title: Construction of the supergravity side explicitly in gauge/gravity dualities

text: Although the motivation of this question comes from the AdS/CFT correspondence, it actually is related to a more general principle of gauge/gravity duality. We know from Maldacena's conjecture that a $\mathcal{N}=4$ SYM theory is dual to a Type IIB String Theory on an $AdS_5 \times S^5$ background. Mathematically, the identification is \begin{equation} \left\langle \int \phi_0 \mathcal{O}\right\rangle_{SYM} = Z[\phi(\mathbf{x},0)=\phi_0] \end{equation} where $\phi_0$ are boundary fields and $\phi$ s are bulk fields while $\mathcal{O}$ are single trace operators of the boundary theory.\ Now, this comes mainly by studying the symmetries of the two theories. (Both of them have the symmetry groups $SO(2,4)$ and $SO(6)$). Also, in the low energy limit, i.e when $\alpha^{\prime} = l_s^2 \rightarrow 0$, we see that the two theories decouples from the system. In the bulk we have a theory of 10D free supergravity. Now, say I start with some gauge theory (not necessarily $\mathcal{N}=4$ SYM theory) on the **boundary which is Minkowski like**. Starting with that theory, how do we construct the correct gravity theory in the bulk that will be dual to the gauge theory on the boundary. What i understand is that there are some things about the bulk metric that we can say from the holographic principle. Since, we started with a gauge theory in Minkowski background, the metric in the bulk should be such that its conformal boundary must be **Minkowski**. My questions is:\ Is there an explicit prescription which will allow me to find out what is the theory in the bulk if I know what my boundary theory is?\ Also, there is this issue of the weak-strong coupling duality. the strong coupling regime of the gauge theory is dual to the weak coupling regime of the gravity side and vice versa. Anyways, we cannot evaluate the LHS in the strong coupling regime since, perturbation theory will just break down. So, we **have** to take the weak coupling limit on the LHS. But, that would imply working in the strong coupling regime on the gravity side. Then, we basically extend the result of AdS/CFT and we can say that we at least know the partition function of the gravity side when it is strongly coupled.


title: Aligning images of starfields

text: I have two images taken within 30 minutes of each other in the same part of the sky. They are very similar but are slightly offset due to the Earth's rotation and other factors. I know: the X, Y coordinates and RA/dec of the center of each image, and the pixel scale of both images (eg. 1.3 arcsec/pixel). I've tried treating the image as though it had a linear coordinate system, but the field of view is wide enough (~12 arcmin) that this isn't really accurate. I want to align these images so the stars overlap. How do I determine the X,Y pixel offset of these images so I can overlay them?


title: Phase shift for Scattering in radial potentials

text: given a radial potential in 3 dimension and its Schroedinguer equation $ -D^{2}U(r) + \frac{l(l+1)}{r^{2}}+V(r) $ here D means derivative with respect to 'r' then if we apply quantum scattering how can we calculate the PHASE SHIFT ?? $\delta $ , for a general potential ?? .. for example with the condition $ V(r) \to \infty$ as $ r\to \infty$


title: Quantum dimension in topological entanglement entropy

text: In 2D the entanglement entropy of a simply connected region goes like \begin{align} S_L \to \alpha L - \gamma + \cdots, \end{align} where $\gamma$ is the topological entanglement entropy. $\gamma$ is apparently \begin{align} \gamma = \log \mathcal{D}, \end{align} where $\mathcal{D}$ is the total quantum dimension of the medium, given by \begin{align} \mathcal{D} = \sqrt{\sum_a d_a^2}, \end{align} and $d_a$ is the quantum dimension of a particle with charge $a$. However, I do not quite understand what this quantum dimension is, or what a topological sector (with I guess charge $a$?) is. It is usually just quoted in papers, such as $\mathcal{D} = \sqrt{q}$ for the $1/q$ Laughlin state, $\gamma = \log 2$ for the Toric code... Could someone please explain? How do I know how many topological sectors a state (? system?) has, and how do I get its quantum dimension? In addition, I am guessing that a topologically trivial state (i.e. not topological state) has $\mathcal{D} = 1$. Would that be right? What makes a state be non-trivial topologically (i.e. have $\mathcal{D} > 1$)? thanks.


title: How to tackle 'dot' product for spin matrices

text: I read a textbook today on quantum mechanics regarding the Pauli spin matrices for two particles, it gives the Hamiltonian as $$ H = \alpha[\sigma_z^1 + \sigma_z^2] + \gamma\vec{\sigma}^1\cdot\vec{\sigma}^2 $$ where $\vec{\sigma}^1$ and $\vec{\sigma}^2$ are the Pauli spin matrices for two particles separately. I think $\sigma_z$ is the z component, I found that $$ \sigma_z = \left( \begin{matrix} 1 & 0 \\\ 0 & -1 \end{matrix} \right) $$ which is 2x2 matrix. I am wondering if the $\sigma_z$ is the same for particle 1 and 2? if so, $$ \sigma_z^1 + \sigma_z^2 = 2\left( \begin{matrix} 1 & 0 \\\ 0 & -1 \end{matrix} \right) $$ Is that right? The most confusing part is $\vec{\sigma}^1\cdot\vec{\sigma}^2$, there are two matrices involved, so how does the dot product work? I am trying solve for the eigenvalues of H, it looks like to me that each $\sigma_z^1$ and $\sigma_z^2$ is 2x2, so there are two eigenvalues, is that correct?


title: The shear stress assumption in Navier Stokes

text: Viscous fluid mechanics starts with the assumption that shear stress is linearly related to velocity by $$\tau = \mu\frac{du}{dy},$$ but it later turns out that some forces in high speed situations go as the square of the speed. This is really a reference request: are there critical discussions about the correctness of the linear assumption in view of experiments? This might be in every good textbook but I just haven't seen it. I'm framing this as an experimental topic because of the limited understanding (http://www.claymath.org) of Navier Stokes in a strict mathematical sense.


title: What's the highest speed one can obtain safely at home?

text: Among common household appliances, things one can make from stuff in the garage or hardware store, and reasonably safe, e.g. within reach of hobbyists and high school kids entering a science fair, what is the highest speed of motion of any bit of matter one can obtain? The bit of matter, or surface points of some object, should be macroscopic - big enough to see, time the motion of somehow, make collisions with other bits of matter. Electrons aren't big enough! Fluids are fine too. Guns, dynamite would be too dangerous. The tips of a fan blade or shutter edges in a camera are fine. Using a drill to make something spin real fast might be okay if all mad scientists involved are careful.


title: If I fall into an evaporating black hole, where do I end up?

text: This question has been bothering me for a while. I have a crude hypothesis... As I understand it, an observer falling into a black hole will cross the event horizon at some specific future (proper) time in, and that it will not be a traumatic event if the black hole is big enough (e.g. tidal forces will be quite mild). Also, the observer will see the universe above "speed up", and can see any future date arrive at a distant point _before crossing the event horizon_. Also, black holes evaporate, which may lead to some caveats about the previous two statements (which do not take evaporation into account). So suppose we have a large black hole, destined to evaporate and vanish in the the year 1050 AD. And suppose I jump into it, equipped with a telescope that lets me observe the Earth. Before I reach the event horizon I will see 1050 AD arrive on earth. At that point I will see astronomers on earth waving flags to indicate that they have seen the black hole vanish. So if I look "down" I will see empty space with no black hole looming. So where am I? If I'm just adrift in space, am I in a cloud of all the other objects that ever fell into the hole? Now for my crude hypothesis: as I fall, and the hole gets smaller, and the curvature near the horizon gets more acute, I'll be racked by tidal forces and blasted by Hawking radiation. Any extended body I happen to have will be disintegrated, so "I" will survive only if I'm an indestructible point, and the cloud of such particles is what astronomers see as the final flash of Hawking radiation. Is this even close to plausible?


title: Interesting Harmonic Oscillator Solution

text: On page 89 of Griffith's QM book, an exact solution to the time-dependent SE equation for the harmonic oscillator is mentioned: $$ \Psi(x,t)=\left(\frac{m\omega}{\pi\hbar}\right)^{1/4}\exp\left[-\frac{m\omega}{2\hbar}\left(x^2+\frac{a^2}{2}(1+e^{-2i\omega t})+\frac{i\hbar t}{m}-2axe^{-i\omega t} \right)\right] $$ He says that this solution was found by Schrodinger in 1927. I am wondering if anyone can tell me where this solution comes from or at least point me to a paper where it is discussed. I haven't been able to find much on it. Thank you for any help.


title: Force from solenoid

text: I'd like to approximate the force from a solenoid, or at the very least find a formula which is proportional to the force so that I can experimentally find the constant for my particular case. Apparently an exact answer to this is hard and involves quantum physics that is a bit beyond me; something about the Ising model. I've found $F =\frac{(NI)^2 μ_0A}{2g^2}$, where $N$ is number of loops, $I$ is current though the solenoid, $A$ is cross-sectional area of the inside of the solenoid and $g$ is the "is the length of the gap between the solenoid and a piece of metal". I found that here. Is this a decent approximation? If so could you more rigorously define $g$, because I don't know what distance this is supposed to be? If it is not, what might I use instead? I found the Gilbert model on Wikipedia that is apparently a good approximation of the force between magnets. Could I use this to approximate the force on a piece of iron from a solenoid?


title: Charge of an electric wire

text: If an electric current is flowing through an electric wire, can we consider that wire charged? The answer is required with a proof. Can we consider the wire to be charged positively or negatively?


title: The speed of an unconstant motion

text: ![ ](http://i.stack.imgur.com/X7ogH.png) There's already the solution for the problem but I still don't understand why the velocity can't be calculated by just $$ a = (v(t))'= -B_0 + B_1t \Rightarrow v(t) = -B_0t + 1/2B_1t^2$$ Also, both my solution and given solution tell that at t=0, the velocity will be 0. It is the point that I don't understand.


title: Tension of rope in the gravitational field of earth

text: > Two balls of mass $m$ each one are connected with mass-less rope with the > same length as the radius of earth. The system is in free fall. Prove that > the tension of the rope when the nearest (to the earth) ball's distance from > the earth surface is $R_E/2$ is: $T = \frac{32}{225} mg$ ![Illustration](http://i.stack.imgur.com/TgZjm.png) What I did is the following: $F_1$ is a gravitation force exerted on the nearest ball by the earth: $F_1=G \frac{M_Em}{(1.5R_E)^2}$ $F_2$ is a gravitation force exerted on the farthest ball by the earth: $F_2=G \frac{M_E m}{(2.5R_E)^2}$ $T=F_1-F_2=G \frac{M_E m}{(1.5R_E)^2}-G \frac{M_E m}{(2.5R_E)^2}=\frac{G M_E m}{R_E^2} \left (\frac{4}{9} - \frac{4}{25} \right)=\frac{64}{225} mg$ However, my answer is somehow twice bigger than what is expected. Where am I wrong? What am I missing?


1. Has martian sunset same spectra than this earthly bluish-violet sunset?    2. What about sunset on Mercury?


title: Vortex in D dimensions soliton

text: ![enter image description here](http://i.stack.imgur.com/bQYEE.jpg) let us consider the two-dimensional configuration shown in Fig. 3.1a. The lengths of the arrows represent the magnitude of φ, while their directions indicate the orientation in the $φ_1 -φ_2$ plane. Regardless of the behavior of the fields in the interior region, it is clear that the fields cannot be continuously deformed to a vacuum solution, such as that shown in Fig. 3.1b. > How can I get the idea of vortex from the above statement?


title: Does the unit of a quantity change if you take square root of it?

text: For example, I have a mass, m = 0.1kg and I square root it, giving me m = 0.316 (3s.f.) does the unit still stay as kg, or does it change?


title: How fair is it to say that all chemistry arises from failures of the ideal gas law?

text: I was reading here about how the ideal gas law assumes point masses and non- interaction. Is it fair to say that all chemistry arises from failures of that? Of course, such a sweeping generalization will be strictly false, but is it on the right track? (I'm reminded of Feynman, "All mass is interaction," although that seems to be getting at something deeper.)


title: How to find angular velocity of a point inner a circumference

text: Let's consider a cicumference that have the center in the origin of axes and rotates around x-axes. Let's stick a bar in a point $A$ of this circumference and at the end of the bar let's stick a mass point $P$. Let's call $\theta$ the angle that the radius $OA$ formes with z-axes and $\phi$ the angle that $AP$ formes with z-axes. How can I find the angular velocity of $P$? At the beginnig I have thought that angular velocity must be the same for all the points inner and on the circumference. And so I have thought that it could be $\dot \theta$. The book gives me this result for the kinetic energy of P: $\frac{1}{2}m(\dot \theta^2+\dot \phi^2+2\dot \theta \dot \phi cos (\theta-\phi))$. Considering that in this case the kinetic energy is only rotational and that general expression of kinetic rotational energy for a mass point is $\frac {1}{2}mr^2\omega^2$ I can't understand the origin of this result... and I can't understand why I have to consider the two angles. Could you help me?


Is there real materials have Lieb lattice structure? Some examples?


title: Connection between quantum physics and consciousness

text: Can someone explain the quantum physics-consciousness connection? In the double slit or quantum eraser experiments, the system behaves as a whole, with some apparent time independent traits. Invoking some kind of time independent but still causal physics, the problem could be simplified. But to include consciousness in the picture, we need better evidence. So I'm going to propose the following version of the double slit experiment. If consciousness was actually involved, a person could create or destroy the interference pattern by just looking at the slit measurement result even after it is recorded. To emphasize my point, imagine that the experiment is spread across 3 days. The first day experiment is done but the results are recorded into computer memory and no one looks at it. The second day someone comes in and just looks at the results of the slit detector. He/she could even encode some message by closing and opening his/her eyes while looking at the results. Since consciousness is involved, it should effect the interference pattern even after such an indirect and delayed measurement (by consciousness). Then on the third day a second person could just look at the interference pattern that was recorded two days earlier, and still see the interference pattern being created and destroyed and even read the encoded message by just looking at the slit measurements on the second day. Does this sound reasonable to you, or what am I missing?


title: Nuclear Salt Water Rockets: viability and follow-up

text: This is the original paper by R. Zubrin proposing the Nuclear Salt Water Rocket design. Basically the design is that a capillar set of pipes store a uranium salt- water solution, inside a cadmium matrix, which helps to keep it below criticality. The fluid is simply flown at high speed into a combustion chamber where it reaches criticality, heats quickly and is expelled from some unspecified nozzle. The paper is from 1991, and i'm sure that the design has been revisited many times, so there must be good critiques and discussion of the problems that would have to be addressed, but i could not find any serious follow-up literature about the viability of the design, just, you know, the usual blog and mailing list informal ranting that you can find from a simple google search. Have there been any simulations or attempts to evaluate the design with more detail in the literature that i'm not aware? The original paper doesn't go into much discussion about the temperatures, which i presume are quite high, and most of the informal viability analysis i've read about focus on that aspect.


title: Why do air bubbles stick to the side of plastic tubing?

text: I'm watching water with air bubbles flow through transparent plastic tubing. The inner diameter is a few mm. Bubbles typically are the same diameter as the tubing, with length about the same or up to several cm long. I am not sure what kind of plastic the hose is made of, but it's quite flexible. The general water flow rate varies but, guessing by eye, is in the range of about 1cm/sec to 10cm/sec. Often the bubbles will stick in one place, or break apart with one portion staying stuck. The next bubble along may dislodge, but then it may stay stuck at the same place. I assume there's some sort of contamination or electrostatic charges to explain this. What exactly is the physics of this? What relations are there between the type of plastic (or other material), properties of the fluid, temperature, flow rate and so on? If the physics is understood well enough, is there a way to simulate this phenomenon numerically, as CFD or something? Does simulation software exist?


title: Can a $\gamma$-ray photon give some of his energy to an atom and accelerate it?

text: I know gamma-ray photon can only give its momentum energy to the electrons of an atom. My question is: Can a photon give some of its momentum to the atom (including its nucleus) to give it heat or speed? If yes, can you tell me how much energy can it give?


title: Usage of helium in MRIs

text: More and more articles pop up on the shortage of helium, and on the importance of it. Its usage in MRI's spring to mind for example. I looked it up and found out that helium is used for its 'low boiling point' and 'electrical superconductivity'. So this gives me a couple of questions:   * How can the amount of helium be depleting? When we use helium (for purposes other than balloons) it stays on earth right? Since it doesn't dissappear, can't we 'recycle' the helium previously used for certain purposes and just use it again?    * We often hear that helium supplies are depleting at alarming rates. This makes me wonder; isn't every element we're using on earth depleting in supplies? Or are there elements which 'arrive' on earth at a faster rate than they're leaving earth? Beforehand I thought of carbon (in relation with the emission of carbondioxide), however, then I figured that the amount of carbon in the atmosphere is increasing while the amount beneath the ground is decreasing, thus making no difference to the amount of carbon on the earth as a whole.   * Why is helium the only element suitable for usage in MRI's? In other words, why are its properties so unique or rare? And what properties are those, besides 'low boiling point' and 'electrical superconductivity' ?    * Is there research being done towards replacing helium in its purposes by a more viable substitute?


title: Blue-shifting as opposed to violet-shifting

text: A recent XKCD comic implies that the sky is blue as opposed to violet due to human physiology, and that animals more sensitive to shorter wavelengths will perceive the Earth's sky as the shortest wavelength that they can comfortably see. Is this phenomenon also responsible for relativistic blue-shifting, as opposed to relativistic violet-shifting? In other words, might different creatures perceive bodies approaching them as violet or even shorter-wavelength colors that humans cannot see?


title: How do Zener diodes maintain the potential across their terminals?

text: My physics book has a topic about Zener diodes being used as voltage regulators in the reverse bias. Well, I'm curious to know how does a Zener diode maintain the potential across its terminals after it has undergone avalanche breakdown? Does it start conducting in full offering almost zero resistance? If so, how can there be a potential gradient across it? Is the principle that for high current change, there is a minimal and negligible change in potential across the Zener? But doesn't it behave as a pure conductor in avalanche breakdown? If so, how is it possible for there to be a drop in potential? After all it allows large amounts of current through it. Finally, can you keep answers somewhat simple?


title: The Equivalence principle of General Relativity and the Doppler Effect

text: I am studying General Relativity and am trying to understand the Equivalence Principle more thoroughly. Basically, it is said that if you are in a uniformly accelerated frame of reference in free space away from any gravitational field, let's say an elevator, there is no way for the observer in the elevator to know whether he is in an accelerated 'box' or a uniform gravitational field. But suppose a person is at rest in free space relative to the surrounding electromagnetic radiation (photons hitting the box are of equal wavelength in all directions). When the box accelerates uniformly, its speed increases in the direction of acceleration and thus the photons on the top of the box (the box is accelerating in the 'top' direction) will hit it with shorter wavelength that photons hitting the bottom. This would create a force imbalance, but we will say there is a rocket on the bottom which increases its thrust in order to maintain uniform acceleration. The person on the inside of the box can detect neither the rocket nor the photons. However, as the speed increases as a result of the acceleration, the box will be squeezed as a result of the speed increase (the force from the redshifted photons plus the rocket thrust always equals the force from the blueshifted photons hitting the top of the box resulting in an ever increasing compressive force). If the observer inside the box continually sends photons from the bottom of the box, which are then reflected back down, the observer will find that the time it takes the photon to make its trip gets increasingly less and thus measures that the box is shrinking (or that time is slowing down at an increasing rate?). The observer in a uniform gravitational field would find no such thing. For small accelerations (or equivalently weak gravitational fields) this difference would be small. For very large accelerations, the effect would be much more noticeable. Furthermore, Einstein showed that light should fall in a gravitational field because in the equivalent accelerating frame of reference, a beam of light could be emitted from the center of the box perpendicular to the direction of motion and it would 'trace' a parabolic path from the perspective of the internal observer as a result of the acceleration. But for a very large acceleration, again, the box would be squeezed, and the observer should therefore measure the light falling further than the observer in the equivalent gravitational field. So my question is, shouldn't effect of the equivalence principle depend on the magnitude of the acceleration/gravitational field and the strength of the surrounding electromagnetic field?


title: Does a battery in a vacuum produce a current?

text: I've always assumed that the strong resistance of air is the reason there is no flow of electrons between the terminals of a battery until a wire is connect. However, in a vacuum there is no resistive substance to impede the flow of electrons. Does this mean that in a vacuum an average AA battery will spontaneously have a flow of electrons jump out of one terminal to the other?


title: What is the value of a quantum field?

text: As far as I'm aware (please correct me if I'm wrong) quantum fields are simply operators, constructed from a linear combination of creation and annihilation operators, which are defined at every point in space. These then act on the vacuum or existing states. We have one of these fields for every type of particle we know. E.g. we have one electron field for all electrons. So what does it mean to say that a quantum field is real or complex valued. What do we have that takes a real or complex value? Is it the operator itself or the eigenvalue given back after it acts on a state? Similarly when we have fermion fields that are Grassmann valued what is it that we get that takes the form of a Grassmann number? The original reason I considered this is that I read boson fields take real or complex values whilst fermionic fields take Grassmann variables as their values. But I was confused by what these values actually tell us.


title: Velocity vs Time Bounce

text: Could someone please explain the trajectory of the ball that is bouncing in this picture... The vertical component of the velocity of a bouncing ball is shown in the graph below. The positive Y direction is vertically up. The ball deforms slightly when it is in contact with the ground. ![Ball track](http://i.stack.imgur.com/nHzSE.png) I'm not sure what the ball is doing and when, what happens at 1s?


title: Help an aspiring physicists what to self-study

text: This is probably not the kind of question you'll often encounter on this forum, but I think a bit of background is needed for this question to make sense and not seem like a duplicate: 2012 has been an annus horribilis in my life. I have lost a lot of close relatives in a sudden surge of cardiovascular diseases in my family. I also discovered I inherited genetic diseases and that I'll probably undergo the same fate sooner or later. One of the people I've lost is my father. We used to talk about physics all the time, and ever since I was 5 I kept telling him my dream was contributing to the field. When I lost him a couple of months ago, I used studying as an emotional outlet. He always emphasized the importance of academic excellence, and for this reason I got obsessed with studying and getting high grades even more than I ever did. I now am 1/80 in a top high school, but I am frustrated enormously. I find that I waste my time at high school, especially since I probably won't have as much time here as many other people do. I find the mathematics and physics boring and easy, and I feel like I'm wasting my time with certain classes which don't interest me at all (for example Latin). So I decided to study physics and mathematics outside of school. My school has been somewhat supportive, granting me a day per week off to do whatever I want, basically. I of course have a considerable amount of free time in addition to that day, since I ace almost every test without too much studying and without making my homework (not because I don't want to, but because I don't need to). I decided to self-study because I decided that life is too short (and mine will be even shorter, if I reach 50 I'd be lucky) to waste time. So my plan is to do at least the first 2 years of undergraduate physics in the 2 years I've got left at my high school. My main objective is to gain a mathematical and physical understanding of quantum mechanics, as advanced as I possibly can. I am currently studying Linear Algebra and Statistics, but I have a problem. I don't know what to study and, especially, in what order to study it. I have read literally read dozens of questions and answers as to what should be the mathematical/physical background for Quantum Mechanics (my future field of interest). But I find these to be too general, and I often am overwhelmed by it. In the same way you can get overwhelmed when you need to clean your house, but it’s so dirty that you don’t know where to start. So I would like your help. My current mathematical background:   * Basic differential calculus and no integral calculus, we will get that later on this year, however, I think it’s best for me to study it myself before we get it at school since it is crucial in physics. To show my level of differential calculus, this is about the toughest homework question we had to solve algebraically: _Given are the functions $f_p(x) = \dfrac{9\sqrt{x^2+p}}{x^2+2}$. The line $k$ with slope $2.5$ touches the function of $f_p$ at point $A$ with $x_A=-1$. Get the function of $k$ algebraically._   * Trigonometry and trigonometric functions. Again, as above, one of the toughest question we had to solve: _Given are the functions $f(x)=-3+2cos(x)$ and $g(x)=cos(x-0.25\pi)-2$. Get the functions $s(x)=f(x)+g(x)$ and $v(x) = f(x)-g(x)$ in the form $y(x)=a+bcos(c(x-d))$._   * Analytic Geometry (conic sections, tangency, bisections, you know the drill).   * And of course everything below this level. I probably forgot some things, but you can ask my in the comments if I know certain fields. We will get a lot more mathematics in the coming years, but I want you to disregard that fact when answering that questions. I want to self-study as much as I can, and my mathematics teacher is very fond of me, so if I know a topic before we get it in class, he will let me do other mathematics that I want (he even said this). So I won’t lose time by self-studying subjects we’ll get eventually, so don’t worry about that.  My current physics background (names of the chapters we discussed):   * Newton’s laws, Mechanical energy/forces    * Pressure and Heath    * Signal processing    * Electric currents (Ohm’s law, Series and parallel circuits, etc.)   * Again, everything below this level too (again, I’m probably forgetting stuff). Here exactly the same thing counts as with mathematics, we will get a lot more physics in the coming year, but again, disregard that. My physics teacher adores me, even more so than my mathematics teacher, so again, he won’t mind if I do something else if I know the material he’s discussing already. This is of higher level than American AP classes and British A-levels, keep that in mind. Now my question is, what mathematics and physics do I need to study, and my importantly, in which order do I need to study it, in order to have a basic understanding of quantum mechanics in 2 years? I know basic is a very general term, but I think you people, as people who studied it themselves, know what is realistic and achievable. I know this might seem like a duplicate of hundreds of previous questions, but it isn’t. All the other people asking this question have gotten answers that I don’t find suitable for me. Mostly the answers are from people who assume that you have to ‘have a basic understanding of this, a basic understanding of that’, etc. But how do I know what ‘basic means’? Also, now that you guys know exactly what I know and what I don’t, you can more finely tune the answers into my personal situation. As I said, currently I am doing Linear Algebra and Statistics, so you can omit those 2 from your answers, and start from the point I finished those 2 (which will be around January).   * p.s. If you want to recommend certain books, be my guest. If it's a good book, than money is no issue, I've saved up enough money throughout the years


title: Residues in QFT propagator

text: It is a well known fact that the location of the pole of a propagator (in QFT) can be interpreted as the physical mass. Is there an interpretation for the residue of the propagator? **Note:** I´m thinking of generalised propagator, not necessarily a propagator of a fundamental field.


title: Modification of Newton's Law of Cooling

text: Yesterday I randomly started thinking about Newton's Law of Cooling. The problem I realized is that it assumes the ambient temperature stays constant over time, which is obviously not true. So what I tried doing was to modify the differential equation into a system of differential equations, and taking the heat capacity of each into account. $$ \frac{dT_1(t)}{dt} = C_2(T_2(t)-T_1(t))\\\ \frac{dT_2(t)}{dt} = C_1(T_1(t)-T_2(t))\\\ T_1(0) = T_{10}\\\ T_2(0) = T_{20} $$ which had the following solution: $$ T_1(t)={\frac {C_{{2}}T_{{20}}+C_{{1}}T_{{10}}}{C_{{1}}+C_{{2}}}}+{\frac {C_{ {2}} \left( -T_{{20}}+T_{{10}} \right) {{\rm e}^{- \left( C_{{1}}+C_{{ 2}} \right) t}}}{C_{{1}}+C_{{2}}}}\\\ T_2(t)={\frac {C_{{2}}T_{{20}} +C_{{1}}T_{{10}}}{C_{{1}}+C_{{2}}}}-{\frac {C_{{1}} \left( -T_{{20}}+T_{{10}} \right) {{\rm e}^{ -\left(C_{{1}} +C_{{2}} \right) t}}}{C_{{1}}+C_{{2}}}} $$ When I plotted the two equations out they seem to be right. They also follow the fact that at any point in time $C_1\Delta T_1=C_2\Delta T_2$. The end behaviour also seem to be correct. However, in cases where $C_1\rightarrow\infty$ you would expect it to behave similarly to Newton's Law of Cooling, but in reality $T_2$ drops its temperature in a very short period of time, which doesn't seem to be right. I tried looking this up but couldn't find much on this topic. If anyone can point out what I did wrong that would be great. I've had very little experience with thermodynamics but I do realize how limited these models are at describing real world scenarios. This was more for the fun of playing with differential equations.


title: Energy source of electrons?

text: I am aware that electrons are moving in an empty space so basically there is no friction to slow it down and its velocity stays the same. However where did the electron get its energy from in the first place(during the creation of the universe"Big bang"). Plus when we "touch" there is no physical contact.The electron's negative charges oppose themselves and repel each other.Basically I'm floating now in my chair.However when electrons push against each other doesn't that mean there is a force acting on the electron and pushing it away from its trajectory.So why doesn't everything fall apart when we sit on a chair or grab a pencil,why wont the electrons fall from trajectory and get caught by the protons.


title: Are there any practical applications of the uncertainty principle

text: I believe GPS works because of extremely small time differences between the satellites. Because of how small the time differences are, it needs to take into account gravity's effect on time. Although gravity warping time seems very far fetched in normal life, there are practical applications that we use every day that would not work if we did not know about it. I was curious if the same applies to the uncertainty principle. Is there something that I use everyday that wouldn't work if we didn't know about the uncertainty principle? Or is it just crazy quantum mechanics?


title: purely hypothetically, could a quantum being appear to be human? I ask for the purpose of research on a novel

text: I am a budding novelist, and im researching a few things for a fictional narrative. the characters are from other dimensions and i want to present a coherent scientific structure. any suggestions orhelp would be appreciated.


title: Is it outside science to figure out the precise string compactification of our universe?

text: String theory has at least $10^{500}$ 6D compactifications. Denef and Douglas proved the computational complexity of finding a compactification which fits the parameters of the low energy effective action like the cosmological constant, the electroweak scale, the Higgs self-coupling, Yukawa couplings and gauge coupling strengths is NP-complete. There are limits on the energies future particle accelerators can reach, and cosmic ray frequencies. The amount of data gleanable from the cosmic microwave background anisotropies is limited, and can only probe up to the energy scale of slow roll inflation. If it is beyond our abilities to determine the exact string compactification describing our universe, is it not wrong to call the compactification _unobservable_? Only things which are experimentally observable, possibly with the aid of theoretical interpolations, and can make experimentally verifiable predictions count as science. Within the realm of physics, are we forbidden to ask which string compactification describes our universe? Must we stick to the effective field theory if we wish to remain physicists and not philosophers?


title: How come evaporation is possible?

text: If I have wet clothes and put them on the ground and let the Sun shine on them, they dry as water evaporates. But water should evaporate at 100 degrees Celsius so how come they go dry if it's just about 30 degrees Celsius? What am I missing?


title: How can water evaporate at room temperature?

text: Boiling point of water is 100 degree Celsius. The temperature at which water in liquid form is converted into gaseous form. Then how it possible for water to evaporate at room temperature?


title: Why does evaporation take place?

text: I was looking at at puddle when I thought that this puddle will evaporate tomorrow but then it occurred to me that the boiling temperature of water (aka to turn into gas) is $100$ degrees under 1 atmospheric pressure of earth. Then I thought the temperature around me was 25 degrees which is way away from temperature of boiling therefore it should in theory or at least in my perspective never evaporate the water, yet nature disagrees and does this so. Therefore I proposed an hypothesis (Still in Middle School so please excuse me) the water turns into gas as a result of the photons "hitting" the atoms and transferring its energy into the water atoms which in turn moves the atoms giving it $KE$ (Kinetic Energy) and this happens over a millions of times which results in water particles gaining this energy and therefore evaporating without substantial notice or feel of it bubbling or showing any sign of it as this happens very slowly. However I as a middle school student am not too considerate or confident on my hypothesis so would you say my hypothesis is correct or wrong and can you explain why this happens?


title: Condensed matter physics for mathematicians

text: What is a good way for me to learn the basics of condensed matter physics? I'd like to get a better understanding of the fundamentals behind recent technological developments like OLEDs, applications of graphene, or get a grip on what the fractional hall effect is about. I don't expect to do research in these areas, but I would like to be able to have meaningful conversations with people working in these areas. My background is this: I have a PhD in pure mathematics, I even studied a little solid state physics (eg. Bloch waves), QFT and a first course in String Theory, but a long time ago now. My mathematics is much stronger than my physics intuition so I'm looking for a more mathematical treatment. My lab experience is precisely zero. I'm more interested in the theoretical ideas than precise details. Are there any books or downloadable lecture notes that might be recommended for someone in my position?


title: Mathematical rigorous introduction to solid state physics

text: I am looking for a good mathematical rigorous introduction to solid state physics. The style and level for this solid state physics book should be comparable to Abraham Marsdens Foundations of mechanics or Arnols mechanics book for classical mechanics or to Thirrings Physics course for quantum mechanics. Any recommendations? **Edit** : As a reaction to Peter Shor's comment, I try to narrow the scope of the question a bit and give some more specific subareas of solid state physics I am in particular interested in:   * semiconductors and applications   * the quantum hall effect   * superconductivity


title: Is the quantum analog of a probability distribution the wave function or the density matrix?

text: Classically, probability distributions are nonnegative real measures over the space of all possible outcomes which add up to 1. What they mean is open to debate between Bayesians, frequentists and ensemble interpretations. A degenerate distribution is the least random distribution with a probability of 1 for a given fixed event, and 0 for everything else. What is the analog of a classical probability distribution in quantum mechanics? Is it a wave function augmented with the Born interpretation for probabilities, or is it the density matrix? Does a pure density matrix correspond to a degenerate distribution?


title: Einstein’s Thought Experiment of a Stationary Box and a Photon

text: > 1) First of all, let us consider a particle of light, also known as a > photon. One of the interesting properties of photons is that they have > momentum and yet have no mass. This was established in the 1850s by James > Clerk Maxwell. However, if we recall our basic physics, we know that > momentum is made up of two components: mass and velocity. How can a photon > have momentum and yet not have a mass? Einstein’s great insight was that the > energy of a photon must be equivalent to a quantity of mass and hence could > be related to the momentum. > > 2) Einstein’s thought experiment runs as follows. First, imagine a > stationary box floating in deep space. Inside the box, a photon is emitted > and travels from the left towards the right. Since the momentum of the > system must be conserved, the box must recoils to the left as the photon is > emitted. At some later time, the photon collides with the other side of the > box, transferring all of its momentum to the box. The total momentum of the > system is conserved, so the impact causes the box to stop moving. > > 3) Unfortunately, there is a problem. Since no external forces are acting on > this system, the centre of mass must stay in the same location. However, the > box has moved. How can the movement of the box be reconciled with the centre > of mass of the system remaining fixed? > > 4) Einstein resolved this apparent contradiction by proposing that there > must be a ‘mass equivalent’ to the energy of the photon. In other words, the > energy of the photon must be equivalent to a mass moving from left to right > in the box. Furthermore, the mass must be large enough so that the system > centre of mass remains stationary. My questions: 1) I'm not able to grasp the concept of centre of mass in paragraph (3). 2) What's the center of mass of the system of the box and photon? 3) If no external forces are acting on the system, does the location of the center of mass remains the fixed? Then what does it mean by the location of center of mass being fixed if the box has moved? 4) What's the relation between the mass being large enough and the center of mass to remain stationary in paragraph 4?


title: Different representations of the Lorentz algebra

text: I've found many definitions of Lorentz generators that satisfy the Lorentz algebra: $$[L_{\mu\nu},L_{\rho\sigma}]=i(\eta_{\mu\sigma}L_{\nu\rho}-\eta_{\mu\rho}L_{\nu\sigma}-\eta_{\nu\sigma}L_{\mu\rho}+\eta_{\nu\rho}L_{\mu\sigma}),$$ but I don't know the difference between them. Firstly, there is the straightforward deduction evaluating the derivate of the Lorentz transformation at zero and multiplying it by $-i$. It's a very physical approach. Another possibility is to define: $$\left(J_{\mu\nu}\right)_{ab}=-i(\eta_{\mu a}\eta_{\nu b}-\eta_{\nu a}\eta_{\mu b})$$ This will hold for any dimension. I find it a bit confusing because we mix matrix indices with component indices. We could also define: $$M_{\mu\nu}=i(x_\mu\partial_\nu-x_\nu\partial_\mu)+S_{\mu\nu}$$ Where $S_{\mu\nu}$ is Hermitian, conmutes with $M_{\mu\nu}$ and satisfies the Lorentz algebra. I think this way is more geometrical because we can see a Lorentz transformation as a rotation mixing space and time. The two last options look quite similar to me. Lastly, we could start with the gamma matrices $\gamma^\mu$, that obey the Clifford algebra: $$\\{\gamma^{\mu},\gamma^{\nu}\\}=2\eta^{\mu\nu}\mathbb{I}$$ (this is easy to prove in QFT using Dirac's and KG's equations). And define: $$S^{\mu\nu}=\frac{i}{4}[\gamma^{\mu},\gamma^{\nu}]$$ It seems that this is the most abstract definition. By the way, how are Clifford algebras used in QFT, besides gamma matrices (I know they are related to quaternions and octonions, but I never saw these applied to Physics)? Are there any more possible definitions? Which are the advantages and disadvantages of each? Are some of them more fundamental and general than the others?


title: Behaviour of individual terms in Einstein-Smoluchowski fluctuation-dissipation relation

text: Consider a bath of Brownian particles at temperature $T$. If we sprinkle some larger particles in this (eg: pollen grains in water or dust motes in air), they'll diffuse with diffusion constant $D$ due to bombardments by the Brownian particles. For the same bombardments, any acceleration of these bigger particles due to an external force will die down to a terminal velocity $v_t=F/\gamma$, where $\gamma$ is a damping coefficient. The relation between their fluctuation and dissipation is given by a fluctuation-dissipation equation: $\gamma D=k_BT$ (Einstein-Smoluchowski relation) I now have a basic question regarding the behaviour of the individual terms on the left side. Suppose I were to slowly change just the temperature of the bath. That would change the product $\gamma D$. But how would $\gamma$ and $D$ separately change? Drawing an analogy from the ideal gas state equation $PV = k_BT$, their individual behaviour might depend on the particular process in which I change $T$. So assume that my system (say a bath of water with pollen grains) remains at atmospheric pressure and at the same volume as I just notch up the temperature of the heat bath. How would $\gamma$ and $D$ change then?


title: Rutherford Scattering Spectra

text: I am using SimNRA to simulate the classical Rutherford Scattering. Playing around with it, I came across some spectra that I cannot explain... First of all, if someone plots the spectrum of scattering angle $\theta=0$ will get one! I expected that I won't get any spectrum at all. Are there any higher order terms apart from $1/\sin^4\theta$? Secondly I get three different kind of spectra. For angles $0<\theta<60$ I get a perfectly gaussian distribution ![enter image description here](http://i.stack.imgur.com/9MiR6.jpg) If I am around $90^\circ (80<\theta<120)$ I get a rhather confussing spectrum which looks like that ![enter image description here](http://i.stack.imgur.com/nrIKN.jpg) For scattering angles $120<\theta<180$ I get a weird spectrum as well ![enter image description here](http://i.stack.imgur.com/Fn9MP.jpg) Why am I getting so different a spectrum for those scattering angles ranges? I've read that the last one is used in RBS to define the thickness of a leyer but why does it have that specific behaviour? And what is this small peak around $90^\circ$? Could that be a recoil gold atom? If so, I cannot understand how a light $4.7\;MeV$ particle can move away a heavy atom... Any help or hint will be more than welcome!!! **EDIT** All the above are for a target of $2\mu m$ thickness and a beam with $500keV$ spread. If I turn off the energy spread the backscattering specrtum has the same flat distribution. ![](http://i.imgur.com/H9IXIY0.jpg) In addition if the target is $5\mu m$, with no energy spread the spectrum is ![](http://i.imgur.com/7kSpoaf.jpg)


title: If starting speed is faster than terminal velocity then what?

text: If an object is say thrown down (vertically) at an initial speed that is faster than its terminal velocity, what would happen to that objects speed? Would it slow down?


title: is the nature of particle beam weapons in science fiction true to the reality of particle physics?

text: I am referring to the use of specific particle types such as an antiproton beam, positron beam, meson beam or muon beam for example in the likes of shows like star trek. I was curious if a beam of a certain particle would produce specific effects or is this purely wishful thinking. I am by no means a physicist so straightforward answers would be great if possible. If the answer could describe what effect each particle would have on materials, which particles are of particular use and which ones are impractical. Would it matter if the particle is normal or antimatter? I understand there is a lot of variables and factors like the way a particle decays, the particle half life and velocity affects its range distance, the energy of each particle, the materials the particles interact with, etc etc so any input is appreciated. To summarize, i am asking in terms of the realistic attribution of weapon effects to certain particle types and also non-weapon effects.


title: How does Earth's interior dynamo work?

text: I'm interested in getting a basic physical understanding of how Earth's magnetic field is generated. I understand that it's a "dynamo" type of effect, driven by convection currents in the molten outer core. These currents cause charges to move, and this generates the field. However, what I can't find a good explanation of is why there is a separation of charges in the first place. Presumably, moving neutrally charged molten iron would have the same effect as moving any other neutrally charged thing, i.e. it wouldn't create a field. And presumably, if the fluid wasn't moving then it would become neutral pretty quickly, since molten iron is a good electrical conductor. So am I right in thinking that the charge separation is the result of positive feedback, in that an intial deviation from neutrality would generate a field, and this would (somehow) cause a greater separation of charges, resulting in a kind of self-maintaining charge separation? Or is there another explanation? In either case, does anyone know of a good resource that explains the basic principles in physical terms? I know that the interior dynamo is a very complex phenomenon, but I'd like something that gives a good physical picture of how the electromagnetic and fluid dynamical phenomena interact, rather than diving straight into partial differential equations.


title: Does the Relative density of water change based on the state it is in

text: Will Relative density of water change based on the state it is in? Ie solid, liquid, gas. What causes this change(if any) in Rd?


title: From Lagrangian to Hamiltonian in Fermionic Model

text: While going from a given Lagrangian to Hamiltonian for a fermionic field, we use the following formula. $$ H = \Sigma_{i} \pi_i \dot{\phi_i} - L$$ where $\pi_i = \dfrac{\partial L}{\partial \dot{\phi_i}} $ In a Lagrangian involving fermionic fields given by, $$ L = \dfrac{1}{2}(\bar{\psi_i} \dot{\psi_j} - \dot{\bar{\psi_i}} \psi_j)$$ a direct computation gives $\pi_{\psi_j} = -\dfrac{1}{2}\bar{\psi_i}$ and $\pi_{\bar{\psi_i}} = -\dfrac{1}{2}\psi_j$. But on adding a total derivative $\dfrac{1}{2} \dfrac{d}{dt} (\bar{\psi_i} \psi_j)$ to the Lagrangian (which can always be done as the action won't change) but $\pi$'s become different. So the Hamiltonian as well changes. How do we resolve the issue ?


title: Regulator-scheme-independence in QFT

text: Are there general conditions (preservation of symmetries for example) under which after regularization and renormalization in a given renormalizable QFT, results obtained for physical quantities are regulator-scheme-independent?


title: What is the significance of the QCD scale parameter $\Lambda$?

text: I see that it appears as a constant in the relation for the running of the strong coupling constant. What is its significance? Does it have to be established by experiment? Is it somehow a scale for quark confinement? If yes, how? I ask because I saw this in Perkins' _Particle Astrophysics_ > _After kT fell below the strong quantum chromodynamics (QCD) scale parameter > ∼ 200 MeV, the remaining quarks, antiquarks, and gluons would no longer > exist as separate components of a plasma but as quark bound states, forming > the lighter hadrons such as pions and nucleons._


title: Having the same number of neutrons

text: Sorry if this is a silly question. If I understand correctly, for two atoms "having the same number of protons" is equivalent to "being of the same element", while "having the same number of protons and the same number of neutrons" equates to "being of the same isotope (of the same element)". But does "having the same number of neutrons" in itself have some significance in physics? And what about "having the same total number of protons and neutrons (but not necessarily with the same summands)"?


title: What does the LHC do afterwards?

text: Astronomical telescopes are now mega projects and cost $1Bn and although they are pitched to solve the current interest of the day they are general purpose machines and with upgrades and new instruments have a life of perhaps 50years. It seems that large accelerator projects are built to answer one question, to find one particle. But since the design must be based around the particle having a particular energy and the cost and timescale being so large - you have to be pretty damn sure that you expect the particle to exist and at the predicted energy. It almost seems that if you had a good enough estimate to build the accelerator then you don't really need to! Is something like the LHC a one trick pony? You turn it on and confirm the Higgs or if not - build a bigger one? Is the LHC really a more general purpose experiment but the Higgs gets the press attention or is it just that the nature of discover in HEP is different and you need to build a single one shot experiment?


title: On use of Hamiltonians for Helium

text: The Hamiltonian of helium can be expressed as the sum of two hydrogen Hamiltonians and that of the Coulomb interaction of two electrons. $$\hat H = \hat H_1 + \hat H_2 + \hat H_{1,2}.$$ The wave function for parahelium (spin = 0) is $\psi(1,2) = \psi_S(r_1, r_2)\dot \xi_A(s_1, s_2)$ with the first being a symmetric spatial function and the second being an antisymmetric one. We can separate this into the normalized function $\psi_S(r_1,r_2) = \frac{1}{\sqrt{2}}[\psi_1(r_1)(\psi_2(r_2)+ \psi_1(r_2)(\psi_2(r_1)]=\psi_S(r_2,r_1)$ For orthohelium the functions look like this: $\psi(1,2) = \psi_A(r_1, r_2)\dot \xi_S(s_1, s_2)$ $\psi_A(r_1,r_2) = \frac{1}{\sqrt{2}}[\psi_1(r_1)(\psi_2(r_2) - \psi_1(r_2)(\psi_2(r_1)]=-\psi_A(r_2,r_1)$ Show the ground state of helium is parahelium. The hint is what happens to the wavefunction. OK, so I start with that the Hamiltonian of the given wave function(s) $$H_1 = \frac{\hbar^2}{2m}\frac{\partial \psi}{\partial^2 r_1^2} = E_1 \psi$$ $$H_2 = \frac{\hbar^2}{2m}\frac{\partial \psi}{\partial^2 r_2^2} = E_2 \psi$$ $$H_{1,2} = -\frac{e^2}{4\pi\epsilon_0 r_{1,2}}$$ OK, I was trying to get a handle on how to get started with this. So I wanted to check if what I have above is "allowed" -- that is, is the second derivative (the nabla, really) of the psi functions treatable this way, since they all have two variables (really two position vectors) in them? Basically this is all about how to set up the initial differentials I would solve. EDIT: One thing I thought of doing was this (for $H_1$): $H_1 = -\frac{\hbar^2}{2m}(\frac{\partial \psi}{\partial^2 r_1^2}+\frac{\partial \psi}{\partial^2 r_2^2})=E_1(\psi_1(r_2)+\psi_1(r_1))$ but again I don't know if that's kosher.


title: How to determine Young's modulus of a wire by Searle's method?

text: The question is: > A student performs an experiment to determine the Young's modulus of a wire, > exactly 2m long, by Searle's method. In a particular reading, the student > measures the extension in the length of the wire to be 0.8mm with an > uncertainty of 0.05mm at a load of exactly 1.0Kg. The student also measure > the diameter of the wire to be 0.4mm with an uncertainty of + or - 0.01m. > Take g=9.8m/s^2(exact). Calculate the Young's modulus. Explain me what is Young's modulus and how it is related to Searle's method. Is there any equation to solve this problem?


title: Why is spacetime curved by mass but not charge?

text: It is written everywhere that gravity is curvature of spacetime caused by the mass of the objects or something to the same effect. This raises a question with me: why isn't spacetime curved due to other forces or aspects of bodies? Why isn't it that there are curvatures related to the charge of a body or the spin of particles or any other characteristics?


title: Electron Charge is 150%?

text: Is there a theory for why the charge of an electron is precisely 50% larger (magnitude) than a quark's? I have usually thought of this the other way around: the charge of a quark being 2/3 (or -2/3) that of an electron. Is there something fundamental in space/time/mass/etc that makes it a nice proportion? (I understand that this question is not worded perfectly to cover the entire list of standard model components. I tried to word it to cover proton charges and positron charges etc but it became too complicated. I opted for simple.)


title: How electrons act under rotating magnetic field?

text: I study Power Engineering in University. Today I asked my lecturer to explain me exactly how atom's electrons act under spinning rotor's magnetic field, that generated dynamic electricity. But he even didn't gave me to finish my question and said : "Explaining it to you is useless, you are not capable of understanding it". I felt very angry hearing these words, like I was an idiot. So I have some thoughts about electrons behavior under spinning rotor's magnetic field and I want your answer, is this theory correct and if it's not than why? **here is my theory:** Consider this is simple atom: ![enter image description here](http://i.stack.imgur.com/JMHTx.jpg) When magnetic field cross atoms, if magnet's north side (+) is near to this atom the center of electron's path (gray circle) will not be on the center of nuclear, it will move toward magnet, because magnet has north side (+) and electron has (-) potential. As I believe the Voltage (U) is distance between center of nuclear and the center of electron's path. And the number of these atoms make amps. ![enter image description here](http://i.stack.imgur.com/5jRY7.jpg) That's why increasing rotor's magnetic field makes more voltage, and that's why air's ionization happens around high voltage wires (voltage like: 220 kv, 500 kv). Nuclear has not enough power hold electron and so this electron moves to new air's nuclear. I think that also explains why Current and Voltage are 90 degrees out of phase. Please read my theory and tell me I am right or not, I really want to understand how this everything is done. **_Also my lecturer told me that every information I can find on internet is written by fools and foolers are reading them. So I want to tell him that he is WRONG!!!_**


title: What would be an appropiate shape for a parachute?

text: I'm trying to design a parachute that minimizes the descent velocity, but I'm not sure what shape I should use. From what I've read, ellipse-shaped parachutes are too aerodynamic and minimize drag, while squares are good enough to maximize drag, but I've read this from very unreliable sources and unfortunately I've never taken a fluid dynamics course so I'm not sure if those answers are right.


title: What causes a spark to move along rods that are not parallel?

text: I took my son to a science museum where they had a gadget that many of us probably saw in movies involving a mad scientist. The gadget had two metal rods about two inches apart at the bottom. The rods were about six feet long, and four inches apart at the top. An electric spark would start at the bottom where the rods are about two inches apart. Then the spark would move up to the end where the rods are about four inches apart. The spark took about three seconds to get from one end to the other. After the spark got to the end, it would start again at the end where the rods are close together. I am fairly sure the spark is caused by a high voltage between the rods. What causes the spark to start at one end, and move to the other end?


title: What law of electro-magnetics explains this?

text: I took my son to a science museum where they had a solenoid oriented vertically with a plastic cylinder passing through the solenoid. An employee dropped an aluminum ring over the top of the cylinder when there was no current going through the solenoid. Then they turned on the current going through the solenoid and they aluminum ring went flying up and off the top of the solenoid. What law of electro-magnetics causes the force on the aluminum ring?


title: What does the notation $|x_1,x_2\rangle$ mean?

text: I would like clarification on an equation in the paper "Free matter wave packet teleportation via cold-molecule dynamics", L. Fisch and G. Kurizki, _Europhysics Letters_ **75** (2006), pp. 847-853, DOI: 10.1209/epl/i2006-10205-7. The paper talks about entangling two particles translationally, meaning that two particles' position and momenta are correlated such that a precise measurement of particle 1 will cause particle 2's spread in momenta to be uncertain, vice versa. So the equation is equation (2) in the paper, $$\langle x_1, x_2 | \Psi \rangle= N e^{-\left({x_+}/{2\Delta x_+}\right)^2}N e^{-\left({x_-}/{2\Delta x_-}\right)^2}$$ where $x_+ = (x_1 + x_2)/2$, $x_- = x_1 - x_2$, and $N$ is a normalization constant. I'm assuming that the $\Delta x_\pm$ are the standard deviations of $x_\pm$. I've never seen bra-ket notation with "$\langle x_1,x_2|$" in it. This confuses me a lot! It doesn't make sense to have $x_1$ (comma) $x_2$. What the heck does this mean? I am interpreting this as the expectation value of the positions of the two entangled particles where $|\Psi\rangle$ is the wave function of two translationally entangled particles. Can someone please help me?


title: What would happen if the Earth was tidally locked with the Sun?

text: I'm thinking of writing a short story set on a version of Earth that is tidally locked to the Sun. I'm not exactly sure how to research the topic. Here's a number of questions about what would happen:   * How hot would the light side get? Are we talking ocean-boiling levels? I imagine that life would eventually flourish, given the massive constant energy source. Is this accurate?     * On that note, I imagine massive thunderstorms along all the coasts due to increased evaporation. How bad would they get? Would the ground ever see the Sun, or only rainfall?   * How cold would the dark side get? Is it conceivable that any life could still exist there? (Life has proven itself quite versitile in the past, i.e. life at the bottom of the ocean.)   * What wind speed would the twilight zone experience? I imagine the atmosphere would transfer heat from one side to the other, but would the wind speeds be bearable? In what direction would air flow?   * I hear that the oceans would recede into disjoint northern and southern oceans if the world stopped spinning. Would this also happen if the Earth became tidally locked?   * Would the Sun create a 'tidal' bulge in the ocean at the apex of the light side? Would this or the above dominate ocean behavior?   * Would we completely lose the magnetic field? Would life be able to survive without such shielding from magnetic radiation?   * Would the Moon eventually unlock the Earth? What state would the Moon have to be in for there to be both a locking between the Sun and the Earth as well as the Earth and the Moon?   * What other radical differences would exist between our Earth and a tidally locked alternative?


title: The discrepancy between the two values in units of sigmas?

text: Assuming gaussian error distributions, how to calculate the discrepancy between the two values in units of sigmas?


title: Wasn't the Hawking Paradox solved by Einstein?

text: I just watched a BBC Horizon episode where they talked about the Hawking Paradox. They mentioned a controversy about information being lost but I couldn't get my head around this. Black hole thermodynamics gives us the formula $$S ~=~ \frac{A k c^3 }{4 h G}.$$ And we also have Einstein's famous $E = m c^2$, which mean that mass can be turned into energy, right? Hence information is either lost or it is preserved, but now in energy-form instead of mass-form. I can't understand why radiation from black holes would be any different than an atomic bomb for example, where mass is also turned into energy?


title: Equality of electric charges of all leptons

text: What does it precisely mean the often repeated statement that the electric charges of all leptons are the same. Let's consider QED with two leptons: electron and muon. The interaction part of the bare lagrangian contains two electron-electron-photon and muon-muon- photon vertices with some coupling constants $e^{bare}_e$ and $e^{bare}_\mu$ respectively. After division of lagrangian into two parts: finite part and counterterms there are mentioned vertices in each part and the coupling constants in front of them are $e_e$, $e_\mu$ (finite/physical coupling constants), $\delta e_e$, $\delta e_\mu$ (become infinite when regularising parameter $\epsilon \rightarrow 0$; $\epsilon$ - deviation from dimension 4 in dimensional regularisation). I suppose that the equality of charges of electron and muon means that 3-point vertex function $\Gamma^{(3)}$ at some fixed point $(p_1,p_2,p_3)$ has the same value for electron-electron-photon and muon-muon-photon vertex (is there any distinguished point?). That should mean that (at least in some renormalization scheme) $e_e=e_\mu$. However, in general, for finite positive $\epsilon$, $\delta e_e \neq \delta e_\mu$ because the masses of leptons are different and we need different counterterms.


What physical quantity has SI unit $\mathrm{kg}/\mathrm{m}$? For example, the physical quantity with SI unit $\mathrm{kg}\cdot\mathrm{m}/\mathrm{s}^2$ is force $F$ and the physical quantity with SI unit $\mathrm{m}/\mathrm{s}^2$ is acceleration $a$.


title: What amount of force is needed to topple a person?

text: Assuming no sliding and that the shoulder is 1.2m from the feet, what force is required to topple a person weighing 70 Kg standing with his feet spread 0.9 m? If possible, please include an explanation about your answer.


title: Why is speed/position relative but acceleration not?

text: I think i understand it now, if found this: link I know that position and speed are relative. There is no such thing as universal coordinates. Then why is acceleration absolute? Is the 3th and 4th derivative of position to time also absolute? Where/how can i see this in the math/laws of motion? For example: you are in a rocket and you have an accelerometer with you, and someone else is standing on an astroid that is not attracted by gravity towards something else, so it has no net force on it. If the engines of the rocket are turned on, then you will see that on the accelerometer, and you will know that it is you that is accelerating, not the astroid. Also if the universe were completely empty, except for a bucket of water. If the water would be pushed towards the edges of the bucket, then we would know that the bucket is rotating around its axis. But what would happen if you could rotate space itself around the bucket? It probably is because $F=ma$ and not something else like $F=m^2a^3+2a+...$, is that correct? Can $F=ma$ be deduces from deeper principles, or is it an experimental law?


title: Homeomorphism between the space of all Ashtekar connections and spacetime?

text: Excerpt from an essay of mine: > Let $\Psi(\varsigma)$ be the wavefunction in the loop representation, where > $\varsigma:[0,1]\to\mathcal{M}$, where $\mathcal{M}$ is spacetime. Then, let > $\mathcal{A}$ be the Ashtekar connection and > $\mathcal{W}_\varsigma[\mathcal{A}]$ be the Wilson loop of the connection > $\mathcal{A}$. Give the connection the action > $S[\mathcal{A}]=\oint_\varsigma -i\mathcal{A}$. One then has > \begin{multline} \Psi(\varsigma)\Psi(\varsigma_{1})=\langle > \mathcal{W}_\varsigma[\mathcal{A}]\mathcal{W}_{\varsigma_1}[\mathcal{A}_1]\rangle=\langle0| > \mathcal{W}_\varsigma[\mathcal{A}]\mathcal{W}_{\varsigma_1}[\mathcal{A}_1]|0\rangle=\\\ > \int D\mathcal{A}\int > D\mathcal{A}\left(\operatorname{Tr}\left(\mathcal{P}\exp\left(\oint_\varsigma > -i\mathcal{A}\right)\right)\right)^2\mathcal{W}_\varsigma[\mathcal{A}]\mathcal{W}_{\varsigma_{1}}[\mathcal{A}_1] > \end{multline} This, as is written in the equation, has the form of a 2 > Wilson loop correlation function. For simplicity, we assume an interaction > of the form $(\mathcal{W}_\varsigma[\mathcal{A}])^2$. Let $|\Theta\rangle$ > be the ground state of loop quantum gravity. Then, one has the S-matrix > elements of loop quantum gravity in the space of all Ashtekar connections as > \begin{equation} > \langle\Theta|\mathcal{W}_\varsigma[\mathcal{A}]\mathcal{W}_{\varsigma_1}[\mathcal{A}_1]|\Theta\rangle=\sum^\infty_{n=0}(-i\lambda)^n\int\mathrm{d}^4x_n\langle0|\mathcal{W}_\varsigma[\mathcal{A}]\mathcal{W}_{\varsigma_1}[\mathcal{A}_1]\prod^n_{j=1}(\mathcal{W}_{\varsigma}[\mathcal{A}^j])^2|0\rangle=\sum^\infty_{n=0}\Theta^{(n)} > \end{equation} Here, $\lambda$ is the coupling constant of LQG. > $\mathcal{A}^j$ stands for different connections, i.e., > $\mathcal{A}^1=\mathcal{B},\mathcal{A}^2=\mathcal{C}$, e.t.c. The Feynman > diagrams come from > $\langle\Theta|\mathcal{W}_\varsigma[\mathcal{A}]\mathcal{W}_{\varsigma_1}[\mathcal{A}_1]|\Theta\rangle=D_F(\mathcal{A},\mathcal{A}_1)+\mbox{\emph{all > possible contractions}}$. Here, ``all possible contractions'' means all > possible pairings between the connections of which the Wilson loop is a > functional. We choose to study only a limited number of contractions. > > We write two contractions for $\Theta^{(1)}$: > 1)$D_F(\mathcal{A},\mathcal{A}_1)\left(D_F(\mathcal{B,B})\right)^2$ > > 2) $D_F(\mathcal{A,B})D_F(\mathcal{A}_1,\mathcal{B})D_F(\mathcal{B,B})$ > > We write four contractions for $\Theta^{(2)}$: 1) > $D_F(\mathcal{A,A}_1)\left(D_F(\mathcal{B,B})\right)^2D_F(\mathcal{B,C})\left(D_F(\mathcal{C,C})\right)^2$ > > 2) > $D_F(\mathcal{A},\mathcal{B})D_F(\mathcal{A}_1,\mathcal{B})D_F(\mathcal{B,B})D_F(\mathcal{B,C})\left(D_F(\mathcal{C,C})\right)^2$ > > > 3)$D_F(\mathcal{A},\mathcal{B})D_F(\mathcal{A}_1,\mathcal{C})\left(D_F(\mathcal{B,B})\right)^2\left(D_F(\mathcal{C,C})\right)^2$ > > 4) > $D_F(\mathcal{A,C})D_F(\mathcal{A}_1,\mathcal{C})\left(D_F(\mathcal{B,B})\right)^2D_F(\mathcal{B,C})D_F(\mathcal{C,C})$ > > Note that this list does not contain every single propagator. In the space of all Ashtekar connections (which is a subset of the space of all principal connections of spacetime), the interactions of LQG are given by finding out all possible contractions of the Feynman propagators and drawing diagrams in the space of all Ashtekar connections. My question thus is: _Is it possible to somehow find a homeomorphism between the space of all Ashtekar connections and spacetime (so that the interactions of LQG can be formulated on spacetime itself)_?


title: Any process or algorithm to estimate color sensitivities or color matching function of a RGB camera?

text: CIE defined color matching functions (CMF) for XYZ and thus RGB colors. But of course, each camera has its own color sensitivities for visible spectrum. Is there any way to estimate this function (or a series of values by wavelengths) for a given camera?


title: How do photon emitters and photon detectors work?

text: The interpretation of the double slit experiment is very strange and i want to understand how they did it before I give up my concept of reality. With the double slit experiment a interference pattern is created even when you emit just one photon. However, when you use photon detectors to find out which slit the photon went though, the photon goes in straight line. So firstly I want to know how do the photon emitters work and secondly I want to know how these photon detectors work. How are you even meant to detect a photon without even touching it? also how big the the slits?


title: Does $R=\dfrac{4q^2}{\pi^2a^2gH^2}$ suggest a square root relationship?

text: I did an experiment involve how R scales with q (the other variables are constant), and I got a relationship like $R\ \propto \sqrt q$ I have been told to evaluate my findings with those in literature, and I found an equation in literature to compare it to. I have been told that the equation $$R=\dfrac{4q^2}{\pi^2a^2gH^2}$$ suggests a square root relation. However, this is what I am not sure about. The equation says $R=\dfrac{4q^2}{\pi^2a^2gH^2}$. I have found that $R\ \propto \sqrt q$. How does the equation suggest there is a square root relationship betwen R and q, if it does? Is it the division? There is also a more complicated version of the equation: $$R=\dfrac{4q^2\sqrt{\frac{\pi^2gd}{8q^2}+\frac{1}{A^4}}}{\pi^2gH^2}$$ Is it the square root in the more complicated equation itself that suggests there is a square root relationship between R and q? edit: here's one of the graphs I've got, with R on the Y axis and q on the X axis ![enter image description here](http://i.stack.imgur.com/0TWSg.png) I then plotted $R \ vs \sqrt q$ to try and get a straight line. ![enter image description here](http://i.stack.imgur.com/R4mUv.png) It seemed rather straight, so in theory, lnR vs lnq should also be straight with a gradient of 0.5. However, this is what I got. (ln R on Y, ln q on X) ![enter image description here](http://i.stack.imgur.com/N7SID.png) The gradient isn't 0.5, but 0.7. I wanted to basically say what I got and what literature suggested (square root relationship) are different.


title: How to calculate the electrostatic force applied on an object that is along the axis of a charged tube?

text: Let's say there is a charged tube(a cylinder with no top and bottom) with radius r and length l, charge q1 which also made out of insulating material. And also if there is an object with charge q2 along the axis of the surface of the tube, how can I calculate the force applied on this object?(the distance between tube and object is d) Here is my calculations: From the Gauss' law: $$ \phi_E= \frac Q{\epsilon_0} $$ I took cylinder as Gaussian surface. $$ E \times 2\pi dl= \frac {\lambda \bullet l}{\epsilon_0} $$ $$ E = \frac {\lambda}{\epsilon_0\times 2\pi d} $$ or $$ E = \frac {q_1}{\epsilon_0\times 2\pi dl} $$ Thus we found the electric field applied to a point d away from the tube.To calculate the force we use $F=qE$. And finally the electrostatic force is: $$ \frac {q_1q_2}{\epsilon_0\times 2\pi dl} $$ However what I found is the force applied to an object d away from the tube but perpendicular to the surface of the tube. Would this calculation be valid for an object which is along the axis of the surface of the tube?


title: Could a fourth dimension be "spatially simulated" one point at a time by using movements?

text: I wonder if time and movement can be translated into a simulation of an extra spatial dimension. At least in one point at a time. I'll try to explain. Imagine a fixed probe which measures the distance to one point on the surface of a cube. As the cube rotates around its center, the probe will measure different distances to its surface. If the probe starts perpendicularly to a side of the cube, the distance will decrease as the cube rotates. After an edge meets the probe, the distance will start to increase. This could be simulated by a 2D rectangle if, when the probe meets its edge, the rectangle momentarily turns 90 degrees. The probe would be fooled to think it is measuring a cube. Could in a similar way a fourth spatial dimension, a hypercube, be simulated by moving a regular cube? Does a hypercube consist of regular cubes, like a cube consists of rectangles?


title: Probability of finding n particles in a volume v

text: I'm trying to calculate the probability of finding $n$ particles in a certain volume $v$ in a system with a total of $N$ particles and total volume of $V$. My problem is that I've tried two approaches which both seem valid to me, but give differing answers. One approach is to use binomial probability, where the probability of success (particle in the volume of interest) is $\frac{v}{V}$. Furthermore, the particles are indistinguishable, so it doesn't matter the order of "successes" and "failures". This gives: $P=(1-\frac{v}{V})^{N-n}\,(\frac{v}{V})^{n}\,\frac{N!}{(N-n)!n!}$ My other approach is to say to start saying that any configuration (remembering particles are indistinguishable) has equal probability and so the probability for our event is simply $P=\frac{\mathrm{\\#\ of\ configurations\ with\ n\ particles\ in\ the\ cell}}{\mathrm{\\#\ of\ configurations}}$. Now from combinatorics, the number of configurations is $\binom{N+\frac{V}{v}-1}{N}$, and the number of configurations with $n$ particles in $v$ is $\binom{N-n+\frac{V}{v}-2}{N-n}$. This gives a probability: $P=\frac{\binom{N-n+\frac{V}{v}-2}{N-n}}{\binom{N+\frac{V}{v}-1}{N}}=\frac{(\frac{V}{v}-1)!\,N!\,(N-n+\frac{V}{v}-2)!}{(N+\frac{V}{v}-1)!\,(\frac{V}{v}-2)!\,(N-n)!}$ Which definitively isn't the binomial distribution, I checked numerically as well. So what is the problem?


title: Does Fluctuation-exchange (FLEX) approximation including spin-orbit coupling (SOC) exist?

text: FLEX method was invented by N.E.Bickers and D.J.Scalapino in 1989 (PRL 62,961; Ann. Phys. 193,206). Later it was extended to multi-orbital system (T.Takimoto,PRB,69,104504). But I don't find FLEX including the SOC term. More generally, can FLEX method be applied to the $SU(2)$ spin-rotation symmetry _broken_ Hamiltonian? And does any one know something about it?


title: Approximating density from pressure and flow

text: I want to get the density of a fluid going through a pipe. I can measure the flow and pressure with a flowmeter and the temperature using a thermometer. With this information, I want to calculate (or approximate) the instantaneous density of the fluid passing through my instruments. How would I go about doing this? I'm looking at a liquid (not a gas), mostly incompressible. Think water with varying amounts of solute (salt, sugar) in it.


title: Topological Phases and Confinement

text: I recently attended a talk in which the speaker defined a topological phase as "A phase which has a gap above the ground state for bulk excitations in the thermodynamic limit." I am interested in what sense then can we think of confinement in non-Abelian Yang-Mills theories as topological phases. What I'm looking for are the analogies; what would the **thermodynamic limit** and the **bulk excitations** mean if we were talking about a YM theory (QCD, for instance). The thermodynamic limit is the number of particles $N\to\infty$, which I suppose we can think of as the number of Feynman diagrams (order of the loops) growing as large as possible. "The bulk" seems a bit more vague (which is perhaps because of my definition), but it seems like bound states of quarks is the appropriate notion for that. So, is it possible to (correctly) say something like "QCD is a topological phase for the standard model"? If not, is there a clear reason why this is not the case?


title: Can open, unsafe nuclear fusion reaction burn the atmosphere?

text: I happened to hear people saying that the nuclear fusion bomb tests could set the atmosphere on fire. I have some serious doubts about that - but I have no facts.   Nuclear fusion reaction requires $15*10^{6}$ kelvins to start. If we produce such temperature in " _open air_ " would the atmosphere become a fuel for further fusion? Shouldn't the whole thing just be torn apart by its terrible pressure?


title: Quantum Mechanical Effects of an object accelerating near speed of light $c$?

text: Consider a space ship, undergoing constant acceleration (which for our purposes means that the same amount of energy is being used per second to increase its speed). According to special relativity the ship will accelerate but in such a way that it will approach the speed of light $c$, but never reach the speed itself or cross it. This means however that the more time we spend accelerating the space ship, the closer and closer it will get to the speed of light. Which basically amounts to: our knowledge, or accuracy of our velocity is going to steadily go up. Heisenberg's Uncertainty principle states that there is limit to how accurately we can know our velocity before we need to give up some information about our position? What does this mean in our context? Say we reach the critical threshold where we are accurate to some 35+ significant digits aka 0.999999999999999999999999999999999999999999999999999999999999999999c is our speed. We need to lose accuracy of our position between our start and endpoint (assuming this trip has a definite endpoint)? Whats happening? Is the ship suddenly making jumps to random locations? Is it spreading out like a wave, where we the faster we go, the less likely we are to know when we reached our endpoint? I'm very very curious. * * * ## EXTENSION: Additional Thought Consider an object in circular orbit around a black hole, outside of the event horizon (the black hole itself is stationary) and these are the only 2 systems present, with the object in question having very negligible mass compared to the black hole. As the object is brought closer and closer to the black hole, the centripetal force it experiences, obviously goes up, causing to orbit at a faster speed. The closer you bring it towards the center of the black hole, the faster the object will orbit, allowing you to bring it arbitrary close to value of C. According to heisenberg as the velocity increases to an accuracy beyond the reduced planck's constant (0.9999...)c the position of the object becomes increasingly unknown. It would start to "smear" out in a circle around the orbit, transforming into a haze which can be observed. Or So I think... Is this correct intuition?


title: What is 2 $\theta$ in X-ray powder diffraction (XRD)?

text: Why are we taking 2$\theta$ instead of $\theta$ in X-ray powder diffraction (XRD). I have found the forum post _2 theta in X-ray Powder Diffraction (XRD)_ , but there is no answer. What is the explanation?


title: Geometry of wireless signal strength

text: How does wireless signal strength correspond to distance? RSSI lies between -100 and 0 (at least, on my computer). Let's say I walk a distance x towards the router, and my RSSI goes from -60 to -50. Now, lets say instead I walk a distance 2x towards the router. Would this imply that RSSI would go from -60 to -40? I'm curious what the relationship of the metrics is, is RSSI linear/logarithmic/etc with respect to distance? I'm a math guy with little physics/engineering background so some help would be very appreciated. Thanks.


title: An electron is subjected to an electromagnetic field using the canonical equations solve

text: So I was given the following vector field: $\vec{A}(t)=\\{A_{0x}cos(\omega t + \phi_x), A_{0y}cos(\omega t + \phi_y), A_{0z}cos(\omega t + \phi_z)\\}$ Where the amplitudes $A_{0i}$ and phase shifts \phi_i are constants. An electron is moving through this field with the constant initial velocity: $\vec{v}_0=\\{v_{0x},v_{0y},0\\}$ The vector field pulls the electron out of a metal that occupies half-space $z<0$ causing photoelectron emission. I need to find a condition on phase \phi_z under which the electron will continue to fly away and never return back and hit the metal as well as the average drift velocity of the electron. I started by finding the Lagrangian: Where I shortened the initial velocity to: $\vec{v_0}=\\{\dot{x},\dot{y},0\\}$ Therefore: $L=\frac{1}{2}m(\dot{x}^2+\dot{y}^2)+\frac{e}{c}\dot{\vec{r}}\vec{A}-U(r)$ This simplifies to: $L=\frac{1}{2}m(\dot{x}^2+\frac{e}{c}[A_x \dot{x}cos(\omega t + \phi_x)+A_y \dot{y} cos(\omega t +\phi_y)]-U(r)$ Next I found the momentum using the canonical relationship: $p=\frac{\delta L}{\delta \dot{q}}$ This gives: $p_x=m\dot{x} +\frac{eA_x}{c}cos(\omega t + \phi_x)$ $p_y=m\dot{y} + \frac{eA_y}{c}cos(\omega t + \phi_y)$ This lets me find the energy via the hamiltonian: $E=H=\frac{p_x^2}{2m}+\frac{p_y^2}{2m}$ When you plug in for $p_x$ and $p_y$ you get: $H=\frac{1}{2m}[m\dot{x}+\frac{eA_x}{c}cos(\omega t + \phi_x)]^2+\frac{1}{2m}[m\dot{y}+cos(\omega t + \phi_y]^2$ where: $\alpha=\frac{eA_x}{c}cos(\omega t +\phi_x)$ $\beta=\frac{eA_y}{c}cos(\omega t + \phi_y)$ We can also say: $p_x=m\dot{x}$ $p_y=m\dot{y}$ Therefore: $H=\frac{1}{2m}[p_x+\alpha]^2+\frac{1}{2m}[p_y+\beta]^2$ Now we can use the relation for the unknown equation: $\frac{\delta W}{\delta q_i}=p_i$ Plugging this into the energy equation we get: $E=\frac{1}{2m}[\frac{\delta W_x}{\delta x}+\alpha]^2+\frac{1}{2m}[\frac{\delta W_y}{\delta y}+\beta]^2$ pulling the y terms out we get: $0=\frac{1}{2m}(\frac{\delta W_y}{\delta y})^2 + \frac{\beta}{2m}(\frac{\delta W_y}{\delta y})$ This simplifies to: $W_y=-\beta y$ Which we plug back into the equation. With algebra we can now simplify the energy equation to: $2mE=(\frac{\delta W_x}{\delta x})^2+\alpha\frac{\delta W_x}{\delta x}+ \Omega$ Where $\Omega=\alpha^2+\beta^2$. My first question is, how do I solve the above equation for $W_x$. My second question is this even the right direction I should be heading? I feel like I have some egregious error that I am not seeing.


title: Accuracy and Error of Atomic Clocks

text: I'm quoting a passage from my notes: > The development of clocks based on atomic oscillations allowed measures of > timing with accuracy on the order of $1$ part in $10^{14}$, corresponding to > errors of less than one microsecond (one millionth of a second) per year. I do not understand what the accuracy of $1$ part in $10^{14}$ means. Does it mean that the atomic clocks can tell us the time accurate and ceratain to $10^{-14}s$? How should I understand this? Moreover, what is meant by the error of one microsecond per year? Is it a kind of uncertainty in measurement? How should I understand it? I googled this topic and found information about the atomic clocks and also reviewed the definitions of accuracy and error; however, I'm not able to make any sensisble connection between the concepts. Please help me, thank you.


title: one loop correlator in ads cft

text: Is there any example of explicit one loop computation for Witten diagrams? It seems like it will be hard to compute for even for a simple $\phi^4$ theory in the bulk.


title: Stimulated Emission

text: In the case of stimulated emission we always see that one photon goes into the gain medium and two photons come out. How can this conserve energy?


title: Is the gravity between objects other than celestials observable?

text: I've always read that all matter has gravity. But, can we observe it? I mean, The Earth pulls us but what about small daily objects? For example, if we release 2 small objects in space, do they get closer? My point of asking this question is that, maybe the gravity comes from the rotation of the Earth, not from its mass. I know I'm wrong by saying that but I want to know about the experiment and observation (aka proof) of that concept.


title: What is negative about negative energy states in the Dirac equation?

text: This question is a follow up to What was missing in Dirac's argument to come up with the modern interpretation of the positron? There still is some confusion in my mind about the so-called "negative energy" solutions to the Dirac equation. Solving the Dirac equation one finds the spectrum of allowed energies includes both positive and negative solutions. What does this negativity refer to? Given that the Dirac equation is symmetric under charge conjugation, the convention to call one positive and the other negative appears perfectly arbitrary. Would it be therefore correct to refer to the electrons as "negative energy positrons" ? In a similar spirit, physicists used to be worried about the "negative energy" solutions decaying into infinity through emission of photons. By the same symmetry argument, this should also be a problem for photons. It is not entirely clear to me how the quantization of the field supresses this issue : is the "photon emission" for the negative state re-interpreted as photon absorption by a positron ? My understanding is that the whole discussion about "positive" or "negative" energy solutions is misleading : what matters is the physical content through the QED interaction hamiltonian, which does not predict this infinite descent. Is this correct? Edit: I think I understand the source of my confusion after the comments. If I get it right, the Dirac sea picture is equivalent to the freedom of choice in the formally infinite vacuum energy one observes after quantization of the QED Hamiltonian. Holes in the sea are positive-energy positrons, equivalent to the action of the positron creation operator on the vacuum. Is this correct?


title: How far does a trampoline vertically deform based on the mass of the object?

text: If a baseball is dropped on a trampoline, the point under the object will move a certain distance downward before starting to travel upward again. If a bowling ball is dropped, it will deform further downwards. What is the nature of the relationship between the magnitude of this deformation and the object's mass? Linear? Square? etc. **Edit:** I would like to add that the heart of what I'm asking is along the lines of this: "If a small child is jumping on a trampoline and the trampoline depresses 25% towards the ground, would an adult who weighs slighly less than four times as much be safe from depressing it all the way to the ground?"


title: Relation between density and pressure for a perfect fluid

text: what is the relation between mass density $\rho$ and pressure $P$ for a perfect fluid ?


title: Why does the thickness of a wire affect resistance?

text: For small thicknesses of wire, it's pretty obvious why resistance affects thickness. (The electronics squeeze to get through). But after a certain thickness shouldn't the thickness become irrelevant? For example if your trying to pour a bucket of water through a straw, the thickness of the straw is obviously gonna be a bottle neck- the bigger the straw, the easier it is for water to get through. But if you try to pour a bucket of water through a tunnel - the size of the tunnel doesn't really matter, because the tunnel is already big. So after a certain thickness shouldn't the thickness stop mattering?


title: Dispersion of Probability Wave Packets

text: A picture in my text book shows a three dimensional wave packet dispersing, "resulting from the fact that the phase velocity of the individual waves making up the packet depends on the wavelength of the waves." Does this mean a particle moving through space has a gradually diminishing probability of being in it's location? Also, why does the wavelength of the the wave change the speed for a probability wave? I thought the dispersion was characteristic of the medium, and I thought for things like vacuums&light, air&sound, and also probability waves and space, that they wouldn't disperse.


title: Combining errors to calcuate the total error (standard deviations)

text: I have a measurement method for which I want to study the measurement error by an error budget. Therefore, I listed all possible errors (error sources) (lets say $x_1, x_2, x_3,\ldots$). For each error $x_i$, I have derived an analytical expression from which I can calculate the resulting measurement error $\epsilon_i$. To calculate the total measurement error epsilon, I specified an interval for the possible values of each error value $x_i$ and performed an Monte-Carlo simulation calculating the measurement error $\epsilon_i$ 100.000 times for each error $x_i$. This means that I have 100.000 error values $\epsilon_i$ for each $x_i$. From this data I can calculate the standard deviation $\sigma_i$ of each measurement error $\epsilon_i$. However, I am interested in the total measurement error epsilon, not in the independent measurement errors $\epsilon_i$. How can I combine each measurement error $\epsilon_i$ to come to the total measurement error $\epsilon$? Suppose the errors $x_i$ are independent.


title: Optimality of product input state in quantum channel

text: Let $\mathcal N^{A_1\rightarrow B_1}_1,..,\mathcal N^{A_1\rightarrow B_1}_k$ be a set of valid quantum evolutions with equal input and output dimensions. And let the effect of a channel on a system $\rho_{A_1A_2}$ be: $$\mathcal N^{A_1A_2\rightarrow B_0B_1}(\rho_{A_1A_2})=\int dB_0|b_0\rangle\langle b_0|\otimes\sum_ip_i\mathcal N^{A_1\rightarrow B_1}_i(\rho_{A_2})$$ where $B_0$ is a random basis which is given as output and $p_i=tr(B_0^i\rho_{A_1})$ is the probability of obtaining outcome $i$ after measuring the system $A_1$ in the $B_0$ basis. If the maps $\mathcal N^{A_1\rightarrow B_1}_1,..,\mathcal N^{A_1\rightarrow B_1}_k$ are random unitaries, I have seen without proof that the optimal input for the coherent information, is a product state between $A_1$ and $A_2$, does anyone know how to prove it? Is that also true for general channels, i.e. non random unitaries?


title: Relativity question: frequency shift under constant acceleration

text: Okay, my buddy tells me this: Let there be a starship, ovoid, and me and my buddy stand each at the extremities of the ship, him below the roof, me on the floor. We start a journey and the ship accelerates at a constant rate, let's say $1g$, in the direction of the axis of the ship. _My buddy says_ : since the ship is accelerating, when he strobe (on off on off) a light from the room, back to me, at a frequency $F$, I'll see a shifted frequency $F' = F + \delta f$. If he switches the light at $1Hz$, I'll see $1.01Hz$ for instance. _My take on the thing_ tells me that in this case, there's no shift: we're both immobile in our referential, and the light will travel at $c$ in this referential. Who's wrong?


title: Does the total mass of an isolated object include the mass stored in its gravitational field?

text: Since neither the object nor its field could exist without the other, it would seem strange not to include the field energy as part of the object. But how exactly does the accounting go? How is the mass of the system divided between the rest mass and the field mass? For a Schwarzschild black hole, the mass appears to be shifted completely to the field. According to Lynden-Bell and Katz, http://adsabs.harvard.edu/full/1985MNRAS.213P..21L, the total energy distributed in the gravitational field of a Schwarzschild black hole is mc^2. In other words, _all_ the mass of the black hole resides outside the event horizon.


title: Why does a coulomb explosion induce fusion?

text: If you strip the valence electrons apart with a very short intense electromagnetic field the remaining core explodes in a so called coulomb explosion. But experiments have shown that under certain cases fusion does occur in a manner, that the ions are accelerated and you get a non maxwellian velocity distribution. Can you explain the process a little bit more and show how it is possible to overcome the nuclear repulsion for fusion? The wikipedia page to this topic is a little bit missleading, so I give you a paper Efficient fusion neutron generation from heteronuclear clusters in intense femtosecond laser fields that this kind of fusion is indeed possible and experimentally proven.


title: What's the right way to calculate charge on a capacitor?

text: How much charge is on each plate of a 4.00-F capacitor when it is connected to a 12.0-V battery? I said 2.4 x 10^-5 C because there are two plates of a parallel plate capacitor. But the key said only 4.8 x 10^-5 C Why?


title: We tend to think the action of a constant force...?

text: This is a school exercise. We tend to think that the action of a constant force produces a constant movement speed as well. How can you explain this situation in accordance with Newton's second law? Because as the force will end up in a certain time that means there will not be a movement of constant speed; I'm correct?


title: Which derivation of drift velocity is correct?

text: In the derivation of drift velocity I have seen two variations and want to know which one's correct.   * $s=ut+\frac{at^2}{2}$   Assume that the drift velocity of any electron in any conductor is : $$\vec{v_d}=\frac{\vec{l}}{t}$$ Due to the electric field the acceleration of electrons in any conductor is: $$a=\frac{-e\vec{E}}{m}$$ Now the distance travelled by an electron after a long time (initial thermal velocity = 0) $$l=\frac{at^2}{2}\implies \frac{-e\vec{E}t^2}{2m}$$ the time between the collisions is $\tau$ $\therefore$ $$l=\frac{-e\vec{E}\tau^2}{2m}$$ thus the velocity is $$\vec{v_d}=\frac{-e\vec{E}\tau}{2m}$$   * In another proof I saw the author using $v=u+at \implies v_d=\frac{-e\vec{E}\tau}{m}$ My question is which of the the two equations of motion can be used in the proof? Can they be used at all.


title: How do we recognize hardware used in accelerator physics

text: When I see a new accelerator in real life or on a picture, I always find it interesting to see how many thing I can recognize. In that way, I can also get a small first idea of how the accelerator is working. Here is a picture, I have taken of LEIR at CERN![LEIR](http://i.stack.imgur.com/d8t7B.jpg) Help me to be able to recognize even more stuff, than I can now(I will post a few answers myself) Suggested answer form:   * **Title**   * **Images**   * **One line description**   * **Link**


title: My book uses helium weight in reaction formula instead of weight alpha particle?

text: I'm suppose to write out reactions where atoms send out alpha radiation and decay. The book uses the 4-2 H, 4 as nucleon number and 2 as proton number, but isn't that wrong? The mass of helium is greater than the alpha particle due to two electrons? Shouldn't they use a different notation for the alpha- particle other than that for helium?