title: In an optical system with a single convex lens, where should the image area be placed in order to take pictures at infinite focus?

text: I'm trying to learn how camera lenses work, and have gotten stuck. Say that we have an optical system consisting of a single lens with focal length _f_ , and we want to take pictures of objects which are infinitely far away. The light rays will in other words be parallel, and when these rays pass through the lens they will be focused at a single point at a distance _f_ away from the lens. But, how can we then take pictures of such objects if all of their rays are focused at a _single_ point? I'm not questioning the laws of optics since obviously they work, but I can't figure out where the error in my thinking is. Could someone please shed some light on this (pun intended)? EDIT: For those who are interested, this question is closely related (although not exactly the same).


title: How hot is aurora?

text: Has anyone done research on how hot aurora is?   I mean if it is plasma it should be hot and since it is emitting mostly green light due to nitrogen (~78%) in the air, could it then be considered that it is at least temperature at which nitrogen turns to plasma?   I have tried looking it up in Wikipedia but could not find any data of temperature.


How many state configurations are possible for $N$ particles in completely different states? I cannot remember if the total number of state configurations for $N$ particles in completely different states is $n^n$ or $n!$. Also, if $n$ particles are in completely different states, does this represent the maximum possible entropy of this system?


title: Could anyone help explain this current voltage graph for an LED in liquid nitrogen?

text: I've been doing my coursework investigating LEDs at various temperatures and I've come across an interesting phenomenon which nobody I've asked has been able to explain thoroughly - wheras at room temperature, the LED gives a standard exponential response, when placed in liquid nitrogen (at -196 C) the graph is pretty strange. This data was recorded using a constant current power supply, and is a combination of three different experiments - it still worked fine after each experiment at room temperature. I've asked a couple of my teachers, and the answers they gave ranged from 'the lattice might change and contract at cooler temperatures' to 'the internal temperature of the LED might increase when it has higher currents'. I was wondering if anyone had any more domain-specific knowledge than my teachers and could help explain this :) Here's the graph of this particular LED at room temperature (around 24 C in this case): ![enter image description here](http://i.stack.imgur.com/geKHM.png) Here's the one in LN2: ![Confusing graph that goes up and down...](http://i.stack.imgur.com/Ka31P.png) If it helps, the LED was Cyan in colour and had a wavelength of roughly 485nm in LN2, and 429nm at 80 C. Many thanks, Tom


title: Periodic Wave formula; need explanation?

text: I was listening to Quantum Mechanics lecture and there were wave explanation; to be exact it is periodic wave .. its formula is v (speed) = lambda (wavelength) X f (frequency cycles/second) the professor said that if for example wavelength is shorter wave, that means the frequency is becoming higher .. and if wavelength is longer wave, that means the frequency is becoming lower ... I don't get it, I thought that if we have an object which has many cycles per second such as 50 c/s, then it has much more opportunity than the 30 c/s to reach as far place as I know... please someone could explain what does professor means?


title: Is the Baum Framptom a valid alternative to Big Bang?

text: In the Baum Frampton model, proposed in 2007, because $\epsilon\lt-1$, after the Big Rip the universe starts again empty. The problem of this model is that inverting the arrow of the time, after a certain number of Big Rips it should have been a single universe from which all the sequence of Big Rips started. Is this model compatible with the astronomical observations of the last years and how is it possible to explain in the contest of this model the anomalies in the background microwave radiation?


title: Proof of Liouville's theorem: Relation between phase space volume and probability distribution function

text: I understand the proof of Liouville's theorem to the point where we conclude that Hamiltonian flow in phase-space is volume preserving as we flow in the phase space. Meaning the total derivative of any initial volume element is 0. From here, how do we say that probability distribution function is constant as we flow in the phase-space? What's the relation between phase space volume and the density function, which instantaneously tells us the probability of finding the system in a neighborhood in phase-space?


title: Minimum separation problem

text: I am trying to solve this problem where two point charges are infinite distance apart(horizontally) but separated by some finite distance vertically. Both are positively charged and projected horizontally, anti parallely with some velocity. I am required to find the minimum distance of separation. How can I set up this problem? I am not able to figure out how to describe the motion of the particles because the forces will act all throughout the motion and there will be perpendicular and parallel components of this force which keep changing direction. I am just a grade 12 student and this is an exam problem. To me this problem seems extremely complex, so am I missing out on something? Does the problem have a trivial solution?


title: Planar motion in central forces

text: In a two body problem under central force, corresponding to a potential $V(r)$(assume one body is massive compared to the other so that its motion is negligible), conservation of angular momentum implies the motion of the body to be in a plane spanned by position r and momentum p vectors. But if we have three bodies, one of them massive, are the motions of other two bodies still restricted to a plane? Now the total angular momentum is $$L = L_1 + L_2 = r_1 \times p_1 + r_2 \times p_2$$ , which is conserved. Mathematically, $L$ could be kept constant while $L_1$ and $L_2$ are changing. Which means we could have motions of the two bodies in two planes at angle to each other, a non-planar motion. Is this allowed in principle? in reality? If not, why? Then, what is reason for the planar motion?


title: Was Titanic's captain's decision correct?

text: I just saw a documentary on Titanic. Someone in that documentary told Captain Smith > "Shall we close the water tight doors, sir?" and the Captain replied > "No, let it be open. Let the water be distributed uniformly throughout ship, > so that we could lower the life boats." Note: The quotes are not exact but very near to exact. * * * ## My Question If and if, the watertight doors were shut, was there a chance of Titanic to escape from its tragic fate? And what exactly is a water tight container? I presume that this is related to physics :)


title: Examples of piecewise smooth dynamical systems

text: I have recently been studying continuous dynamical systems whose phase space can be divided into a number of regions. Inside each of these the flow is smooth, but there is a discrete jump in the flow just at the boundaries. In the mathematical description, the right hand side of the differential equation is different for different regions of the phase space of the dynamical variables. **Note:** I don't mean something trivial like systems which exhibit smoothness in different regions of _physical_ space separated by boundaries, like differently heated gases in partitions, or water in contact with vapour etc. The different regions I mention are regions in the **phase space** of the dynamical systems. So imagine a set of continuous-time differential equations defining a flow which is segregated in its phase space into regions in which the evolution of the equations is piecewise smooth. I also don't mean _phase transition_. There is no variation of order parameter or bifurcations here. The piecewise smoothness exists in the dynamical phase space for a _fixed_ value of the system parameters. I have been studying them in an engineering context of a mechanical device in which there is a sudden change in the velocity of a moving part when it hits something. But it struck me that such piecewise smooth systems should be found in many scenarios, from other areas of physics, maybe certain quantum phenomena, to biological systems that can be studied with the theory of dynamical systems. Some examples of the kind of systems I am looking for are:   * Quantum mechanics: the Muffin-Tin potential is a quantum model where the potential (the right side of the differential equation) is approximated to be piecewise defined.   * Classical mechanics: the hard impacting oscillator (oscillator with a rigid wall at an end restricting the amplitude, like the devices I was studying).   * Theoretical computer science: Hybrid automata and reachability problems which are further piecewise linear. I am curious to apply my understanding of the mechanical system to such systems. **So, what are other dynamical systems in nature which exhibit piecewise smooth behaviour?**


title: Are old fashion photomultipliers used in LHC experiments?

text: Are any photomultipliers used in any of LHC detectors? Or are only semiconductor devices used? If yes, then where?


title: Carrying water on person, or on the frame when bicycling

text: So, the question is as follows: What is the difference in work exerted by the rider in the two following scenarios? a) Rider + bike. Water carried in a holder on the frame   b) Rider + bike. Water carried in a backpack (for simplicity, assumption that weight of holder = weight of backpack and backpack is perfectly aerodynamic, not increasing drag) I would assume that in case a), as the weight is closer to the bike's centre of mass, it would require the least amount of extra work, compared to b), where the weight is more distant from the centre of mass. Thanks in advance.


title: What's the difference between exchange spin wave and magnetostatic spin wave?

text: So far I've heard of three kinds of spin waves   * Magnetostatic spin waves (MSW)   * Dipole-exchange spin waves (DESW)   * Exchange spin waves (ESW) What's the difference?


![enter image description here](http://i.stack.imgur.com/2cO6y.png) What is principle of solution behind this induction problem (problem 29)? The problem can be find in here (problem 29). Problem is: The five separate figures below involve a cylindrical magnet and a tiny light bulb connected to the ends of a loop of copper wire. These figures are to be used in following question. The plane of the wire loop is perpendicular to the reference axis. The states of motion of the magnet and of the loop of wire are indicated in the diagram. Speed will be represented by v and CCV represent counter clockwise. (look at the picture in here( problem 29)) 29) In which of the above figures will the light bulb be glowing? (a) I, III, IV (b) I, IV (c) I, II, IV (d) IV (e) None of these.


title: How do I find minimum constant deceleration so that object does not pass distance d in time t?

text: I'm working on a problem for an online judge site. I've boiled down the problem to this calculation: given a vehicle with an initial velocity $v$, how can one calculate the minimum constant deceleration $a$ required to make sure the vehicle does not travel more than distance $d$ by time $t$? The constant deceleration is always applied until the vehicle either stops or reaches distance $d$. Stopping the vehicle at distance $d$ is allowed, if no other deceleration $a$ exists that slows the vehicle just enough so that it does not reach distance $d$ until time $t$. I tried starting with \begin{equation} d = vt - \frac{at^2}{2} \end{equation} and solving for $a$, giving $a = 2\frac{vt - d}{t^2}$ but that doesn't work by itself since it allows for the vehicle to extend past $d$ and return to it by going backwards (negative velocity), which violates the requirement that the vehicle not pass distance $d$ until time $t$. So I applied $v = v_0 - at$ to calculate the maximum constant deceleration $a$ that produces non-negative final velocity at time $t$: $a \le \frac{v_0}{t}$. And if the calculation above exceeds this maximm deceleration then I apply the stopping distance deceleration formula: $a = \frac{v^2}{2d}$ (it's ok that this doesn't take exactly time $t$). But my intuition tells me there should be some cases where the vehicle can be slowed without stopping it using constant deceleration over distance $d$, but I'm not seeing how to apply the suvat equations to do this. Maybe there is no way unless $\frac{v}{t} = 2\frac{vt - d}{t^2}$?


title: What are the details around the origin of the string theory?

text: It is well-known even among the lay public (thanks to popular books) that string theory first arose in the field of strong interactions where certain scattering amplitudes had properties that could be explained by assuming there were strings lurking around. Unfortunatelly, that's about as far as my knowledge reaches. > Can you explain in detail what kind of objects that show those peculiar > stringy properties are and what precisely those properties are? > > How did the old "string theory" explain those properties. > > Are those explanations still relevant or have they been superseded by modern > perspective gained from QCD (which hadn't been yet around at the Veneziano's > time)?


title: Electron revolving in an atom

text: When an electron revolves around the nucleus in P or d-orbitals why does not it collide with the nucleus.   I mean to say that the shape of the orbital narrows near the nucleus , so shouldn't it collide with the nucleus?


title: Diffeomorphism Invariance of General Relativity

text: I'm sorry I know this has been asked before, but I'm still a bit confused. I understand that an active diffeomorphism $\varphi:M\to M$ can be equivalently viewed as a coordinate transformation so that since the equations of general relativity are tensorial $\varphi^*g$ will be a solution to Einstein's equations if $g$ is. However I don't see how that same reasoning doesn't imply that other physical theories are diffeomorphism invariant. What's the difference between general relativity and other physical theories, like classical mechanics? Why can't diffeomorphisms be viewed as coordinate transformations in both (or am I confused?).


title: Computing a "best-fit" of discrete points from a multipole expansion, i.e. invert the multipole moments

text: Take a field $\phi(\bf{x})$ created from a charge distribution contained within a radius $R$. The multipole expansion in spherical harmonics $Y_{\ell,m}$ outside of $R$ is approximated by: $$ \phi({\bf x}) \approx \frac{1}{4\pi \epsilon_0} \sum_{\ell=0}^{\ell_{MAX}} \sum_{m=-\ell}^{\ell} \frac{4\pi}{2\ell +1} \alpha_{\ell, m} \frac{Y_{\ell,m}(\theta, \phi)}{r^{\ell +1}} $$ Given only the finite set of the multipole moments $\alpha_{\ell,m}$, is it possible to find a charge distribution of $N$ discrete charges $(q_i, r_i, \theta_i, \phi_i)$ that "best-fit" this potential? Right now I'm using a simple minima finder over the $3^N$ variables (see note 1), but I'm wondering if there is any prior work/observations on inverting the multipole moments. **Note 1** If the positions of the charges are fixed, the charge magnitudes $q_i$ are simply linear combinations: $$ \alpha_{\ell, m} = \sum_i^N q_i Y^*_{\ell,m}(\theta_i, \phi_i) r_i^\ell $$ thus simple linear algebra gives the best-fit charge _magnitudes_. **Note 2** It's easy to see that this charge distribution need not be unique, indeed if $\ell_{MAX}=0$ any combination such that $\sum_i^N q_i \propto \alpha_{0,0}$ should work. This is ok, I'm more concerned with finding good approximations than the absolute best fit. **Note 3** If the charge magnitudes are real (which is a given for a physical problem), the number of unique moments are reduced by roughly a factor of 2 since: $$ \alpha_{\ell,-m} = (-1)^m \alpha_{\ell,m}^* $$


title: Violation of Newton's 3rd law and momentum conservation

text: Why and when does newtons 3rd law violate in relativistic mechanics? Check this link http://www.animations.physics.unsw.edu.au/jw/Newton.htm.


title: How do individual photons make up an EM wave?

text: I'm trying to understand the connection between the wave model and the particle model for light. It's understood that the energy of a photon is given by E=hf, but from my understanding of fourier analysis, the only kind of wave that has a precise frequency is a plane wave. The plane wave is an idealization, since no real wave permeates all of space and time. So imagining a more realistic EM pulse, the frequency spectrum will have some kind of spread depending on the shape of the pulse. Is the pulse a single photon? Or is it a collection of photons, each with different frequencies? In the photoelectric effect, it's usually described as a single photon with sufficient energy being absorbed, kicking the electron out of its orbit. Let's imagine the pulse is symmetrically centred around the frequency with energy exactly equal to the metal's work function. What exactly happens to such a pulse? Will the whole pulse be absorbed, since its average frequency has energy of the work function? Or will the half of the pulse that has the higher frequency be absorbed, leaving the rest to reflect or what-not?


title: Atomic Orbitals, Singlets, Triplets

text: Why are singlet states representative of the S orbitals, and similarly why are triplet states representative of P orbitals? I learned about these states as resulting from having a total spin of a total amount-but I don't see why these orbitals would have spin limits.


title: Do monochromatic waves carry information?

text: The answer is negative according to http://www.mathpages.com/home/kmath528/kmath528.htm > It should also be remembered that a perfectly monochromatic wave carries no > information, and therefore is not subject to any limitation on the phase > velocity. Can someone clarify?


title: Running: Determine how much more energy is needed per extra kilogram of weight

text: (I recently asked this on maths but was directed here) I have recently become a runner and having a keen interest in kinematics I'm very interested in the maths/physics of my running. Can someone please help me determine a function that will enable me to determine how much harder I have to work to carry my 10kg overweight body over a given distance? I was pretty sure I could just use `work=weight x distance` for the flat roads, but someone has pointed out that my time would be meaningless in such a scenario, but I'm having difficultly determining this for the hills too. In addition I need to include time as a variable so I can work out how my time affects my energy expenditure.


title: How to find the phase constant?

text: I was given this velocity-vs-time graph of a particle in simple harmonic motion: ![enter image description here](http://i.stack.imgur.com/NHuuX.png) I determined the amplitude to be $A = 1.15$ m, which Mastering Physics confirmed is correct. Then I was asked to find the phase constant. I tried a few things but finally gave up and asked Mastering Physics for the answer, which is: $\phi_0=2.62$ rad. Since I had this equation in my notes, $$v_x(t)=-\omega A sin(\omega t+\phi_0)=-v_\max sin(\omega t+\phi_0)$$ First I found the angular frequency $\omega$: $$\omega = {2\pi \over T}={2\pi \over 12 s}=0.523$$ Multiplied by $A=1.15$, it matches the $v_{max}$ of 0.60 m/s shown in the graph. Then I tried this, using $t$ = 0 s: $$v_x(t)=-\omega A sin(\omega t+\phi_0)$$ $$-0.30=(-0.523)(1.15)sin(\omega (0)+\phi_0)$$ $$-0.30=(-0.60)sin(\phi_0)$$ $${-0.30 \over -0.60}=sin(\phi_0)$$ $$0.5=sin(\phi_0)$$ $$sin^{-1}(0.5)=\phi_0$$ $$0.523=\phi_0$$ But that just gave me the angular frequency value, not the correct answer (which is 2.62 rad). Did I do something wrong in the calculation, or am I confusing units? My calculator was on radians mode. Note: My algebra skills are pretty weak, so I'm thinking that's where the mistake might be... ?


title: is there any relation between resistance and magnetism?

text: i was holding a resistance wire( **insulated** ) coiled up and both of its ends were connected to a Ohm-meter it was showing 18 ohms while the circuit was on i pulled the coil from both ends and made that a straight wire with barely a knot in between and observed that the reading of Ohm meter **dropped** to 16.5 Ohms. Is there any relationship between resistance and magnetic field? or was it just because the ohm-meter is having errors?


title: How to simplify e to power of j.t

text: I have exam tommorow and cant get this figured out :( dont blame me but please answer this question. I Want to simplify this term: 3 times ((e topowerof 5jt) + (e topowerof -5jt)) Thanks.


title: Twisted supermultiplets

text: What is a twisted supermultiplet, in a generic supersymmetric theory? Which ordinary fields belong to one of such twisted supermultiplets? I am confident with the idea of a supermultiplet or a superfield, in $\mathcal{N}=1,2,4$. However, I don't know what the term _twisted_ stands for. Also, why is it useful to consider such things?


title: Radioactive decay / binding energies

text: If my understanding is correct, the binding energy determines a nucleus' stability and the greater the binding energy, the more stable the nucleus (e.g iron-56). The mass of the sum of nucleons that make up a nucleus is greater than the mass of the nucleus that they come from. What I would like to know is where the energy comes from to initiate the decay process? Presumably energy is required (similar to a chemical reaction where energy is required to break bonds)? I could be on the wrong track here but if someone would be so kind as to answer this for me, it would be greatly appreciated.


title: Complementarity between the laws of physics?

text: Is this following proposal plausible, worth considering, or dismissable as lunatic fringe science? What if the universe isn't really what we think it is but some universal quantum computer where we read off one output at a time? This universal quantum computer is initialized to a superposition of program instructions -- which are identified with the laws of physics -- and after reading off one output after another, one finds there are in general two different kinds of questions we can ask; general relativity type questions, where we get GR type answers, and quantum mechanical type questions, where we get QM type answers. But we have to choose between one or the other, and whichever we choose, we get the answer appropriate for that domain. The place where both are united is the Planck scale, but unfortunately, the Planck scale as way too inaccessible to us, and is hence unobservable. Can this explain the confusion about UV/IR mixings? String compactifications are computer programs determining the parameters of the effective field theory describing the domain of answerable experiments. A complementarity between the laws of physics themselves. GR/QM complementarity. Is there any flaw with my proposal?


title: How can I calculate how long to keep my beverage chilled?

text: The problem is simple, I suppose... I have a beverage at room temperature (let's say 23°C) in an aluminum can (it's a Barq's root beer.) I want to put it in a freezer (let's say it's at -17°C). How long do I have to wait for it to reach a chilled temperature (let's say 8°C). I need to know for a couple reasons. One, I don't want the can to explode. Two, I don't want it to get too cold for my teeth (perhaps I shouldn't be drinking pop? :-) ) Three, I want don't my coworkers to steal my nice, cold beverage on the first hot day of the year. If I could find the right equations, I would just plug in the numbers, but I can't seem to find the right ones.


title: What is potential energy truly?

text: I have a problematic question for which I have been unable to attain a satisfactory answer. What is potential energy truly? - I have read about how potential energy can be seen as the "highering" of an object within a field, be it gravitational, electromagnetic, etc. In a way, an object is lifted to a higher position, and when it eventually falls down again energy is released back. The object does not even have to be "lifted" beforehand, as with the case of asteroids. However, since energy cannot be destroyed, it must somehow be stored, yet how does energy truly store itself within a field? It seems hard for me to grasp that energy can physically be stored within the positioning of an object? I can't really get my head around how a field conserves energy within it. Kinetic energy on the other hand, can be, for me, intuitively understood, as the object really is "moving" and energy manifests itself through motion. Can anyone elaborate on this, perhaps give reasoning as to why a field can in fact "store" energy? Please point out any fallacious or naive views I seem to hold.


title: The question about quantization of free EM field

text: Let's have the free EM field theory with Coulomb gauge: $$ \partial^{2}A_{\mu} = 0, \quad A_{0} = 0, \quad (\nabla \cdot \mathbf A ) = 0. $$ One of the ways of quantizing the field is the following. The expression for energy $$ \tag 1 W = \frac{1}{2}\int (\mathbf E^{2} + \mathbf B^{2})d^{3}\mathbf r $$ with a given solution $$ \mathbf A = \int (\mathbf a (\mathbf k) e^{-ikx} + \mathbf a^{*} (\mathbf k ) e^{ikx} ) \frac{d^{3}\mathbf k}{\sqrt{(2 \pi )^{3}}}, \quad \omega_{\mathbf k} = |\mathbf k| $$ is rewritten in a form $$ \tag 2 W = \frac{1}{2}\int \left(\omega^{2}_{\mathbf k}\mathbf Q^{2}(\mathbf k) + \mathbf P^{2}(\mathbf k)\right)d^{3}\mathbf k , $$ where $$ \mathbf Q = \mathbf a + \mathbf a^{*}, \quad \mathbf P = -i\omega_{\mathbf k}(\mathbf a(\mathbf k) - \mathbf a^{*} (\mathbf k )). $$ So, the question: why do we quantize $(2)$, not $(1)$? I.e., why do we postulate $$ [Q_{i}(\mathbf k ) , P_{j} (\mathbf l ) ] = \delta_{ij}\delta (\mathbf k - \mathbf l ), $$ not $$[E_{i}(\mathbf x ) , B_{j} (\mathbf x ' ) ] = \delta_{ij}\delta (\mathbf x - \mathbf x' )? $$


title: Functional differential equations with multiple solutions in physics?

text: Are there any system in physics that are represented by a functional differential equation that has multiple but a finite number of solutions?


title: Trace of Fermion Loops in Effective Field Theories

text: I'd like to know whether we need to take the trace of fermion loops in effective theory in the same way that we need to do so for renormalizable theories. At first thought, it seems obvious that should need to since we are dealing with fermions, but I recently saw a calculation and the professor did not bother to take the trace. Here the particular problem at hand: The theory at hand is, \begin{equation} {\cal L} = \bar{\psi} ( i \partial _\mu \gamma ^\mu - m ) \psi + \frac{1}{2} ( \partial _\mu \phi ) ^2 - \frac{1}{2} M ^2 \phi ^2 + g \phi \bar{\psi} \psi \end{equation} At low energies we have, \begin{equation} {\cal L} = \bar{\psi} ( i \partial _\mu \gamma ^\mu - m ) \psi + \frac{ a }{ M } ( \bar{\psi} \psi ) ^2 + ... \end{equation} Doing matching between the theories at tree level we find that,$ a = g ^2 $. We then move on to calculate the diagram, $\hspace{6cm}$![enter image description here](http://i.stack.imgur.com/aTqFk.png) which the professor wrote as (omitting the external legs), \begin{equation} \frac{ i g ^2 }{ M ^2 } \int \frac{ \,d^4k }{ (2\pi)^4 } \frac{ k_\mu\gamma^\mu + m }{ k ^2 - m ^2 } \end{equation} Not taking the trace of the diagram changes the final result by a factor of $ 4 - \epsilon $ (using dim-reg). On the one hand, this makes sense since we didn't need to take a trace in the high energy theory where the corresponding diagram is $\hspace{6cm}$![enter image description here](http://i.stack.imgur.com/X4khT.png) on the other hand these are fermions after all. Is the professor just being sloppy here or do you not need to take the trace for some reason I am missing? For more context please see my notes under Effective Field Theory in this link (Eq. 4.6)


title: Why dynamic Casimir effect does not appear in static gravity field?

text: Dynamic Casimir effect tells us that a constantly-accelerated mirror should emit radiation due to interaction with vacuum. Following principle of equivalence, a similar mirror placed in static gravitational field should emit either. So will planet which surface is covered with a mirror emit any radiation and if no, why not?


title: What fundamental principles or theories are required by modern physics?

text: We have been taught that speed of light is insurmountable but as we know an experiment recently tried to show otherwise. If the experiment did turn out to be correct and confirmed by others, would it make physics to be rethought of? What other concepts are fundamental to physics, which, if disproved would need radical rethinking? If this sounds too juvenile and/or misinformed, please understand that I am a layman, having nothing, professionally or academically to deal with science, directly, and this question is out of curiosity. I have developed a liking to "science stuff" and been reading popular science variety of literature lately. This question was also prompted by what Sheldon Cooper had to say in one of the episodes (I was watching a rerun).


title: Is it a total or an explicite time derivative in the Schrödinger equation?

text: I am always dubious when I need write Schrödinger equation: do I write $\partial / \partial t$ or $d/dt$ ? I suppose it depends on the space in which it is considered. How?


title: Perturbed stress-energy tensor in a cosmological context?

text: In the theory of cosmological pertubations, we can write the metric of a null- curvature expanding Universe as : $ds^2 = -c^2\left(1+2\frac{\psi}{c^2}\right)dt^2 + a^2 \left(1-2\frac{\phi}{c^2}\right)\left(dx^2+dy^2+dz^2\right)$ From that metric we can compute the perturbed Einstein tensor $G_{\mu\nu}$. Now, I search what is the form of the stress energy tensor associated to this in order to write the Einstein equation : $G_{\mu\nu}=\frac{8\pi G}{c^4}T_{\mu\nu}$. So, starting from the fact that we consider a perturbed fluid : $\rho=\bar{\rho}+\delta\rho$ $P=\bar{P}+\delta P$ what is the $T_{\mu\nu}$ associated to that fluid ? * * * Remark : Here are some indications (see 4.4.1) : http://www.damtp.cam.ac.uk/user/db275/Cosmology/Chapter4.pdf but I do not understand very well why the author write the $T_{\nu}^{\mu}$ instead of the $T_{\mu\nu}$ (so if someone knows why, please explain it to me).


title: Nature of Perturbed state in Perturbation Theory?

text: I'm interested in the Nature of Perturbed state in Perturbation Theory. The first order perturbed state is given by $$\psi^{(1)}_{n}=\Sigma_{m}a_{m}\psi^{(0)}_{m}.$$ Where $$a_{m}=\int\psi_{1}^{(m)*}\psi_{n}^{(0)}d\tau.$$ Now as the Hamiltonian changes while applying perturbation. It's basis should change,My question is, why it is not changing when we expand perturbed state. More precisely what is the logical step behind expanding perturbed state as sum of unperturbed states?


title: Switching from an accelerated frame of reference to a locally inertial reference system

text: _Using the equivalence principle, show that the interval for an accelerated observer ($\textbf{g}$ uniform and constant) has the form_ $$ ds^2|_{\text{first order in }\textbf{g}}=(1+2\varphi/c^2)c^2dt^2-d\textbf{x}^2+o(1/c^2), $$ where $\varphi=-\textbf{x}\cdot\textbf{g}$.


title: Why does the wave description say that probability oscillates, while the phase interpretation says constant amplitude?

text: The wave description of a particle illustrates an oscillating probability of the particle being found in any point in space. When a particle travels, it carries along with it a phase that oscillates in the complex plane. When the particle travels, the amplitude never changes although the phase does, which doesn't matter. The only property that matters is the amplitude. Why does the wave description say that probability oscillates and the phase interpretation says constant amplitude? What am I missing?


title: Kinetic energy of a cylinder

text: It is a long cylinder (you can approx $R=0$), and it has a fixed point in one os its ending points, it's rotating on a plane and I have to calculate the kinetic energy from reference systems situated in the center of mass, and both extremes. If I calculate it in the fixed point, it has only rotational energy: $$\frac{1}{6}ML^2\dot{\varphi}^2$$ If I choose the center of mass, the moment of inertia changes but I have to add to that the energy of that moving point $1/2Mv_{CM}^2$, and the result is the same. My problem is that when I choose the moving extreme, the result changes: The moment of inertia is the same as with the fixed point: $1/3ML^2$, and the speed at that point is $1/2ML^2\dot\varphi ^2$, so adding them gives you something different from the $1/6$ result, and they should be the same, ¿What am I doing wrong?


title: Question about Wigner's friend

text: The Wigner's friend thought experiment can be used to understand non-realism in quantum mechanics. For anyone not familiar, the thought experiment involves two researchers observing an experiment at different times, let's say it's an electron spin. The question is, how should the second researcher to observe the experiment treat the system between his observation and his friend's observation? In a non-realist interpretation we would say that the second researcher should continue to time evolve the system with the Schrodinger equation until he observes the system himself, but now he must include his friend as part of the quantum system as well. So, the two observers see the collapse of the wavefunctions at different times, which is fine because the time of wavefunction collapse is not given by a linear Hermitian operator so they need not agree about it. My question concerns what happens if the additional time evolution that the second researcher observes changes the probability. For instance, say there is a 50% probability of spin up when the first researcher observes the experiment. Can this probability change from the second observer's perspective in the time between observations? If it can, how do we explain the fact that if they repeat the experiment 100 times the first researcher would expect to see spin up 50 times and the second researcher would expect to see something else. If it can't, what is the purpose of the additional unitary time evolution that the second researcher uses to describe the situation, couldn't he just use a description where the wavefunction collapses when the first researcher observes the experiment and get the same answer?


title: Why it is called a Newton Sphere? (Velocity map imaging)

text: In velocity map imaging (photo-dissociation and photo-emission), the ejected particles form a newton sphere. I didn't really get the concept why it is called a "newton sphere" and also why at the poles the distribution is $\cos^2(\theta)$?


title: Lagrangian for a system of particles

text: If a system of particles attracting each other under inverse square force, then prove that $$ 2<T> \+ <V> = 0.$$


title: What is the difference between weight and mass?

text: My science teacher is always saying the words "weight of an object" and "mass of an object," but then my physics book (that I read on my own) tells me completely different definitions from the way these words are used in my science class... so which is right? What is the difference between the weight of an object and the mass of an object?


title: Higgs maths video Brian Greene

text: What do you think of the video below? ==> http://www.youtube.com/watch?v=KWj00MCqSxs Is the explanation in this video largely correct, or missing something? Is the mathematical description in the video of the higgs mechanism right? He said that phi is re-written as a m + h. What does it mean? Is the m in this video the vacuum expectation value? Thanks for all answers!


title: Anti-reflective coating effect on total internal reflection?

text: If the surface of prism is coated with an anti-reflective coating, (specifically an index matching, refractive index gradient, or moth-eye structure), and light impinges at a greater than critical angle (i.e. would otherwise be reflected by total internal reflection), at what angle does the light exit the prism?


title: How close are humans to creating gold from other elements?

text: Gold is made when dying stars collide. How close are humans to doing that? Or is it even possible to do that unless gravity is absurdly high?


title: Path Integral Quantization

text: I was reading through my notes on the path integral quantization of bosonic string theory when a general question about path integral quantization arised to me. The widely used intuitive explanation of a path integral is that you sum over all paths from spacetime point $x$ to spacetime point $y$. The classical path has weight one (is this correct?), whereas the quantum paths are weighed by $\exp(iS)$, where $S$ is the action of the theory you are considering. In my current situation we have the Polyakov path integral: $$ Z = \int\mathcal{D}X\mathcal{D}g_{ab}e^{iS_p[X,g]}, $$ where $S_p$ is the Polyakov action. I have seen the derivation of the path integral by Matrix kernels in my introductory QFT course. A problem which occured to me is that if the quantum paths are really "weighted" by the $\exp(iS_p)$, it only makes sense if $\mathrm{Re}(S_p) = 0$ and $\mathrm{Im}(S_p)\neq0$. If this were not the case, the integral seems to be ill-defined (not convergent) and in the case of an oscillating exponential we cannot really talk about a weight factor right? Is this reasoning correct? Is the value of the Polyakov action purely imaginary for every field $X^{\mu}$ and $g_{ab}$? Secondly, when one pushes through the calculation of the Polyakov path integral one obtains the partition function $$ \hat{Z} = \int \mathcal{D}X\mathcal{D}b\mathcal{D}ce^{i(S_p[X,g] + S_g[b,c])}, $$ where we have a ghost action and the Polyakov action. My professor now states that this a third quantized version of string theory (we have seen covariant and lightcone quantization). I am wondering where the quantization takes place. Does it happen at the beginning, when one postulates the path integral on the grounds of similarity to QFT? I am looking for a well-defined point, like the promotion to operators with commutation relations in the lightcone quantization. Finally, in a calculation of the Weyl anomaly and the critical dimension, the professor quantizes the ghost fields. This does not make sense to me. If the path integral is a quantization of string theory, why do we have to quantize the ghost fields afterwards again?


title: Where does the extra equation come from to determine the forces from an object on a table?

text: I have a question about basic statics but somehow I cannot manage to find the answer on my own (btw, this is not a homework. It's been so many years since school for me...). The problem is very simple: we have an object with weight $D$ at a given location on table wit with four legs ($F_1$ to $F_4$). What is the force applied on each leg? (for simplicity, I'm just using the same labels $F$ and $D$ for both the location and the force) ![object on a table](http://i.stack.imgur.com/ZXhRB.png) $W$, $H$, $x$, $y$ and $D$ are given. To find the forces on each leg, as far as I remember, I have to consider two general equations: $\sum F=0$ and $\sum M=0$. So I have: $$ F_1 + F_2 + F_3 + F_4 - D = 0 $$ Also, considering the moments round the point $F_1$: $$ W(F_2+F_3) - xD = 0 $$ $$ H(F_3+F_4) - yD = 0 $$ But this just give me 3 equations! I missing one more equation and cannot figure it out.


title: Proper time in Nordstrom gravity

text: This wikipedia article claims that there are two interpretations of Nordstrom's scalar theory of gravity: 1) A scalar field theory on flat space. The reason why an apple falls is that its mass is coupled to $\phi$, which causes it to deviate from a geodesic. 2) A theory of curved space-time. $R = k T$. The metric of space-time is $g = \phi^2(x) \eta$. The falling apple travels on a geodesic. My question is, why are these considered two interpretations of the same theory, and not two separate theories? Although both give the same predictions for the trajectories of a particles, shouldn't the time elapsed on a watch differ by a factor of $\sim \phi$ between the two theories? In other words, suppose that a particle gets displaced a small amount $\delta x$ in space-time. The proper time elapsed on its watch is according to 1) is $\delta \tau^2 \sim \eta \delta x \delta x$, and in 2) isn't $\delta \tau^2 \sim \phi^2 \delta x \delta x$? On the other hand, wiki says: > which metric is the one which according to Nordström can be measured locally > by physical experiments? The answer is: the curved spacetime is the > physically observable one in this theory (as in all metric theories of > gravitation); the flat background is a mere mathematical fiction


title: Tensor fields on space-time

text: I'm reading landau electrodynamics . For the investigation of the dynamics of a charged particle in an electromagnetic field , the author introduces a four vector potential field on spacetime . The action is $\frac{-e}{c^{2}}\int A^{\mu }dx_{\mu }$ I want to do the same for higher rank tensors $T^{\mu,\rho,..}$ contracted with other higher rank tensors . I think that derivations will be very similar to the four vector field but more complicated . Are there any interesting consequences of these theories ?


title: Polarization of light

text: So for my experimental optics class, I had to create a device that would emit horizontally polarized light such that its intensity is independent of an incoming linearly polarized beam of arbitrary polarization direction. What would be the most efficient method? I thought of using an xy analyser which would split the beam into two orthogonal beams, and use two quarter wave plates to turn the vertical beam into a horizantal one whilst the total intensity is still constant. Then I would just reflect both beams and since they are both horizontal now, they could superimpose with the same total intensity as the original beam.


title: Harmonics in an open organ pipe

text: How to solve the following question? > _An open organ pipe has two adjacent natural frequencies of 500 and 600 Hz. > Assume the speed of sound in the air 340m/s. The length of the organ pipe > is?_ What is **open organ pipe**? It means two open ends? Like this? Or one closed and one open? ![enter image description here](http://i.stack.imgur.com/DmRsb.jpg) Should I use this formula? $L=\frac{(2n+1)v}{4f}$


title: Isn't this wikipedia equation of navier-stokes actually wrong?

text: There is a wikipedia page about NS Existance and Smoothness It seems to me that the Navier Stokes equations is wrong?   (because in one side of equal sign unit is $\frac {m}{s^2}$ but in other side it is $\frac {kg.m}{s^2}$) Navier Stokes Equation: ![enter image description here](http://i.stack.imgur.com/UMfVC.png)


title: Direction of angular velocity

text: Angular velocity is the rate of angular displacement about an axis. Its direction is determined by right hand rule. According to right hand rule, if you hold the axis with your right hand and rotate the fingers in the direction of motion of the rotating body then thumb will point the direction of the angular velocity. The direction of angular velocity is above or below the plane. But what does it mean? I mean in linear velocity the direction of velocity is in the direction of motion of body but what does it mean that body is moving in one direction while the direction of its angular velocity is in another direction?


title: What is Equilibrium Equivalent Concentration (EEC)?

text: Recently I am doing a research about radon measurement. During the process I met two concepts named Equilibrium Equivalent Concentration (EEC) and Potential Alpha Energy Concentration (PAEC), but until now I am not so clear about the definition of EEC, who know this clearly? Could you be nice to tell me exact definition of Equilibrium equivalent concentration? and what is the different between EEC and PAEC?


title: A foundational question about harmonic motion : 2 pipes with different length and frequencies

text: How to solve the following question? > **A pipe open only at one end has a fundamental frequency of 256Hz. A second > pipe, initially identical to the first pipe, is shortened by cutting off a > portion of the open end. Now, when both pipes vibrate at their fundamental > frequencies, a beat freqency of 12Hz is heard. How many centimeters were cut > off the end of the second pipe? The speed of sound is 343m/s.** _The ans is > 1.5cm._ Using the formula :$L=\left(\frac{2n+1}{4}\right)\left(\frac{v}{f}\right)$ The length of the first one $=\left(\frac{2\left(1\right)+1}{4}\right)\left(\frac{343}{256}\right)= 1.0048828m$. Then, The frequency of second one $= 256 + 12 = 268Hz$ Therefore, the length of the second one $=\left(\frac{2\left(1\right)+1}{4}\right)\left(\frac{343}{268}\right) = 0.95988806m$ The length difference $= 1.0048828 - 0.95988806 = 0.04499474m = 4.499474cm$ How to calculate the right answer? * * * _**Thank you for your attention_**


title: Convolution kernel of poisson equation by FFT

text: I'm trying to solve poisson equation using FFT. In genral it is a convolution of the charge density with potential well of point charge ( Green's function of laplace equation ) which is $1/r$ I'm not quite sure about fourier transform of this convolution kernel. the 2D fourier transform of $1/r$ is $ \frac{1}{\sqrt{k_x^2+k_y^2}}= 1/k_r$ acording to this: http://sepwww.stanford.edu/public/docs/sep103/jon3/paper_html/node3.html while in many papers about FFT solution of poisson equation is used kernel like this: $ \frac{1}{(4-2cos(2*\pi*k_x)-2cos(2*\pi*k_y))} $ For example here: http://www.physics.buffalo.edu/phy410-505-2004/Chapter6/ch6-lec2.pdf In one dimension the kernel is $1/k^2$ **Basicaly, I would like to make clear what is the kernel in 1D, 2D and 3D case both in real space and Fourier space**


title: Why does the action have to be hermitian?

text: The hermiticity of operators of observables, e.g. the Hamiltonian, in QM is usually justified by saying that the eigenvalues must be real valued. I know that the Lagrangian is just a Legendre transformation of the Hamiltonian and so is also hermitian. I am wondering if there is some other reason that the Lagrangian must be hermitian in QFT, this also made me question, why is it necessary that the action be real? am I missing some other underlying principle here?


title: Gaussian integration and dimension argument

text: I made a mistake recently regarding the Gaussian density, by putting the determinant of the variance to the power $\frac{d}{2}$. Would the following argumentation be valid to highlight it should be to the power $\frac{1}{2}$? : The argument of the exponential must be dimensionless, so whatever unit is $x$ in, the variance has dimension $x^2$. $$ dimension( ~{ x^t \Sigma^{-1} x}) = 0 $$ in the density formula, dividing by the determinant yields therefore a quantity with dimension $x^{-2*d}$. $$ dimension~ ( \frac{\alpha}{\det A} ) ~=~ ~x^{-2*d}$$ But since you want a density, that means if you multiply that by a volume (of dimension $x^d$) you get a dimensionless number. $$ dimension ~( \frac{\alpha}{\det A} . dx ^d ~) ~=~ 0$$ So one should raise the determinant to the power $\frac{1}{2}$


title: Photoelectric effect high frequency limit?

text: The photoelectric effect has a low frequency limit below which nothing is observed. Increasing the frequency energy is enough to free an electron. Continuing to increase the frequency the material will start to be more transparent and interact weakly with radiation. My question is in what range is the Einstein photoelectric effect true.


title: Max temperature that can be obtained from radiation source when nonlinear interaction with matter is important?

text: Using a black body radiation as furnace it is usually said that the max achievable $T$ is limited to the T of the source. So a source of 3000K can not be used to heat a body above 3000K. A long wavelength source let say $3000\ \mathrm{K}$ or nearly $1$ micron is considered. Having enough power and using a lens to focus, the intensity at the focus could exceed the binding potential of any core electron in any atom or some at least. The nonlinear interactions are important. So this 3000K source is ionizing radiation in the volume where the intensity is high enough to create "high" temperature plasma. (A small scale plasma experiment similar to the national ignition facility.)   1. What can we say about the entropy and the temperature in that volume? Is the temperature in the system much higher than $3000\ \mathrm{K}$? Let say intensity is high enough to doubly ionize He gas and create "hot" plasma.   2. Is there limit for the max temperature?   3. Once the matter is completely ionized would increase in intensity lead to higher T? My question is not strictly related to lasers but here are few numbers for a laser source showing that the energies are "relatively small" for high nonlinear interactions: For a laser with $1\ \mathrm{mJ}$ at $1000\ \mathrm{nm}$ (3000K) beam diameter $20\ \mathrm{mm}$, $40\ \mathrm{fs}$ pulse duration, using $F=15\ \mathrm{cm}$ at the focus the peak intensity and field are $Ip=6.5 10^{16}\ \frac{\mathrm{W}}{\mathrm{cm}}, E= 7*10^9\ \frac{\mathrm{V}}{\mathrm{cm}}$. This is roughly on the scale what will completely remove the electrons from He.


title: Solutions of damped oscillator differential equation

text: I am reading about damped harmonic motion in my Physics book (Gerthsen Physik) and there are two things that irritate me: # Stokes friction It says that Stokes friction would be $$F = -m \gamma \dot{x}.$$ To me, it does not make any sense why it depends on the mass of the object. It means that the deceleration does not depend on the mass. Or is it just since it is proportional to the volume and the volume usually depends on the mass. # linear combination of solutions Then the differential equation is set up, I fully understand it, except the confusion about the friction. $$ m \ddot{x} + \gamma \dot{x} + \omega_0^2 = 0 $$ With $$\delta = \frac{\gamma}{2}$$ and $$\omega = \sqrt{\omega_0^2 - \delta^2}$$ the solution to the $$x(t) = \exp(\lambda t)$$ approach are $$\lambda = -\delta \pm i \omega t$$. One of the plus or minus variants are a valid solution, but the linear combination of both are all solutions. I understand that. The book then says that the general solution is the combination of real and imaginary parts. Alternatively, one can add a phase angle and have one "compact" solution like this: $$ x(t) = x_0 \exp(-\delta t) \exp(i ( \omega t + \phi ) ) $$ The $\exp(i \phi)$ certainly makes up a linear combination of real and imaginary parts, but how is it a combination like $$c_1 \exp(i\omega t) + c_2 \exp(-i\omega t) ?$$


title: Fluid properties

text: I cannot understand the following sentence, > Without flow , however, shearing stresses cannot exist, and compression > stress or pressure is the only stress to be considered. (from _A Physical Introduction to Fluid Mechanics_ , by A. J. Smits, John Wiley & Sons, 2000) Could someone explain this to me?


title: about satellite is it possible to get signal from two satellite at a time?

text: 1. Is it possible to get signal from two satellites at the same time?   2. If we go far away from a certain country to another, and be able to continue to recieve the same signal?


title: Are Black Holes set to take over the Harmonic Oscillator in the 21st century?

text: A few years ago I attended a talk given by Andy Strominger entitled Black Holes- The Harmonic Oscillators of the 21st Century. This talk, http://media.physics.harvard.edu/video/?id=COLLOQ_STROMINGER_091310 was quite formidable and has had a lasting impression on my physics studies since. I'd consider the talk transformative in my approach. Now, imagine hearing the power of the harmonic oscillator model for the first time before taking an EM or QM class and simply having a spring mass system in mind... Fast forward a few years and consider learning the eigenfunctions of the harmonic oscillator Hamiltonian ${\Omega_n}$ are an orthonormal basis set for $\mathcal{H}=L_2(\mathbb{R^d},d\mu)$. A statement equivalent to: 1) Schwartz Space $\mathscr{S}$ being Fourier invariant 2) The Fourier inversion theorem holding Now, the power of the harmonic oscillator is coming to fruition in what I can solve in physics. Admittedly, there is still much in the field that is beyond me at this point that is well represented by the harmonic oscillator model. This leads me to first ask what are the best known fits for the harmonic oscillator model? However, this question is not the main focus of this post as I am also deeply interested in models extending beyond the harmonic oscillator paradigm. Namely, what are examples of objects currently being modeled as Black holes? It's interesting (for me, hopefully more) to ask this question in the "firewall" era, given that I attended this talk before the now famous AMPS paper. So, in summary I am asking for examples of physical objects modeled by harmonic oscillators, or black holes, relevant in the field today. It would be interesting to compare the usage of the models in light of the theme of Andy Strominger's talk.


title: Electrostatics-Coulomb's Law

text: Coulomb's Law is an experimental result $$F=k_e\frac{q_1 q_2}{d}.$$ How did or What did Coulomb do to verify this law as it is applicable for point charges at rest and in reality charges won't be at rest?


title: Formulas for mass of nucleus and atom. Looking for clarification

text: I've been searching the internet for a formula for nuclear mass which accounts for various interactions present. I've encountered the following formulas: $$m = Zm_p + (A-Z)m_n - \frac{B}{c^2}$$ $$m = Z(m_p+m_e) + (A-Z)m_n - \frac{B}{c^2} -\frac{b}{c^2}$$ $$m = Z(m_p+m_e) + (A-Z)m_n - \frac{B}{c^2}$$ I have a feeling that the $m$'s and $B$'s aren't the same thing in different formulas. In the first one it looks like we're ignoring the electrons, and $B$ appears to be the binding energy between two nucleons. So $m$ is the mass of the nucleus? The second one includes the electrons, as well as $b$ which seems to be the binding energy between the nucleons and electrons? I'm not entirely sure on what $b$ is. The capital $B$ looks like it's the same thing, but $m$ means the mass of the entire atom? The third one is a mystery, it includes the mass of the electrons, but not the little $b$. Perhaps it's an approximation to the second formula? I'm looking for clarification regarding these formulas. What are the conditions regarding their usability? What do the $m$'s and $B$'s stand for?


title: Noether theorem with semigroup of symmetry instead of group

text: Suppose You have semigroup instead of typical group construction in Noether theorem. Is this interesting? In fact there is no time-reversal symmetry in the nature, right? At least not in the same meaning as with other symmetries ( rotation, translation etc). So why we construct energy as invariant of such kind of symmetry, not using semigroup instead? Is is even possible? There is a plenty of references about Lie semigroups - i looks like it is active field of mathematics. Is there any kind of Noether theory constructed within such theories? Could You give any references to, ideally introductory texts?


title: Modelling the movement and jumps of a chalk while drawing a dashed line on a blackboard

text: You probably know that if you try to draw a line using a piece of chalk on a blackboard , under some conditions (for example, $\alpha<\frac{\pi}{2}$ in the picture below) you will have a dashed line pattern instead of a continuous line. My question is : (how) Can you model this special movement of the chalk ,and specially , **find the length of line segments and the distance between them** (which are the _visible_ characteristics of the motion)? Note:gravity is present. (I think it affects the solution, at least in some models) ![the chalk](http://i.stack.imgur.com/jy0C5.png) The pattern: ![enter image description here](http://i.stack.imgur.com/8awdr.png)


title: Motion of rigid body system in absense of work

text: In the absence of work on the system, is there a closed form equation for the motion of a set of constrained rigid bodies (let's say, using Revolute (ie: simple pivot) constraints)? If the bodies are fixed to the ground and subject to gravity, there definitely isn't. For instance, the double pendulum. But if no work is being done (ie: there are no non-constraint forces) does the motion allow a closed form? If so, how do you find it given the initial velocities and configuration of the system? The center of mass of the system would have a linear velocity, for instance, as a consequence of Newton's second law. But the individual bodes themselves could have some very twisty motions. I'm thinking it would have a form similar to the forward kinematics equation for a robot arm. That has the form $[\Theta_1] * (r_1 + [\Theta_2] * (r_2 + \ldots))$, where $r_i$ is the displacement from one link to the next and $[\Theta_i]$ is the rotation transform of a certain link. EDIT: as an example, consider something like: ![frisbee with pendulums](http://i.stack.imgur.com/Eielj.jpg) Are the motions of the central body and all the little pendulums expressible in a closed form assuming there's no gravity or motors or anything?


title: Is there a typo in this modified Lennard-Jones potential?

text: The standard 12-6 Lennard Jones potential is given by $$U(r_ij) = 4\epsilon\left[ \left(\frac{\sigma_{ij}}{r_{ij}}\right)^{12} - \left(\frac{\sigma_{ij}}{r_{ij}}\right)^{6} \right]$$ where $\epsilon$ and $\sigma$ are the parameters. I am reading a paper (Pohorille, A.; Jarzynski, C.; Chipot, C. _J. Phys. Chem. B_ **2010** , _114_ , 10235-10253), and my mind is very cloudy right now. In the paper, in **equation (65)** on page 15 (p. 10249), the authors define a so-called soft-core potential by modifying the original Lennard-Jones potential using parameters $\lambda$ and $\alpha$: $$U(r_{ij}; \lambda) = 4\epsilon\left(1-\lambda\right)^{n}\left\\{ \frac{1}{\left[\alpha\lambda^{2}+\left(\frac{\sigma_{ij}}{r_{ij}}\right)^{6}\right]^{2}}-\frac{1}{\alpha\lambda^{2}+\left(\frac{\sigma_{ij}}{r_{ij}}\right)^{6}}\right\\} $$ Then the paper says: > ... the original Lennard-Jones potential is fully recovered at $\lambda = > 0$. But _I think_ that if I plug $\lambda = 0$ into the equation above, I get: $$U(r_ij) = 4\epsilon\left[ \left(\frac{r_{ij}}{\sigma_{ij}}\right)^{12} - \left(\frac{r_{ij}}{\sigma_{ij}}\right)^{6} \right]$$ which is not the original Lennard-Jones potential. Is there something wrong with my reasoning, or is there a typo in the paper?


title: Mathematical Modelling for Efficiency of PV thermal technology

text: As the efficiency of PV thermal or wind energy decreases with time(from the installment). Is there any mathematical model that gives its relation with time. i.e., whether it is a linear decrease or some other function of time. I need to use the result of this in my research on energy markets.


title: Is instantaneous velocity an abstraction?

text: In introductory analysis, the discussion the derivative emphasizes that while average rates of change are measurable, _instantaneous_ rates of change are a "limiting abstraction". While this makes perfect sense from the formal view of analysis, I wonder how it maps to the view of physisists. Are instantaneous rates of change, such as velocity, only ever inferred from observable average rates of change (measured or imagined over arbitrarily small intervals), or can they be directly observed?


title: WKB approximation to loop diagrams

text: I'm a bit confused with the terminology here. This paper claimed to use WKB method to calculate the usual loop diagrams. Notice that the vertex is approximated by expanding around the classical (on- shell) solution of the massive particle propagator. However this method doesn't look like the usual WKB I know. AFAIK in WKB (eikonal) approximation we are supposed to expand the propagator around the particle limit of the solution, but NOT the classical field limit. So, how do you describe the WKB method in QFT? Is it practical enough to give us quick intuitive calculation on our system? (probably in this sense, the WKB+1loop is the crudest semiclassical approximation we can do -> first order in $\hbar$ and coupling constant)


title: Motivation for maximum Renyi/Tsallis entropy

text: The Conditional limit theorem of Van Campenhout and Cover gives a physical reason for maximizing (Shannon) entropy. Nowadays, in statistical mechanics, people talk about maximum Renyi/Tsallis entropy distributions. Is it just because these distributions are heavy tailed? Is there any motivation (or physical significance) for maximizing Renyi/Tsallis entropies?


title: Conservation of energy in transformers

text: How does energy remain conserved in a transformer if emf is increasing, or decreasing? Does the current decreases to accomodate? Does Ohm's law still hold here? Although we know, Ohm's law is not universal.


What is the smoking gun signature of string theory? Suppose we have a complete and consistent model of quantum gravity with a zero or negative cosmological constant, but all we are given is its complete S-matrix. Using only this information, how do we figure out whether or not this model corresponds to a string theory compactification? If the cosmological constant is positive, what can we use in place of the S-matrix?


title: Mathematically, what is color charge?

text: A similar question was asked here, but the answer didn't address the following, at least not in a way that I could understand. Electric charge is simple - it's just a real scalar quantity. Ignoring units and possible quantization, you could write $q \in \mathbb{R}$. Combination of electric charges is just arithmetic addition: $ q_{net} = q_1 + q_2 $. Now to color charge. Because there are three "components", I am tempted to conclude that color charges are members of $\mathbb{R}^3$. However, I've read that "red plus green plus blue equals colorless", which seems to rule out this idea. I can only think that either:   * red, green and blue are not orthogonal, or   * "colorless" doesn't mean zero color charge (unlikely), or   * color charges don't combine in a simple way like vector addition In formulating an answer, please consider that I know some mathematics (vectors, matrices, complex numbers, calculus) but almost nothing about symmetry groups or Lie algebras.


title: Are orbiting planets an  example of perpetual motion?

text: I understand that perpetual motion is impossible, but I became confused when a friend of mine told me that the orbits of the planets is an example of perpetual motion. Will earth's orbit ever end, and whats wrong with his statement. I'm asking for a simple answer to 'disprove' my friends 'theory'.


title: Crystal visualization software for visualizing lattice with reciprocal vectors drawn in same image

text: I'm looking for a free crystal visualization program, preferably for Linux, that can visualize the common lattice structures in 3D interactively (rotatable with mouse) and draw in the same picture the axes of the direct lattice base vectors. It would be nice if it could also draw the reciprocal lattice and Brillouin zones and Wigner-Seitz cells on top of that. Most importantly, I'd like to have the axes in the same image. The background for this is that I want to easily visualize if my text book says something like "this is the case for NaCl in the [111] direction".


title: Does the AdS/CFT correspondence for thermal states really imply time evolution for evaporating black holes is unitary?

text: You always hear theoreticians proudly proclaim the AdS/CFT correspondence implies time evolution for evaporating black holes is unitary. But if you examine the argument carefully, you find AdS black holes can't completely radiate away because the outgoing Hawking radiation always fall back into the black hole in an asymptotically anti de Sitter geometry. To get a black hole which completely evaporates away, we need a zero or positive cosmological constant.


title: Rayleigh-Taylor Instability dependence on acceleration direction

text: I'm trying to bolster my understanding of the Rayleigh-Taylor instability, and I've gotten stuck on the point of which fluid (more or less dense) is being accelerated into the other. Cases of uniform acceleration make sense (e.g. plane-parallel fluids under gravity; or being accelerated by a piston), but my understanding falters in more exotic configurations. Consider the following: A wind tunnel(-like) setup---without gravity---in which one fluid (density $\rho_1$) is at rest, while another ($\rho_2$) is blown into the first with a constant velocity. Because the two substances are colliding --- they'll both feel an acceleration. But because there is no acceleration between the **reference frames** of $\rho_1$ and $\rho_2$, how do you tell which configuration is stable, and which is not?


title: Why is a paramagnetic material attracted into a magnetic field?

text: I want to start by saying I've seen this topic: Attraction and repulsion of Magnetic materials and its supposed duplicates and it hasn't helped me very much. In the book Introduction to Electrodynamics by Griffiths, it says the following: "In general, when a sample is placed in a region of nonuniform field, the paramagnet is attracted into the field, whereas the diamagnet is repelled away." (section 6.14, p.273, 4th ed.) I don't understand why this should be true. The book states that the reasons for paramagnetism and diamagnetism are due to quantum mechanics, but the book seems to imply there is a classical reason as well. The only thing I can think of is the equation that in a nonuniform magnetic field, $F=\nabla(m \cdot B)$. Now if the $B$ field is getting weaker if you continue in the direction the field is in at that point (and stronger if you go the reverse direction), then for a paramagnetic material, with $m$ lined up with $B$ the direction of $F=\nabla(m \cdot B)$ is into the field, and for a diamagnetic field, $F=\nabla(m \cdot B)$ is away from the field. But I don't see any reason why the $B$ field gets weaker as you continue along the direction of the $B$ field. Any help would be very appreciated.


title: When should one apply the unitary time evolution operator?

text: When is it appropriate to use $\hat U$, the unitary time evolution operator? For example, say I had a system in a certain potential that is changed to a different one at time $t = 0$. Would it be valid to operate $\hat U$ on the new wavefunction (at $t = 0$) to calculate the wavefunction at later time $t > 0$?


title: Transferring CFT correlations from $\mathbb{R}^3$ to $S^3$

text: There seems to be a simple method to transfer a CFT's correlations from $\mathbb{R}^3$ to $S^3$ but I am not understanding why it is supposed to work. The idea is that somehow because, $ds^2_{S^3} = \frac{4}{(1 + \vert \vec{r} \vert^2 )^2}ds^2_{\mathbb{R}^3 }$ it follows that all that one needs to do is that replace all occurences of $\vec{r} - \vec{r}'$ by $\frac{2(\vec{r} - \vec{r}') }{\sqrt{ (1 + \vert \vec{r} \vert ^2 )(1 + \vert \vec{r}' \vert ^2 ) } } $   * Why is this supposed to work?    * What is the necessary relationship between the two CFTs for this to work? (like if one uses this on the free scalar field theory then it will turn out that w.r.t the Laplacians on $\mathbb{R}^3$ or $S^3$ the two 2-point functions satisfy _different_ P.D.Es respectively and hence they come from two different Lagrangians - since we anyway knew that the Lagrangian for the conformally coupled scalar on the flat space is not the same as the Lagrangian for the conformally coupled scalar on $S^3$)   * Is it part of some general idea that would work between other pairs of manifolds?


title: A quanta of time

text: A question of Quantum Time: Does a minimum interval of time cause wave-like behavior? If we think about the uncertainty principle, could it derive from a quanta of time? Does plank’s constant somehow derive from the quanta of time? Hypothesis: When delta t is larger than a quanta of time, then particles behave like particles. When delta t tries to be smaller than a quanta of time, then we have uncertainty because delta t cannot be smaller than a quanta of time. So when the delta t is smaller than a quantum of time, we don’t have the granularity in time to be certain of momentum, or location of a particle at a later time smaller than a quanta. Interference waves seem to be caused by particles that compress to the point where they influence each other. E.g. an atom of H2O behaves like a particle until it is organized with many atoms of H2O at a certain point of density where the atoms all influence each other. At this point the action/reaction caused be adjacent contact creates an interference wave of movement over time. Another way to say this is that a certain minimum level of distance between two particles, the unit of distance cannot be condensed further, and interference waves are generated. Could this happen with time as well. Could it be that that time can only move in a quanta, and this causes the interference wave? Additionally, could this help explain the limit of the speed of light? If time has a minimum value (a quantum), then how does an observer measure speed for distances traveled faster then the minimum value of time?


title: Electric force on spherical surface

text: I have a doubt about electrical forces on surfaces, for instance, on the surface of a sphere. I'll explain my point: let's say we have some spherical surface of unit radius and there's one point charge over it. My idea then is to represent the surface by the unit sphere $S^2\subset\mathbb{R}^3$ and then I say that on point $a\in S^2$ there's some charge $q$. That's fine, then I add some other charge $q_1$ at another point $a_1$ and I want to know what's the electrical force on $q_1$ due to $q$. Since we're thinking of the sphere embedded in $3$-space it's fine to thake the vector $v=a_1-a$ and say that the distance is the euclidean norm of this vector. Then I use Coulomb's law and everything works fine. But my thought is: both charges are on the sphere, so it seems to me that we could really forget about the ambient space and just use intrinsic measures, for instance, calculate the distance given by the geodesic that joins the points $a$ and $a_1$. The point is, this gives rise to another distance, one that doesn't depend on the ambient space. So, this distance works also to calculate the force ? If not, is there a reason to depend on the ambient space ? Because for me it seems that everything is just happening on the sphere, without connection to the remaining of $3$-space. Sorry if that's just a bunch of nonsense, it's just that I really don't know how to figure this out by myself.


title: Time to emit electrons

text: Explaining the photoelectric effect, our teacher told us that back in that time, they calculated the time that an electron would take to be emitted from an atom if the energy got to the atom in a continuum way. We calculated it for potassium and a source of 1 Watt at 1 meter, giving approximately a couple of minutes. He told us then that the real time could not be calculated because of how small it was, being it $t<10^{-9}$ seconds. I don't know if he meant they couldn't do it back then, or it's still impossible nowadays becuase of how small that time is? Because that number $10^{-9}$ looks big to me having in mind the times they were able to (incorrectly) measure at the CERN when all that neutrino stuff happened. Is there a smaller bound?


title: Quantum computing problem

text: Suppose that a qubit is in the state $|\varphi\rangle=a|0\rangle+\sqrt{1-a^2}|1\rangle$, where $a\in[-1,1]$. If we first perform a standard basis measurement on this qubit and then perform a $\\{|u\rangle,|u^\perp\rangle\\}$-basis measurement, where $$|u\rangle=b|0\rangle+\sqrt{1-b^2}|1\rangle \text{ for some }b\in[-1,1],$$ what is the probability that the outcome of the second measurement is $|u\rangle$, in terms of $a$ and $b$? Edit: I've found a solution, it should be $ab+\sqrt{(1−a^2)(1-b^2)}$.


title: X-ray diffraction (is it possible?)

text: Is it possible to extract full lattice structure of a crystal from any of X-ray diffraction methods? (especially that the diffraction pattern is 2-Dimensional most of times)


title: Intervals as infinitesimals of same order (Landau & Lifshitz)

text: I don't understand the following statement in Landau & Lifshitz, _Classical Theory of Fields,_ p.5: > _$ds$ and $ds'$ are infinitesimals of same order. [...] It follows that > $ds^2$ and $ds'^2$ must be proportional to each other: $$ds^2 = a \, > ds'^2.$$._ I don't get why the proportionality applies, and why does it apply to the squares of the infinitesimals.


title: Killing Vectors of BTZ black hole and their calculation in general

text: I was wondering what are the Killing vectors of BTZ black hole and how to guess them easily? Will it be the same as of AdS? What then will be Killing vectors for AdS-Schwarzschild e.g.?


title: Charge gained due to photoelectric effect

text: ![enter image description here](http://i.stack.imgur.com/PmoZO.png) Here I think, one beam will knock out just one electron. So, I am not able to even understand what the question says. Please someone give a hint as to what the question asks... As source of the question isn't reliable, it may be wrong too (?)


title: Translator Operator

text: In _Modern Quantum Mechanics_ by Sakurai, at page 46 while deriving commutator of translator operator with position operator, he uses $$\left| x+dx\right\rangle \simeq \left| x \right\rangle.$$ But for every $\epsilon > 0$ $$\langle x+ \epsilon \left| x \right\rangle = 0.$$ Therefore this limiting process $$\lim_{\epsilon \rightarrow 0} \left| x+ \epsilon \right\rangle = \left| x \right\rangle$$ does not make sense for me. I couldn't derive commutator relation without using these.


title: Is there a theorem that says that QFT reduces to QM in a suitable limit? A theorem similar to Ehrenfest's theorem?

text: Is there a theorem that says that QFT reduces to QM in a suitable limit? Of course, it should be, as QFT is relativisitc quantum mechanics. But, is there a more manifest one? such as Ehrenfest's theorem which dictates the transition from QM to Classical Mechanics. I am curious as the basic object in QM is wave function whereas the basic object in QFT is quantized operator, which looks very different, and wondering how the comparison works. (Of course, I am aware that QFT can calculate corrections for Coulomb potential energy, which may seem to be a transition from QFT to QM, but I want one more manifest than that.)


title: Poisson structure on moduli space of CFTs

text: The moduli space of CFTs with central charge 26 forms the classical phase space of bosonic string theory, in some sense. Similarily the moduli space of SCFTs with central charge 10 forms the classical phase space of type II superstring theory We might expect this should make these moduli spaces symplectic, or at least Poisson (super)manifolds. Is there such structure on them? More generally, the only geometric structure on CFT moduli spaces I encountered is the Zamolodchikov metric. What other structures do they have?


title: What would happen if some signal could move faster than light?

text: The two postulates of STR doesn't say that any signal cannot move faster than light. It also doesn't assert that any signal except light cannot have velocity equals to that of light. So at the very beginning of analysis of STR we cannot say anything about the argument of "light having the ultimate(fastest) speed". Infact STR shows that we cannot accelerate a material particle to achieve and exceed the speed of light but we cannot just deny the existence of particles which could be generated similar to photons which at their creation can have speed equals to or greater than $c$ because they have achieved this speed at their birth they are not been accelerated to achieve this speed. So Suppose there exists a hypothetical signal named schite having velocity=infinity but the speed of light is still same for all the inertial observers "would the events A and B be dissimultaneous in train's frame to validate constancy of speed of light in the Einstein's rail thought experiment"? Moreover could we use this schite signal to synchronize clocks? In other words would time dilation and length contraction exist? * * * I have heard little bit about tachyons. It is said using tachyons we can transmit signals at higher speed than $c$ if (assuming hypothetically) such particles exists can we use them to synchronize clocks?


title: Is string theory local?

text: By locality I mean something like the Atiyah-Segal axioms for Riemannian cobordisms (see e.g. http://ncatlab.org/nlab/show/FQFT). I.e. to any (spacelike) hypersurface in the target we associate a Hilbert space and to any cobordism an S-matrix. I am familiar with the S-matrix prescription for the target being $\mathbf{R}^n$ and the hypersurfaces being the asymptotic time infinities. Can one extend that to any cobordism? Does locality appear only when we integrate over the worldsheet conformal structures and sum over all genera, or can we see it even for a fixed conformal structure? I believe this is what string field theory is about, but why would one expect locality from the (perturbative) string theory point of view?


title: Can an induction coil heat two layers of metal?

text: Imagine we have an induction coil which is strong enough to heat a sheet of metal. We can put a sheet of ferromagnetic metal close to the coil at distance $h_1$, and it gets heated to temperature $t_1$, or at distance $h_2 > h_1$ so that the sheet gets heated to temperature $t_2 < t_1$. I want to know what happens if we have two identical sheets at once, one at distance $h_1$ and one at distance $h_2$, on the same side of the coil, with some insulator between the sheets (the insulator does not conduct electricity, is not ferromagnetic, and does not conduct heat well). Will the sheet farther away from the coil heat up at all, or will the closer sheet shield it from the electromagnetic field in some way? What will the temperatures $t_1'$ and $t_2'$ of the sheets in this experiment be, higher, lower or the same as $t_1$ and $t_2$? Does the answer to the above change if we have a small conducting connection between the two sheets of metal, e.g. a wire which touches both the close and the far one, but most of their surface is still separated? The application for this question: I am thinking of getting a cast iron waffle iron to use on my induction stove, and I am trying to imagine how this will function. By the way, I know that I will get some heat conducted through the waffle itself, and I will probably turn it anyway so both plates get hot, but please ignore these effects when answering the question and tell me the effects of induction only.


title: Recommendations for Statistical Mechanics book

text: I saw Book recommendations   No reference to Statistical Mechanics there. I learned thermodynamics and the basics of statistical mechanics but I'd like to sit through a good advanced book/books. Mainly I just want it to be thorough and to include all the math.   And of course it's always good to give as much intuition about the material. Some things I'd be happy if it includes (but again it mostly just needs to be a good clear book even if it doesn't contain these) are:   1. As much justifications for the postulates if possible, I'm very interested in reading more about how Liouville's theorem connects to the postulates   2. Have examples of calculating partition functions, hopefully not just the partition function for the ideal gas. Thank you


title: Statistical mechanics prerequisite

text: I`ll be taking senior level and graduate course in statistical mechanics in a month. I was wondering what would be the best statistics and probability textbook to prepare for it? I`m currently reviewing classical mechanics and thermodynamics. I forgot quite a bit. Should I focus on a book that teaches mainly probability or both statistics and probability? I'm getting flags for this being a possible duplicate. I already have the statistical mechanics book I'll be studying from. But it doesn't have a decent enough review of statistics and probability which is what I'm trying to prepare for.


title: Uncertainty in path integral formulation

text: In Feynman's path integral formulation, in order to calculate the probability amplitude, we sum up all the possible trajectories of the particle between the points $A$ and $B$. Since we know precisely that the particle will be at $A$ and $B$, does it mean that the uncertainty of the momentum is infinite?


title: What is the effect of the compact extra dimensions of the heterotic theories?

text: The five super-string theories are generally said to be 10-dimensional. However the Heterotic theories combine a bosonic left mover (which lives in 26 dimensions) with a super right mover (which lives in 10 dimensions). The extra 16 dimensions compactify into a torus with either $\frac{\operatorname{Spin}32}{\mathbb Z_ 2}$ or $E_8\times E_8$ symmetry. So these theories differ from the others by this compact torus. What is the effect of the this torus physically?


title: Blocking electro-magnetic waves between two near field antennas

text: Its been a while since I studied physics, so pardon my ignorance. I have two near field antennas. The range of the antennas is about 2-3 cm. I am looking for a way where I can block and unblock the communication between them. One of the way that I tried is to place a flat tightly woven wire and if I short the two ends of the wire, I was able to block the communication. My assumption is that it is effectively changing the frequency of the signal and thus blocking the signal. Is that what is happening? However, if the radio signal has good strength, the coil is not doing a good job. Is there any other better solution that I can achieve blocking/unblocking with electrical/electronics (without mechanical parts)?


title: Movement of man and ladder and their center of mass

text: ![pulley,man](http://i.stack.imgur.com/UJhDi.png) Suppose there is a massless frictionless pulley. A rope over it carries a mass $M$ and on other side carries a ladder of mass $(M-m)$ and a man on that ladder, of mass $m$. Now the man starts climbing the ladder at a velocity $v$ upwards($v$ in ground frame). The ladder and the block also move, but I cannot happen to imagine his hands going down on the ladder legs and the ladder going down in the ground frame. I need to somehow get to the movement of center of mass. My view is that since momentum should be conserved(he is climbing the ladder with distinct downward forces that span over very small time, say $dt\rightarrow 0$, so impulse should be $0$). This means that $v_{cm}$ should be $0$, since its expression has total momentum of system in the numerator, which was initially $0$ and hence finally will be $0$. This means the center of mass should not change, which somehow seems absurd, as the man is moving up. Could anyone make clear what goes down and whether $y_{cm}$ changes or not? (And yes cm means center of mass.)


title: Forbidden trajectories in path integrals

text: In Feynman's path integral formulation we add all the possible trajectories of a particle to get the probability amplitude. What are forbidden trajectories? Not differentiable? Is this related to the forbidden states of Pauli exclusion?


title: Bondi's k factor

text: Consider two observers, travelling away from each other, after meeting (at which time they sync their clocks). $O$ sends a photon towards $O'$ at times $t$, received by $O'$ at time $t'=kt$, where we have defined Bondi's k factor. In the following reasoning I know I have made a mistake: So by Milne's radar definition of simultaneity, if the photon is instantly sent back from $O'$ and received by $O$ at $t_2$, then: $$t'=\frac12(t+t_2)=kt\Rightarrow t_2=2t'-t=(2-\frac{1}{k})t'$$ But $t_2$ is meant to equal $kt'$ Could someone explain to me where I have I gone wrong here? EDIT: Basically I am trying to show that $t_2=kt'$ working from only the two assumptions that: (i) they both observe the speed of light as c (ii) Only relative motion is observable ![enter image description here](http://i.stack.imgur.com/4Ct1t.jpg)


what is the relationship of sound volume to atmospheric pressure? if I was in a plane with a cabin pressure equal to 8000 feet would the volume drop be noticable?


title: How come random matrices can predict energy spectra of heavy atoms?

text: Some of the applications of random matrices is to find the spectra of heavy atoms in nuclear physics which are usually difficult to find otherwise. How can starting from randomness of some kind, such as random matrices, lead to prediction of that sort? Any idea why this work? Random matrices are used in string theory too, may be some string theorist here can shed some light on this topic Revo


title: Simple model of edge states for a two-dimensional topological insulator

text: Quantum spin Hall states or, topological insulators are novel states of matter that have insulating bulk and gapless edge states. Are there any simple models that show these features? See e.g. the article > Xiao-Liang Qi and Shou-Cheng Zhang, _The quantum spin Hall effect and > topological insulators._ The pdf file is available here. They provide an effective Hamiltonian in the box on page 36 for a mercury telluride topological insulator. It's easy to find the eigenstates and dispersion. How do you find the gapless edge states as shown in fig. 2b of the same paper?


title: Can the distance of a quasar be determined accurately?

text: As noted in A doubt about the age of the universe, the wiki about quasars still contains the following misleading sentence: "The highest redshift quasar known (as of June 2011) is ULAS_J1120+0641, with a redshift of 7.085, which corresponds to a proper distance of approximately 29 billion light-years from Earth." But even if the "strange 29 billion" is replaced by the "correct 12.9 billion", the fact remains that the actual measurement is "a redshift of 7.085". The "proper distance" is only a different way to express that measurement. It's not clear to me how "accurately" this describes the distance of the quasar, because the quasar surrounds a black hole and rotates quite fast. So there are at least two additional sources for the redshift, but how significant is their contribution?


title: Relativistic Lorentz force law

text: If we consider the the relativistic Lorentz force law: $$\frac{d}{dt} (m\gamma \vec{u})=e(\vec{E}+\vec{u} \times \vec{B})$$ How can we deduce: $$\frac{d}{dt} (m\gamma c^2)=e \vec{E} \cdot \vec{u}$$ Clearly dotting with $\vec{u}$ will give us the RHS. Which leaves us: $$\vec{u} \cdot \frac{d}{dt} (m\gamma \vec{u})=e \vec{u} \cdot \vec{E}$$ Could anyone help explain how to proceed and if this is the correct method? EDIT: If it helps: with reference to these notes i'm working through: http://www.maths.ox.ac.uk/system/files/coursematerial/2012/2393/8/WoodhouseLectures.pdf Page 86, eq (178), the paragraph underneath states 'The ﬁrst equation (which follows from the second)', this is what i'm trying to prove (a warning, the notes are riddled with errors..).


title: Hyperbolic, parabolic, elliptical PDE related to under-, critical- and overdamped in harmonic osciallation

text: A damped harmonic oscillator has three cases for the damping: underdamped, critically damped and overdamped. With partial differential equations, I know the hyperbolic wave equation, the parabolic heat equation and the elliptical Laplace equation. Since underdamping means oscillation and therefore waves and the heat diffusion looks like a critically damped fluid, I was wondering, if those have anything in common.


title: Probability in Quantum Mechanics

text: Do you need to take a probability/statistics course for Quantum Mechanics, or is the probability in quantum mechanics so rudimentary that you can just learn it along the way? I'm in doubt as to whether I should take the course or not, is it needed? What probability is actually in QM? What I know already: Arithmetic mean, median, mode, sample & population variance, standard deviation, discrete and continuous random variables, binomial, poisson and gaussiand distribution and finally the expected value. If you would like to add things I need which aren't listed, that would be wonderful. I'd hate taking the course knowing almost everything I need apart from a couple of things.


title: Can temperature be defined as propensity to transmit thermal energy?

text: I was recently surprised to learn that defining temperature isn't easy. For a long time, it was defined operationally: how much does a thermometer expand. Also surprising, temperature isn't a measure of thermal energy: 0 C water has more thermal energy than 0 C ice, but the same temperature. I came up with this definition, and I'd like to know if it's correct: Temperature is the propensity of something to give off heat, that is transmit thermal energy. Equivalently, it is the resistance of something to receive heat. Is that correct? Please justify your answer.


title: What is the highest energy position for a double pendulum? And for which energy positions is it chaotic?

text: Math/physics teachers love to break out the double pendulum as an example of chaotic motion that is very sensitive to initial conditions. I have some questions about specific properties:   1. For a simple double pendulum (which I'll define as two equal masses joined by two massless arms of the same length) is the highest initial energy state when both masses are completely raised vertically, as intuition would suggest?   2. http://en.wikipedia.org/wiki/Double_pendulum#Chaotic_motion contains a plot of conditions for which the double pendulum "flips" or does not flip. My question is: does a "flipping" of either top or bottom pendulum imply subsequent chaotic motion, or are there certain energies for which we can fully predict the motion and flips? Conversely, if we are in low energies where neither top nor bottom pendulum can possibly flip, can we still have a chaotic system or do all those energies have predictable motions?


title: Calculating Moment Of Inertia

text: The problem I am working on is: > _A uniform, thin, solid door has height 2.10 m, width 0.835 m, and mass 24.0 > kg._ > > _(a) Find its moment of inertia for rotation on its hinges._ > > _(b) Is any piece of data unnecessary?_ I am not really certain of what I am doing at all. So, here is my attempt, following the outline given in the book. I know from the book that the height isn't necessary to know for this problem, but I am not really certain why. As for part (a), I don't really know how to approach the problem. Do I need calculus techniques? Is it one of the integrals that I am suppose to use?


title: Where does the $10^{500}$ estimate for the number of stringy vacua come from?

text: This number is thrown around a lot so I'd like to understand its origin. I know that it counts compactifications of string theories on (not only) Calabi- Yau manifolds and probably also some other ingredients. > 1. What precisely are those ingredients that differ among the stringy vacua? But as far as I know no one knows the precise number of Calabi-Yau 3-folds (it might still well be infinite). Is the estimate based on the number of presently found 3-folds? > 2. Where do the estimates come from?


title: why can't the missing mass (evidence for dark matter) be matrioshka brains/dyson spheres?

text: i know that the claim is that nucleosynthesis models say that the observed abundance of deuterium is too high for dark matter to be baryonic, but couldn't deuterium be a waste product of civilizations? you say that postulating civilizations is not the simplest explanation, but isn't it far more complex to postulate that we are unique (or an invisible form of matter with exotic physics for which we have no evidence)? even from a single seed civilization, galactic colonization times are in the hundred million year range, so we should expect some evidence of environmental manipulation everywhere we look. given the need for civilizations to hide from predators, isn't eliminating EM interactions/emissions the best strategy?


title: Why are the classical electron radius, the Bohr radius and the Compton wavelength of an electron related to each other?

text: Using the definition of the fine-structure constant $\alpha = \frac{4 \pi \epsilon_0 \hbar c}{e^2}$ and the Compton wavelength of an electron $\lambda_c = \frac{h}{m_e c}$ the classical electron radius $r_e$ and the Bohr radius $a_0$ can be expressed like $$r_e = \alpha \frac{\lambda_c}{2\pi}$$ $$a_0 = \frac{1}{\alpha} \frac{\lambda_c}{2\pi} $$ This means e.g. that the classical electron radius can be expressed in terms of the Bohr radius as $r_e = \alpha^2 a_0$. Isn't that peculiar? Why should the classical radius of the electron and the distance of an electron to the nucleus in an atom be related to each other? And why are both multiples of the Compton wavelength?


title: Information faster-than-light and GR vs. QM

text: 1. What is meant by the statement that information cannot travel faster than light?   2. If I write down something on a paper, isn't there according to QM a non-zero probability that an identical paper can spontaneously fluctuate into existence a lightyear away?    3. Can't a person successfully guess every character that I write, every time?    4. Is this why QM and GR are incompatible?    5. If not, which principle or axiom of QM must be modified to be compatible with GR?    6. Is there any difference between ordinary particle travel and particles fluctuating in and out of existence along what looks like the world-line of a particle?   7. Other than different probabilities?   8. Can a Ferrari spontaneously fluctuate into existence according to QM?    9. What happens then to Energy conservation?


title: At what rate does a rotating black hole lose mass via Hawking Radiation?

text: I initially thought it was inversely proportional to the mass, but I think that's wrong because temperature is inversely proportional to mass. If someone could give the formula(s) for finding this that would be fantastic. I've searched around and just found stuff about Swartzchild radius etc.


title: Historic derivation of Wien's law

text: Every book I've read, including a lot of websites, Wikipedia, etc, say that Wien derived this: $$\rho_\nu(T)=\rho(\nu,T)=\nu^3f\left(\frac{\nu}{T}\right)$$ Being $\rho_v(T)$ the spectral enegy density of a black body for a given temperature and electromagnetic wave frequency. And everywhere it's mentioned that he proved this using thermodynamical arguments in a paper from 1893. I haven't been able to find that paper or that thermodynamical argument, which is what I'm interested in. I've been looking for a few days already. Does anybody know how he did this?


title: Trouble with the Lorentz law of force: Incompatibility with special relativity and momentum conservation?

text: In Physical Review Letters, there was a paper recently published: > Masud Mansuripur, _Trouble with the Lorentz Law of Force: Incompatibility > with Special Relativity and Momentum Conservation,_ Phys. Rev. Lett. 108 > (2012) 193901, arXiv:1205.0096. The paper considers the case of a stationary point charge at a fixed distance from a stationary magnetic dipole in some frame, stating that the magnetic dipole neither experiences a torque nor force from the stationary charge. Upon transforming to another frame, they argue that the magnetic dipole is transformed into a magnetic dipole and electrical dipole. This combination will therefore experience both an electrical force, and magnetic torque from the field of the moving charge. The authors therefore claim that there is a contradiction and therefore a problem with the consistency of the Lorentz force and special relativity. Is the paper correct, or has the author overlooked something?


title: Finding the Electric Field (and other information, besides)

text: The problem I am working on is: > Two parallel plates having charges of equal magnitude but opposite sign are > separated by 29.0 cm. Each plate has a surface charge density of 33.0 nC/m2. > A proton is released from rest at the positive plate. > > (a) Determine the magnitude of the electric field between the plates from > the charge density. > > (b) Determine the potential difference between the plates. > > (c) Determine the kinetic energy of the proton when it reaches the negative > plate. > > (d) Determine the speed of the proton just before it strikes the negative > plate. > > (e) Determine the acceleration of the proton. > > (f) Determine the force on the proton. > > (g) From the force, find the magnitude of the electric field. So far, I've only attempted to solve part (a). For part (a), seeing as they didn't provide the dimensions of the plates, I assumed them to be infinite-sized; furthermore, they both have to have the same area (although, I can't justify why this is so). So, for plate 1, which we'll consider positive: $\sigma = 33.0 \cdot 10^-9 C/m^2$ Since they don't say otherwise, that the plates do not have uniform charge distribution, we'll assume that the charge is in fact uniformly distributed; this is reasonable, seeing as they give one surface charge density for the entire plate. $\sigma = \frac{dq}{dA} \rightarrow dq = \sigma \cdot dA \rightarrow \int dq = \sigma \int dA \rightarrow q_1 = \sigma \cdot a^2$, $q_1$ being the charge of the entire plate. A similar process can be done for the second plate, which we'll regard as negative: $q_2 = \sigma \cdot a^2$ Hence, $q_1 = - q_2$ This is the only way of starting that I could think of; and as you may see, it wasn't very helpful. How should I approach this problem? * * * As per Michael Brown's suggestion, should I generate a Gaussian plate, infinite in size, and place it between the two sheets? Then, find the magnitude of the electric field from sheet, and multiply it by two?


title: Does the Standard Model have a Landau pole?

text: I have seen the statement that the Standard Model has a Landau pole, or at least it its believed that it does at $\sim 10^{34}$ GeV. Has this actually been proven (at least in perturbation theory, as in QED) or what kind of evidence is there to support this?