiv\left(1+i\vec{\delta\varphi} \cdot\hat{\textbf{L}}\right) f(\vec{r}) $$ $\hat{\textbf{L}}$ is the angular momentum operator. However, the state vector $\Psi$ can be of any "geometrical" origin itself (scalar, $1/2$-spinor, vector, ...) and undergoes the transformation as well - there is an addition $\delta\Psi$ similar to $\delta\vec{r}$: $$\Psi'(\vec{r}') = \Psi(\vec{r}') + \delta\Psi(\vec{r}') \equiv \left(1+i\vec{\delta\varphi}\cdot\hat{\textbf{S}}\right) \Psi(\vec{r}') $$ $\hat{\textbf{S}}$ is the operator of spin. Note that the argument $\vec{r}'$ is the same on both sides. Since $\Psi$ is not necessarily a vector, each of three $\hat{\textbf{S}}$ is not necessarily a matrix of rotation. Full transformation is the combination of two: $$\Psi(\vec{r}') = \left(1+i\vec{\delta\varphi} \cdot\hat{\textbf{L}}\right) \left(1+i\vec{\delta\varphi}\cdot\hat{\textbf{S}}\right)\Psi(\vec{r}) = \left(1+i\vec{\delta\varphi} \cdot(\hat{\textbf{L}}+\hat{\textbf{S}})\right)\Psi(\vec{r}) $$ $\hat{\textbf{J}}\equiv\hat{\textbf{L}}+\hat{\textbf{S}}$ is the "total" angular momentum. There are $0$, $1/2$, $1$, $3/2$. ... representations of the group of rotations. For "angular" part we have a restriction that position vector is an observable so after $360 ^\circ$-rotation it must return to its original value. This is why only integer eigenvalues of $\hat{\textbf{L}}$ are allowed. Since state vector $\Psi$ does not directly correspond to an observable, there are no such restrictions on $\hat{\textbf{S}}$ .


title: Difference resultant aerodynamics force on an airfoil and a flat plate

text: From basic airfoil theory the following free body diagram can be determined for a two dimensional asymmetric airfoil: ![enter image description here](http://i.stack.imgur.com/MlDJx.png) Here the direction of the resultant force is governed by the geometry of the airfoil section. However, I'm unsure on how the direction of the resultant force is affected when instead of an airfoil a two dimensional flat plate is considered.   Do the resultant force $R$ and normal force $N$ just overlay each other as shown below (my suspicion)? ![enter image description here](http://i.stack.imgur.com/wZ1wC.png)


title: Is information propagated across a medium in any other way than waves?

text: Is information propagated in any other way than waves? Please distinguish "propagation across a medium" from information "storage within stable states of matter", which might difuse or interact chemically. Information might be stored in stable configurations of matter, which might diffuse, or interact chemically (odor,DNA), but these might be orders of magnitude weaker, in range and dissipation. Is there a domain in physics comparing wave vs non-wave propagation. The two most known are sensory: sound and electromagnetic propagation. I think gravity probes are still searching for waves in this medium. Why does nature prefer waves for long distance calls? Perhaps because it involves a minimum dissipation of energy?


title: Ergodicity of the Drude model

text: The Drude model of electric conduction in solids deals with independent free electrons subject to _random_ collisions with the crystal lattice (the direction where the electrons are scattered after a collision is random). A simplified model is the Lorentz gas, where the collision are deterministic. If I understand correctly, it was shown by Sinai that this model is ergodic. What about the original Drude model: is it ergodic? (Are there references on that?) A side question is whether this would have a physical significance?


title: what would be the maximum depth of tunnel?

text: Perhaps more of an engineering question, but with the subject recently in the news about hyperspeed maglev tunnels I'll chance it here. Say you set up a network of line of sight underground tunnels for your maglev transportation system operating roughly 50' above sea level through the American continent. These tunnels would not contour to the curvature of the earth, rather be more akin to a knitting needle piercing the tangent of an orange, shortening travel distance by transecting the arc. My question is, given real world conditions, material strengths and costs, geothermal, fault and groundwater issues, what is the maximum depth a reinforced tunnel could exist under earth's gravity? Since both ends of the tunnel segment are at equal elevations (disregarding surface features) the effect would be a gravitational "rope swing". Couple that with evacuating the tube of air to vacuum or near vacuum levels and you have a very energy efficient delivery system. How a vacuum would increase stress loads on the tube I'm not really sure. Some very crude calculations of my own put a surface distance of 180 miles requiring a center tunnel depth of about 1.1 miles, which would be a very workable distance between stations.


title: Is there a fundamental difference between the statistical methods of science, comparing medicine to physics?

text: Is there a fundamental difference between the statistical methods of science, comparing medicine/biology with small sample sizes(n < 10^2 or 10^3) to the statistics applied in Quantum Mechanics (h: order 10^34) or statistical mechanics (N: order 10^23)


title: Do systems with level crossings have unstable eigenbases?

text: It's folklore dating back to von Neumann and Wigner that time-dependent Hamiltonian systems tend not to have level crossings of their energy eigenvalues. However, we can of course consider smoothly varying Hamiltonians which have been engineered to have level crossings. These don't even have to be complicated: for instance, any Hamiltonian of the form $$ H = -\sum_{(i,j)} \sigma^{(z)}_i \sigma^{(z)}_j + \epsilon P t $$ on a 1D spin chain, for $0 < \epsilon \ll 1$ and any Hermitian operator $P$, is an example: the all-up state and the all-down are ground-states for $t = 0$, though for $t \ne 0$ such symmetry is typically broken, so that for $|t| \ll 1$ we expect to have eigenstates close to the all-up and all-down states but with distinct eigenvalues. After some investigation, I've come to suspect the following: > **Conjecture.** If $H$ has a level crossing between energy levels $E_0, E_1$ > at some time $T$, and $\Pi$ is the projection onto the span of the $E_0$- > and $E_1$-eigenstates then there are well-defined (continuously varying) > eigenvectors through the level crossing only if $\Pi [H(T), \dot H(T)] = 0$ > -- that is, if the change in the Hamiltonian at the level crossing is only a > change in the values of the two crossing eigenvalues, for a common pair of > eigenvectors. Specifically: if this equality does not hold, then any > (unitary) time-dependent change-of-basis operator from the standard basis to > the energy eigenbasis of $H$ at time $t$ which is continuous for some > neighborhood $t \in (T,T+\epsilon]$ will oscillate infinitely rapidly as $t > \to T$.   * Is this true generally? (If not, can you point to a counterexample?)   * Is this a known result, and is there a reference that I can refer to where this question is treated clearly, and more-or-less formally for bounded operators (e.g. on finite-dimensional systems)?


title: Does this person get electrified?

text: Does a person get electrified when his feet are on the floor, his right hand is holding $220~V$ and his left hand holding a piece of wood or any other insulator?


title: How small can diffraction grating be?

text: Like in title, how small can diffraction grating be? And, of course be "fully operational". I mean, littles diffraction grating possible that still works as full-sized one.


title: When two spheres of equal charge make contact, why does the larger sphere gain more charge?

text: Suppose that two spheres, $S1$ and $S2$, with radii $R1$ and $R2$ resp. have the same charge uniform charge $Q$ and $R1 > R2$. After they are forced to come in contact, why does $S1$ gain more charge? Why don't they both continue to have equal charges?


title: How to model an apartment's airconditioning?

text: **Background:**   This reminds situation reminds be of an episode from the _The Big Bang Theory_ , although it is quite different from it. My roommate and I have similar temperature preferences and also share the objective of saving money on electricity. The optimal solution towards this objective is what we have differing opinions about. **The problem:**   In my opinion, during the daytime when no-one's around, the temperature setting should be at least 4 °F more than the normal setting (given an ongoing summer 90+ °F). I would go far as to argue that during the daytime the cooling should be completely switched off. My roommate's counter-argument is that assuming a PID controlled model, the overshoot caused by the cooling in the evening, in an effort to get the temperature down from the day's high, will more than nullify the electricity saved during the day thereby resulting in higher costs. The controller is a programmable digital thermostat with 4 separate settings for `sleep`, `wake`, `leave` and `return` for two regimens namely `weekday` and `weekend`. **Specifically:**   Instead of hand-waving I was looking for a mathematical model to simulate the heating-cooling of a typical apartment. A quick Google search yielded no useful models (most I could get my hands on are macro-models related to multiple commercial units). I am looking for a model — with elements like the `heat exchange` (i.e. the ac), the `heat sink` (i.e. the apartment) and the `environment` — a general model which can be used for a _first-order_ approximation. Assume that the cost of electricity is constant throughout the period under consideration (even if it isn't, a cost vs time graph can be easily incorporated in the model).


title: If we could build a telescope to view the cosmic neutrino background, what would we see?

text: If we could build a neutrino telescope capable of viewing relic neutrinos that decoupled after the big bang, with a similar angular and spectral resolution that is possible now for the CMB, what would we see? (And I know this will be technically incredibly difficult.) How would the C$\nu$B differ because of the finite neutrino mass and earlier decoupling? Would there be additional diagnostics and insights that are unavailable from the CMB? How big would the fluctuations in neutrino temperature be compared with the CMB? Would these fluctuations give us the neutrino mass or tell us more about inflation? NB: related questions are Why are we blind for the era before the recombination? , Seeing beyond the CMBR with neutrinos? and Is it possible to look into the beginning of the Universe? but none these ask specifically, or have answers, about what could be seen or probed by the C$\nu$B if we could examine it in detail.


title: Pseudoforce and friction: confusion

text: Consider the following situation. A block of mass $M$ is resting on a rough horizontal ground, and a frame is moving towards the right horizontally with an acceleration $a$. Suppose the coefficient of static friction on the ground on which the block is standing is $ \mu > a/g$. Clearly the block is moving towards the left w.r.t. the frame with acceleration $a$. Suppose we try to explain this from the non-inertial frame. Then we add a pseudoforce $Ma$, towards the left, on the block. But observe that $Ma < \mu Mg$ as $\mu g > a$. So the block shouldn't move, due to static friction. How is this possible?


title: size of an image formed by a concave mirror

text: > a short linear object of length b lies along the axis of a concave mirror of > focal length f at a distance u from the mirror.what is the size of the > image? what i have done so far: since the object is of length b, consider the point of the object closest to the mirror A and the point farthest from the mirror B. for the image of point A, 1/vA + 1/(-u)=1/(-f) for the image of point B, 1/vB +1/(-(u+b))=1/(-f) therefore vA=fu/(f-u) and vB=f(u+b)/(f-(u+b)) so size of image is vA-vB = b[(f^2)/(f-u-b)(f-u)] my question: can this be approximated to b[(f^2)/(f-u)(f-u)] or will that be wrong?


title: How was transformed an integral below?

text: I know how transform an integral below, $$ \iint f(\mathbf v_{1})f(\mathbf v_{2})d^3\mathbf v_{1}d^3\mathbf v_{2}, $$ using relative speed coordinates: we just use $$ m_{1} \mathbf v_{1} + m_{2}\mathbf v_{2} = M\mathbf V, \quad \mathbf v = \mathbf v_{1} - \mathbf v_{2} , $$ and then we may use spherical coordinates. But if I have an integral like $$ \iint f(\mathbf r_{1})f(\mathbf r_{2})d^3\mathbf r_{1}d^3\mathbf r_{2}, $$ I don't know how to transform it by using a spherical coordinates of center of masses. In Pathria's book called "Statistical Mechanics" I saw a transform that I need, but I don't understand how it was made. Can you help me? ![](http://i.stack.imgur.com/EyLh8.jpg) . And it's not a homework!


title: Scale invariance in sandpile model and forest fire model

text: I asked a similar question but the wrong way here. Because my intention was to ask about non thermodynamic system, i will be more specific:   * What is the relation between critical behaviour and the scale invariance in these two model (sandpile, forest fire)? What i can't figure out is the meaning of "infinity correlation length" in these two model.


title: Introduction to Physical Content from Adjoint Representations

text: In particle Physics it's usual to write the physical content of a Theory in adjoint representations of the Gauge group. For example: $24\rightarrow (8,1)_0\oplus (1,3)_0\oplus (1,1)_0\oplus (3,2)_{-\frac{5}{6}}\oplus (\bar{3},2)_{\frac{5}{6}}$ (Source: SU(5) GUT Wikipedia article) While I do understand the Basics in representation theory from a mathematical viewpoint, as well as Gauge Theory (up to this point), I've been looking High and Low for some good article on how to understand what the above formula means physically? Specifically I don't understand the following:   * I'm having a bit of a problem with the notation. $(1,1)$ denotes the tensor product of a **1** and **1** of $SU(3) \times SU(2)$ in this case, does the subscript $()_0$ belong to the $U(1)$ part? Or did I completely misunderstand something?   * How to arrive at the above transformation? How to choose the right hand side of the **24** transformation, it seems random to me   * The physical content. $(8,1)_0$ looks to me like gluons, because of the 8, $(1,3)_0$ like W and Z bosons and $(1,1)_0$ like the photon. But these are all guesses I made according to the numbers I see and the fact that the SM should arise from $SU(5)$ breaking. How would one know this? And what are the other 2 components? Any reference is also greatly appreciated, especially one that focuses on precisely this.


title: Electricity & Magnetism - Is an electric field infinite?

text: The inverse square law for an electric field is:   $$ E = \frac{Q}{4\pi\varepsilon_{0}r^2} $$ Here: $$\frac{Q}{\varepsilon_{0}}$$ is the source strength of the charge. It is the point charge divided by the vacuum permittivity or electric constant, I would like very much to know what is meant by source strength as I can't find it anywhere on the internet. Coming to the point an electric field is also described as: $$Ed = \frac{Fd}{Q} = \Delta V$$ This would mean that an electric field can act only over a certain distance. But according to the Inverse Square Law, the denominator is the surface area of a sphere and we can extend this radius to infinity and still have a value for the electric field. Does this mean that any electric field extends to infinity but its intensity diminishes with increasing length? If that is so, then an electric field is capable of applying infinite energy on any charged particle since from the above mentioned equation, if the distance over which the electric field acts is infinite, then the work done on any charged particle by the field is infinite, therefore the energy supplied by an electric field is infinite. This clashes directly with energy-mass conservation laws. Maybe I don't understand this concept properly, I was hoping someone would help me understand this better.


title: Question about moment of inertia and velocity

text: First off, I swear this is not homework. I'm doing some practice problems because I got an exam coming up. I'm stuck on this one: ![alt text](http://i.stack.imgur.com/w7KMz.png) I figured I would use energy conservation for this problem. So since the thing is not moving initially, I tried doing $mgh=1/2 I\omega^2+1/2 mv^2$, but that doesnt give me the right answer. Any ideas?


title: How does that Boltzmann distribution interact with entropy?

text: In an ideal gas, the Boltzmann distribution predicts a distribution of particle energies $E_i$ proportional to $ge^{-E_i/k_bT}$. But, doesn't entropy dictate that the system will always progress towards a state of maximum disorder? In other words the system evolves towards a macro- state which contains the maximum possible number of indistinguishable micro- states. This happens when all particles have the same energy, which seems to contradict the Boltzmann distribution. I'm pretty sure I've misinterpreted entropy here, but I'd be please if someone could explain how!


title: Will the Hubble constant reach zero asymptotically in the far future?

text: In the current accelerated expansion universe model will the Hubble constant reach zero asymptotically in the far future?


title: Is mass of a particle changed when it is charged?

text: If a particle of mass $M$ is given an electric charge $Q$, will its mass change?


title: Why does the water level equalize in a series of tubes?

text: Say I have a series of tubes (not the internet) looking like this, where `w` represents water:               |    |      | |     |    |      | |     |wwww|      |w|     |wwww|      |w|     |wwww+------+w|     |wwwwwwwwwwwww|     +-------------+      Why is it that if I put some water in either side, the water level changes on the other side until they equalize? Why does this work even if I put water on the thinner side? And why is it that when you tilt the whole contraption, the water level is still even, although slanted?


title: Constraint force on a rod

text: I really hope someone will take a quick look at the following, I would just love to better understand it... This exercise is from Arnold's "Mathematical Methods of Classical Mechanics", p. 97 in the chapter on d'Alemberts principle: A rod of weight P, tilted at an angle of 60° to the plane of a table, begins to fall with initial velocity zero. Find the constraint force of the table at the initial moment, considering the table as (a) absolutely smooth (b) absolutely rough (In the first case, the holonomic constraint holds the end of the rod on the plane of the table, and in the second case, at a given point.)" ![](http://i.stack.imgur.com/Mq3d0.jpg) * * * I must admit, I am pretty unsure on how to do calculations using this "fancy" mathematical kind of physics. First off, I'm lost with (a). But I'll have a go at (b): As far as I understood, d'Alemberts principle states, that if $M$ is a constraining manifold, $x(q)$ is a curve in $M$ and $\xi$ is a vector perpendicular to $T_xM$, then $x$ satisfies Lagrange's equations $\frac{d}{dt} \frac{\partial L}{\partial \dot q} = \frac{\partial L}{\partial q} \qquad L = \frac{{\dot x}^2}{2} - U(x)$ iff for the following inner product, we have $\left(m \ddot x + \frac{\partial U}{\partial x}, \xi \right) = 0$ I guess that's about right so far? The constraint would in this case essentially be $\mathbb S^1$, since the rod moves on a circe around the point in contact with the table. Can we assume that all the rod's mass is at the center of mass? Would we then have $U(x) = -gx_2$ in this case (where $x_2$ is the vertical component of $x$)? If yes, then $\partial U / \partial x = -ge_2$. Where $e_1, e_2$ are the horizontal and vertical unit vectors, respectively. At the inital moment, we have $x = \cos(60°) e_1 + \sin(60°) e_2$, so by d'Alembert's principle we must have $\left(m \ddot x + \frac{\partial U}{\partial x}, \cos(60°) e_1 + \sin(60°) e_2 \right) = 0$ or written differently $m \ddot {x_1} \cos(60°) + m \ddot{x_2} \sin(60°) - \sin(60°)g = 0$ * * * So: Is this correct so far? Or am I way off? Is (a) handled any differently up to this point? Thanks for reading (and hopefully thanks for your helpful reply)! If anyone could recommend a good problem book (with solutions), in which this kind of mathematical approach is used (I don't know if this is how physicists would actually compute stuff??), I would greatly appreciate it. Kind regards, Sam


title: Probability vs. degree of belief in facts of nature ("Plausibility")

text: I just came across a line in a paper: "Assume the probability that a Lagrangian parameter lies between $a$ and $a + da$ is $dP(a) = $ [...]." This reminded me again of my single biggest qualm I have with the "Bayesian school" - that they assign _probabilities_ to facts of nature, say $P(m_\mathrm{Higgs} = 126\,\mathrm{GeV}$). I understand that you can assign a probability to the outcome of an experiment when you perform it many times. I would also say that you don't actually have to do the experiments, since you can reason about ensembles. You can consider the probability that there is an earthquake tomorrow without really having multiple earths. I would not say however that you can assign a probability to a fact of nature, like a natural constant. It is either such, or such, and there is no dynamics changing it. Bayesians think of probability as "degree of belief", and in that case of course you can assign probabilities to arbitrary hypotheses. $P (\mathrm{SUSY\;exists})\approx 60\%$ and the such. I myself like to keep the two cases distinct. I say "probability" when I'm talking about dice, or quantum mechanical decays and the such, and "plausibility" when I'm talking about degrees of belief. The probability that SUSY exists is either 1 or 0. The plausibility is a very sociological thing and depends on who you ask, and what experimental and theoretical inputs you consider. The nice thing is that the maths for plausibility is basically the same as for probability, so no new formulas to learn ;-). I've been using that distinction in conversations for a while, and nobody seemed to find it strange, so I think this concept is pretty widespread. However, I can't find any literature on it (beyond the usual frequentist/Bayesian debate). My question is, are there any references / essays / lectures on the two different kinds of "$P\,$"? What's the degree-of- belief-$P$ (my "plausibility") usually called? What are the implications of keeping the two distinct?


title: Why are there both Static and Kinetic Friction?

text: When dragging an object, there is a greater start-up force than the force it takes to keep it moving. Why is this? Why are there two different values for friction?


title: Why do skidding wheels have less traction than non skidding wheels?

text: I do not understand in terms of coefficients of friction why this happens? From my understanding the ground and the tire still have the same coefficients of friction right? So why is there less traction when it is stopped then when it is moving?


title: What plane is Area in, in this equation?

text: I have the equation $$F = (NI)^2\mu_0\frac{\text{Area}}{2g^2}$$ Which calculates the force of an electromagnet. I was wondering if anyone knew what plane Area is taken from? I have heard it is the cross sectional area of electromagnet(which doesn't make sense to me because it already uses the number of turns, N and the current, I), I have heard is is the cross-sectional area of plane **b** , and I have heard it as the cross sectional area of plane **a**. I have also heard it as plain "Area". A ferromagnetic cylinder being acted on by an electromagnet.(Assume this cylinder is being drawn into a solenoid circle-side first) ![enter image description here](http://i.stack.imgur.com/0HY5G.png) Personally, I think it uses the cross sectional area of plane **b**. Being a physics problem, can anyone confirm whether or not I should be using the area from plane **b**?


title: The effect of A/C and global warming

text: I had been thinking about the way an air conditioning system moves heat from one place to another. The unit runs and drops the temp. in the building and raises the temp. outside. Also there is mechanical energy that converts to heat (the compressor and fan motors) and is also put outside. If this is the case, is there an effect caused by A/C systems running on the outside temp.? Can the total amount of heat being moved from inside to outside contribute to global warming? Consider how many systems there. As more and more A/C systems are installed and more and more buildings built, can't this lead to (if it hasn't started already) a higher temp outside? I am not a scientist; if this question is wrong, I would like to know why.


title: Wick Rotation, interpretation of $\bar{p}^2$ vs the usual $p^2=m^2$

text: Suppose we use the metric $(+,-,-,-)$ thus the momentum squared is $p^2 = p_0^2-\vec{p}^2 = m^2>0$ Defining $p_E:=\mathrm{i}\cdot p_0$ and $\bar{p}:=(\,p_E,\vec{p})$ with Euclidean norm $\bar{p}^2 = p_E^2+\vec{p}^2$. Here's my question: If we plug in $\mathrm{i}\,p_0$ instead of $p_E$ we see that $\bar{p}^2 = -p^2 = -m^2$? So $\bar{p}^2$ is negative? Also if $\bar{p}^2 = -p^2 = -m^2$ is true...is it always so? Since $p^2$ is a Lorentz invariant, but how do we interpret $\bar{p}^2$ if it's equal to $-m^2$ What I want to get to is the following: Let $\mathcal{L}_{int} = \frac{1}{2}g\phi_1^2\phi_3+\frac{1}{2}h\phi_2^2\phi_1$ with: $p_3 = p_1+p_2$ and $M>2m$ Suppose we have a triangle loop with incoming momentum $p_3$ of mass $M>2m$ and two outgoing identical particles $\phi_2$ and $\phi_2$ momenta $p_1$ and $p_2$ each of mass $m$ (sorry bad notation but $\phi_1$ is not related to $p_1$). The incoming particle $\phi_3$ splits into two light ones (two $\phi_1$'s) of mass $\eta$ each and each of these connects to two $\phi_2$'s. Thus we have the following momenta flowing inside the loop: $k$, $ k-p_2$ and $k+p_1$. After a Wick rotation we get the following integral: \begin{equation} \int{\,\frac{\mathrm{d}^4\bar{k}}{(2\pi)^4} \frac{1}{\bar{k}^2+m^2}\frac{1}{\left(\bar{k}-\bar{p}_2\right)^2+\eta^2}\frac{1}{\left(\bar{k}+\bar{p}_1\right)^2+\eta^2}} \end{equation} Now to evaluate these let's use Schwingers trick: \begin{equation} \frac{1}{\bar{k}^2+m^2} = \int_0^\infty{\mathrm{d}s\,\mathrm{e}^{-s(\bar{k}^2+m^2)}}, \end{equation} but for this to be OK we need to have $\mathrm{Re}(\bar{k}^2+m^2)>0$ and similarly for the other propagators, and this is where I get confused. It seems they don't satisfy this condition depending on how we interpret $\bar{k}^2$ and $\bar{p}^2$ and so on... On the other hand if all the squares in the denominator of the integrand are taken as positive then the convergence condition is trivially satisfied. So please help me understand what I'm doing wrong and if you guys can show how to satisfy the positivity condition. Thanks in advance.


title: Epistemologically, can we ever deduce the 'true' theory of everything?

text: Operationally, we can only know about the results of experiments and observations. From them, we can conclude our world is one which is complex enough to allow for computers. After all, we're using computers right now. Our universe has the property of Turing completeness, meaning we can embed any possible computer program on it subject to space, time and energy requirements. Unfortunately, any theory which is Turing complete can always be emulated on any other theory which is also Turing complete. What consequence does this have upon our ability to deduce the 'ultimate' theory of everything?


title: $\frac{1}{(1-x)_+}$ type distributions and parton distribution functions

text: I am trying to get to grips with Altarelli-Parisi-type equations. In chapter 17 of Peskin/Schroeder, they first develop the equations for a similar problem in QED. Equation $(17.123)$ introduces the sum rule $$ \int_0^1 dx ( f_e(x,Q) - f_\bar{e}(x,Q) ) = 1 $$ where $f_e$ and $f_\bar{e}$ are the distribution functions of electrons and antielectrons 'inside' an electron. **I'm trying to prove that this is independent of $Q$.** The evolution equations are ($(17.120)$ in Peskin/Schroeder) $$ \frac{d}{d\log Q} f_e(x,Q)= \frac{\alpha}{\pi}\int_x^1 \frac{dz}{z} \left( P_{e\leftarrow e}(z) f_e(\frac{x}{z},Q) + P_{e\leftarrow\gamma}(z)f_\gamma(\frac{x}{z},Q)\right) $$ $$ \frac{d}{d\log Q} f_\bar{e}(x,Q)= \frac{\alpha}{\pi}\int_x^1 \frac{dz}{z} \left( P_{e\leftarrow e}(z) f_\bar{e}(\frac{x}{z},Q) + P_{e\leftarrow\gamma}(z)f_\gamma(\frac{x}{z},Q)\right) $$ where the relevant splitting functions is given by (equation $(17.121)$ in Peskin/Schroeder) $$ P_{e\leftarrow e}(z) = \frac{1+z^2}{(1-z)_+}+\frac{3}{2}\delta(1-z) $$ Using $\frac{d}{d\log Q}$ on $(17.123)$ gives (the part involving $P_{e\leftarrow\gamma}(z)$ cancels): $$ \frac{\alpha}{\pi}\int_0^1 dx \int_x^1 \frac{dz}{z} P_{e\leftarrow e}(z) \left( f_{e}(\frac{x}{z},Q) - f_\bar{e}(\frac{x}{z},Q) \right) $$ Insterting $(17.121)$ and using that $$ \int_0^1 dx \frac{f(x)}{(1-x)_+} = \int_0^1 dx \frac{f(x)-f(1)}{(1-x)} $$ I get $$ \frac{\alpha}{\pi}\int_0^1 dx \int_x^1 \frac{dz}{z} \left(\frac{1+z^2}{(1-z)_+}+\frac{3}{2}\delta(1-z) \right) \left( f_{e}(\frac{x}{z},Q) - f_\bar{e}(\frac{x}{z},Q) \right) $$ $$ = \frac{\alpha}{\pi}\int_0^1 dx \left( \int_x^1 dz \left( \frac{1+z^2}{(z-z^2)}\Delta(\frac{x}{z},Q) -\frac{2}{1-z} \Delta(x,Q) \right) \+ \frac{3}{2} \Delta(x,Q) \right) $$ This expression is singular and it seems that the singularities in the first two terms should cancle. However, I'm at a loss what to do here. My idea was to extract the singularity in the first term, but that seems like i'm doing it backwards (and I haven't figured out how to do it). Any hint would be appreciated, I'm not looking for full solutions.


title: Why is the speed-of-light "the upper limit" rather than the speed of "particle type X"?

text: Basically, I can't stop wondering why light (the photon) is so special, compared to all the other particles known (and unknown) to modern day physics. Could it be that there exists an upper limit on speed that is instead a property of another "entity" (particle / wave), other than the photon? A question obviously inspired by the recent _very slightly possible_ superluminal neutrino findings


title: Surface normal on the earth to the sun at a given point in time

text: How complicated is it to calculate a surface normal on the spherical approximation of the earths surface pointing towards the sun at a given point in time? What I try do is to highlight a small area on a world map where the radiation from the sun tangential hits a imaginary sphere around the earth. This is just to get an idea where this area was at the given time, so this information has not to be very accurate. As a mater of fact I will just incorporated this information in the map if it is easy enough to calculate.


title: Will adding heat to a material increase or decrease entropy?

text: Does adding heat to a material, thereby increasing electrical resistance in the material increase or decrease entropy? Follow up questions:   Is there a situation were Heat flux ie. thermal flux, will change entropy?   Does increasing resistance to em transfer prevent work from being done?


title: Mean free path of a photon in a fiber

text: Is there a way to calculate, or a reference table I can to look up which provides the average distance a photon travels before it encounters an electron and is absorbed or re-emitted in a fiber optics cable? I asked a fiber manufacturer but all they could come up with is the index of refraction. I'm just looking for an approximation.


title: center of mass Hamiltonian of a Hydrogen atom

text: I'm working through Mattuck's "A Guide to Feynman Diagrams in the Many-Body Problem", but I'm stuck on a bit which I feel should be trivial. In section 3.2 (p 43 in the Dover edition) he gives a Hydrogen atom as an example of a system which can be considered to have a p-dependent potential term. In his words, "the Hamiltonian of the center of mass motion of a Hydrogen atom is $H = p^2 / (m + m_e)$ where $m$ = proton mass and $m_e$ = electron mass. (First question: shouldn't there be a factor 2 in the denominator here?) This may be broken up into $$H = \frac{p^2}{2m} - \frac{m_e}{(m_e+m)m}p^2$$ and the second term treated as if it were a perturbating potential." Second question: I can't for the life of me figure out how he breaks it up like that, whether I include the aforementioned factor 2 or not.. Both multiplying by $\frac{m - m_e}{m - m_e}$ and ignoring terms quadratic in $m_e$, and deriving the first order taylor expansion around $m$ (in both cases inclluding the factor 2 which I believe should be there) give $$\frac{p^2}{2m} - \frac{p^2m_e}{2m^2}$$ which looks kind of close, but not quite it. Of course this is not a hugely significant part of the text but it's really bugging me so a bit of help would be greatly appreciated...


title: Scale invariance plus unitarity implies conformal invariance?

text: What has the reaction been towards the recent paper claiming to have a proof that scale invariance plus unitarity implies conformal invariance in 4d?


title: Gauss' Law for Magnetism Derivative Form: With or without volume integral?

text: I've been reading through FLP Vol. II, and he has proven that as the flux through a closed surface is: $\ \int_{surface} \mathbf{F} \space \mathrm{d}\mathbf{a} $, according to the divergence theorem, the flux through a surface can be defined as: $\ \int_{volume} \nabla \cdot \mathbf{F} \space \mathrm{d}V $, where $\ \mathbf{F} $ is any vector field, and the volume is that which is enclosed by the surface. Previously he had stated as a word equation that: $\ \text{Flux of } \mathbf{B} \text{ through any closed surface}=0. $ I would therefore assume that $\ \int_{volume} \nabla \cdot \mathbf{B} \space \mathrm{d}V = 0$, however Gauss' law for magnetism states that: $\ \nabla \cdot \mathbf{B} = 0$. Does that mean that $\ \nabla \cdot \mathbf{B} = 0$ and $\ \int_{volume} \nabla \cdot \mathbf{B} \space \mathrm{d}V = 0$ are equivalent statements, or am I making a fundamental error somewhere?


Is it true that the mass of air in normal atmospheric pressure over 1$m^2$ is 10326 $kg$? I calculated it from pressure formula $p=\frac{F}{A}$. Let $m=?, A=1 m^2 and p=101300 Pa$. $p=\frac{F}{A} \Leftrightarrow ...\Leftrightarrow m= \frac{pA}{g}=\frac{101300 \cdot 1 }{9,81} = 10326 kg $. So why we don't feel that weight, because if $A=160 cm^2 = 160 \cdot 10^{-4} m^2$( the area of normal man s feet), then $m=\frac{pA}{g}=\frac{101300 \cdot 160 \cdot 10^{-4} }{9,81} =165 kg$. Am I missing something? Why we don't feel that weight?


title: Performing work on a box of gas by lifting it, and first law of thermodynamics

text: What happens if we lift a box of ideal gas? Work is done to the box but no heat is getting into it. So does it's internal energy increase by the amount of work done? Or is it that lifting is not counted as work done to the system? Or is it the case that work turns into heat in a box which quickly flows out to the surrounding, thereby resulting in no net internal energy change?


title: Ideal gas and diatomic gas with same temperature

text: If a box of ideal gas and another box of diatomic gas are in thermal equilibrium,   1. does it mean that the average translational energy of ideal gas particle (A) is the same as that of diatomic gas particle (B)?    2. or does it mean that A is equal to the sum of the average translational energy (B) + average vibrational energy (C) + average rotational energy (D) of diatomic gas particle?   3. Equilibrium is achieved when A = B = C = D?    4. or is it A = B + C + D?


title: Unitarity in QFT and measuring unitarity

text: I am trying to make sense of statements about unitarity in this popular science article about Nima and Jaroslav's new idea. My first query is that it is claimed that unitarity is a pillar of quantum field theory: > _Locality and unitarity are the central pillars of quantum field theory_ In second quantization, however, I recall nothing of unitarity, and one can construct quantum field theories that are not unitary e.g., in the Standard Model without a Higgs boson, the $WW\to WW$ elastic scattering matrix element is greater than unity for $\sqrt{s}\gtrsim1 \textrm{TeV}$. Is it correct that unitarity is a pillar of quantum field theory? My second query regards a described experiment for measuring unitarity. It is suggested that one repeatedly measures the final state of a scattering process: > _To prove [unitarity], one would have to observe the same interaction over > and over and count the frequencies of the different outcomes. Doing this to > perfect accuracy would require an infinite number of observations using an > infinitely large measuring apparatus, but the latter would again cause > gravitational collapse into a black hole._ Take, e.g., a simplified experiment in which one repeatedly measures the state $$ \psi = a \phi +b\chi. $$ What would the results for repeated measurements of the state be for $|a|^2+|b|^2>1$? and $|a|^2+|b|^2<1$? Upon measurement, I would be entangled with the state. I could only see one outcome per measurement. Maybe an omniscient god could see that states were destroyed for $|a|^2+|b|^2<1$ and created for $|a|^2+|b|^2>1$, if he saw all worlds in a many worlds interpretation. But I would see nothing odd? So how is this experiment supposed to work?


title: When did we learn that stars die?

text: As we all know, the stars we see in the night sky might already be dead. I was wondering though, when was this fact or conclusion commonly established? Today, most people (let's assume with an above average education) would probably be aware of this fact. When is the earliest time when the same could be said? I am particularly interested if the same could be said for the time period revolving around the period 1850 - 1900. I know that the speed of light was approximated fairly accurately in the 17th century. Knowing this (finite) speed, it's not hard for me to draw the conclusion that the source of the light I see may not be there anymore. Would this be an easy conclusion to draw a hundred years ago however? Maybe they thought stars don't die?


title: Ambiguity in Beta Functions (2-loop)

text: Beyond one-loop, the beta function of a QFT is scheme dependent. I would like to understand better this ambiguity. The easiest thing to say is that you haven't calculated something physical, so of course it does not need to be scheme independent. However, the anomalous dimension of operators is I think an observable quantity since we may measure critical exponents in the lab, and the anomalous dimension results from the same sort of calculation. Moreover, I can relate the beta functions to the trace anomaly. Schematically, $\langle \partial_\mu j_{dilation}^\mu \rangle=\langle T^\mu_\mu \rangle \approx \beta$ (See Peskin 19.5 for the case of QED). If i couple some field to the trace of $T$ I think I should be able to turn this anomaly into a cross-section for some process which would be measurable (think of ABJ anomaly and $\pi^0 \to \gamma \gamma$ for instance). So the questions is: 1) Is it known how the terms in the beta function may differ between regularization schemes? If I try to calculate the couplings at the fixed point using different schemes, will I get the same answer (I am aware the location of the fixed point is not physical, but if I use the same field variables I could imagine this being scheme independent)? How may I see that although the beta function and location of the fixed point are ambiguous, the anomalous dimensions are not? 2) How would this ambiguity cancel out if I have a theory where I can turn the trace anomaly into a prediction of a scattering amplitude? Or can this simply not be done? Any clarification or suggestion for references is appreciated.


title: Are the collected papers of A. Einstein, entirely translated into English?

text: I need some Einstein's papers translated into English. There is one complete collection of his papers but only a few of them have been translated in English. The rest are in their original language -- German, which, unfortunately, I can't read. Have you seen anywhere all of his paper translated in English?


title: Is there an analogue of configuration space in quantum mechanics?

text: In classical mechanics coordinates are something a bit secondary. Having a configuration space $Q$ (manifold), coordinates enter as a mapping to $\mathbb R^n$, $q_i : Q \to \mathbb R$. The primary thing is the manifold itself and its points. On the contrary, quantum mechanics for classical coordinates has operators $\hat q_i$. And I never encountered some sort of "manifold abstraction" for the space operator. Is there a coordinate-free approach to the space operator in a non-relativistic quantum mechanics?


title: Using Lorentz Invariance of Charge To Calculate Current Density

text: I'm attempting a problem from Zwiebach: A First Course in String Theory and am completely stuck. Could anyone give me a hint? The problem is as follows. Consider $S$, $S'$ two Lorentz frames with $S'$ boosted along the $+x$ axis. In frame $S$ we have a cubic box with sides of length $L$ at rest. The box is filled by a material, also at rest, of uniform charge density $\rho$. In $S$ we assume that the charge density $\underline{j}=0$. Use the Lorentz invariance of charge to calculate the charge density $\rho'$ and current density $\underline{j}'$ in $S'$. Verify that $(c\rho,\underline{j})$ a 4-vector. The charge density is easy. Indeed $L^3\rho = Q = Q' = L'^3\rho'=\frac{L^3}{\gamma}\rho'$ so $\rho' = \gamma \rho$. I know I'm right here because this agrees with what we'd expect from a 4-vector under Lorentz boost. To do the current density I tried to use $0=\frac{\textrm{d}Q}{\textrm{d}t}=\frac{\textrm{d}Q'}{\textrm{d}t'}=\int_S\underline{j}'.\textrm{d}\underline{a}=j'_xL^2$ so $\underline{j}'=0$ since $j'_y=j'_z=0$ clearly must be zero. I know this is wrong though, because it doesn't agree with what I'd expect from a 4-vector! What am I doing wrong? And is this the right way to go about this question? Many thanks in advance!


title: undergrduate physics courses in American colleges

text: I am just curious as a non-US person: how does undergraduate physics course progress in U.S. colleges? Do they go right into classical mechanics books, or do they teach introductory courses first, then specialize on classical mechanics, electrodynamics etc. in the second year? First-year, second-year, third-year distinction would be much appreciated.


title: Is dimensional analysis used outside fluid mechanics and transport phenomena?

text: Most dimensionless numbers (at least the ones easily found) used for dimensional analysis are about fluid dynamics, or transport phenomena, convection and heat transfer - arguably also sort of fluid mechanics. My understanding of dimensional analysis is the following: Derive dimensionless numbers from the description of a system, find the ones physically meaningful, and use them to compare different situations or to scale experiments. Is this possible in other fields, like classical mechanics, and their engineering applications? Example: describe a horizontal beam by: $$ X=\frac {\text{forces acting on the beam}} {\text{forces beam can withstand without plastic deformation}} $$ Both parts of the ratio being functions of shape, density, gravity, material constants etc. My assumption is yes, it's possible, but most fields outside the sort-of fluid mechanics described above are easy enough to calculate without dimensional analysis.


title: Measurements for thermal diffusivity of graphene?

text: We have known for a long time that graphene has in-plane thermal conductivity ranging between 2000 and 4000 $W m^{-1} K^{-1}$. But in order to model heat transport on a sheet of graphene, we need more than the conductivity: we also need specific heat in order to obtain the thermal diffusivity that is used in the equation. I couldn't find any measurement results online for this quantity. I've only seen some crude estimates based on phonon transport, but even so, no specific figures.