What would be likely to completely stop a subatomic particle assuming it was possible?

Suppose that completely stopping a subatomic particle, such as an electron, could happen under certain conditions. What would be likely ways to get an electron to be perfectly still, or even just stop rotating the nucleus and collapse into it by electromagnetic forces? What would likely be required, below absolute-zero temperatures? Negative energy? Or could a 0 energy rest state not exist in any form, of any possible universe imaginable? Let's say there was a magnetic field of a certain shape that we could postulate that is so intensely strong that if we put an electron in the center of it, it could not move at all in any direction. Would the energy requirement of the field be infinite? What would be the particle's recourse under this condition? Further, suppose it were possible and one could trap an electron and stop all motion completely. What would this do to Heisenberg's Uncertainty Principle and/or Quantum Mechanics, because its position and momentum (0) would both be known? If it can be done, is Quantum Mechanics no longer an accurate model of reality under these conditions? Could we say QM is an accurate model under most conditions, except where it is possible to measure both position and momentum of a particle with zero uncertainty? # Clarification: Please assume, confined in a thought experiment, that it IS possible to stop a particle so that is has 0 fixed energy. This may mean Quantum Mechanics is false, and it may also mean that under certain conditions the uncertainty commutation is 0. ASSUMING that it could physically be done, what would be likely to do it, and what would be the implications on the rest of physics? # Bonus Points Now here's the step I'm really after - can anyone tell me why a model in which particles can be stopped is so obviously not the reality we live in? Consider the 'corrected' model is QM everywhere else (so all it's predictions hold in the 'normal' regions of the universe), but particles can be COMPLETELY stopped {{inside black holes, between supermagnetic fields, or insert other extremely difficult/rare conditions here}}. How do we know it's the case that because the uncertainty principle has lived up to testing on earth-accessable conditions, that it holds up under ALL conditions, everywhere, for all times?


How to learn physics?

I am an engineering student (CSE) in India..But recently I have developed a strong love to physics..I want to learn physics and understand it in deep..I know physics is the search of deep fundamental laws of nature.That means I need to start from first principles and extrapolate from it?How should i learn physics?


Is it more efficient to stack two Peltier modules or to set them side by side? And why? I have a small box that I want to cool down about 20 K below ambient -- cold, but not below freezing. (I want to keep my camera cool, so I'm putting in this cool box. The camera looks through a flat glass window on one side of the box). The heatsink I have on hand is about twice as wide as the widest Peltier module I originally planned on using. So there's room to put 2 Peltier modules side-by-side under the heatsink. Or I could center a stack of 2 Peltier modules under the heatsink. Which arrangement is more efficient? I have to cut a bigger hole in the insulation for the side-by-side arrangement, so the unwanted heat that "back-flows" through the side-by-side arrangement is worse. On the other hand, other effects are worse for the stacked arrangement. (Is http://electronics.stackexchange.com/ a better place to post questions about Peltier coolers?)


Identical fermions in the same quantum state

If we are to take two Hydrogen atoms and subject them to the same potential, then wouldn't both Hydrogen atoms be in the same exact quantum state? This bother me because no two identical fermions can be in the same quantum state! This seems to contradict the principle. This applies to any two elements or molecules that are subjected to the same potential. Say these two Hydrogen atoms are located 1m from each other, then would the only way to distinguish them would be their spatial location? What is the technical term for two seemingly identical things to be distinguishable by their location?


Is the firewall paradox really a paradox?

The firewall paradox is a very hot topic at the moment (1207.3123v4). Everyone who is anybody in theoretical physics seems to be jumping into the action (Maldacena, Polchinski, Susskind to name a few). However, I am unable to see the paradox. To me Hawking's resolution of the information paradox (hep-th/0507171) also resolves the so called firewall paradox. Hawking never says that the information is not lost on a fixed black hole background. In fact, he says the opposite. He says (page 3): "So in the end everyone was right in a way. Information is lost in topologically non- trivial metrics like black holes. This corresponds to dissipation in which one loses sight of the exact state. On the other hand, information about the exact state is preserved in topologically trivial metrics. The confusion and paradox arose because people thought classically in terms of a single topology for spacetime." To surmise, in quantum gravity you don't know if you actually have a black hole or not, so you have to include the trivial topologies, including those when there isn't a black hole there, in the amplitude. Only then do recover unitarity. It seems to me that the error of the AMPS guys is that they use a fixed black hole background and assume conservation of information (i.e., late time radiation is maximally entangled with early time radiation). It is no wonder they are lead to a contradiction. They are simply doing the information paradox yet again. They give a menu of implications in the abstract: (i) Hawking radiation is in a pure state, (ii) the information carried by the radiation is emitted from the region near the horizon, with low energy effective field theory valid beyond some microscopic distance from the horizon, and (iii) the infalling observer encounters nothing unusual at the horizon. But the obvious solution is that (i) is wrong. The radiation, within the semi- classical calculation in which they calculate it (i.e., not quantum gravity), is non-unitary. So my question is, why is this a paradox? Something so obvious surely can not be overlooked by the ``masters'' of physics. Therefore I'd like to hear your opinions.


Why do we require manifolds to be a topological space?

Roughly speaking, we define a manifold $M$ to be covered by a set of charts $\\{(U_i , \varphi_i)\\}$ such that locally the $n$-dimensional manifolds looks like $\mathbb{R}^n$. One of the conditions is that all the $U_i$ are open sets of the topology of the manifold. Why do we require the manifold to be a topological space? And why do we want $U_i$ to be open sets? What are the implications of these requirements on physics? (It appears to me that without these conditions the manifold still looks locally like $\mathbb{R}^n$.) Edit to make my question more concrete: are there any physical theories that use manifolds that are not topological spaces? For instance, what would happen to general relativity if the spaces are diffeomorphic to each other, instead of homeomorphic (see answer below by Robin Ekman)?


Current Density

I'm trying to understand the definition given on my electromagnetism course for the current density. More specifically, I want to know why, as defined below, the current density is given the name "current density." On my course, the current density is $\vec{j}(t,\vec{x}):=\rho (t,\vec{x}) \vec{v} (t,\vec{x})$ where $\vec{v} (t,\vec{x})$ is the velocity field governing how the charged particles move. I'm trying to get some intuition for what this quantity is. To give an example of what I'm talking about, in classical mechanics where you have momentum equal to mass multiplied by velocity, the definition makes sense intuitively because momentum is the oomph you will feel if an object hits you, and you feel that oomph more if either the mass or velocity of the object increases. So I have a really tangible idea of what momentum is. Wikipedia describing current density: "In electromagnetism, and related fields in solid state physics, condensed matter physics etc. current density is the electric current per unit area of cross section." This justifies calling it a density (as it's an area density by defn.). I'm trying to understand what a current density could be, and in my head I've got an idea of a cross-sectional area with some fluid flowing through it (it's the same scenario in which I picutre Gauss' Law). I'm not sure what current at a point is, so I don't really understand what a current density could be! Then I need to relate this to the definition I've been given on my course somehow. Thanks for any help!


Resistances connected in series

When resistances connected in series then why through each resistances the different potential difference occurs?


If we connect a source of electricity in a large water body ,will it be dangerous?

I was wondering whether a person will be electrocuted if he is in large pool or a sea which is connected through a source of electricity . As sea water is saline it will conduct electricity so will it be dangerous !?


Why Aren't Saturn's Rings Clumping into Moons?

While reading with my son about how a Mars-like planet collided with the early Earth that resulted in our current moon, it said the initial debris also formed a ring, but that ring ended up getting absorbed by the Earth and the Moon. I couldn't answer his question then why Saturn still has rings. Shouldn't Saturn's rings be clumping into Moons or getting absorbed by Saturn's gravity?


Ghost fields in particle physics

In my particle physics lecture, ghost fields were briefly mentioned. As far as I understand, these come up when computing cross sections by the path integral method, to compensate for equivalent contributions due to gauge freedom. I'm still not sure how these equivalent contributions come up in the calculations. Say you want to compute the cross section of an electron anti- electron annihilation into a photon near a massive nucleus. I assume you would then take the sum of the path integrals over all possible Feynman diagrams corresponding to that interaction. QED has charge symmetry (i.e. $U(1)$), so the physics stay the same if the two charges are swapped. Where concretely do we have to introduce these ghost fields, and why can we not simply combine the Feynman diagrams that are graph-isomorphisms of each other into a single equivalence class and count the contribution of each equivalence class only once into the final probability amplitude. Please keep the answers simple, as I'm not a physics major.


Classical Mechanics & Coordinates

1. What is the meaning generalised coordinates in Classical Mechanics?   2. How is Lagrangian formalism different from Hamiltonian formalism?   3. How are they related to Hamilton's Principle?   4. How are they related to Euler-Lagrange equation?


Quantum Mechanics, Uncertainty Principle-- help understanding notes

There is a section of my notes which I do not understand, hopefully someone here will be able to explain this to me. The notes read (after introducing the uncertainty operator): > > If the state $\chi_A$ is an eigenstate of $\hat O_A$ then the uncertainty > is zero and we measure it with probability 1. However, if $\hat O_B$ is > another observable which does not commute with $\hat O_A$, then the > uncertainty in any simultaneous measurement of the two observables will be > infinite. I understand the first sentence, but I can't see how to justify/prove the second one. Can someone tell me how the second sentence is justified, please?


If we suddenly lost track of time, how would we know what time is now?

If we hypothetically lost all watches and all devices that keep track of time, how would we say what is the current time? Or we actually don't and time is just a convention?


Is my boss wrong about our mechanical advantage from our pulley system?

I work on a drilling rig as a roughneck and we had a lecture today (at the office) about mechanical advantage in pulley systems. Now, I know that my boss is well educated in oil drilling, but my instincts tell me that he may have this one wrong. A drilling rig works sort of like a crane in that it has a tall structure supporting a pulley system. There is a large winch permanently installed on the base platform and then it goes over the top of the structure (the crown of the derrick) and down through a floating sheave--this has a few wraps to give us more mechanical advantage. I am including pictures to help describe the situation. ![Floating Sheave \(called the "blocks"\)](http://i.stack.imgur.com/naok2.jpg) Here the picture shows the floating sheave (the blocks) which we use to do most all of our operations. Most importantly, we use it to pick up our string of pipe that is in the ground. ![The set up of a drilling rig](http://i.stack.imgur.com/nW0mS.gif) As seen in this picture, the blocks hold the weight of the string of pipe. Now he told us that if the pipe get stuck in the hole (maybe it snags something or the hole caves in), that we lose all of our mechanical advantage. He said that is why the weight indicator will shoot up and go back down after it is freed. He said that because when the pipe is snagged in the hole then we are not dealing with a free floating sheave anymore and that is what is required to have a mechanical advantage. I disagree with this because even if it is not free, there is still a mechanical advantage such that (say the normal mechanical advantage is 6 to 1) our pulling force is multiplied by 6. I would like somebody to confirm this for me. First picture taken from www.worldoils.com on June 21, 2013 Second picture taken from www.PaysonPetro.com on June 21, 2013


The sound of coffee

> **Possible Duplicate:**   >  Why does the sound pitch increase on every consecutive tick at the bottom > of a filled cup of coffee? A colleague suggested this experiment this morning :   * Get a cup of coffee   * Spin the coffee in the cup using a spoon   * Tap the bottom of the cup with the spoon _Observations_ : As the coffee slows down, the sound produced by the tapping gets higher in frequency. I'm very curious about this, does anyone have an explanation ?


How did Fizeau make his famous speed-of-light experiment?

I heard once in a TED talk how Fizeau measured the speed of light in the 19th century. Here is the link https://www.youtube.com/watch?v=F8UFGu2M2gM You can read about it here in Wikipedia: http://en.wikipedia.org/wiki/Fizeau%E2%80%93Foucault_apparatus A short summary: He placed a kind of rotating wheel in front of a beam of light, and a mirror far away from these two things. The beam of light passes between two teeth of the rotating wheel, reaches the mirror and goes back from the original source. As the wheel is spinning very fast, during the time that the light has been travelling, the wheel has rotated a tiny bit, but enough to impede the passage of time through the point where it entered. Knowing the distance from the mirror as well as the speed at which the wheel is rotating, the speed of light can be easily calculated. The experiment is better explained in wikipedia, here I wrote a simplified version of it. I loved the experiment, because it seemed fairly easy to reproduce, so I ordered a green laser pointer on Amazon, which can reach up to 10 km. As a proof of principle, I went with a friend in the night to a place where there is a good visibility. We began setting a mirror somewhere 500 metres away from the laser, but, even from that far, the light had scattered so much that it was impossible to collect the light with the mirror. The laser is a very powerful one, of those that you can see the whole beam (usually used in astronomy). If I can't repeat the experiment using a laser like this, how on earth could Fizeau do that in the 18th century employing a much more rudimentary source of light and placing the mirror much farther away? It says in Wikipedia that the distance between them was like 8 km.


Why the shape of rainbow is semicircular after rain why not the whole atmosphere is colorful?

I have a very simple question. Everyone must have seen the rainbow after rain. According to the theory the rainbow is created due to the passing of sunlight from small drops of water in the atmosphere(means by dispersion of light). Now my what I want to know is that after rain the rain drops are present in the entire atmosphere. So the whole atmosphere should look colorful. Why only a semicircular shape is formed (or is colorful).


Determining the Wave Function From Initial Conditions

This is Problem 2.6 (b) in Griffiths, _Intro to QM:_ > A particle in an infinite square well has its initial wave function an even > mixture of the first two stationary states: > > $\Psi(x,0) = A[\psi_1(x) + \psi_2(x)]$. Here is the part of the problem that I am having a little trouble with: > (b) Find $\Psi(x,t)$ and $|\Psi(x,t)|^2$. Express the latter as a sinusoidal > function of time, as in Example 2.1. To simplify the result, let $\omega > \equiv \frac{\pi^2 \hbar}{2ma^2}$ According to the answer key, even after $t=0$, the wave function continues to be a mixture of the first two stationary states. Why is that? I am having a little difficulty understanding this. Why can't it be a new 'mixture?' Are my questions sufficiently clear?


Heat transfer to a cube from one of the faces

Heat Flux= 10 MW Inlet water temperature = 28 degree centigrade for cooling purpose through a tube passing through the centre of retangular block of length 50 mm,height 30 mm and width 30 mm inner diameter of tube=10mm and outer diameter=12 mm tube is made of copper. Find out the heat transfer coefficient of water at 42 bar and given temperature Find out Nusselt number and Reynold's number.


Colliding bubbles in hyperspace

Assuming the following   1. A universe is the surface from a bubble in hyperspace. Inside a bubble there is nothing, only the surface represents a universe. The size of the bubble is time.   2. Dark matter is from bubbles different then our own bubble that only shares gravity.   3. The big bang are bubbles colliding, creating a other bubble. My problem is if bubbles do only share gravity why would they have a collision, I expect that they would just pass trough each other without creating a big bang?


Renormalizability of the Polyakov Action

I was told today that the Polyakov action for a $p$-brane is (superficially) re-normalizable iff $p\leq 1$. Of course, when I went to check for myself, I screwed up my power-counting, and I'm having trouble seeing why. We work in units with $c=1=\hbar$, so that $L=T=M^{-1}$. In these units, any action must have dimension $1$, so from looking at the Nambu-Goto action, $$ S_{\text{NG}}:=-T_p\int d\sigma ^{1+p}\sqrt{-g}, $$ we see that $[T_p]=L^{-(1+p)}=M^{1+p}$. From the Polyakov action, $$ S_{\text{P}}:=-\frac{T_p}{2}\int d\sigma ^{1+p}\sqrt{-h}h^{\alpha \beta}\partial _\alpha X^\mu \partial _\beta X^\nu G_{\mu \nu}(X), $$ we see that the the coupling constant of the interaction of of the scalar fields $X^\mu$ with $h_{\alpha \beta}$ is precisely $\frac{T_p}{2}$, which has dimensions $M^{1+p}$, and so is going to be (superficially) re-normalizable iff $1+p\geq 0$ . . . But this, of course, is not the result I was looking for . . . Where is my mistake?


Does a universe experiencing "heat death" have a temperature?

As defined by Wikipedia: > The heat death of the universe is a suggested ultimate fate of the universe > in which the universe has diminished to a state of no thermodynamic free > energy and therefore can no longer sustain processes that consume energy > (including computation and life). Heat death does not imply any particular > absolute temperature; it only requires that temperature differences or other > processes may no longer be exploited to perform work. Does it even make sense to describe temperature in the system described? If so, would it be a very cold system or a very hot system?


The surface area to volume ratio of a sphere and the Bekenstein bound

I am trying to relate the surface-area-to-volume-ratio of a sphere to the Bekenstein bound. Since the surface-area-to-volume-ratio decreases with increasing volume, one would surmise that, per unit of volume, a small space is richer in information than a large one. How can this be and how can this bound work for black holes of various sizes? Thank you very much Mr. Rennie. I appreciate and have investigated your answer. It turns out that I am familiar with the AdS/CFT correspondence and have sufficient understanding of the math (just barely) to be intrigued with the conjecture and, of course, the holographic theory. If the correspondence only works for a certain diameter black hole, the conjecture seems, to me, weak because of the changing surface-area-to-volume-ratio of a sphere. For myself, it would appear to be, likely, a mathematical curiosity or fluke. However, if, through some aspect that I do not understand, the correspondence holds for varying diameters, in fact, all diameters of black holes, then it seems quite astonishing, indeed. After searching for some time, I have once seen the amount described as trivial and possibly in another instance, that it may have something to do with informational redundancy. I’m afraid I cannot site these references as they were far too brief to be of any help.


Conformal fields on compactified manifolds? An apparent paradox!

I would appreciate it if someone tells me how a cft on a compactified manifold (e.g. by means of periodic boundary conditions) can be meaningful? The global conformal invariance is broken due to the scale over which the manifold is compactified (e.g. the period). The local conformal invariance of course still classically exists, but for the simple case of a field on a cylinder for instance, the trace of the stress-energy tensor becomes non-zero (in proportion with the central charge); hence apparently compactification of the space-time manifold is in contrast with having conformal symmetry.


Collision between a photon and a massive particle

Just a small question regarding collisions. Imagine a head-on collision between a photon and a particle with mass that moves with a non-relativistic speed, the particle was on its ground state, completely absorbs the photon, and moves to its next energy level. Is it always the case that the particle ends up with a non-relativistic speed after the collision? Something more specific: > To study the properties of isolated atoms with a high degree of precision > they must be kept almost at rest for a length of time. A method has recently > been developed to do this. It is called “laser cooling” and is illustrated > by the problem below. In a vacuum chamber a well collimated beam of Na23 > atoms (coming from the evaporation of a sample at 103 K) is illuminated > head-on with a high intensity laser beam (fig. 3.1). The frequency of laser > is chosen so there will be resonant absorption of a photon by those atoms > whose velocity is v0. When the light is absorbed, these atoms are exited to > the first energy level, which has a mean value E above the ground state and > uncertainty of (gamma). Find the laser frequency needed ensure the resonant > absorption of the light by those atoms whose kinetic energy of the atoms > inside the region behind the collimator. Also find the reduction in the > velocity of these atoms, ∆v1, after the absorption process. Data E = > 3,36⋅10-19 J Γ = 7,0⋅10-27 J c = 3⋅108 ms-1 mp = 1,67⋅10-27 kg h = > 6,62⋅10-34 Js k = 1,38⋅10-23 JK-1


Solar neutrino problem in 1968 and experimental verification of neutrino oscillation in 2001. Why the huge delay?

Solar neutrino deficit was first observed in the late 1960's. And theory of neutrino oscillation was developed in 1967. But,in 2001, the first convincing evidence of solar neutrino oscillation came in SNO. Why did it take nearly 35 years to verify the neutrino oscillation? reference: http://en.wikipedia.org/wiki/Neutrino_oscillation


Could sphaleron-induced proton decay also cause vacuum decay?

I will say right away that I don't mean standard-model sphalerons, I mean the sphalerons of some extension of the standard model. The reason to even think about this is last year's paper by Frampton and Hung (discussion), which proposes that the Higgs mass might be 126 GeV "because" the timescales for vacuum decay and for instanton-induced proton decay are the same. More precisely: the timescale of vacuum decay is unknown, but if it were the same as the timescale for the spontaneous occurrence of a sphaleron in a proton, that would indicate a Higgs mass near the observed mass. (This is not as remarkable as it may sound: the Higgs mass is right on the edge of making the vacuum genuinely stable, so _any_ extremely long lifetime for the vacuum corresponds to a Higgs mass near the value that we see.) They don't have a coherent causal model. But one way for it to work, would be if proton decay _caused_ vacuum decay. This ought to make for problems with early-universe cosmology, but put that to one side for now, maybe there's a way around that. What I would like to know is whether the idea of proton decay causing vacuum decay even makes sense. What I envisage is a standard model extension with new scalars that appear in the sphaleron action _and_ in an extended Higgs-sector potential. Perhaps the new scalar has a VEV of its own, the proton-destroying sphaleron action is minimized when that VEV changes, but the new VEV will in turn destabilize the existing Higgs VEV and drive it into a new, true minimum. The effect of an extra scalar on electroweak vacuum stability is a current topic of research, and section 4 of this paper shows sneutrinos nominally contributing to the sphaleron action in an extension of the MSSM (though here their contribution is negligible).


charge moves if you scuff the rag with your shoes

Why is that when you scuff with your shoes on, charges move (since electrometer moves back and forth), but if you don't have your shoes on, the electrometer doesn't move. Here's the corresponding video detailing the event. http://www.youtube.com/watch?v=cJSp8v0YJrA It is at about 47 min of video from beginning. I don't get the logic behind it.


Clarification on Wald's book

I have a question concerning the Wald's book: General Relativity. In the appendix E, he derived the Einstein equation by considering the surface term (GHY). I do not understand what he said after the equation (E.1.38). Actually he considers that $h^{bc}\nabla_c(\delta g_{ab})=0$, because we fix $\delta g_{ab}=0$ on the surface, but therefore why the other term in (E.1.38) is not null, the term $h^{bc}\nabla_a(\delta g_{bc})$. They look the same for me, and after some algebra, where we replace the covariant derivative by the one compatible with the metric on the surface we should have a total derivative term on the surface that we can integrate away. Thanks in advance


Spherical harmonics

Given the following potential: $$V(\theta,\phi)=\frac{Q}{a}\left(\sin\theta \cos\phi+\frac{1}{2}\cos^2\theta\right)$$ on the surface of a sphere of radius $a$ I am trying to solve Laplace's Equation outside the sphere (where there aren't any charges). I know the general solution to Laplace's Equation outside the sphere is given by: $$\phi(r,\theta,\phi)=\sum_{l=0}B_l r^{-l-1}P_l(\cos\theta).$$ I am not quite sure how to proceed as am very new to spherical harmonics. Does the next step involve expressing the given potential as a Legendre polynomial? I'd appreciate some guidance.


Damping Coefficient for a Hydraulic-Controlled Door

I am not very good with physics terms, so please treat me as an ignorant. I am trying to calculate a damping coefficient dynamically for a hydraulic- controlled door that opens and closes due to hydraulic pressure (opening/closing). The formula where I need my damping coefficient is: viscous_fric_mom = $C \times \omega \times 2/\pi$; The $\pi/2$ division is because the maximum angle that the door can be opened at is $\pi/2$ (i.e. 90°), $\omega$ is the angular velocity of the door which is in $\rm rad/s$. And $C$ is the damping coefficient, which I need to calculate dynamically. The system specifies that my damping coefficient unit is in $\rm N m s$! I thought it would be $\rm N s /m$, because usually it is `Force*time/distance`. Apparently I am wrong. Could someone suggest what I should consider for calculating this damping coefficient? I am really bad at math and do not know any better of doing it. **UPDATE** I am trying to create a software model of a hydraulic-operated door. The door will open given that the effective hydraulic jack pressure have been applied. Same goes true when the door is closing i.e. an effective hydraulic jack retraction pressure must be applied. I use two part integration (integration of acceleration and velocity) to get the current angular position of the door, i.e. from the locked position. My current operating assumption is that the door will have either $\pi/2$ or $-\pi/2$ acceleration (i.e. opening or closing). If I integrate that, I get the velocity and double integration will give me the position. If I take the angular position and feed it back to my damping coefficient calculator, that should work, right?


Does anybody know of any good sources that explain (generically) how we form Lagrangians/Actions/Superpotentials for different field content?

I regularly find that I'll understand where the field content in a particular physics paper comes from, but then a Lagrangian or action or superpotential is stated and I don't know how it's derived. Is there a set of general rules for building a Lagrangian/action/superpotential if you already know the field content of the theory? Any suggestions of sources that explain how to do so would be very welcome as I'm having little trouble finding much that helps.


Is quantum field operator $\psi$ same as quantum field $\psi$?

So in QFT, quantum field operator $\psi$ is there. $\psi$ seems to take the role of wavefunction in QM, which now acts upon vacuum state. Then, in lagrangian of various quantum field theories, $\psi$ appears, but now it is called quantum field. So is quantum field operator no different from quantum field here? If it is not different, then how can $\psi$ can have scalar $|\psi|$ in any assumption as it only operates upon vacuum state? Assuming that vacuum state is represented by some $n \times 1$ vector (let us forget about infinite-dimension for now), operator should be of the form $n \times n$.


Partially filled orbitals and strongly correlated electrons

Interesting behavior of strong correlation between electrons occur in metals with partially filled d or f orbitals (transition metals). Why these strong correlations do not appear with elements with incomplete p or s orbital for example ?


Proof for angar speed equation

I'm struggling to find a solution to a project for college. I have looked through my textbook countless times and googled for days, but I just can't seem to figure it out. The question is that we need to prove the angular speed of a hanging rotating weight is given by the expression: W = root(g/Lcos.theta) Please can you help me or push me in the right direction. Any help would be appreciated. Thanks in advance


Calabi-Yau manifolds and compactification of extra dimensions in M-theory

I just finished learning M(atrix) theory and the basics of the compactification of extra dimensions. The extra 6 dimensions of superstring theory can be compactified on 3 Calabi- Yau manifolds (because 6 real dimensions means 3 complex dimensions). However, when it comes to M-theory, one cannot compactify on 3.5 Calabi-Yau manifolds, so after compactifying 6 dimensions, where does the extra 1 dimension go? Is it just compactified on a circle, or something like that?


Stabilizer formalism for symmetric spin-states?

This question developed out of conversation between myself and Joe Fitzsimons. Is there a succinct stabilizer representation for symmetric states, on systems of _n_ spin-1/2 or (more generally) _n_ higher spin particles? By a "stabilizer representation", I mean that:   * every symmetric state (or some notable, non-trivial family of them which contains more than just product states) is represented as the unique +1-eigenstate of some operator or the unique joint +1-eigenstate of a list of operators, where   * each element of this set of stabilizing operators can be succinctly described, as an operator on the larger Hilbert space ( _i.e._  not only as a transformation restricted to the symmetric subspace itself), and    * where the stabilizing operators transform in a nice way in the Heisenberg picture under symmetric local unitaries ( _i.e._  unitary transformations of the form _U ⊗n_). Ideally, one would be able to efficiently describe all sorts of transformations between various symmetric states; but one cannot have everything. The constraint of being a unique +1-eigenstate of the list of stabilizing operators could also be made subject to the constraint of being a symmetric state. (For instance, many states on _n_ spin-1/2 particles are stabilized by a σz operator on a single spin, but exactly one symmetric state is stabilized by that operator. Not that I would expect such an operator necessarily to arise in the formalism...) Does a representation with the above properties (or one close to it) exist?


Introduction to neutron star physics

I enjoy thinking about theoretical astrophysics because I want to understand black holes. Given that no one understands black holes, I like to ponder the nearest thing to a black hole: a neutron star! I have searched around the web for pedagogical discussions of the structure of neutron stars such as this link from NASA: http://heasarc.nasa.gov/docs/objects/binaries/neutron_star_structure.html, but none seems to be at an advanced enough level for my liking. The problem is that I do not know what literature I should read in order to learn more. What is the current state of neutron star research? What are some good review articles? More specifically, I am curious about theoretical predictions for "starquakes" referenced in the link above, and how they would look to an observer on Earth. I would also be interested in understanding what happens to gas falling into a neutron star -- specifically, if Sol was spiralling to its death by neutron star.


How to get started in Astronomy (UK based)

I have always been interested in space and astronomy (in my youth - I wanted to be an astronaut). However for various reasons, I never quite got started. I now want to get started - small but steadily. I live in a city, ergo: light pollution is a problem - however, I would like to get a telescope (maybe a second hand one but a good make), maybe join a local astronomy club ? In an ideal world, I will get a "good" telescope which has the following attributes   1. Can be extended to make progressively more powerful   2. Can use one of the opensource astronomy packages (I am a programmer!)   3. Allow me to take photographs  Can anyone provide me with a series of steps to help me finally turn my dream of observing the skies into reality? Note: I am aware that it is very likely that I will have to start with a small, hand held telescope - but I like the idea of a telescope that will "grow with me" - if that is at all possible.


Is there a [set of] rules/patterns that apply to elements

> **Possible Duplicate:**   >  Do current models of particle physics explain the chemical properties of > elements/compounds? Is there a rule/pattern, or multiple rules and/or patterns, which describe the properties of an element given the number of protons and neutrons it has. For example a carbon atom which has 6 protons and 6 neutrons, howcan you work out what it will act like, reactivity, if there was a bar of carbon, how could they be arranged and what would its properties be like strength. But the rules would work for any number of protons and neutrons.


What is the mechanism of magnetic core saturation?

Why does a magnetic core saturate? What is its actual mechanism?


Books for learning Mathematics in Physics?

Currently I'm doing Advanced Classical Mechanics courses. I'm finding it hard to understand due to the lack of knowledge in linear algebra, multi variable calculus and other chapters. Can anyone suggest a mathematical book which is dedicated to teaching all the math that is used in physics?


Book for multivariable calculus

Hi I want to start learning multi variable calculus specifically for learning electrodynamics. What are some good text books?


Books on representation theory

> **Possible Duplicate:**   >  Best books for mathematical background? I'm looking for a textbook on the group/representation theory for a student- physicist. The main questions of interest are representations of SO, SU and the Lorentz/Poincare group in various dimensions (or maybe some other topics that i've missed but important for particle physics).


Mathematics for physics resources

Looking for book suggestions. I have a good basic understanding of many mathematical concepts but I want to improve to a physicist level. This is mainly because I want to study atmospheric physics (fluid dynamics, gases, quantum mechanics, optics, etc.).


Do eye-glasses decrease the Depth of focus?

I was reading up about Depth of Focus and wondered if glasses affect depth of focus. If yes, is it noticeable to the user?


Why does a faraday cage protect you from high currents?

In an electrostatic case it is clear that that in a space enclosed with a conductor (without charge in it) the electric field is zero. This is often demonstrated in physics shows like on the following image: ![enter image description here](http://i.stack.imgur.com/mKkdX.jpg) However it you have the lighting a current is flowing through the air and through the cage. So wie are not in the electro- **static** situation anymore since we have currents, i.e. moving charges. How can one account in the explanation properly that we have moving charges? Some people say that the fact that the man inside the cage is safe doesn't have to do anything with faradays cage, it's simply because the the cage is a better conductor. Sometimes also the skin-effect is mentioned. So what's true. It would be great do get a detailed and correct explanation of this. Do you have any good references?


work done by tension

## The problem statement, all variables and given/known data Consider the following arrangement: ![enter image description here](http://i.stack.imgur.com/Jge7p.jpg) Calculate the work done by tension on 2kg block during its motion on circular track from point $A$ to point $B$. ## The attempt at a solution We know that work done by a force is product of force and displacement. We know the displacement of point of application as 4m. How to find the work done by the tension as it is not constant it is variable! 2nd attempt(calculus approach) The 2 kg block moves along the circle, so its speed is Rdθ/dt. It pulls the string, the length of the string between point O and the block can we obtained with simple geometry at any position θ (ignoring the size of the pulley). The total length of the string is unchanged, so the speed of the 1kg block is dL/dt. http://imgur.com/PbpBpOB Can u help after this


How big is an inertial frame?

How big is an inertial frame? Consider a huge rod which is rotating about a fixed point in a plane, its length is 1 light year. Thus light from its end closer to the fixed point to the end farther from the fixed point takes one year to reach. Now the angular velocity of the point closer to the fixed point is much slower than the angular velocity of the farther point. Thus the end closer to the fixed point has a relative velocity to the end farther away. At some point in time, the clocks at the two ends are synchronized by sending a light signal to both ends from exactly from the centre of the rod (half a light year from any point). My question is, are the two ends of the rod in the same inertial frame? Is it possible for 1 object to be in two inertial frames simultaneously? A simpler question. A rod is 1 light year long and travelling at a a velocity c/2 along its length. There will be length contraction of the rod by gamma. Now imagine a light source at one end of the rod. The light from one end takes one year to reach the other end (as measured by either end), when the rod is stationary. Now since the distance is contracted when the rod is moving, so must the time be different (as compared to an observer looking at this rod) for the two ends of the rod as light must be measured with the same velocity on both ends. The time must be same at both ends by symmetry, so they are indeed in the same inertial frame. However, if there is length contraction, they will not measure one light year, they should each measure less than one light year. An external observer observing this rod would see the distance as less than one year. But to measure the same velocity of light, must also measure the same time as the ends of the rod, thus implying the observer is in the same frame as the rod? The second part of my question is this. In zero gravity, a person holds a ball in his hand. Thus the person and the ball are in the same inertial frame. Then the person "throws" the ball and as a result, both and ball and the person now acquire a relative velocity to each other (action and reaction). Are the two in the same inertial frame? The third part of my question is this. In zero gravity, can a ship and an observer on the ship be in the same inertial frame? That is, do they share acceleration? Assuming the ship is a plane, when the plane accelerates, the person is floating around, hence the person and the ship cannot share acceleration unless the person is strapped to the ship.Thus in zero gravity, every object must have its own inertial frame?


How does quantum world affect us and why should I care about it?

We live in a world which is much larger than quantum world. The laws of quantum physics are not valid. While I am pressing the keys on my laptop, I have 100% certainty that I am writing what I really want. Then how is such microscopic world going to affect me? The quantum world is entirely different, perhaps best suited for those Aliens. Then why should I worry about uncertainty and other stuffs happening at infinitesimally small scale?


A simple question on $SU(2)$ gauge transformations in Wen's papers on projective symmetry group (PSG)?

Recently I am studying the projective symmetry group (PSG) and the associated concept of quantum order first proposed by prof.Wen. In Wen's paper, see the last line of Eq.(8), the local SU(2) gauge transformation for spinor operators is deﬁned as $\psi_i\rightarrow G_i\psi_i$, where $\psi_i=(\psi_{1i},\psi_{2i})^T$ are fermionic operators and $G_i\in SU(2)$. Why we define it like this? Since as we know, the Shcwinger fermion representation for spin-1/2 can be written as $\mathbf{S}_i=\frac{1}{4}tr(\Psi_i^\dagger\mathbf{\sigma}\Psi_i)$, where $\Psi_i=\begin{pmatrix} \psi_{1i} & -\psi_{2i}^\dagger \\\ \psi_{2i} & \psi_{1i}^\dagger \end{pmatrix}$, and $G_i\Psi_i$ which is the same as the above transformation $\psi_i\rightarrow G_i\psi_i$ is in fact a spin rotation of $\mathbf{S}_i$, while $\Psi_iG_i$ does _not_ change spin $\mathbf{S}_i$ at all. So in Eq.(8), why we define the SU(2) gauge transformation as $G_i\Psi_i$ rather than $\Psi_iG_i$?


Literature on the time reversal operator

Time reversal symmetry seems to be a very useful concept and is mentioned in a good number of papers I recently came across. Most of the time people claim that a certain system or Hamiltonian is time reversal invariant and deduce certain properties without going further into details. I have a hard time grasping the entire time reversal symmetry thing and would like to read more on the subject in general. So I was wondering whether anybody had good literature references? Preferably with a lot of applications and examples. I know the mathematical definition, but its applications usually elude me.


Determining the decay constant for a damped wave

My question is short and simple. If a damped, travelling wave (say on a string) could be described as $y\left( x,t\right) =Ae^{-\gamma x}\sin \left( kx-\omega t\right)$ how could/would one determine the decay/dampening constant $\gamma$ mathematically?


Same momentum, different mass

The question is: if   * A bowling ball and ping pong ball    * are moving at **same momentum**   * and you exert **same force** to stop each one   * which will take a longer time? or some?    * which will have a longer stopping distance?  * * * So I think I can think of this as: $$F = \frac{dp}{dt} = m \cdot \frac{v_i - 0}{\Delta t} = \frac{p_i}{\Delta t}$$ Since both have same momentum, given same force and momentum, time will be the same? Is this right? * * * Then how do I do the stopping distance one?


Applications of the Linearized Einstein Field Equations (EFE)

Look up _linearized Einstein field equations_ anywhere and the first thing you'll see will be a discussion of gravitational waves. Using the linearized EFE's is pretty handy when studying gravitational waves, but it doesn't seem like they are used anywhere else! Is this true? If not, what are the other applications?


Expanding Universe Balloon Analogy - Anything Similar for Time?

It is difficult to imagine the infiniteness of space and how it itself is expanding rather than the universe expanding into something else. A helpful analogy is that of drawing little dots (representing galaxies or some other sub-universal structure) onto a deflated balloon and then blowing it up. The surface expands in all directions, with each dot moving away from every other dot. Although the analogous surface (the outside of the balloon) is effectively 2 dimensional, it's possible to imagine its translation into 3 dimensions. As for time, though, I have a hard time picturing its "before / during / after" states, and I realize those words aren't even accurate. Time supposedly began at the Big Bang and may end at the Big Crunch. But I'm wondering if anyone knows of an analogy for time, similar to the balloon analogy that applies to space. Is there a way to imagine time in some comprehensible way?


Who foots the (magnetic) energy bill?

Gravitational attraction and electrostatic attraction/repulsion are intrinsic properties of matter, any particle (electron, proton) for some unknown reason can produce KE at a distance. But magnetic attraction/force is not an intrinsic property of matter, a charged particle generates a magnetic field/flux and a magnetic force only when it is moving: higher velocity = much higher force. The definition of KE says that it is _'work done to accelerate an object'_ , energy spent only to make it move, and does not mention the generation of other forces/energy or doing 'extramural' work. _[Electrons moving in a current produce magnetic induction that makes an electric appliance do work, who/what is spending the necessary energy to produce such work? Voltage (the difference of potential) coming from the mains provides the energy to accelerate the electrons, not the energy to blend your fruit. Same voltage would produce equivalent v/Ke if the electron is accelerated in a vacuum (such as in a synchrotron, where there is no fruit). But this is difficult to prove.]_ **Edit** : I will not reply to any comment. They are attributing to me statements I never made. Now, to avoid technical complications ( _voltage_ , wattage etc,) that could trick me, forget the previous example, **let's consider another, simpler case:** An electron is travelling at high speed (say,0.9 C). If is moving near another electron, (proton, positron or a live wire) _it can make anything move, acquire KE_ , it **can do work**. When v approaches c, the attractive magnetic force gets so great that it _equals the huge electrostatic repulsion._ All the issues you have raised (intrinsic spin, magnetic field, electromagnetism etc., some comments have been deleted) are irrelevant here. If those properties exist they exist even when the electron is at rest. When the electron is at rest, it still has spin, nevertheless it is **not able any more** to do work. If this is an irrefutable fact, then, how come it can do work when it is moving, where is the necessary energy coming from?


Why can't a piece of paper (of non-zero thickness) be folded more than $N$ times?

Updated: > _In order to fold anything in half, it must be $\pi$ times longer than its > thickness, and that depending on how something is folded, the amount its > length decreases with each fold differs._   >  – Britney Gallivan, the person who determined that the maximum number of > times a paper or other finite thickness materials can be folded = 12. Mathematics of paper folding explains the mathematical aspect of this.   I would like to know the physical explanation of this. Why is it not possible to fold a paper more than $N$ (=12) times?


Why does the speed of light within a solid depend on frequency?

Different frequencies of light travel at different speeds through solids, which along with Snell's law allows for rainbows. Has this phenomenon of variable speeds been predicted through derivations? What does it tell us about the interactions that occur when light travels through a solid?


How "things" radiate electromagnetic radiation?

How things radiate electromagnetic radiation? I don't ask why they radiate (higher temperature than 0K) but how they radiate this electromagnetic waves?


Double-slit experiment formulated in a quantum circuit

What would the double-slit experiment or something analogous to it look like implemented as a quantum circuit/program? Also, what about the delayed choice quantum eraser, how would that look in a quantum circuit? For example, if implemented in this quantum computing simulator: http://www.davyw.com/quantum/?example=Toffoli . I couldn't find any direct answer via Google, so even good search terms would be useful.


Space time curvature due to electric charge or magnetic charges

since we know that gravitational force is nothing but a curvature in space- time. I have a similar analogous for the electric or magnetic charges. Similarity is that both electromagnetic and gravitational forces follow the inverse square law. consequence of this can be that we can have a another interpretation of electron orbital motion causes by curvature in space due to electric charge (as mass of electron is very small so STR have no significance here). But problem is that we have two different type of charges (+ve and -ve) which gives repulsion and attraction, whereas in gravitation force mass always gives the attraction forces. Please correct me if I am wrong.


ratio between work and heat

I am really stuck on a problem in my textbook: Water is heated in an open pan where the air pressure is one atmosphere. The water remains a liquid, which expands by a small amount as it is heated. Determine the ratio of the work done by the water to the heat absorbed by the water. MY ATTEMPT: We are given that: $P = 1.013 \cdot 10^5 Pa$ We then have: $$\frac{W}{Q} = \frac{P \Delta V}{cm \Delta T} = \frac{P \beta V_0 \Delta T}{cm \Delta T} = \frac{P \beta m \Delta T}{ cm \rho \Delta T} = \frac{P \beta}{c \rho} = \frac{1.013 \cdot 10^5 \cdot 207 \cdot 10^{-6}}{4186 \cdot 1} = 5 \cdot 10^{-3}$$ But according to the textbook, the solution should be $4.99 \cdot 10^{-6}$. If anyone can help me by pointing out what I'm doing wrong here, I would be extremely grateful!


Explanation for self-rupture glass is needed

I witnessed a phenomenon that I couldn't conclude its cause. Please bear with me for the length of the recall, for I merely want to include any details that might help us to investigate. I had a **cooking glass lid** sat on a wooden shelf that is **away from** the stove and oven and other heating objects. The shelf is nailed on the wall and is situated just above my eye level, and a counter top is also on the same side of the wall where the shelf is installed. Now here comes the surprise. In a winter afternoon 2011, my room had almost the same temperature as an autumn morning, and while I was cutting my lettuce on that counter top which I pointed out in above passage, **a pounding sound** , as if a heavy car door slam or a tree trump falling on top of the roof, knocked its introduction from the shelf that was just above my eye level. First, I thought I may had knocked something around me off(which I didn't believe that for there wasn't anything around me to knock off); then I thought it may be my neighbor next door dropping a heavy box; last, I suspected somewhere my roof top collapsed. But it was my third suspicion directed me to meet that glass lid I mentioned above, and **I found it had ruptured completely like glacier creaked BUT still having all broken pieces bounded without any pieces scattering toward random direction! Only the nob of the lid popped out partially.** Before this happened, I hadn't used that lid for cooking for years, and I didn't removed it from any heating object nor there was something on top of the lid that day, and I believe what the lid had maybe just an invisible layer of dust. I was glad my face hadn't been stung by any glass residues, but ponder what really happen to that glass lid and why it ruptured without collapsed. Below, I attached 2 pictures of the scene from that day. If you have any similar experience or know the theory behind it, may you please drop me an explanation to this incidence? Thank you in advance. I'm new and need reputation to post pictures. I definitely post that 2 pictures once I earn enough reputation point.


Need to use lasers,mirrors and prisms for a project.How to make the light visible and presentable at the same time?

Alright so I really want to put some art into science for my project.project. I'm thinking of creating a beautiful image,preferably in air.I read about the visible lime-green lasers but i still want to be able to use other colors,and I need to make it visible.How can i make "fog" for it so that a pretty image is seen yet at the same time the inspectors won't be suffocating.Any ideas on how should I make it and what other things I can use to create pretty images?Please explain in details as I might not know some terms in optics.Thanks in advance!


phase transformation of liquid under motion

I was testing in my lab with water and found that it starts to solidify when it is stationery at 0 degree centigrade.but when I move the liquid with some velocity it dosent change its phase.my interpretation of this happening because when I move liquid it has kinetic energy which dosent allow it to crystallize.but at certain velocity and further lowered temperature it should start crystallising again.is there any graph which shows at what velocity and temperature it crystallize with atmospheric pressure?as in phase transition of water under temperature and pressure?


How are atomic bonds created?

From what I have learned in my chemistry course, Electrons with similar quantum numbers but with opposite spin are attracted to each other. What does this mean when there is a covalent bond being formed between lets say hydrogen and fluorine? I can think of four different results: A) A bond is not formed until an unpaired fluorine electron meets an unpaired hydrogen electron with the same energy shift (in this case, 0). B) The fluorine or hydrogen electron forces the other electron to have the same energy shift. C) The electrons don't change, but pair up regardless of quantum numbers. D) The electrons from both atoms enter into a different state that I have not learned yet. Thanks in advance for any help received!


Gravastars: Are they observationally distinguishable from Black-Holes?

Are observations of Hawking radiation at the acoustic event horizon in Bose- Einstein condensates consistent with Gravastars? To reconcile the second law of thermodynamics with the existence of a black hole event horizon, black-holes are necessarily said to contain high entropy while Gravastars not at all. An Event-Horizon forming out of a collapsing star's intense gravity sufficient enough to force the matter to phase change transforming into Bose-Einstein condensate would be such that nearby matter would be re-emitted as another form of energy, and all matter coming into contact with the Event-Horizon itself would become incorporated. So, it seems reasonable to wonder if Black-Holes are distinguishable from Gravastars since Gravastars appear to be better emitters, and Black-holes better entropy sinks. What do observations of Hawking radiation from acoustic black holes from Bose-Einstein condensate seem to suggest?


Hilbert space of a free particle: Countable or Uncountable?

This is obviously a follow on question to the Phys.SE post Hilbert space of harmonic oscillator: Countable vs uncountable? So I thought that the Hilbert space of a bound electron is countable, but the Hilbert space of a free electron is uncountable. But the arguments about smoothness and delta functions in the answers to the previous question convince me otherwise. Why is the Hilbert space of a free particle not also countable?


Ambiguity in number of basis vectors

The dimension of the Hilbert space is determined by the number of independent basis vectors. There is a infinite discrete energy eigenbasis $\\{|n\rangle\\}$ in the problem of particle in a box which can be used to expand a general state $|\psi\rangle$ as: $$|\psi\rangle=\sum\limits_{n=0}^{\infty} C_n |n\rangle$$ Here, the basis implies that the Hilbert space has **countably infinite** number of basis vectors. But one could well expand $|\psi\rangle$ in any other basis as well, say the continuous position basis. Then that superposition, $$|\psi\rangle=\int dx \psi(x)|x\rangle$$ is also possible. But this basis $\\{|x\rangle\\}$ is continuous and **uncountably infinite**. Such sets cannot be equal because they have different cardinality. Then my questions is how many independent basis vectors are really there in this Hilbert space?


Is the universe a quantum computer - is light speed barrier a computational constraint

There is currently a debate ongoing on leading maths blog Gödel’s Lost Letter, between Gil Kalai and Aram Harrow, with the former arguing that building a quantum computer may not be possible due to noise propagation, and the latter arguing to the contrary. I am wondering if there is any argument to show that building a quantum computer is possible, by virtue of showing that quantum computation is evident in the physical world. So the question is: (A) Are there any known examples of physical interactions where macro level state transitions could be determined to only be in correspondence with an underlying quantum computation? I.e. similarly to Shor's algorithm being exponentially faster than any known classical factoring algorithm, are there any examples of known physical processes, for example perturbation stabilization in a very large particle cluster, that could be shown, assuming P<>NP, to only be efficiently solved by a quantum computation. Some, I admit highly speculative, additional questions would then be: (B) Is the speed of light barrier possibly a natural computational limit of our particular universe, so that for the computational complexity class of quantum mechanics, working on an underlying relational network-like spacetime structure, this is the maximum speed that the computational rules can move a particle/wave representation through a network region of the lowest energy/complexity (i.e. a vacuum)? (C) Is quantum mechanics an actual necessity for the universe to follow classical physical laws at the macro level? The informal argument being that in many-to-many particle quantum level interactions, only the capability of each particle to compute in parallel an infinite or quantum-quasi-infinite number of paths is what allows the universe to resolve a real-time solution at the macro level. Requesting references to research along these lines, or any arguments to support or contradict these speculations.


How would a semi-classical particle react to the double slit experiment?

As far as I know, there is a smooth transition between quantum and classical regimes, so that even classical particle like a massive object has a wavefunction associated with it. However, the double slit experiment can either show quantum character, where the particle supposedly passes through both slits and interferes with itself, or classical character, where the particle passes through one slit with no interference. What would a semi- classical particle do when faced with this situation? In addition, I have a problem imagining a semi-classical particle in general. What is a good example of one?


Does an interaction of entangled particles with each-other cause decoherence?

I'll apologize in advance if this is not an appropriate place for my question. My background is not in physics, and my understanding of quantum mechanics is extremely rudimentary at best, so I hope you'll be forgiving of my newbish question. Given a system of entangled particles (eg, 2 or more electrons), possibly in a superposition state: if the particles interact with each-other, what effect does this have on their quantum state? Is their state now determined (but perhaps unknown until observed)?


Derivation of Brillouin-Wigner theory for coupled subpaces

I recall faintly from my quantum theory lecture that there was a really neat way to derive Brillouin-Wigner perturbation theory for the special case of two coupled subspaces that involved a geometric series in reverse. I know that the beginning was to have the Hilbert space split into subspaces 0 and 1 so that the Hamiltonian reads $$H = H_{00} + H_{01} + H_{01}^\dagger + H_{11}$$ where the two indices indicate what subspace goes "in" and what subspaces comes "out" of each of the components. If $P$ is a projector onto subspace $0$ and $Q$ a projector onto subspace $1$, this means, for example, that $$H_{00} = PH_{00}P, H_{01} = PH_{01}Q$$ and so on. In matrix notation $$\begin{pmatrix} H_{00} & H_{01} \\\ H_{01}^\dagger & H_{11}\end{pmatrix} \begin{pmatrix} \psi_{0} \\\ \psi_1\end{pmatrix} = E \begin{pmatrix} \psi_0 \\\ \psi_1\end{pmatrix}$$ Now we can formally solve the Schrödinger equation for $\psi_1$ only: $$|\psi_1\rangle = \frac{1}{E - H_{11}} H_{01}^\dagger |\psi_0\rangle$$ and insert that back into the SG for $\psi_0$ to obtain $$H_{00} |\psi_0\rangle + H_{01} \frac{1}{E-H_{11}} H_{01}^\dagger |\psi_0\rangle = E|\psi_0\rangle$$ Now I know that some cool trick with geometric series and an inverted matrix was going on to immediately write down the expansion of $|\psi\rangle$. Usually in the literature one first shows the iterative formula and THEN notes that this is a geometric series, but here it was done in the reverse, somehow an operator of the type $1/(A-B)$ with $A$ and $B$ matrices, was found and then expanded in the geometric series, but no matter how I massage the equations, I cannot seem to make it work, because there are so many different points where one could substitute one form into the other etc. Any ideas on how this works? I'm also relatively sure that the work was done using operators, not for "individual" matrix elements. Personally, I find this more elegant because it makes it clearer what's going on on a more abstract level instead of dealing with the low-level matrix elements. It could also be that at this point already the approximation $E \approx E_0$ was made where $E_0$ is a typical energy of subspace $0$, assuming that this subspace is near-degenerate and well-separated from subspace 1.


What distance has the soldier travelled?

> A troop $5$ meters long starts marching. A soldier at the end of the file > steps out and starts marching forward at a higher speed. On reaching the > head of the column, he immediately turns around and marches back at the same > speed. As soon as he reaches the end of the file, the troop stops marching, > and it is found that the troop has moved by exactly $5$ meters. What > distance has the soldier travelled? I thought that some info is lacking until my friend showed it in a book. How to get the answer?


What would be walking speed in low gravity?

In $1g$ the average adult human walks 4-5 km in an hour. How fast would such a human walk in a low gravity environment such as on the Moon $(0.17g)$ or Titan $(0.14g)$? Let's ignore the effects of uneven terrain (regolith or ice/snow/sooth); suppose our human walks on hardened pavement.


Do we move at the speed of light relative to light?

My understanding of relativity isn't very sophisticated, but it seems to me that relative to a photon moving at the speed of light, we are moving at the speed of light. Is this the case?


Explain Heat Transfer

I would like to know what are these formulas used for. There is no intro about it in my book at all, and I am reading Heat Transfer book. ![enter image description here](http://i.stack.imgur.com/nHffh.jpg) If needed Q. can be edited.


Comparing effect of electric and magnetic dipoles on their fields

So at the end of one of my prof's lectures he gives us something to think about: > Both electric and magnetic dipoles tend to line up with their respective > fields. > > Materials made out of electric dipoles cause the electric field that turns > the dipoles to be **reduced**. > > Materials made out of magnetic dipoles cause the magnetic field that turns > the dipoles to be **increased**. > > Why are the two types of dipoles different in this regard? Now, this isn't a homework assignment or anything. It is just something to ponder on. I'm curious why this happens.


Energy conservation and quantum measurement

Consider a particle in a potential well. Let’s assume it’s a simple harmonic oscillator potential and the particle is in its ground state with energy E0 = (1/2) ℏω0. We **measure** its position (measurement-1) with a high degree of accuracy which localises the particle, corresponding to a superposition of momentum (and therefore energy) states. Now we **measure** the particle’s energy (measurement-2) and happen to find that it’s E10 = (21/2) ℏω0. Where did the extra energy come from? In the textbooks it’s claimed that the extra energy comes from the act of observation but I wonder how that could work. Measurement-1 which probed the position of the particle can’t have delivered to it a precise amount of energy, while measurement-2 might just have been passive. No doubt there is entanglement here between the particle state and the measuring device but where, and which measurement?


Weak measurement and Hardy's paradox

How the notion of _weak measurement_ resolves Hardy's paradox?


Why is it hard to extend the Feynman Checkerboard to more than 1+1 dimensions?

The Feynman Checkerboard Wikipedia article states: "There has been no consensus on an optimal extension of the Chessboard model to a fully four- dimensional space-time." Why is it hard to extend it to more than 1+1 dimensions?


Mysterious spectra?

In my blog post _Why riemannium?_ , I introduced the following idea. The infinite potential well in quantum mechanics, the harmonic oscillator and the Kepler (hygrogen-like) problem have energy spectra, respectively, equal to 1) $$ E\sim n^2$$ 2) $$ E\sim n$$ 3) $$ E\sim \dfrac{1}{n^2}$$ Do you know quantum systems with general spectra/eigenvalues given by $$ E(n;s)\sim n^{-s}$$ and energy splitting $$ \Delta E(n,m;s)\sim \left( \dfrac{1}{n^s}-\dfrac{1}{m^s}\right)$$ for all $s\neq -2,-1,2$?


Why does the milk frother on my coffee machine make so much noise?

I have a Sunbeam home espresso machine with a steam wand. The steam roars out straight from the end of the wand. When it's first placed in the cold milk it really screams! Once the milk has a bit of a whirlpool action going it's much quieter, so I guess that the noise is because of fast-moving steam hitting stationary cold milk. Something something fluid dynamics?


Why Landau Level quantization is observed only in low temperature and strong magnetic field in real experiment?

I know that Quantum Hall Effect and Fractional Quantum Hall Effect origin from Landau Level quantization. In magnetic field, the energy of in-plane(plane perpendicular to magnetic field) degree of motion is quantized, which is $E=(n+1/2)\hbar\omega$, $n$ is integer. In experiment, both QHE and FQHE are observed in low temperature and strong magnetic field, suggesting that landau level quantization is observable under the condition $k_B T<<\hbar\omega$, where $k_B$ is Boltzman constant. I am not sure why this condition is important in experiment.


How to prepare a desired quantum state?

Given a quantum state function, we can Fourier expand it in terms of stationary states of the Hamiltonian. So if we want to build that same quantum state approximately all we need to do is to superpose stationary states with proper amplitudes. Assuming that we can prepare such stationary states individually, how is their superposition done experimentally? As a specific example, how to prepare experimentally, an ensemble described by the quantum state that is a superposition of 4 stationary states of the particle in a box (or the hydrogen atom) that corresponds to n = 1, 2, 3, and 4 say ?


Pulley problem - unable to understand the concept

Given:   * The pulley is moving towards the right.   * All blocks have different masses. (The pulley and the strings are massless.) ![Pulley](http://i.stack.imgur.com/unSZF.png) What I don't understand:   * Is the tension the same for both A and B?   * If the tension is the same, then both blocks should have different accelerations, but this is not true?


What do we mean when we talk about Gibbs Free Energy?

Before I start, I'm aware that this question may be better suited on the Chemistry or Biology site, but it's my belief that physicists are more likely to have a clear understanding on what certain terms mean, so by all means move the question if you feel like it will get a better response elsewhere. Okay. In Chemistry we learn about this thing called the Gibbs Free Energy (which I understand is borrowed from Thermodynamics). It's pretty simple. $\Delta G < 0$, and the reaction is spontaneous. $\Delta G > 0$, and the reaction is not spontaneous. Other terms in the equation for Gibbs Free Energy are the total enthalpy change, which I interpret as the amount of energy that the system either takes in or releases, and also the temperature and total change in entropy. Observe these graphs of an ambiguous 'Energy' plotted against the progress of the reaction: http://upload.wikimedia.org/wikibooks/en/a/a6/Gibbs_free_energy.JPG http://www.citruscollege.edu/lc/archive/biology/PublishingImages/c05_10.jpg http://images.tutorvista.com/cms/images/101/exothermic-and-endothermic- reaction.png The idea is the same. Some reactions take in 'Energy,' and the curve ends higher than where it began. Some reaction release 'Energy,' and the curve ends lower than it began. All reactions seem to require an 'Activation Energy' which prevents the reaction from occurring spontaneously. Notice how the Y-axis has different names, such as Gibbs Free Energy, PE of molecules, and PE. Is Gibbs Free Energy the same or different from PE? I'm not sure anymore. Also, in one graph, the change in Energy is portrayed as $\Delta G$, so a decrease implies spontaniety, and increase implies nonspontaniety. Yet both require an activation energy to proceed. One more thing to notice is the change in terms. In a Biology context, the terms are Endergonic and Exergonic. In Chemistry, it is Endothermic and Exothermic. Why different terms for the same idea? I would very greatly appreciate an explanation for this, which has been bugging me for a while.


Wald problem 11.4

Consider a stationary solution with stress-energy $T_{ab}$ in the context of linearized gravity. Choose a global inertial coordinate system for the flat metric $\eta_{ab}$ so that the "time direction" $(\frac{\partial }{\partial t})^{a}$ of this coordinate system agrees with the time-like killing vector field $\xi^{a}$ to zeroth order. (a) Show that the conservation equation, $\partial^{a}T_{ab} = 0$, implies $\int _{\Sigma}T_{i\nu} d^{3}x = 0$ where $i = 1,2,3$, $\nu = 0,1,2,3$, and $\Sigma$ is a $t = \text{constant}$ hypersurface (therefore it has unit future-pointing normal $n^{\mu} = \delta ^{\mu}_{t}$). (there is also a part b but it is trivial given the result of part a so I don't think there is any need to list it here) I am very lost as to where to start for this question. Usually for these kinds of problems, you would take the local conservation equation $\partial^{a}T_{ab} = 0$ and use the divergence theorem in some way but that doesn't seem to be of any use here given the form of $\int _{\Sigma}T_{i\nu} d^{3}x = 0$ (it isn't the surface integral of a vector field over the boundary of something nor is it the volume integral of the divergence of a vector field over something - it's just the integral over $\Sigma$ of a scalar field $T_{i\nu}$ for each fixed $i,\nu$). The only thing I've been able to write down that might be of use is that since the linearized field equations are $\partial^{\alpha}\partial_{\alpha}\gamma_{\mu\nu} = -16\pi T_{\mu\nu}$, we have that $\partial^{t}\partial^{\alpha}\partial_{\alpha}\gamma_{\mu\nu} = \partial^{\alpha}\partial_{\alpha}\partial^{t}\gamma_{\mu\nu} = 0 = \partial^{t}T_{\mu\nu}$ where I have used the fact that in this global inertial coordinate system with stationary killing field $\xi^{a} = (\frac{\partial }{\partial t})^{a}$, the perturbation cannot have any time dependence. This then reduces the conservation equation to $\partial^{\mu}T_{\mu\nu} = \partial^{i}T_{i\nu} = 0$ where again $i=1,2,3$. I really haven't been able to make much progress from here though. I would really appreciate any and all help, thanks.


Elastic Collision And Momentum

The question I am working on is, "Two blocks are free to slide along the friction-less wooden track shown below. The block of mass $m_1 = 4.98~kg$ is released from the position shown, at height $h = 5.00~m$ above the flat part of the track. Protruding from its front end is the north pole of a strong magnet, which repels the north pole of an identical magnet embedded in the back end of the block of mass $m_2 = 9.40~kg$, initially at rest. The two blocks never touch. Calculate the maximum height to which $m_1$ rises after the elastic collision." ![enter image description here](http://i.stack.imgur.com/Uz4To.gif) This question comes from webassign. On webassign they have a feature called "Watch It;" this feature allows you to see a person solve a problem nearly similar to this one. I feel as though there is an error in what the person says in the video. The person says that the mechanical energy of the block- earth system is conserved, but that wouldn't be true; if we were looking at the block-block system--that is, $m_1$ and $m_2$--the mechanical energy would be conserved. By looking at just the block-earth system, there would be a loss in kinetic energy, because when the two blocks "collide," they apply a force over a distance(work), causing a change in kinetic energy of each block, because the kinetic energy of the moving block is transferred into the other block, which is why the first block doesn't return to its initial height. Is this correct? If not, what am I misunderstanding? As a result of this contention with what the person said in the video, I am not very certain on how to solve this problem. EDIT (attempt to solve): Energy analysis: $m_1:$ $PE_i=mgh=KE_f$, where $h$ is the height from which it drops. $KE_i=0~J$; $PE_f=mgh_0$, where $h_0$ is the height is rises to after the collision $m_2:$ $KE_i=PE_i=PE_f=0~j$; $KE_f=\frac{1}{2}m_2v^2_{f,2}$ Momentum Analysis: $m_1:$ $\vec{p}_{i,1}=m_1\vec{v}_{i,1}$; $\vec{p}_{f,1}=m_1\vec{v}_{f,1}$ $m_2:$ $\vec{p}_{i,2}=m_2\vec{v}_{i,1}$; $\vec{p}_{f,2}=m_2\vec{v}_{f,2}$ When I set up an equation for change in mechanical energy, and an equation for conservation of momentum, I get two equations, with a lot of unknowns. What did I do wrong?


Deborah Number for harmonic excitation

I think I do not understand well the concept of Deborah number. It is presented in the sources available to me as the ratio between the relaxation time of a fluid and a characteristic time scale of the flow. In other sources the denominator of the ratio is occupied by the "observation time scale". Firstly, I struggle in front of the definition of a "relaxation time" for a "real world" fluid, characterised by a continuous spectrum of relaxation times. But the worst is to come, so let us assume a fluid with a clearly defined relaxation time is at hand. If an harmonic excitation with, say, frequency $\omega$, is applied to the viscoelastic and the flow observed for a duration of time $t$, how to define Deborah number? The relaxation time is fixed as a material property, but what to use in the denominator? The time scale linked to the frequency applied, or the observational time scale?


Constructive Interference of Electromagnetic Waves

So I was wondering if Electromagnetic wave has the same property of interference as normal waves. I understand that both the electric and magnetic parts of the wave would have to be in the same position at the same time. To negotiate the fact that only one part of the wave would match up at one time due to the fact that light can't go faster than it's self I had the idea for three waves all intersecting at the same point. All of these waves would be of different wavelengths so that they do not interfering with each other before the main point. If I knew of a good easy way off making a fbd I would but i am rather new to really doing physics Ioutside of a high school classroom. So I was wondering if constructive interference worked on electromagnetic waves and if it does whether or not is decrease's the wavelength (increasing the energy).


Should I abandon my thought experiment about time?

I'm trying to think about special relativity without "spoiling" it by looking up the answer; I hope someone can offer some insight - or at least tell me I'm wrong. Suppose I have an ordinary clock in front of me and I push it back with my hands. The force applied to the clock causes it to retreat away from me and after the push, it will travel away with uniform velocity. Suppose further, I can always see the clock clearly no matter how far away it is. Since the speed of light is constant, the light coming from the clock must travel a longer distance to reach my eye as it moves away. This would make time appear to slow down? If, on the other hand, the clock is moving towards me, the distance the light must travel to reach my eye becomes shorter and shorter, thus time would appear to speed up?


Can a number of gravitational slingshot stop a planet?

The answers to Where does the extra kinetic energy come from in a gravitational slingshot? state that in a gravitational slingshot the object being accelerated "steal" speed from the planet (or moon). Does that mean that an excessive number of g-slingshot could stop a planet?


Which fraction of light is refracted from a source of light under a lake?

I was trying to solve this problem: "A punctiform source of light is standing inside a lake, at a height h of the surface. f is the fraction of the total of energy emitted that escapes directly from the lake, ignoring the light being absorbed in the water. Given n, the refractive index of water, determine f." I understand that, since the maximum refraction angle is 90°, there is a maximum incident angle. The next image explains the principle: http://img441.imageshack.us/img441/9573/lake01b.png In it, the yellow incident light rays diverge (since the air is less dense than the water), until, at one point, the refractive angle is 90°. Then, the rays stop refracting. At this point, I applied Snell Law:               n1 * sin i = n2 * sin r     n * sin i = 1 (it's air) * sin 90     sin i = 1/n      Now let's analyze the following triangle: http://img690.imageshack.us/img690/8421/lake02.png As you can see, the triangle is formed by: _90 - i_ , _90 - i_ and _a_.               a + 90 - i + 90 - i = 180     a = 2i      The fraction of light that made out of the lake is _a_ over the total circle, that is, 360°. So:               f = 2i/360 = i/180     i = arc sin (1/n)     f = (arc sin (1/n))/180      However, the answer I have for this exercise (and it does seem to be right, because it is from a University*) is $f = \tfrac12 - \tfrac{1}{2n} \sqrt{n^2 - 1}$. And I don't know what I did wrong. It is very important for me to solve this exercise, and I hope someone would have a hint of what I am doing wrong. *It is a very old test (1969), and there is no resolution anywhere (just the final answer). * * * Second try, using Solid Angles: At is the total Area of the light sphere of radius h:               At = 4 * pi * rt²     rt = h     At = 4 * pi * h²      Ap is the partial area of the circle of light that gets out of the water:               rp = h / (tg(90 - i))          tg (90 - i) = sen (90 - i)/ cos (90 - i)     sen (90 - i) = sen 90*cos i - sen i*cos90 = cos i     cos (90 - i) = cos 90*cos i + sen 90*seni = sen i     tg (90 - i) = cos(i)/sen(i) = 1/tg(i)          rp = h*tg(i)     Ap = pi * rp²      So f must be Ap/At:               Ap/At = (pi * rp²) / (4 * pi * rt²)     f = h² * tg²(i) / (4 * h²)     f = tg²(i)/4      Still not there.


Coupled Oscillation Simulation

I'm looking for an online coupled oscillation simulation. The best I have got so far is this --- https://phet.colorado.edu/sims/normal-modes/normal-modes_en.html But I'm looking for something which has more options like changing the mass of the objects, changing the spring constants, cutting off the springs. Please suggest if you have come across better simulations. PS : I apologize if this question does not belong here. I need a coupled oscillation simulator for my work.


Scalar Particles, Flavor Changing Processes and Gauge Symmetries

Let's consider an extended version of the Standard Model (SM) with a new Yukawa operator of the form $$ \sum_\ell g_\ell\bar{\ell}\ell \phi ,$$ where $\ell$ is any lepton of the SM and $\phi$ is a new real spin-0 particle, which is assumed to be a singlet of $SU(2)_L$. This new term breaks the $SU(2)_L$ symmetry, but I'll not try to justify its existence. * * * Now, my question:   * I want to compute the loop correction to the vertex $\mu e\phi$, which does not exist in the original theory. One possible contribution for this term is show in the figure below (where I also suppose that neutrinos are massive). Does something guarantee that this loop computation will give a finite result in the framework presented here?  $\hspace{6.5cm}$![enter image description here](http://i.stack.imgur.com/rdP80.png),   * If this is not the case, what conditions must be imposed to the lagrangian in order to have finite contributions? Is it enough to have a Hamiltonian with dimension $d\leq4$ operators? Or is it essential to have a perfectly defined gauge theory?


How does rest mass become energy?

I know that there's a difference between relativistic rest mass. Relativistic mass is "acquired" when an object is moving at speeds comparable to the speed of light.Rest mass is the inherent mass that something has regardless of the speed its moving at. When fusion happens, a certain percentage of the rest mass of two hydrogen atoms is converted into energy. I understand that $E = m_{rest}c^2$. **How does the rest mass of an object turn into energy?** Does this involve particle physics? Doesn't this suggest that rest mass can be viewed as some sort of potential energy? It seems to me that it's a sort of locked energy that has to undergo a certain process to expel its energy to surrounding particles. Also, **Why does only a small percentage of rest mass turn into energy? Why not all of it. What dictates how much rest mass gets converted into energy?**


What are good examples to demonstrate Einstein's mass-energy relation

According to Einstein's mass-energy relation mass and energy are interchangeable. Can you provide some examples where:   1. Mass gets converted into energy.   2. Energy gets converted into mass.


How does energy convert to matter?

To my understanding, matter and energy are one and the same. Shifting from $E$ to $M$ in Einstein's famous equation requires only a large negative acceleration. If $M$ really is $E/c^2$, does that make matter the solid state of energy? I've read a lot about positron-electron collisions at high energies creating larger particles, and there is obvious matter conversion in fusion and fission reactions, but I can't find anything describing the physics of the conversion from energy to matter, rather than the interactions of what is already matter. Specifically, the thing I'm getting hung up on is the reason energy would take on a solid state in the first place. If energy is represented by waves, how does it become particles? If gravity is determined by mass, and mass is nothing more than static energy, does that make gravity a static- electromagnetic force?


What is the role of the speed of light in mass-energy equivalency?

Where does $c$ squared come into play in the equation $E=mc^2$. Multiplication obviously but how does energy equal mass times the speed of light?


Why are color values stored as Red, Green, Blue?

I learned in elementary school that you could get `green` by mixing `blue` with `yellow`. However with LEDs, TFTs, etc. you always have RGB (red, green, blue) values? Why is that? From what you learned in elementary `yellow` would be the 'natural' choice instead of `green`.


Uses of Vectors In Real Life

I always wonder how vectors are used in real life.Vectors and decomposition of vectors,dot and cross products are taught in the early stage in every undergraduate physics course and in every university.My question is how and where are vectors used? **Do physicists really use vectors in every day life? If so where?** I'am Looking for Motivation for learning Vectors.


A roadmap for learning standard model of particle physics

Assuming that a person has understanding of theory of Lie groups, Lie algebras and basic quantum mechanics, what is the simplest route to gain a basic understanding of the SM of particle physics? Are there any particular books suited for people with this background?


Book to read before "introduction to gauge field theory" by Bailin and Love

A teacher has recommended me to read this book in order to prepare for a project I am doing. Anyway, I feel that I should need a book in order to prepare for this one. Any suggestions?


How much energy from extreme coldness?

Let's say I have: 1: one mole of extremely cold ideal gas   2: unlimited amount of ideal gas at temperature 300 K   3: one ideal heat engine Can I generate for example 1 MWh of mechanical energy using those three things? Alternative formulation: When temperature of the cold gas approaches 0 K, what does the amount of generated energy approach? (there is no other heat sink than one mole of cold ideal gas)


Why does a capacitor discharge a percentage of the original energy in the same time?

If I charge a capacitor ($220\mu{F}$) using a 6V battery, and then measure the time it takes to discharge 90% of the initial energy over a resistor (${100k}\Omega$), and then charge the same capacitor using a 12V battery and measure the time it takes to discharge 90% of its initial energy again (over the same resistor). Why are both times the same? Especially given that the second time there is 4 times more starting energy that the first time. ($E=\frac{1}{2}CV^2$.)


How long does it take a object captured by a star falling to the center?

If the captured object do not have tangential velocity, it's just the free- fall time. But when it has, it may take longer time to fall in, right ? The function should be $\ddot{r} = -GM/r^2 + (v_0r_0/r)^2 / r = -GM/r^2 + v_0^2r_0^2 / r^3$ , where v_0 is the initial tangential velocity . After one integration, it becomes $\dot{r}^2/2=GM(1/r-1/r_0)-v_0^2r_0^2(1/r^2-1/r_0^2)/2 $ . I don't know how to deal with it. But I guess there is a analytic solution. Anyone knows something about it ?


Momentum paradox

A cistern rail car is standing on infinitely slippery ice. The cistern is filled with water and it has an outlet in the form of a thin vertical pipe (spout) at the left end, so when the valve is open the water can escape vertically downward (in the car frame of reference). Initially the system is at rest, the valve closed. Then we open the valve and the cistern starts moving (presumably to the right). However the water dropping out must be moving (slipping on the ice) in the same direction as the cistern, according to the spout geometry. So we end up with everything moving in the same direction (say, to the right) in spite of initially having zero momentum. How to resolve this paradox? What will be the motion of the cistern in the process of water leaking out? How will the water on the ground and the cistern be moving after all water leaks out? ![enter image description here](http://i.stack.imgur.com/lJYsB.jpg)


How do you integrate an expression over a variable in the limit of an integral?

I am trying to follow the steps to solve the integro-differential equation that arises from a plasma sheath problem given in this paper. This is the step I can't follow: > $$\epsilon_o\frac{d}{d\varphi}\biggl(\frac{E^2}{2}\biggr) = > \sqrt{\frac{m_e}{2e}}\frac{j_{eo}}{\sqrt{\varphi}} - > \sqrt{\frac{m_i}{2e}}\frac{j_{eo}}{\lambda_I}\int_\varphi^{\varphi_w}\frac{\varphi'/\varphi_I > - 1}{\sqrt{\varphi' - \varphi}}\frac{d\varphi'}{E'},\tag{4}$$ > > where $\lambda_I =1/\sigma_o n_a$ is the ionization mean free path for the > electron energy $E_I$. The integration of Eq. (4) over $\varphi$ leads to an > integral equation for the electric field $E(\varphi)$, > > $$\begin{multline}\frac{\epsilon_o}{4j_{eo}}\sqrt{\frac{e}{2m_e}}(E_w^2 - > E^2) = \bigl(\sqrt{\varphi_w} - \sqrt{\varphi}\bigr)\\\\- > \frac{1}{\lambda_I}\sqrt{\frac{m_i}{m_e}}\int_\varphi^{\varphi_w}\biggl(\frac{\varphi'}{\varphi_I} > - 1\biggr)\sqrt{\varphi(z') - > \varphi(z)}\frac{d\varphi'}{E(\varphi')},\tag{5}\end{multline}$$ The paper claims that instead of solving the integro-differential equation numerically from the form above, both sides of the equation can be integrated with respect to $\varphi$. I am not sure if it is valid to integrate both sides of this equation with respect to this variable since it appears in the lower limit of the integral. Can someone explain how you would handle integrating both sides when $\varphi$ is in the the limit of the integral? Thanks! Or if anyone has an argument for why a mistake might have been made in this step that would be helpful too.


Could each non-dependent physical contant represent dimentions, and our universe be a point on this n-dimentional structure?

For example say the gravitation constant instead of equaling G, was actually a range bounded between 0 and infinity. Our Universe would be at a point on this range (equal to our G value) where things could physically exist and produce stars, galaxies and life. Yet in the gravitation constant's dimension it is able to be lower and higher then our G value. All other physical constants being equal, this would allow the sort of full universe we recognise to exist on only a very tiny part of the full range of the gravitation constant's dimension. The rest of the dimension would be full of non-universes that would not be able to support anything. The other non-dependent physical constants perhaps also represent dimensions. And there might even be a set of useful life supporting universe's on a continuous 'line' inside this n-dimensional structure.


What is the optical power level of common fiber optics lasers?

In the book Nonlinear Optics, it is stated that the nonlinear effects start to become a problem in WDM systems (around 1550 nm) after about 1 mW of optical power. However, I measured the optical power at transmission of a transmitting laser of a 10GBASE-ZR on a short, 2 km link, and found that it was 2.5 mW. What is the operating optical power of telecommunications lasers? Does it differ for inter-continental links compared to short links, i.e. between buildings on a campus? If they are above 1 mW, are nonlinear effects a concern?


Why do we stop using optics for photons above a certain energy?

I'm reading about how the soon-to-be-launched NuSTAR is on the cutting edge of focusing x-rays, which captures 5 to 80 keV radiation by focusing them with optics that have a 10.15 meter focal length onto 2 sets of 4 32×32 pixel detector arrays. These are particular "hard" (high energy) x-rays, which is a part of what makes the task difficult and the NuSTAR telescope novel. If I understand correctly, imaging gets particularly difficult with electromagnetic radiation beyond a certain energy, as true gamma rays (above 100 keV) are detected with a family of radiation detectors that sense the Compton scatter or photoelectric absorption with an electrical pulse that is (in a naive sense) insensitive to the originating direction or location within the detector. It should be obvious that imaging can still be done with the use of an array of detectors, each constituting a single pixel, and these capabilities may improve with time as semiconductor detector technology evolves. So the critical distinction I'm trying to establish is between x-rays and gamma rays. It would seem that we focus x-rays and do not focus gamma rays. For a very good example of researchers _not_ focusing gamma rays, consider Dr. Zhong He's Radiation Measurement Group at UM, who do actual imaging of a gamma ray environment (the UM Polaris detector). They use a grid of room temperature semiconductors laid out bare in a room and use back-processing of the signals to triangulate a sequence of scatter-scatter-absorption reactions in 3D space. This is a lot of work that would be completely unnecessary if you could focus the gamma rays like we do for a large portion of the EM spectrum. Both of the technologies I reference, the NuSTAR telescope and the UM Polaris detector, use CdZnTe detectors. Functionally they are very very different in that the telescope uses optics to capture light from just a few arc-seconds of the sky. My question is what is the specific limitation that prevents us from focusing photons above a certain energy? It seems this cutoff point is also suspiciously close to the cutoff between the definition of x-rays and gamma rays. Was this intended? Could future technology start using optics to resolve low-energy gamma rays?


Why are the experiments performed by Felix Ehrenhaft on magnetic monopole and magnetic current so much ignored?

_It's really shocking that the following question was voted down twice although the question is yet to be answered by anyone._ So please read the following and give an answer why physics community has taken a double standard in the following case. We know about Michelson's experiment. We also know that despite of the null result how many times the experiment was revisited with the hope to get a positive result. Unfortunately, this is not the case for Ehrenhaft's experiments (http://www.rexresearch.com/ehrenhaf/ehrenhaf.htm). We have already performed a lot of experiments to find a real or artificial magnetic monopole but no monopole is found yet through any experiment. Furthermore, while studying literature on magnetic monopole, I have found that very few researchers even recognized the work of Ehrenhaft who published more than 60 papers on this topic. I don't understand why his work is so much ignored in the mainstream physics. **Is there a large number of references** (I don't expect this number to be comparable to the number of experimental attempts made in case of Michelson's experiment. I only expect a significant number of repetitions which can firmly conclude about the experimental outcome.) **which show that his experiments were revisited thoroughly, carefully and honestly but his claimed outcome could not be reproduced?** Please give your opinion about the following possible answers:   1. Michelson's NULL result was favorable to Einstein's Relativity, while Ehrenhaft's result was not.   2. Ehrenhaft's magnetic monopole was not in agreement with Dirac's theory of magnetic monopole.


Quantum mechanics book that emphasizes momentum space approach?

I have a strong background in Fourier analysis, and I'm looking for QM resources that can build on that. Is there a book around or a little above the level of Griffiths that has that kind of emphasis? Any other resources are good too.


Time travel outside of light cone without causality violation

If one is able to travel into the past but at a spatial distance that puts him outside of his own past light cone would this be considered a causality violating trip? Looking at a Minkoski diagram, it would seem that one ought to be able to travel to a spatially displaced past without producing causality violations. In fact, I'm not sure you could even say whether this was the past of not for sure. ![Generic Minkowski diagram](http://i.stack.imgur.com/U34I5.png) For instance: If you found a one way wormhole, that took you outside of your own lightcone but into the past, could you say for sure that you traveled through time? Would this be a form of timetravel that did not violate causality? Note; I'm not necessarily suggesting that it is an FTL trip, it may be considered to be instantaneous teleportation, but in the reverse direction of x'.


Force as change in momentum vs. change in velocity

Is there ever a situation where the distinction between $F = m \frac{dv}{dt}$ and $F = \frac{dp}{dt}$ is important? I can't think of a situation where one is true and not the other (assuming only conservation of momentum). * * * Edit: Obviously it is important to take a changing mass into account (e.g. for a rocket) when you're considering a full time evolution, i.e. $F(t) = m(t) \frac{dv}{dt}$ (or in the relativistic case, perhaps something like $F(t) = m(t) \frac{d}{dt} \left( \frac{p}{m} \right)$ with $m$ the rest-mass). And perhaps there is a nontrivial relationship between the rate of change of mass, and the forces being exerted (again, e.g. with a rocket --- where the mass loss is tied to the propulsion). What is not clear is that there should ever be a $F = v\frac{dm}{dt}$ term. * * * Edit 2: **My understanding of the solution:**   There should _not_ be a $dm/dt$ term, as pointed out by @garyp. The change in momentum expression is, however, more accurate because $p \neq mv$ in general (e.g. in relativistic cases, or when considering massless systems). It would seem that either one must take the caveat that $dp/dt$ _cannot be used for mass-varying systems_ , or take the much less conceptual or aesthetically pleasing expression that $F = m \frac{d (\gamma v)}{dt}$ ( _which still only applies to classical systems_ ).