What does the sum of two qubits tell about their correlations?

> How much can I learn about correlations between two quits by measuring the > sum of their values? What is the best way to formalize such a question? Below is my original, longer formulation of the same question in spin language. I have an ensemble of pairs of spin-$1/2$ particles. All I can do is to measure total $S_z$ for each pair, each time learning one of the three standard outcomes: $+1$, $0$ or $-1$. Based on the knowledge of this probability distribution $P(S_z)$, what can be said about the density matrix of my ensemble? In particular, what is best measure for correlations between the two particles in these circumstances? I understand that I won't be able to distinguish whether the correlations are classical or necessarily quantum-mechanical, but struggle to come up with a measure better than a classical correlation coefficient $$r=\frac{\langle s_1 s_2 \rangle-\langle s_1 \rangle \langle s_2 \rangle}{\sqrt{\langle (s_1- \langle s_1 \rangle)^2 \rangle \langle (s_2- \langle s_2 \rangle)^2 \rangle }}$$ where $s_1$ and $s_2$ are the $\pm 1/2$ spin projections (or $0$, $1$ qubit values) of the 1st and the 2nd particle, respectively, so that my only observable can be expressed as $S_z=s_1+s_2$. I then minimize $r$ over all possible $\langle s_1 \rangle$ and $\langle s_1 \rangle$ consistent with measured $P(S_z)$ and declare that the outcome is the minimal possible correlation in the pair. Is there a more elegant way? **UPDATE:** Classically, if we take the definition of zero correlations as $\langle s_1 s_2 \rangle =\langle s_1 \rangle \langle s_2 \rangle$, then it implies that $s_1 = +1/2$ and $s_2=+1/2$ with some _independent_ probabilities $p_1$ and $p_2$, so that the sum is distributed bimomially $P(+1)=p_1 p_2$ and $P(-1)=(1-p_1)(1-p_2)$. These two equations have a solution with $0\leq p_1, p_2 \leq 1$ only if $$P(+1) < 1-2 \sqrt{P(-1)}+P(-1)$$ Graphically, this condition is satisfied within the region bounded by a black line in the plot below. The rest of the available probability space is the region of excess correlations ($r_{\text{min}}>0$, depicted in red). ![enter image description here](http://i.stack.imgur.com/BjYIe.png) **Updated question:** is there a quantum loophole in such analysis? I suspect not but would really like to come up with a shorter/cleaner argument. **UPDATE-2** My selected spin/qubit degree of freedom is one of the many variables characterizing the state of the two particles. Can we be sure there exists no pure product state that can lead to $r_{\text{min}}>0$? See a related active question on fermionic entanglement of two particles.


Is phosphorene a topological insulator?

All of the other "-ene" materials (silicene, germanene, stanene, and even graphene) are topological insulators. Phosphorene and these other materials all have honeycomb lattices and would appear to have similar properties. However, I don't see any papers referring to phosphorene as a topological insulator, so is phosphorene just a semiconductor? Thanks.


How did the Bardeen-Brattain point-contact transistor work? What are the conditions under which a Schottky barrier injects minority carriers? In a bipolar junction transistor the forward-biased emitter injects minority carriers into the base, which are then swept across the reverse-biased base- collector depletion layer. At first glance a point-contact transistor might work the same way: the rectifying Schottky barrier might inject minority carriers from the metal emitter into the doped silicon base which would then be swept across the reverse-biased base to metal-collector depletion layer. Except that all the textbooks I can find say things about Schottky barriers like, > Under normal operating conditions, the minority carrier current is orders of > magnitude smaller than the majority carrier current. (Sze, _Semiconductor > Devices_ , 1985). Or, > the forward current ... is due to the injection of majority carriers from > the semiconductor into the metal. The absence of minority carrier injection > and the associated storage delay time is an important feature of Schottky > barrier diodes. Although some minority carrier injection occurs at high > current levels, these are essentially majority carrier devices." (Streetman, > _Solid State Electronic Devices_ , 1980). So what gives? The point contact transistor could only possibly work if a gold to n-type germanium point contact is injecting. Is it perhaps the case that at high current _densities_ minority carrier injection is actually quite common? (But point-contact diodes should have very high current densities, and thus if this is the explanation then we don't have a good explanation for their good capacitance.) There is a lot of erroneous information about how point-contact transistors work floating around on the web. In the "maker" community the suggestion (probably from the article Henderson, PB; Home made transistors, _Wireless World_ , Jan 1954, pp. 20-23) is that somehow melting phosphor-bronze point contacts into N-type germanium does a P-type doping, so that point contact transistors are actually just simple PNP junction transistors. But phosphor is an N-type dopant, so this seems an unlikely explanation. Jeri Ellsworth repeats this assertion in her Youtube video. The Wikipedia page, the PBS page and the Nobel prize page(see frames 19-26 of the animation) all claim that the point contacts somehow magically create a hole based inversion-layer _across the entire surface of the crystal_. (To be fair to all of them, Bardeen/Brattain's original papers in Physical Review (74(2):230-232, July 1948) made extensive arguments about surface states that may have confused the issue). In Bardeen's Nobel acceptance speech, (as published in Science, 126:105-112, Jul 19, 1957 (paywall), however, he gives the traditional explanation of a Schottky rectifying contact and then says (page 109) > _If $\chi$ (the energy barrier) is sufficiently large, the Fermi level at > the interface may be close to the valence band, implying an inversion from > n-type conductivity in the bulk to p-type near the contact. The region of > hole conduction is called, following Schottky, an "inversion layer." An > appreciable part of the current flow to the contact may then consist of > minority carriers in this case, holes. An important result of the research > program at Bell Laboratories after the war was to point out the significance > of minority carrier flow._ So I've got a Nobel laureate saying one thing, and two eminent IEEE fellows saying something that seems contradictory.


Does the Earth rotate the same encased in ice during the height of an Ice Age as it does when the bulk of it's water is liquid and always in motion?

Ice Age vs. Now. Does the Earth rotate at the same rate when encased in ice during the height of an Ice Age as it does when the bulk of it's water is liquid and always in motion?


Does weather obstruct microwaves and radiowaves?

I read an article ('How airplanes use radar' at Bright Hub.com) that stated that airplanes use radio waves for communication because they aren't obstructed by weather. Is this true?


Does nature tetrate?

We see addition, multiplication and exponentiation in the natural formulae that make up physics. However, do we ever see tetration (repeated exponentiation) or higher hyper- operators in nature? (Not counting cases where the tetration step could be trivially eliminated.)


Mechanical energy problem - please help

Quite new to all this - simple question. "A ball starting from rest rolls down a hill and reaches 10m/s at the bottom. If the same ball starts rolling down the hill from an initial velocity of 5m/s, will it's velocity at the bottom be: (A) Less than 15m/s; (B) Equal to 15m/s; (C) Greater than 15m/s." The answer I have marked in the book this question comes from says (A). However, I am unable to reason to that conclusion from the way the question is stated. Here is my reasoning, please point out where it is wrong. If the same ball (the same mass) traveses the same distance down this same hill (the same change in height), the gravitational potential energy must be the same at the point on top of the hill, regardless of its initial velocity. Therefore, on rolling down the hill, by the time it reaches the bottom, this potential energy would have transferred to kinetic energy - indicating (B)? Thanks for your time. Tim.


Calculating the rate at which a car turns

This is going to be difficult to describe, as I have no real concept of this sort of physics, so please bear with me. If this is too much or beyond the scope of this website, I apologize. I am making a simple-ish driving simulator and I need a formula that I can use to determine the rotation of the vehicle for every frame. If the car's rotation is 90.0 degrees and the wheels are pointing 20.0 degrees to the right, that should mean I can use a simple polar offset to send it a distance of _a_ toward 70.0 (90.0 - 20.0). This does not work when the car is not moving, or when the air resistance is large, like when the car is turning at high speeds, and I am not sure how I should be dealing with such factors. What I have works for now, but I need a more accurate solution. **What I have** What I have barely turns the car at low speeds, and allows it to turn on a dime at high speeds. This is the algorithm that I am currently using in pseudo-code.                   if (vehicle.engineCurrentRPM > vehicle.engineMaxRPM) vehicle.engineCurrentRPM = vehicle.engineMaxRPM;         if (vehicle.engineCurrentRPM < -vehicle.engineMaxRPM) vehicle.engineCurrentRPM = -vehicle.engineMaxRPM; // This allows it to go full speed in reverse, which I can fix myself.          // The engineMaxRPM is from a Bugatti Veyron and is equal to 6000.0.     // That is 600.0 lower than the redline, and I did that on purpose.          vehicle.engineCurrentTorque = (vehicle.engineCurrentRPM * (922.0 / 2200.0));     // A simple formula that approximates the torque-rpm relationship. I only include this because torque _might_ be relevant.          differentialRatio = 4.0;     drag = 0.39;     gearRatio = 3.64;     grip = 1.0;     wheelRadius = (2.0 * 0.0254);     // 20 inch wheels converted to meters and set to 10%. I don't know why this is necessary, but it seems to be in all of the formulas I have seen     weight = 1995.806; // 4400 pounds * 0.453592 to get kilograms.     power = (vehicle.engineCurrentTorque * vehicle.engineCurrentRPM) / 5252.0; // HP = Torque x RPM ÷ 5252          // The variable "power" is unused, but there if you need it.          airResistance = 0.05 * (1.0 - drag);          tWheels = vehicle.engineCurrentTorque * gearRatio * differentialRatio;     f = (tWheels / wheelRadius);     vehicle.acceleration = (f / weight) * grip;          vehicle.object.rotation.z -= vehicle.wheelAngle * vehicle.acceleration * time;     float angle = vehicle.rotation.z;     angle *= M_PI; // The pi constant.     angle /= 180.0;          vehicle.velocity.x -= vehicle->acceleration * sin(angle) * time;     // The world is upside-down for some reason, so subtracting is necessary. This will be fixed!     vehicle.velocity.y += vehicle->acceleration * cos(angle) * time;     vehicle.velocity.z += 0.0;          vehicle.position.x += vehicle.velocity.x;     vehicle.position.y += vehicle.velocity.y;     vehicle.position.z += vehicle.velocity.z;          vehicle.velocity.x *= (1.0 - airResistance);     vehicle.velocity.y *= (1.0 - airResistance);     vehicle.velocity.z *= (1.0 - airResistance);


Electric Field Homework Question

Can a charged particle move through an electric field that exerts no force on it?


Stress calculations in a perforated paper

![Unperforated Paper](http://i.stack.imgur.com/y1H0E.jpg) You have a sheet of paper (torn out of a good quality foolscap notebook) as shown above, and you start pulling it apart with both your hands (forces indicating by the blue arrows). Its difficult to tear the paper apart this way --- it has a very high tensile strength, I assume. If you try this at home, you are bound to fail. ![Perforated paper](http://i.stack.imgur.com/75tzW.jpg) Now if you make tiny perforations in the paper (indicated by red circles), pull it apart the same way, you'll notice that its very easy to tear apart the paper. And the line of tearing will definitely have a few circles on them. Is this happening because of the high stresses that are formed around the circle? Can someone give a good mathematical as well as verbal explanation of this phenomenon.


Would sea water moderate a nuclear submarine meltdown?

My son asked me: > If nuclear submarines get sunk or blown up, what is to stop them going into > nuclear meltdown? I thought about it and came to the conclusion that because they're in the sea the water itself would act as a coolant, stopping nuclear meltdown. Is this correct, as I wouldn't want to teach him nonsense?


What is Getting in the Way of Testing D-Wave?

I know there are other questions i.e. Do quantum computers manufactured by D-Wave Systems, Inc. work? , What can the D-Wave quantum computer do? , etc. But I can't seem to find my answer. What is getting in the way of determining if D-Wave is a quantum computer? Why can't we just analyze the way they built, and see if it's quantum? Is there a problem with patents getting in the way? Why can we _still_ not tell?


What's the avarage temperature of the Universe?

I'm sorry if this was asked before but with all the stars and dark matter and all the other stuff, curently in the Universe, what's the avarage temperature of the Universe? Is it like extremely high or extremely low?


If a window in a plane cabin breaks will a bunch of water vapor form?

If a plane has some windows broken mid-flight (ie open) will the plane suddenly be full of water vapor because of the difference of pressure?


Commutator of Lorentz boost generators : visual interpretation

I have always struggled to visualize the correctness of the commutation relation for the generators of the boost in the Lorentz group. We have $$[K_i,K_j] = i \epsilon_{ijk} L_k$$ I fail to picture this. For definiteness' sake, let's take a point $\vec{x}$ in my coordinate system, lying in the $O_{xy}$ plane. The difference between boosting first in the $x$ and then in the $y$ direction, and boosting first in the $y$ and then in the $x$ direction ammounts simply to a rotation in the $Oxy$ plane, since $$[K_1,K_2] = iL_3$$ I must be doing something wrong when I draw this on a paper. I realize that this holds at the infinitesimal level, but a simple drawing should be able to realize this principle 'at first order' shouldn't it ? My problem is that when I draw it, I end up with the same situation regardless of which direction I boost into first.


How do you calculate the radiant for Kreutz Sungrazer comets? where in the sky do I look?

According to wikipedia: > All known members of the [Kreutz Sungrazer] group up until 1965 had almost > identical orbital inclinations at about 144°, as well as very similar values > for the longitude of perihelion at 280–282°, with a couple of outlying > points probably due to uncertain orbital calculations. This longitude of 280 deg seems to point in the direction of the Orion constellation. However, the radiant would seem to be 180 degrees away (opposite) since I expect the radiant to point to the location of the apehelion. I'm also wondering if there is any reason behind the apparent mirror image similarity (or otherwise some connection) between the sungrazers and the orionids, I found this image on pinterest http://www.pinterest.com/pin/129197083031621626/ ![enter image description here](http://i.stack.imgur.com/Z6JZr.jpg) and this picture is actually what peeked my interest in wanting to know where in the sky the radiant for the sungrazers is. FWIW the approximate RA of the orionids radiant seems to be roughly that of Saiph's, or 05h 47m 45 seems to translate to a longitude of -93 deg = 267 deg which, as I noted above, lies close to the perihelion of the sungrazers (http://math.mercyhurst.edu/~griff/courses/webmapping/RA_convert.php). [--edit: oops it appears the orientation of the pic is "upside" down, the red star I mistook for Saiph there in Orion constellation actually seems to be Betelguese whose longitude is -91 deg = 269 deg. Comparing with the stars shown in Stellarium, the radiant seems to lie right in the middle part of Gemini, at RA of 7 hours = longitude of -75 deg = 285 ] [--edit 2: http://ssd.jpl.nasa.gov/sbdb.cgi?ID=dK11W030;orb=1;cov=0;log=0;cad=0#orb contains a jpl orbit diagram for comet lovejoy--an example of a Kreutz sungrazer; the orbit diagram shows that the RA of the APHELION--not perihelion-- of lovejoy is aligned roughly with gemini (or jupiter currently), so now I am completely confused and am wondering if there is a typo in the wiki sungrazer page, or what the misunderstanding on my part is. ] If anyone can point me to where in the sky the sungrazer radiant (such as nearest star/s) is it would be really appreciated from me.


Why does a tail of a balloon poodle not collapse?

I was just wondering. How can it not collapse? There is pressure in the tail top and in tail itself. How can that little "bridge" (with that I mean the part that in not inflated, see image 2) and tail top sustain and not collapse?      ![enter image description here](http://i.stack.imgur.com/xj6kj.jpg)   **1\. Image of balloon poodle** ![enter image description here](http://i.stack.imgur.com/hj9lY.png)   **2\. Scheme of the balloon tail with a schematic "pressure"** As you can see in _Image 2_ I guess that the pressure is something like that. My best guess is that the surface tension or strenght of the "bridge" is strong enough to sustain. But I would like to know what you guys think!


Calculate trajectory of projectile with gravity from disk planets in 2D?

I'm making a 2D game where you control a spaceship and fly between disk-formed planets of differing size (mass/pull). The planets are small and you can fly near several at once. I need to calculate the gravity produced by each of these planets pulling on the ship including the velocity of the ship. I figured out a way to approximate it:   1. Calculate the position of the rocket in the next frame based on the angle and velocity   2. From this position, calculate the position of a point pulled towards the planet where the larger the planet, the further this point is towards said planet   3. Do this for each planet then average each of these points to find the new location for the rocket this frame   4. Rotation doesn't matter in my case, but I would calculate it by comparing this point's location and the previous point's location.   5. Repeat for every frame. This may be a good approximation, and accuracy isn't a huge deal in my case, but I am wondering if there is a better way to do this with physics formulas. I haven't taken a physics class yet. The other main problem with this solution is that it must calculate it step by step or frame by frame. If there is a formula to do this the path could be calculated from the very start before.


Ultracold atoms and General theory of Relativty

I am looking for good reviews for the subject of Ultracold atoms and it's application in test of General theory of relativity. I am planning to pick up this topic as a semester project, can somebody guide me ? I am basically looking forward to some good books and papers which gives me a flavor of theoretical aspects of this subject. I came across an website which says about Quantum gauge theories and Ultracold atoms and that has interested me a lot. http://cordis.europa.eu/search/index.cfm?fuseaction=proj.document&PJ_RCN=10276613


Understanding how clouds form: warm air near the earth's surface?

I'm trying to understand how clouds are formed. I've read that it begins with air near the earth's surface heating up. But I'm hearing two different reasons for how this happens:   1. Air pressure near the earth's surface is high and this causes water vapor near the earth's surface to heat up, since the high pressure causes the kinetic energy of the water vapor molecules to increase.   2. The sun heats up the earth's surface, and this heats up the water vapor near the earth's surface. So what is the truth?


Nonlinear combination of velocities implies no absolute time?

Landau 1961 begins with a brief presentation of special relativity. This question is about the validity of a certain argument that they use in building up the foundations of the subject from scratch, so I think I should first sketch what they consider foundational. They define an inertial frame as one in which Newton's first law holds. They telescope Einstein's two 1905 postulates into a single principle of relativity, which is that the laws of physics are form-invariant w.r.t. changes of inertial frame. They assert (based on unspecified experiments) that forces are propagated at a finite speed. They also seem to assume (presumably also based on experiment) that this speed is a maximum and universal for all interactions. Given a maximum velocity, they argue that time is not absolute. The argument is that in Galilean relativity, velocities combine linearly, which is incompatible with a maximum velocity and with the Michelson-Morley experiment. Also, Galilean relativity posits absolute time. Therefore absolute time must be wrong. This seems like a clear logical fallacy to me. (A implies B, and A also implies C. B is false, so C must be false.) [EDIT] Oops, crucial mistake in my original statement of the question. The parenthetical originally ended with "C must be true." Should have ended with "C must be false." Is there any way of salvaging this argument? Am I misunderstanding it? Landau and Lifshitz, The classical theory of fields, 2nd ed., 1961


What is the infinitesimal work done when the force is given by the gradient of a scalar function that depends both on position AND time?

The title is slightly confusing but I didn't know how else to phrase my question. Basically, this is the situation: When the force applied to a particle is given by the gradient of a scalar function that depends explicitly on JUST the position of the particle, that force is conservative and $T+V$ is conserved.   However, if it depends explicitly on both position AND time, then $T+V$ is still defined but not conservative. In my notes, this is proven: $$F(r(t),r)\cdot \dot{r} dt = -\nabla V(r(t),t)\cdot \dot{r}dt = \left( -\frac{dV(r(t),t)}{dt}+\frac{\partial V(r(t),t)}{\partial t}\right)dt$$ What I don't understand is the second line, working it out a few times but I never get the correct result.


d=2 O(3) sigma model becomes "conformal antiferromagnet"

In Advanced topic in quantum field theory / M. Shifman on page 251 the author discusses the fact that the theta term is topological and does not affect the equations of motion. Then he said: > "In particular, at $\theta=\pi$ the O(3) sigma model becomes conformal and > describes a ferromagnet rather than antiferromagnet. Unfortunately, I cannot > dwell on the aspect in this text" Where he refers to the term $\frac{\theta}{8\pi}S^{a}(\partial_{\mu}S^{b})(\partial_{\nu}S^{c})\varepsilon^{\mu\nu}\varepsilon_{abc}$. I would like to see here more details about this point. Is it a uniqe phenomena for $\theta=\pi$?


What does quantum phenomena exist as prior to observation?

It's been said that according to the Schrodinger equation, independent of observation, particles exist in a state of a wave function, which is a series of potentialities rather than actual objects. Then the act of observing, causes a wave of potentiality to collapse to a state of matter. Is this true? Also, are these quantum experiments anomalies, and not how nature works, or do these quantum experiments, like double slit and quantum erasers, actually reflect how nature works even in the absence of human manipulations.


Cases in which angular velocity and angular momentum point into same direction

I know that angular momentum $\vec{L}$ and angular velocity $\vec{\omega}$ of a rigid body doesn't point into the same direction in general. However if your body spins around a principal axis, $\vec{L}$ and $\vec{\omega}$ point into the same direction. In which other situations is this is exactly or at least approximately true? Please give reasons or a mathematical derivations why it is true in a given situation.


Intuitive explanation for the de Broglie / Planck relations

A friend asked me to explain "why" a particle's energy is proportional to it's frequency, i.e: $$E=h\nu$$ The reason this result is so un-intuitive, is that in the macroscopic world, A wave's energy (e.g electric power dissipated through a resistor) is not frequency dependent but rather amplitude dependent. Of course a "pure" photon doesn't really have an amplitude due to it's description as a plane wave, but that's beside the point. **So in short** : Is there an easy explanation of "why" the energy is only dependent on frequency? can this be "derived" from other principles (say, the particle-wave duality)? * * * **N.B.** I'm aware there are no real "reasons" for any law of physics other then the fact that they agree with experiment, but laying those laws on a smaller number of basic principles is always nice when possible.


Can two electrons never touch each other?

The Coulomb's force is given by $$ F = {k q^2 \over r^2} $$ When $ r \rightarrow 0 $, $ F \rightarrow \infty $ Does this mean two electrons never touch each other?


What determines the speed required to pull a table cloth?

I was watching this show "Street Genius" on National Geographic and the host Tim Shaw demonstrated an experiment about Inertia, What he did was, He tied one end of a table cloth to a car through a long rope and started driving the car, when the rope was taut, the cloth was pulled and the things on the table were still on the table. What are the calculations involved in it and what determines the speed required to pull the table cloth without disturbing the things on it?


Probability density of detection of collinearly emitted photons in two detectors

**Update** :   As proposed by @dmckee, I added equation numbers and improved the display of some equations. The answer by @Trimok inspired me to look at coordinate systems which are not specific to the detector locations, but more general, and this indeed made things clearer since $\delta(\cos \theta + 1)$ is a $\delta(\cos \theta + \cos 0)$ in disguise. We have a relation between a Dirac delta of a ray in spherical coordinates, $\delta^2$, and a Dirac delta of a point, $\delta^3 = \delta(r-r_0)\delta(\theta-\theta_0)\delta(\phi-\phi_0)/(r^2 \sin \theta)$, such that $\delta^2 = \int_0^\infty \delta^3 \operatorname{d}r_0 = \delta(\theta-\theta_0) \delta(\phi-\phi_0) / (r^2 \sin\theta) \tag{E1}$ (Using this definition, one can easily show that the sifting property of $\delta^2$ holds and the result is a line integral of some $f$.) The major problem (of understanding) which I am having is now reduced to this: We have two expressions for $\delta^3$, which are $\delta(r-r_0)\delta(\theta-\theta_0)\delta(\phi-\phi_0)/(r^2 \sin \theta) \tag{E2}$ and $\delta(r-r_0)\delta(\theta-\theta_0)\delta(\phi-\phi_0)/(r_0^2 \sin \theta_0) \tag{E3}$ These are totally symmetric in $x$ and $x_0$ (note that $\delta$ is an even function). However, for $\delta^2$ we have $\delta(\theta-\theta_0)\delta(\phi-\phi_0)/(r^2 \sin \theta) \tag{E4}$ and $\delta(\theta-\theta_0)\delta(\phi-\phi_0)/(r^2 \sin \theta_0) \tag{E5}$ which are not symmetric any longer. So what makes the $r$ stand out compared to $r_0$? One explanation I worked out with the help of a colleague is directed towards the fact that there is no $r_0$ after integration, so having any $r_0$ would "overdetermine" the set of rays through the origin. Does anyone have a better (for example, more intuitive) explanation why the symmetry is lost here? Thank you! **Original question** :   This is a long one. I have been working on this for quite a while now, so I summarized most of what I found out. I am stuck at a certain point, however, and it would be great if someone could help me out. Assume the following situation: Two perfectly collinear photons $p_1$ and $p_2$ are emitted in the origin of some Cartesian coordinate system. We consider two infinitesimal detectors $d_1$ and $d_2$, with surfaces $\operatorname{d}\\!A_1$ and $\operatorname{d}\\!A_2$ directed towards the origin so that we do not have to deal with angles. The solid angles of these detectors be $\operatorname{d}\\!\Omega_1$ and $\operatorname{d}\\!\Omega_2$. I now express some basic detection probability densities, where $p_id_i$ means that "$p_i$ is emitted towards $d_i$".   $\frac{\operatorname{d}\\!P(p_1d_1)}{\operatorname{d}\\!\Omega_1}=\frac{1}{4\pi} \tag{1}$ $\frac{\operatorname{d}\\!P(p_1d_1)}{\operatorname{d}\\!A_1}=\frac{1}{4\pi (r_1)^2} \tag{2}$ $\frac{\operatorname{d}\\!P(p_2d_2)}{\operatorname{d}\\!\Omega_2}=\frac{1}{4\pi} \tag{3}$ $\frac{\operatorname{d}\\!P(p_2d_2)}{\operatorname{d}\\!A_2}=\frac{1}{4\pi (r_2)^2} \tag{4}$ I think there is no doubt that these probabilities are well-defined and the integrated detection probabilities in a spherical $4\pi$-detector equals 1, no matter what its radius is. So a photon is surely emitted towards a detector surrounding the origin. Now, for some reason we want to calculate $P(p_2d_2 | p_1d_1)$, the conditional probability of $p_2$ emitted towards $d_2$, given that $p_1$ has been emitted towards $d_1$. Due to collinearity of the photons, the individual events are not independent: $P(p_2d_2 | p_1d_1) \neq P(p_2d_2)$. I now propose that $\frac{\operatorname{d}\\!P(p_2d_2 | p_1d_1)}{\operatorname{d}\\!\Omega_2} = \frac{\delta(\cos \theta_2(d_1)+1)}{2\pi}, \tag{5}$ where $\theta_2(d_1)$ is the polar angle of the position of $d_2$ expressed in spherical coordinates with polar axis from the origin through $d_1$. $\delta$ is the Dirac function, so obviously, this expression equals 0 if $\cos \theta_2(d_1) \neq -1$, that is, when $\theta_2(d_1) \neq \pi$. We can verify that the integral of this probability over a spherical $4\pi$-detector $d_2$ equals 1: with $\operatorname{d}\\!\Omega_2 = \operatorname{d}\\!\cos \theta_2(d_1) \operatorname{d}\\!\phi_2(d_1)$, we find $\int \frac{\operatorname{d}\\!P(p_2d_2 | p_1d_1)}{\operatorname{d}\\!\Omega_2} \operatorname{d}\\!\Omega_2 = \int \frac{\delta(\cos \theta_2(d_1)+1)}{2\pi} \operatorname{d}\\!\Omega_2 \tag{6}$ $= \int_0^{2\pi} \frac{1}{2\pi} \operatorname{d}\\!\phi_2(d_1) \cdot \int_0^\pi \delta(\cos \theta_2(d_1)+1) \operatorname{d}\\!\cos \theta_2(d_1) = 1. \tag{6}$ So $p_2$ is still surely emitted towards a detector surrounding the origin, no matter where $p_1$ is emitted towards. Further, we would like to verify that the law of total probability is fulfilled by integrating $P(p_1d_1 \wedge p_2d_2) = P(p_2d_2 | p_1d_1) \cdot P(p_1d_1)$ over a spherical $4\pi$-detector $d_1$. This works regardless of which formulation of $\operatorname{d}\\!P(p_1d_1)$ (with respect to $\operatorname{d}\\!\Omega_1$ or $\operatorname{d}\\!A_1$) we use -- I chose $\operatorname{d}\\!A_1 = (r_1)^2 \operatorname{d}\\!\cos \theta_1(d_2) \operatorname{d}\\!\phi_1(d_2)$ here. Note, however, that integration over $d_1$ changes the polar axis used in the formulation of $\operatorname{d}\\!P(p_2d_2 | p_1d_1)$ above, so we need a different, but equivalent formulation. This is more than a change of coordinate system (from one polar axis to another), because $\theta_2(d_2)$ would not help us a lot (it's 0). Instead, one can use $\frac{\operatorname{d}\\!P(p_2d_2 | p_1d_1)}{\operatorname{d}\\!\Omega_2}=\frac{\delta(\cos \theta_1(d_2)+1)}{2\pi}, \tag{7}$ which uses both a different variable in the argument of the $\delta$ as well as a different coordinate system: $\theta_1(d_2)$ is the polar angle of detector $d_1$'s location in spherical coordinates with polar axis from the origin through $d_2$. **Note that both expressions of $\frac{\operatorname{d}\\!P(p_2d_2 | p_1d_1)}{\operatorname{d}\\!\Omega_2}$ are symmetric in the indices 1 and 2.** Finally, we arrive at $\int \frac{\operatorname{d}\\!P(p_1d_1 \wedge p_2d_2)}{\operatorname{d}\\!A_1 \operatorname{d}\\!\Omega_2} \operatorname{d}\\!A_1 = \int_0^{2\pi} \int_0^\pi \frac{1}{4\pi (r_1)^2} \cdot \frac{\delta(\cos \theta_1(d_2)+1)}{2\pi} (r_1)^2 \operatorname{d}\\!\cos \theta_1(d_2) \operatorname{d}\\!\phi_1(d_2) \tag{8}$ $= \frac{1}{4\pi} \int_0^{2\pi} \frac{1}{2\pi} \operatorname{d}\\!\phi_1(d_2) \cdot \int_0^\pi \delta(\cos \theta_1(d_2)-1) \operatorname{d}\\!\cos \theta_1(d_2) = \frac{1}{4\pi} = \frac{\operatorname{d}\\!P(p_2d_2)}{\operatorname{d}\\!\Omega_2} \tag{8}$ So, the law of total probability is fulfilled: the probability of $p_2$ being emitted towards $d_2$ can be partitioned into all conditional probabilities of $p_2$ being emitted towards $d_2$ _given that_ $p_1$ has been emitted towards $d_1$, weighted by the probability that $p_1$ is _actually_ emitted towards $d_1$. So far, so good. Congratulations for reading up to this point :) As expressed above (and verifiable relatively easily), it does not matter which formulation of $\operatorname{d}\\!P(p_1d_1)$ I use -- both $\operatorname{d}\\!\Omega_1$ and $\operatorname{d}\\!A_1$ work fine. However, when I try to base my calculations on $\frac{\operatorname{d}\\!P(p_2d_2)}{\operatorname{d}\\!A_2}$ instead of $\frac{\operatorname{d}\\!P(p_2d_2)}{\operatorname{d}\\!\Omega_2}$, hence, I try to calculate $\frac{\operatorname{d}\\!P(p_2d_2 | p_1d_1)}{\operatorname{d}\\!A_2}$ instead of $\frac{\operatorname{d}\\!P(p_2d_2 | p_1d_1)}{\operatorname{d}\\!\Omega_2}$, I arrive at the following problem. So I started with $\frac{\operatorname{d}\\!P(p_2d_2 | p_1d_1)}{\operatorname{d}\\!A_2}=\frac{\delta(\cos \theta_2(d_1)+1)}{2\pi (r_2)^2}, \tag{9}$ and (6) is still easily verified (integrating over $\operatorname{d}\\!A_2$) since the two additional $(r_2)^2$ simply cancel out. However, working out (8) is now more complex when it comes to changing from one variable/coordinate system to another. I figured that $\frac{\operatorname{d}\\!P(p_2d_2 | p_1d_1)}{\operatorname{d}\\!A_2}=\frac{\delta(\cos \theta_1(d_2)+1)}{2\pi (r_2)^2} \tag{10}$ is the expression which fulfills total probability (I can show this, if necessary), but now I wonder why this is not symmetric with the first expression in the indices 1 and 2 (as it was above). Specifically, I wonder where the connection to a Dirac of a ray along Oz in spherical coordinates is, which is $\frac{\delta(\cos \theta - 1)}{2\pi r^2}$ (compare http://www.physicsforums.com/showthread.php?t=215263) More generally, I am working on expressing the probability density in 3D, so I expect to obtain something like $\operatorname{d}\\!\Omega_2 \cdot \delta_{L(x_1, x_2)}(x_0)$ or $\operatorname{d}\\!A_2 \cdot \delta_{L(x_1, x_2)}(x_0)$, where $L(x_1, x_2)$ is the line segment from $x_1$ and $x_2$, and $x_0$ is the point of emission. So my specific questions are:   * Why are the expressions involving $\operatorname{d}\\!A_2$ not symmetric, while those involving $\operatorname{d}\\!\Omega_2$ are?   * Specifically, where is the connection to the Dirac of a ray in spherical coordinates, which involves an $r^2$ and which I would expect to be changed when converting from one coordinate system to another.   * How (else) can I obtain a 3D probability density involving the three locations, which can be used to integrate over all three volumes in order to describe the number of coincidences detected from emissions from a volume in two finite detectors?


Energy Required To Push Up a Slope

I'm having a bit of trouble with this problem: A $5.0\:\text{kg}$ block is moved up a slope with constant velocity of $2.5\:\text{m/s}$. The amount of force required to push the block up the slope is $20.0\:\text{N}$. The slope is $45.0\:\text{cm}$ high, $150\:\text{cm}$ horizontally on the ground, and its hypotenuse/slope is $125\:\text{cm}$. How much energy is required to push the block up the slope? Angles are not supposed to be used in order to obtain the answer. (I initially thought that the energy required would just be equal to the potential energy at the top of the slope, i.e. $mgh$. However, I'm not sure if there are other factors that are needed to be taken into consideration?)


Why does observation collapse the wave function?

In one of the first lectures on QM we are always taught about Young's experiment and how particles behave either as waves or as particles depending on whether or not they are being observed. I want to know what about observation causes this change?


Is angular momentum always conserved in the absence of an external torque?

Consider either the angular momentum of the earth around the sun or equivalently swinging a ball horizontally on a string. I know that with respect to the point of rotation of the swinging ball, angular momentum is constant. Consider now an origin outside of the 'orbit'. It seems to me that the angular momentum is different as the orbit progresses but there is no external torque (the physical scenario is unaltered!). The easiest way to see this is to consider when the ball is at opposite ends of the circle - the radius changes but the velocity only flips sign. In addition to this, how about we consider an event where we cut the string. Now place our reference point on the line of motion when the ball flies away in a straight line. By definition the ball will have zero angular momentum with reference to any point on this line ($m|r||v|sin\theta=0$) but certainly before the event it had some angular momentum. The main question is in the title, but essentially what am I missing conceptually about this problem?


Can one prove the full spin-statistics theorem from the spin 0, 1/2 and 1 cases?

Using second quantization for scalar field, spinor field and vector fields, we can get commutation and anticommutation relations for the birth and destruction operators of the fields, which leads us to the Bose or to Fermi statistics. Is it possible to expand these results on a field of arbitrary spin (integer or half-integer), using the basic idea that each field can be built by combination of spinor $\frac{\hbar }{2}$ fields?


What is a two-photon process?

I am reading some introductory materials on modern optics, in which they mention two-photon processes everywhere. I know fundamental optics and a bit on quantum mechanics. Can anyone explain in a plain way what is a two-photon process or any simple introduction to a related topic.


If I take a handful of salt and wait for an infinite time will it become a single crystal?

That pretty much says it. Suppose I have some powder of $NaCl$. It is kept in contact with itself in vacuum. You are free to remove all the disturbances that bother you. Is that true that, well, there exist a ($\mbox{very}^\mbox{very}$ large) amount of time $T$ that for every moment $t>T$ you will have a single beautiful crystal with 99% probability.


How do I build an alpha emitter?

I would like to build an alpha emitter. I suppose I had some vague idea of knocking the electrons off of some helium. But upon further examination the idea does strike me as...naive? My question is then, how do I go about building an alpha emitter, cheaply, and with household/hardware store items. (Besides springing for a smoke detector.)


Infinite acceleration?

Why is acceleration regulated by mass? In a frictionless environment, why doesn't an object move at infinite acceleration if force is applied on it? Force causes movement, so unless there is an opposing force there shouldn't be any reason for the force to cause infinite acceleration. Why does mass act like an opposing force that limits the acceleration caused by an force to less than infinity?


Does the earth's gravity affect horizontal movement?

In a frictionless environment, would the weight of an object affect how hard it is to move it horizontally? Since without friction, there is no horizontal force resisting your applied force, would the object be easy to move? I am aware of inertia but don't see any reason for it to play a role considering there is no opposing force to your applied force.


Why does the force of gravity get weaker as it travels through the dimensions?

Some theories predict that the graviton exists in a dimension that we of course can't see, and that is why the force of gravity is so weak. Because by the time gravity has got from the dimension in which the graviton exists to our dimension it has lost much of its strength. For a better understanding of this theory, I think this might help: ![](https://lh4.googleusercontent.com/-M3Z7HBllZys/T5JT- dGobZI/AAAAAAAAFIE/I_XJUA3M0pI/randall_750.jpg) Bigger: https://lh4.googleusercontent.com/-M3Z7HBllZys/T5JT- dGobZI/AAAAAAAAFIE/I_XJUA3M0pI/randall_750.jpg?imgmax=1600 Why does the force lose strength as is travels through the other dimensions?


Normal ordering

If I understood correctly there are two terms called normal ordering:   1. $:c c^\dagger: = c^\dagger c \hspace{.5cm}$so shifting all creation operators to the left and all annihilation operators to the right.   2. $:c^\dagger c:=c^\dagger c- \langle c^\dagger c \rangle\hspace{.5cm}$ so subtracting the contraction Both of these are called normal ordering (I think) and use the same notation. But what is the connection? If I got something wrong, please correct and explain!


How do you determine the value of the degeneracy factor in the partition function?

In the partition function, expressed as $$Z = \sum_j g_je^{-\beta E_j}$$ I'm wondering what determines the $g_j$ factor. I've been trying to look around the internet for an explanation of it but I can't find one. I guess it is the number of degenerate states in a given energy level? How do you determine how many degenerate states there are? A simple example involving how to determine the $g_j$ factor would be great. Thanks for the help!


Microscopic Theory of superfluidity

In comparison with superconductivity what is the microscopic theory of superfluidity? Who has done it?


Inertial Mass of a scalar field

Does it make sense to talk of the inertial mass of a scalar field? By the equivalence principle, it must be equal to its gravitational mass. We know that the scalar field contributes towards the stress-energy tensor, so, shouldn't it have an inertial mass too?


How could we define speed without time?

This is a bit of a brain twister but it's a very serious question. Please be precise in your answer. I do not believe time exists. I believe it is simply an illusion of our perception, the perception of seeing things around us change. The faster things change the more time seems to go by. In some sort, change is the result of the dissipation of energy. Thus what I believe is that what we are really observing is the energy being dissipated and created all around us and we call it time. Since time doesn't exist (for me). That would mean that the equation of [ speed = distance / time ] is impossible. But nevertheless the concept of speed is fundamental for physics. What could we replace this equation with to no longer have speed dependent of time but of energy?


Can a sound like a sonic boom change a object like a wall? And how the light on the walls will be changed after a sound like this?


What is ``thermal" about a thermal quotient of EdS and EAds?

This is in continuation of my previous question and is in reference to this paper. I guess that the authors are interested in $S^n$ and $\mathbb{H}^n$ since these are the Euclideanized versions of $dS^n$ and $AdS^n$ respectively. But I am not clear about the physical motivations of the quotients that they take. They take the following quotients,   * The sphere $S^{2n+1} = SO(2n+2)/SO(2n+1)$ can be thought to be given as the zero-set in $\mathbb{C}^{n+1}$ of the equation, $\sum_{i=1}^{n+1} \vert z_i \vert ^2 = 1$. Here they want to identify $\phi \sim \phi + \frac{2\pi}{n}$ for some $n \in \mathbb{N}$ where $\phi$ is the phase of $z_1$. And this quotient is given by $\mathbb{Z}_n\backslash SO(2n+2)/SO(2n+1)$   * On $\mathbb{H}^{2n+1} = SO_0(2n,1)/SO(2n)$ they take the metric as, $cosh^2\rho dt^2 + d\rho^2 + sinh^2 \rho d\Omega_{2n-1}$ and take the quotient as $t \sim t + \beta$ and then the quotient space is given as $\mathbb{Z} \backslash SO_0(2n,1)/SO(2n)$ . Here $\frac{2\pi}{n}$ and $\beta$ are connected by the same analytic continuation as what takes one from the metric on $S^{2n+1}$ to the metric on $\mathbb{H}^{2n+1}$ Clearly in the paper they keep referring to \beta as the inverse temperature but I can't clearly see what is the thermal picture which will correspond to evaluating the partition function on the these quotients as above. If there were a particle at inverse temperature $\beta$ on spatial $\mathbb{H}^{2n}$ then I would think that one would calculate the partition function on the manifold $\mathbb{H}^{2n}\times S^1$ such that the $S^1$ is of radius $\beta$. Is this somehow related to the quotient of $\mathbb{H}^{2n+1}$ described above in the second bulled point? (...the same could have been asked about $S^{2n} \times S^1$ also..)


At what point does everything become nothing?

I understand that the universe, which I'll call "everything", is expanding and it used to be much smaller. But I keep hearing assertions about a universe coming from nothing. If you rolled the clock back, at what point would everything become nothing?


Lepton masses in the Standard Model

Some simple questions regarding leptonic masses in the Standard Model (SM):   1. Why there is not an explicit mass term in addition to the effective mass term that arises from the Yukawa terms after spontaneous symmetry breaking (SSB).   2. The lepton mass term is: $$m_{l_i}\sim y_i\cdot v \sim y_i\cdot \frac{m_H}{\sqrt{\lambda}}$$ with $m_{l_i}$ the lepton mass, $y_i$ the Yukawa coupling, $v$ the vacuum expectation value of the Higgs field, $m_H$ the Higgs mass, and $\lambda$ the self-coupling of the Higgs.   So the running of leptonic masses in the SM are given by the running of $y_i$, $m_H$, and $\lambda$. Thus the lepton masses are not protected to radiative corrections $\beta(m_{l_i})\neq 0$ for $m_{l_i} = 0$. This is in contrast with explicit lepton mass terms in theories without SSB where Chiral symmetry protect fermions' masses from radiative corrections. Is this correct? If not, why?


Relation among anomaly, unitarity bound and renormalizability

There is something I'm not sure about that has come up in a comment to other question: Why do we not have spin greater than 2? > It's a good question--- the violation of renormalizability is linked > directly to violation of unitarity, which was exploited by Weinberg > (surprise, surprise) to give an upper bound of something like 800 GeV on the > fundamental Higgs mass from the W's and Z's unitarization. The breakdown of > renormalizability is a wrong one-loop propagator correction to the gauge > boson, and it leads to a violation of the ward-identity which keeps the non- > falling-off part of the propagator from contributing. It's in diagrammar (I > think), it's covered in some books, you can ask it as a question too. I know what unitarity bound is the user talking about, but I don't know what is the violation of the Ward identity that he mentions. I guess that it is the _global_ $SU_L(2)$ symmetry but I have never seen anything relating the unitarity bound and this anomaly. The general issue is the following: Assume a Yang-Mills term and the coupling of the charged vector field to a fermionic conserved current under a global symmetry. Then one adds an explicit mass term to the vector field so that one breaks the gauge symmetry by hand, but not the global part that gives the conserved current (the gauge symmetry that goes to the identity in the boundary entails constraints). Then, according to the user (at least what I understood), when one takes into account loops effect the global part is also broken. Therefore, the mass term is breaking the redundancy part of the symmetry by hand (at the classical level) and it is also breaking the global part at the quantum level. I would be grateful if somebody is able to clarify this to me. References are also welcome.


When and how were relative distances to the planets first measured?

I understand that the absolute distance to a planet can be measured using earth-baseline (e.g., diurnal) parallax, and that the first reasonably accurate such measurement was made for Mars by Cassini (and his assistant Richer) in 1672, and then, famously (with some modifications) for Venus by Halley, Cook, and others and others during its 1769 transit; but when were earth-baseline _relative_ distance measurements first made for each of the planets? Interpretation of such measurements requires assumptions about the nature of the orbits of planets, but some of the most compelling evidence for the validity of these assumptions is Kepler's third law of planetary motion, the persuasiveness of which itself rests on observations of relative distances. In fact, all of my texts say, in effect, that the third law "fit" the observed pattern of orbital period and relative distance. Where did these distance measurements come from? What relative distance measurements were available to Kepler in 1619? When was earth-orbit-baseline parallax used to make these measurements for each of the planets?


What is Si-delta doping?

I want to know what the delta means in this case. I know the Si means the element used, by some way to doping. I guess the delta means that using some elements to create holes in semiconductor made of Si. Am I right?


Could Dark Matter be a manifestation of tachyons tunnelling between black holes?

Among the many Dark Matter candidates, I wondered if there are any along the lines of the title. The inner horizon of a black hole, with its mass inflation, and/or the high spacetime curvature, seems a violent enough place for the strangest things to happen that can anywhere, and tachyons (if these exist) seem to have enough counterintuitive properties that they might perceive far greater space volumes accessible inside black holes than outside, so that any created inside and straying out would readily tend to find their way back into the same or another black hole, with some energy change while in transit that would manifest itself outside the hole(s) as "dark mass" (assuming this was a fairly steady state process). Piling speculation on speculation, one might further envisage that a tachyon which exited a black hole and lost enough energy outside (and hence gained speed) for long enough might greatly increase its chance of staying outside and thus manifest itself as dark energy. The latter would also explain why the expansion of the universe is accelerating: If the average distance between black holes (between galaxies for example) is increasing with cosmic expansion, then on the preceding assumption slightly more tachyons can be expected to remain outside black holes, and so on. It seems all this would make most sense (if at all) if tachyons appeared somehow "spread out" in normal space, like a field, as opposed to conventional sub-luminal particles. Apologies if this is all too vague and speculative for some peoples' liking. I am not axe-grinding, quite the opposite, just curious to hear any informed comments on an idle fancy which is most likely untenable.


Fracture because of high-speed rotation

I was watching a rerun of an early MythBusters episode, where they look at whether CDs in high-speed drives can explode / fail simply because of being rotated too fast. The following are some calculations I did: Let the thickness of the CD be $t$, made of material with density $\rho$. Consider a small element of the CD at a distance $r$ from the center. The element subtends an angle $d\theta$ at the center, and is of length (along the radius) $dr$. The centripetal force needed to pull this in at angular velocity $\omega$ is $m r \omega^{2}=\rho \left(r d\theta dr t\right) r \omega^{2}$. The CD has a central spindle hole of radius $r=1.5\ cm$, and is itself of radius $R=12\ cm$. The force then exerted on an elemental area subtending angle $d\theta$ at the inner edge of the CD is (integrating the previous quantity from $r=r$ to $R$) $\frac{\rho \left(R^3 - r^3\right) d\theta t \omega^2}{3}$, which means the inside edge has pressure approximately $\frac{\rho R^3 \omega^2}{3r}$ (dividing the previous quantity by the area $t r d\theta$, and assuming $R^3$ dominates $r^3$). Fracture now happens when $\frac{\rho R^3 \omega^2}{3r} > T$, the fracture strength of polycarbonate plastic (which being brittle, is approximately its ultimate tensile strength). Substituting $\rho = 1.22\ g\ cm^{-3}$, and $T = 75\ MPa$ (all approximate values obtained from Wikipedia), I got $\omega > 17000\ rpm$. My question is, is the calculation correct? Or is there something I'm missing or misinterpreting? The value is the same order of magnitude as what MythBusters reported -- fracture at $\approx 25000\ rpm$. This is heartening, but it would be nicer to know that the number is off only because of incorrect inputs (strength of plastic, density etc.). As nice as MythBusters sometimes is, it would be nicer still if they did a little more hard science. * * * Correction: As akhmeteli points out below, the CD dimensions are incorrect. The central spindle hole has radius $7.5\ mm$, and the CD itself has radius $60\ mm$. There's also an arithmetic mistake in the above calculation. The updated answer is therefore $\approx 24000\ rpm$. That would ordinarily make me very happy, but as both user758556 and ja72 argue, this calculation would be useful as nothing more than a sanity check. What's really happening is a little too complicated for these arguments to properly explain.


Elastic potential

I have a doubt: elastic potential energy is given by: $U=\frac{k}{2}x^2+K$ but does elastic potential exist? (for example: potential gravitational energy is given by $U=mgz+K$ and gravitational potential is given by $V=-U$) ($K$ is the constant of integration)


What are the implications for quantum gravity if the LHC sees no higgs?

Following the news in serious "non-hype" physics blogs I`ve learnd that as things are now one needs a lot of patience and more data to learn what happens in the higgs sector. There are already a number of good questions and discussions about what the absence of a higgs would mean for the standard model or more general for theoretical physics on this site. So I want to "extrapolate" this a bit and ask about what would be the implications for the different quantum gravity approaches if no higgs boson(s) (are) seen at the LHC ?


Mass of empty AdS$_5$

Five dimensional empty AdS$_5$ space has mass $$ E = \frac{3 \pi \ell^2}{32 G}. $$ > Is the above equation correct? Let's do some dimensional analysis to confirm. In natural units, in 5 dimensions $[G] = -3$ where $[...]$ is the mass dimension. Also $[\ell]=-1$. Therefore $\left[ \frac{\ell^2}{G} \right] = 1$. So the dimensions seem to work out fine. Here's my second question: > The limit of $\ell \to \infty$ of $AdS_5$ space is flat space. Isn't it > weird that the mass diverges in this limit? I would have presumed that mass > should vanish in this limit, since flat space has vanishing mass? Are we > using two different definitions of mass? EDIT: Due to some requests in the comments, I will include the derivation of the formula above. I use the boundary stress tensor given by Brown-York (derived from the Einstein action alongwith the Gibbons-Hawking boundary term. $$ t_{ij} = \frac{1}{8\pi G} \left[ K_{ij} - \gamma_{ij} K + \frac{2}{\sqrt{-\gamma}} \frac{\delta S_{ct}}{\delta \gamma^{ij}} \right] $$ Here $K_{ij} = \nabla_{(i} n_{j)}$ is the extrinsic curvature. $n^\mu$ is the unit normal vector to the boundary. $\gamma$ is the boundary metric and $K = \gamma^{ij} K_{ij} $. $S_{ct}$ is the counterterm action included to make all the B-Y charges finite, which are defined as $$ Q_\xi = \int_{\cal B} d^d x \sqrt{\sigma} u^i \xi^j t_{ij} $$ Here $\sigma_{ab}$ is the metric on a spatial hypersurface and $u^i$ is a time-like unit vector normal to the hypersurface. $\xi^j$ is a Killing vector of the boundary metric. Now, for $AdS_5$, the counterterm action is given by $$ S_{ct} = -\frac{3}{\ell} \int d^4 x \sqrt{-\gamma} \left( 1 + \frac{\ell^2}{12} R(\gamma) \right) $$ The B-Y tensor is then $$ t_{ij} = \frac{1}{8\pi G} \left[ K_{ij} - \gamma_{ij} K - \frac{3}{\ell} \gamma_{ij} + \frac{\ell}{2} \left( R_{ij} - \frac{1}{2} \gamma_{ij} R \right) \right] $$ We can now work in the Fefferman-Graham coordinates for $AdS_5$ space where the metric is $$ ds^2 = \frac{\ell^2 d\rho^2}{4\rho^2} - \frac{(1+\rho)^2}{4\rho} dt^2 + \frac{\ell^2 ( 1 - \rho)^2}{4\rho} d\Omega_3^2 $$ Thus $$ t_{ij} = - \frac{ \rho }{4\ell \pi G} \left( \gamma_{ij}^{(0)} + \gamma_{ij}^{(2)} \right) + {\cal O}(\rho^2) $$ where $$ \gamma_{ij}^{(0)} dx^i dx^j = -\frac{1}{4} dt^2 + \frac{\ell^2}{4} d \Omega_3^2 $$ $$ \gamma_{ij}^{(2)} dx^i dx^j = - \frac{1}{2} dt^2 - \frac{\ell^2}{2} d \Omega_3^2 \\\ $$ We also have $$ u = \frac{2 \sqrt{\rho}}{1+\rho} \partial_t,~~ \xi = \partial_t,~~\sqrt{\sigma} = \frac{\ell^3 (1 - \rho)^3 }{8 \rho^{3/2} } \sin^2\theta \sin \phi $$ Plugging all this in, we find that the B-Y charge corresponding to the Killing vector $\partial_t$ is $$ Q_t = \frac{3 \pi \ell^2 }{32 G} $$ This is where I got the formula from. I interpret this as the mass of $AdS_5$ space. Disclaimer - I have intentionally left out several computations to reduce the length of the problem. I have not referred any paper and all computations have been done by me.


What is the longest distance that can be jumped after swinging from a rope?

In the movie _Mission Impossible 3_ , the main character Ethan Hunt tries to enter a building in Shanghai by swing through the sky, as shown below: ![enter image description here](http://i.stack.imgur.com/1CxqB.png) The jump consists of 2 sections, the red part, which is an arc, and the orange part, which is a parabola. A question naturally arises: **_For a target building with a given height, what is the furthest horizontal distance it can be placed such that we can jump to its roof using this method?_** Before answering this question, we must consider the following fact: ![enter image description here](http://i.stack.imgur.com/Vppya.png) Observing the path when we release at different heights, we see that if we release very late, we can reach great heights, but with reduced horizontal distance. If we release early, we increase horizontal distance, but with reduced height. Because of this trade off, the area that can be reached is restricted within the yellow line。 The question is then: (1) **What is the equation x(z) that describes the orange line above?** (2) **For a given point (x,z) that is inside the region bounded by the orange line, what is the proper release height h(x,z) such that the parabola will go through (x,z). The answer should be a set of h. This is the same as asking what is the proper release height if we want to reach a building with height z and is located at x distance away from the center building.** For the ease of discussion, I propose that we use the follow coordinate system and notation : ![enter image description here](http://i.stack.imgur.com/82StP.png) * * * Here are some calculations that are already done: At the release point, $$\text{Upward shooting angle}=\psi$$ $$v=\sqrt{2gh}$$ $$h(\psi)=r\cos\psi-H$$ $$\psi(h)=\arccos\frac{h+H}{r}$$ Upward and horizontal speeds are: $$v_{u}=v\sin\psi$$ $$v_{h}=v\cos\psi$$ Time for Ethan to fly through the orange part is: $$t=\frac{2v_{u}}{g}=\frac{2v\sin\psi}{g}$$ Thus $$d_{h}=v_{h}\cdot t$$ $$=v\cos\psi\frac{2v\sin\psi}{g}$$ $$=\frac{v^{2}}{g}\sin2\psi$$ If we were to use $h$ as the variable, then $$d(h)=\frac{v^{2}}{g}\sin\left(2\arccos\frac{h+H}{r}\right)$$ $$d(h)=\frac{\left(\sqrt{2gh}\right)^{2}}{2g}\left(\frac{h+H}{r}\right)\sqrt{1-\left(\frac{h+H}{r}\right)^{2}}$$ $$d(h)=h\left(\frac{h+H}{r}\right)\sqrt{1-\left(\frac{h+H}{r}\right)^{2}}$$


What is the exact gravitational force between two masses including relativistic effects?

I was wondering if there is a closed-form formula for the force between two masses $m_1$ and $m_2$ if relativistic effects are included. My understanding is that the classic formula $G \frac{m_1 m_2}{r^2}$ is just an approximation (which is good enough for probably even going to the Moon), but what would the "correct" formula be according to the general theory of relativity? Does a closed formula even exist? For example for an idealized situation of just two spherical masses with homogenous mass distribution?


Emf of a rod sliding on two conducting rails

The following is the question. ![enter image description here](http://i.stack.imgur.com/2v9Ro.png) Actually, I do not know how to find the answer. I have tried to find out the solution or hints from the lecture note given to me but it is quite difficult to understand. The following is what I have tried. ![enter image description here](http://i.stack.imgur.com/YBctA.jpg) Please give me some hints. Thank a lot. (Ans : 0.14V)


Physical applications of the mathematical curvature

I was studying multivariable calculus last semester and had one of the topics talking about a curvature, but we had no applications on it. So how does it help in physics? E.g. curvature of curve: $k=|T(t)|/|r'(t)|.$


Why doesn’t gravity break down in a large black hole?

By popular theory gravity didn’t exist at the start of the Big Bang, but came into existence some moments later. I think the other forces came into existence a little latter. When a black hole crushes matter to a singularity (infinite density), at some point shouldn’t the forces cease to exist including gravity? Kent


At what frequency would a quartz crystal shatter?

Would it be possible to emit a sound at such a frequency that a quartz crystal inside of a watch would shatter? (Rendering the watch useless)


Using the Normalization Condition with Wavefunction

I'm very confused with this problem and I was looking for some guidance. $$\psi(x) = Ae^{ikx}e^{-x^2/2a^2}$$ Use the normalization condition to find A. So I understand that you use the normalization condition where $$\int_\infty^\infty|\psi(x)|^2 dx = 1$$ But the integral is really complicated and doesn't really lead me anywhere. Can anyone suggest a hint? Am I missing something? $$\int_\infty^\infty (Ae^{ikx}e^{-x^2/2a^2})^2dx$$ $$\int_\infty^\infty (A^2 e^{2ikx}e^{-x^2/a^2})dx$$


Photons hitting the Black Hole Singularity

I was watching one of Dr. Susskind's Lectures on String theory and he was talking about black hole singularities (see the 40 minute mark of this lecture http://www.youtube.com/watch?v=SkEvsxg5Tu4&list=PL3E633552E58EB230). I'm curious, given that the singularity becomes a 'time' and not a 'place', if at the moment you cross the horizon, you emit a photon, according to the explanation given, does this mean that it would take an infinite amount of time for the photon to hit the singularity? And this would also mean that the faster you move after crossing the horizon, the longer it would take to hit the singularity (which seems kind of cool)? I'm just curious to know if I'm interpreting the explanation correctly.


Reciprocating vs. rotating machines

Why is the RPM achieved on a reciprocating machine in general lesser than that obtained in a rotating machine? For example, an IC engine typically provides an RPM of about 2500 while a turbine can go as high as 80,000 RPM. The inertia and the repetitive change in direction of motion in case of reciprocating machine is one aspect, but is there an analytical justification/limitation on the RPM?


Can you speed up radioactive decay of plutonium?

We all know the issue of deep geological repositories for fuel rods. Is there a currently feasible way to speed up the rod's decay to render them harmless in less than 10 years?


Logic behind topological orders

Long-range entanglement (LRE) is the main feature of topological orders. The string-net condensation model was constructed to exhibit LRE. But the many-body systems of such models do not look like any earthly materials at all but are closer to quantum gravity models. In quantum gravity, nobody sees the detailed structure beyond the Planck scale. But in condensed matter, the microscopic structures are "crystal" clear. Then how can such strange models be used to explain LRE in laboratory materials?


How to calculate the force acting on a magnet due to a solenoid

Can I assume both magnet and solenoid as magnetic dipole and use Coulombs law to find the force acting the magnet?


Where to learn Temperature Dependent Conductivity induced by Electron-Phonon Interaction?

I want to learn how to calculate the temperature dependent conductivity induced by electron-phonon interaction. I know in low temperature, the resistance in metal $\rho$ is proportional to $T^5$, $T$ is the temperature. In high temperature region, $\rho$ is linear to $T$. Could someone tell me which text book and which chapter can I find related calculations? Thanks in advance.


Entropy of a chain

A chain has N segments which can be oriented in either the x or y directions. For each segment oriented along y, there is an energy penalty of $\epsilon$. We also know the end segment is at $(L_x, L_y)$. How can we define the entropy S as a function of E,N,Lx, and Ly? I know $S=k*ln(\Omega)$, but is can you actually solve for the possible arrangements of the chain? I'm assume overlapping and crossing cannot be done.


Is light so important in special relativity?

I'm an amateur physics enthusiast with no formal university education in Physics. So my question might sound very naive, so forgive me. I had this question in the back of my mind since the wrong discovery of neutrinos travelling faster than light. My understanding of special relativity is as following. You can measure the physical parameters like length, time, mass of objects in a frame of reference (A) from another frame of reference (B) having a constant relative velocity with A by only looking into the information signals coming from A to B. Since the light(electromagnetic radiation) is the fastest known carrier of information between two frames and since light has a constant relative velocity with any moving inertial frame, the length/time measurements between moving inertial frames are formulated based on its velocity. Back in 2011 when the 'faster than light' neutrinos were reported one point I heard in popular science reports was that it will invalidate the special theory of relativity. But why is it so? Any particle faster than light would simply interchange the constant $c$ in the special relativity with some new constant(provided this particle also has a constant relative velocity with inertial frame). Doesn't the special relativity theory still holds with the information being exchanged with this new fast particle instead of light? As long as these particles don't travel with infinite velocity all measurements from moving frames will have the time dilation and length contraction. Is my understanding right? Edit 1: I see another question here and the answer to that seems like my understanding is right. Special relativity theory can be derived from any particle travelling with a constant speed.


Is there Pair production in between charged plates

In classical electromagnetic theory, If parallel plates are charged oppositely and placed close to each other, there will be no charge will not flow from one plate to another. How does this situation change if one considers Quantum electrodynamics? Can the electric field in between the plates cause pair production? What is the probability, if it happens? How does one apply the formalism of quantum field theory to such a question ? I am rather new to the subject.


Confined in a box, what is the average distance between a particle hitting a side?

This particle, alone in the box, at a constant velocity $v$, travels in straight lines until hitting a side of the hollow cube, bouncing off in a _random_ direction independent of incident angle (I know it's not momentum conserving, but I'm looking for a statistical average, not a description of a deterministic system). Let the experiment go on for (essentially) infinite time, and (I assume) the average time between collisions (and so distance) wil converge to a value. _What is it_? And what about a hollow sphere? **Edit** :   How to begin this confuses me, so I'll consider the simpler scenario of the particle having to hit on an edge of the original square (that is, its velocity had no $z$-component) from its original position $(x_0,y_0)$. Consider each of the 4 edges in turn, by trigonometry deduce the angle between $(x_0,y_0)$ and the vertices ($\theta_1$ and $\theta_2$), and express that edge in polar form: $r(\theta)=\frac{y}{sin(\theta)}$, if $y$ is the shortest distance between $(x_0,y_0)$ and the edge. Integrate $\int_{\theta_1}^{\theta_2}|r(\theta)|d \theta$ to find the sum of the distances, and divide by the edge's length to find the average. My problem is that I think I'm overcomplicating to the extent that when I extend this to 3 dimensions, it won't spit out a meaningful answer: even in the 2-dimensional case there's $ln(\arccos(...)$ in there that makes me think there's a much more elegant solution. I'm sorry if I'm still being vague, but I've never encountered ways to get around these problems in the first place.


Intensity in rainfall and global warming

Why we should observe an increment on the mean intensity in rainfalls and an increment on mean dry days with global warming?


Is Keshe's technology for real?

http://www.youtube.com/watch?feature=player_embedded&v=UrN99RELqwo They clain they can create free energy using plasma reactors. Is this fake? > M.T. Keshe said that several entities, individuals, groups, and governments, > have been able to replicate the energy technology phenomenon by referring to > the books and patents available from the KesheFoundation.com website. One > scientist allegedly could power three towns pounds with his device.


Charge conjugation in Dirac equation

According to Dirac equation we can write, \begin{equation} \left(i\gamma^\mu( \partial_\mu +ie A_\mu)- m \right)\psi(x,t) = 0 \end{equation} We seek an equation where $e\rightarrow -e $ and which relates to the new wave functions to $\psi(x,t)$ . Now taking the complex conjugate of this equation we get \begin{equation} \left[-i(\gamma^\mu)^* \partial_\mu -e(\gamma^\mu)^* A_\mu - m \right] \psi^*(x,t) = 0 \end{equation} If we can identify a matrix U such that \begin{equation} \tilde{U} (\gamma^\mu)^* ( \tilde{U} )^{-1} = -\gamma^\mu \end{equation} where $ 1 =U^{-1} U$. **I want to know that, why and how did we do the last two equation. More precisely, I want to know more details and significance of the last two equations**.


Supersymmetric Nonrenormalization Theorems

I'm looking for approaches to nonrenormalization theorems in supersymmetric QFT which are as much as possible mathematical, elegant and involve few heavy straightforward computations


Gauge invariance for electromagnetic potential observables in test function form

This is a reference request for a relationship in quantum field theory between the electromagnetic potential and the electromagnetic field when they are presented in test function form. $U(1)$ gauge invariance becomes a particularly simple constraint on test functions for smeared electromagnetic potential operators to be gauge invariant observables. This is such a simple constraint that I think it must be out there, but I have never seen this in text books or in the literature, presumably because we mostly do not work with test function spaces in QFT; instead we use operator-valued distributions directly, where, however, gauge fixing is a perpetual nuisance. For the electromagnetic potential operator-valued distribution smeared by a test function $f^\rho(x)$ on Minkowski space, $\hat A_f=\int_M \hat A_\rho(x)f^{\rho*}(x)\mathrm{d}^4x$, to be an observable that is invariant under $U(1)$ gauge transformations $\hat A_\rho(x)\rightarrow\hat A_\rho(x)-\partial_\rho\alpha(x)$, we require that $\int_M \partial_\rho\alpha(x)f^{\rho*}(x)\mathrm{d}^4x$ must be zero for all scalar functions $\alpha(x)$. Integrating by parts over a region $\Omega$ in Minkowski space, we obtain, in terms of differential forms, $$\int_\Omega d\alpha\wedge(\star f^*)=\int_{\partial\Omega}\alpha\wedge(\star f^*)-\int_\Omega \alpha\wedge(d\\!\star\\! f^*),$$ which will be zero for large enough $\Omega$, and hence for the whole of Minkowski space, for any smooth test function $f^\rho(x)$ that has compact support and is divergence-free, $d\\!\star\\! f=0$. [If we constrain the gauge transformation function $\alpha(x)$ not to increase faster than polynomially with increasing distance in any direction, it will be enough for the test function $f^\rho(x)$ to be Schwartz and divergence-free.] So we have proved: > **Theorem:** The smeared electromagnetic potential $\hat A_f$ is a $U(1)$ > gauge invariant observable if the test function $f^\rho(x)$ is smooth, of > compact support, and divergence-free. The divergence-free condition on $f^\rho(x)$ ensures that the commutator for creation and annihilation operators associated with the electromagnetic potential $\hat A_f=\mathbf{\scriptstyle a}^{\,}_{f^*}+\mathbf{\scriptstyle a}^{\dagger}_f$ , $$[\mathbf{\scriptstyle a}^{\,}_f,\mathbf{\scriptstyle a}^\dagger_g]=-\hbar\int \tilde f^*_\rho(k)\tilde g^\rho(k)2\pi\delta(k_\nu k^\nu)\theta(k_0)\frac{\mathrm{d}^4k}{(2\pi)^4},$$ is positive semi-definite (which is necessary for us to be able to construct a vacuum sector Hilbert space), and that because $\delta f=\delta g=0$ we can construct, on Minkowski space, $f=\delta F$, $g=\delta G$, where $F$ and $G$ are bivector potentials for the electromagnetic potential test functions $f$ and $g$. In terms of $F$ and $G$, we can write $a^{\,}_F=\mathbf{\scriptstyle a}^{\,}_{\delta F}$ , $a_G^\dagger=\mathbf{\scriptstyle a}^\dagger_{\delta G}$ , which satisfy the electromagnetic _field_ commutator $$[a^{\,}_F,a_G^\dagger]=-\hbar\int k^\alpha\tilde F_{\alpha\mu}^*(k) k^\beta\tilde G_\beta{}^\mu(k)2\pi\delta(k_\nu k^\nu)\theta(k_0)\frac{\mathrm{d}^4k}{(2\pi)^4}.$$ Consequently, turning around the usual relationship because we are working with test functions instead of directly with quantum fields, we can regard test functions for the electromagnetic field as potentials for test functions for the electromagnetic potential. Because of the restriction that electromagnetic potential test functions must have compact support (or that gauge transformations must be constrained if electromagnetic potential test functions are taken to be Schwartz), electromagnetic potential observables are less general than electromagnetic field observables if electromagnetic field test functions are taken to be Schwartz (as is most commonly assumed), or equivalent if electromagnetic field test functions are taken to be smooth and to have compact support. So, references? EDIT (October 24th 2011): Noting the Answer from user388027, and my comment, a decent reference for what constraints are conventionally imposed on gauge transformations would be welcome. I would particularly hope for a rationale for the constraints from whatever theoretical standpoint is taken by the reference.


Does the cube stay on the loop-de-loop?

Suppose a cube of mass $m$ enters a loop-de-loop of radius $r$, travelling halfway around and reaching the top. At the top, it has velocity $v$, and the second half of the loop is frictionless. How can I determine whether or not it will continue around the loop, or fall off? According to the notes I took during class I can check to see if $v^2/r < g$, but I don't understand why that works. Can someone explain this? How would I determine if the cube stays on the loop if the second half of the loop isn't frictionless?


Pressing a fluid through leaks

Say I have a cylindrical container of some volume $V$, that I fill with water or some other fluid with dynamic viscosity $\mu$. Here, the bottom of the container is milled such that some fraction, $p$, of its surface area, $A$, is removed to allow passage of fluid. One could, for example, drill pores until this criterion is met. After positioning a plate at the top of the container, I press down on the plate with some force $F$ to compress the fluid. As a function of $F$, how quickly will I drain the fluid through the bottom of the container? To calculate this, what additional measurements/parameters, such as surface hydrophobicity/etc., will I require, and how does this change with the minimum cross-sectional area of the pores I mill in the bottom of the container?


High school double lens optics question

This is a first year high school homework question (in the Finnish high school), and I'm having serious trouble solving it. I apologize for possible non-standard terms: I'm doing the translation from Finnish to English and all of the words may not be how they would be in an English physics book. This was originally posted to math SO. The following paragraph no longer applies after the move: (I am aware that this is the Math and not the Physics stackexchange. I decided to ask this here, because I believe that my trouble lies mainly in the mathematical execution and not the physical understanding. Math also seems significantly more active, thus improving my chances of getting an answer. If that is required or strongly suggested, I will copy this to Physics and either remove it from here or leave it, whichever is desired.) The question: > A convex lens creates a real image from an object on the main axis, $85.0$ > cm from the object. When a concave lens is placed between the object and the > convex lens, $65.0$ cm from the object, the image moves to $140.0$ cm from > the object. What is the focal length of the concave lens? No other information is given. Knowledge of the equation $1/a + 1/b = 1/f$ is assumed, as well as the way multi-lens systems work (the image of a previous lens is the object of the next lens) I have approached the problem by creating three lens equations: $1/a_1 + 1/b_1 = 1/f_1$, where $a_1$ is the distance from the object to the convex lens, $b_1$ is the distance from the convex lens to the first image, and $a_1 + b_1 = 85$ cm $1/a_2 + 1/b_2 = 1/f_2$, where $a_2 = 65.0$ cm is the distance from the object to the concave lens and $b_2$ is the distance from the concave lens to the fake image produced $1/a_3 + 1/b_3 = 1/f_3$, where $a_1$ is the distance from the fake image to the convex lens, $b_3$ is the distance from the convex lens to the (second) real image and $f_3 = f_1$, since it's the same lens In addition, there are a few geometrical equations that can be extracted: $$\begin{align*} &a_1 + b_3 = 140 \text{ cm}\\\ &a_2 - b_2 + b_3 + b_3 = 140 \text{ cm} \end{align*}$$ leading to: $$b_2 = a_2 + a_3 - a_1$$ I know that the key is to calculate $b_2$, but it's proving to be difficult. The farthest I have got by combining the $b_2$ equation above and $f_1 = f_3$ is $$b_2=\frac{a_3(140\text{ cm}-a_1)}{a_1(85\text{ cm}-a_1)}\cdot 85\text{ cm}-75\text{ cm}$$ However, there are still $2$ unknowns, which is $2$ too much. I have the feeling that I am not seeing some geometrical equation that would somehow allow me to calculate the values for $a_1$ and $a_3$. It is also possible that the wording allows the assumption that the second real image is equal in size to the first real image, but 1) I can't see that helping too much and 2) from the way it's worded in Finnish I wouldn't dare make that assumption in an exam, for example. The correct answer, according to the book, is $f_2 = -27.3$ cm. Any hints what to try are welcome, as well as a more complete solution. I have tried to solve this for at least two hours, checking every geometrical relations I can think of and just generally trying to manipulate the various equations in a way that would help me get forward.


Are there 'higher order moments' in physics?

This may be a rather noob question but please let me clarify: I'm struggling to understand the use of the word 'moments' w.r.t., probability distributions. It seems after some research and poking around it seems to have been derived from physics when trying to solve/prove something about/related to binomial distribution and the method was called method of moments. I've asked the corresponding question here: http://stats.stackexchange.com/q/17595/4426 Now 'Pearson' (one of the very famous statisticians) comments: > We shall now proceed to find the first four moments of the system of > rectangles round GN. If the inertia of each rectangle might be considered as > concentrated along its mid vertical, we should have for the sth moment round > NG, writing d = c(1 + nq). Here are some of the details of the proof (as in the above post): ![enter image description here](http://i.stack.imgur.com/VtJ6d.png) Now Pearson talks about calculating the 'rth' moment and uses a derivative function to do so: ![enter image description here](http://i.stack.imgur.com/07Cfx.png) Question: I'm not aware of such a function from my knowledge of elementary physics. What kind of moments are being calculated here? How do you calculate 'higher order moments'? Is there any such thing? Basically looking to clarify something in statistics but was historically alluded to physics and hence just want to get it ironed out :) **UPDATE:** Intent of question: What I want to know is does the above derivation have anything at all to do with the concept of moments in physics and how is it related? Since the 'word' moment (and its intent) seem to be borrowed from physics when the author is making the derivation. I personally want to know if something like this does exist in the field of physics and how are these two derivations (and 'moments' related)


How to interpret the appearance of time units in the units of a physical quantity?

Or phrased more abstractly, how to interpret the appearance of time dimension $[time]$ in the dimension of a physical quantity? For example, the dimension of pressure is $[mass] [length]^{-1} [time]^{-2}$ corresponding to the SI-unit _Pascal._ When you calculate pressure, you do not have to know any time. It is simply _force_ per _area_ – a division. There are many units that have time as component in them.


How to find the electric field at a point based on a uniformly charged surface

What is the general solution to finding the electric field at a point based on some (or multiple) charged surfaces. I know that we can perform a line/surface integral if a charge is close to a wire or a surface. Furthermore, seeing as the surface is uniformly charged, we can use a Gaussian surface to eliminate the integral and write it as: $$ E_{point} = \frac{\lambda}{2\pi\epsilon_0 R} $$ Where $ \lambda $ is the constant charge and $ R $ is the distance of that charge from the wire. Does this formula change when we have a charged _surface_ instead of a wire? And what if there is more than one wire/surface acting on the charge? Are these formulas additive?


projective camera: back-projecting a point on the image plane into 3-space

suppose I got a projective camera model. for this model I would like to back- project a ray through a point in the image plane. I know that the equation for this is the following: $$ y(\lambda) = P^+ \pmb{x} + \lambda \pmb{c} $$ where $P^+$ denotes the pseudoinverse of the camera matrix P. P has a dimensionality of 3 by 4. $x$ is the point on the image plane in homogenous coordinates. Hence it's dimensionality is 3 by 1. $c$ the center of the camera in 3-space in homogeneous coordinates. (note that this equation is taken from the book "Multiple View Geometry in Computer Vision" page 162.) Now I don't fully get this equation. I get that $P^+ x$ results in a point on the line we are looking for. Hence we have two points that we can use for constructing a line. However I don't get the parametrization using $\lambda$. Why is the equation not in the form like: $$y(\lambda) = (1-\lambda) \pmb{a} + \lambda \pmb{b}$$ Any help in understanding the original equation of the resulting ray would be appreciated! :D


How should I think about reciprocal lattice and Miller indices?

When I hear someone talking about a (100) plane or a (111) plane or an (hkl) in general, my first thought is, is the system cubic. The reason I think this is because I tend NOT to think of the planes associated with their miller indices, but the vectors that are normal to planes. In cubic systems, [hkl] vectors are normal to their corresponding (hkl) planes. Hence, direct space (hkl) planes are parallel to their reciprocal space counterparts. In cubic systems, these relationships are true. I think this is a bad way to think of the relationship between Direct and Reciprocal space, as not all systems are cubic and I fumble when dealing with other sorts of lattices. Does anyone have some advice of a more fundamental way to view the relationship between the two spaces? Preferably, one that is quite intuitive. I realize they are Fourier Transforms of one another. I am sorry if this is a silly question. For example, in an HCP system, how should I envision the (hkl) plane? It is not normal to the [hkl] vector? is there no simple intuitive way to see such things? I can certainly do the math I suppose.


In black hole complementarity, does Bob literally observe Alice's death?

I'm interested in the picture of black hole complementarity where an observer (Bob) far away from the black hole watches another observer (Alice) fall in, and develops a very different picture of events. I want to stay within the framework of black hole complementarity, and limit this to _Bob's reference frame_. Mainly:   1. Can Bob practically watch Alice die from her emitted photons?   2. Alternatively, is time dilated so much in Alice's reference frame that Bob would have to wait until the end of the universe to watch this? I _think_ there are arguments against #2, and I might only have this confusion because of arguments I've read which were not fully informed, which I may now be repeating. There's also the matter of cause of death for Bob to write on the coroner's report. It seems that conflicting accounts exist, although they ultimately go back to the firewall concept. > The asymmetric mass, from the point of view of the exterior observer, smears > itself symmetrically on the surface when it gets close. second source: > The AMPS proposal is that what Alice encounters at the horizon does not look > like empty space at all — rather B and A are unentangled, which means that > Alice sees many energetic particles. Monogamy of entanglement is rescued, > but not poor Alice. She is incinerated by an intense wall of fire as she > attempts to pass through the event horizon. While these graphic accounts of Alice's death sound concrete, they're not _necessarily_ inconsistent with point #2 above. If time dilates within certain asymptotic properties, then it could look somewhat like Zeno's paradox. There are ways this could prevent Alice from ever _fully_ merging with the surface of the black hole. But could it prevent Bob from ever being able to confidently able to say that Alice died? Of course Bob has to recalibrate his video camera as it shifts ever-more toward the IR part of the spectrum, but does that video ever show Alice boiled to death or dismembered and smeared into the surface?


White bread toasted seems hotter than brown bread?

Ok this is a bit of a random question but I 99% of the time use white bread for toast and I always toast my bread the same amount and every time I take it out it's too hot to handle and the butter melts instantly on it. But this morning I used brown bread for the first time in forever and I noticed that even after getting it to the same amount of "toastiness" it was a lot cooler and I could easily handle it. Is it something that's in the bread white bread or the fact that it's more dense that causes it to be hotter?


Is the gauge fixing $\partial_\mu A^\mu + \gamma A_\mu A^\mu=0$ used in the literature and does it have a name?

In an exercise for a course on Gauge Theories, I was asked to derive the action of QED with the method by Faddeev and Popov, using the following gauge- fixing function: $$F(A) = \partial_\mu A^\mu + \gamma A_\mu A^\mu, $$ where $\gamma$ is constant. I was wondering whether this function is simply a construction for the exercise or it is actually used in QFT, and how it is referenced in literature.


How does phonon cause two electrons to attract each other?

We know that like charges repel each other. But my professor claimed that two electrons can attract each other as well. What he said was that due to screening an electron traveling at some speed wont repel another electron but will in some case they attract each other due to weak phonon exchange. What does that mean? What is phonon exchange? Do two electrons really attract each other?


U(1) Charged Fields

I don't quite understand what is actually meant by a field charged under a $U(1)$ symmetry. Does it mean that when a transformation is applied the field transforms with an additional phase? More specifically, why is it called "charged"?


Is There Any Work Done By Normal Reaction Force even if point of application of force is not getting displaced?

Suppose that I am standing on the ground and I suddenly jump and rise up to any certain height.So accordingly I would have gained some speed i.e. Kinetic Energy during the process of jumping.According to Work Energy Theorem Total work done on by body is equal to this kinetic energy.Now it is obvious that gravity has no tendency to provide me upside velocity and besides my body's center of mass has not been so much displaced during the process of just jumping.The only other force acting on me is Normal Reaction Force which will provide me the upward change in momentum but here at every instant of the action of the normal reaction force point of application of normal reaction force ( my feet's point of contact with ground ) are at rest during jumping so according to the defination of work done Normal reaction should not have performed the work.What is the funda here ? Please get me out of this confusion.


How do I calculate output voltage from a given supply voltage and resistance?

I have searched everywhere but I can't seem to find out how I can calculate the output voltage (or voltage over a component) from the resistance of a resistor (or other component) and the supply voltage. Everything seemed to be about Ohm's Law, which looks like it could help but I don't know how. In the case I need to solve, there is a supply of 5v and I need to get the resistance needed to lower the voltage for an LED (the current and resistance at optimum voltage I can get if needed) to 3v. I will definitely need to do this operation again so if you could give something like an equation, rather than the necessary resistance.


Finding the terminal velocity of a magnet dropped in a solenoid

> We have to find proportionality of the terminal velocity with the factors of > the system: ![enter image description > here](http://i.stack.imgur.com/OItlm.png) > > Plot: a small dipole(mass $m$) with dipole moment $\mu$ is dropped in a long > solenoid (radius $r$, Resistance $R\: \mathrm{\Omega}$) along it's axis. The > gravity is acting downwards. I found it to be $$v\propto \dfrac{mR}{\mu^2r}$$ Here is how: > The force in downward direction $F_{down}$ must be $mg$ and in upward > direction $F_{up}$ must be magnetic force. $$F_{up}\propto \dfrac{\mu > I_{induced}}{r}$$ > > $$I_{induced}\propto\dfrac{ \text{electric flux change}}R \propto\dfrac{\mu > \pi r^2 v}R$$ Flux change depends directly on the velocity, so added $v$ in the numerator. But the expression is not correct. The proportionality on the $r$ radius is not correct. I think there is a problem in the induced current factor.


Likelihood of the creation of a single unbound quark in the collision of very high energy particle beams

I am going over old exam and am not understanding the logic behind the answer given in the mark-scheme. > A beam of protons and antiprotons attain energies of 1400 GeV in a > synchrotron. Why is it that, even at the very high energies of this particle > beam, the collisions are unlikely to lead to the creation of a single > unbound quark. The mark-scheme, very confusingly, answer this as follows: > as quarks separate the force between them is constant; > > (so) energy required to release single quark increases with distance; > > these energies exceed the threshold for meson (pair) production; Each part, separated by a semicolon, gives one point (so that you understand why its so oddly formatted). I get that we should simply know that the energy transferred between the quarks gets bigger with distance (work = force * distance), and that force is constant, but why does it jump to the conclusion that these energies exceed what we already have? And why is the meson (pair) production relevant?


Hilbert-Schmidt basis for many qubits - reference

Every density matrix of $n$ qubits can be written in the following way $$\hat{\rho}=\frac{1}{2^n}\sum_{i_1,i_2,\ldots,i_n=0}^3 t_{i_1i_2\ldots i_n} \hat{\sigma}_{i_1}\otimes\hat{\sigma}_{i_2}\otimes\ldots\otimes\hat{\sigma}_{i_n},$$ where $-1 \leq t_{i_1i_2\ldots i_n} \leq 1$ are real numbers and $\\{\hat{\sigma}_0,\hat{\sigma}_1,\hat{\sigma}_2,\hat{\sigma}_3\\}$ are the Pauli matrices. In particular for one particle ($n=1$) it is the Bloch representation. Such representation is used e.g. in a work by Horodecki arXiv:quant-ph/9607007 (they apply $n=2$ to investigate the entanglement of two qubit systems). It is called decomposition in the Hilbert-Schmidt basis. The question is if there is any good reference for such representation for qubits - either introducing it for quantum applications or a review paper? I am especially interested in the constrains on $t_{i_1i_2\ldots i_n}$.


error propagation and collision in ideal gas

When dealing with gas, a statistical approach is needed because   1. For N particles, you have to solve 6N equations which cant be done analytically. To know our time step for numerical solving, you can estimate the frequency of a collision. For an ideal gas, this frequency is around $10^9$Hz which gives $10^{32}$ equations per second for 1mole of gas at room temperature.   2. Errors propagate quickly and soon it is completely impossible to know anything about the particles you are studying. To show the second point, consider a collision between 2 molecules as shown on this picture ![collision of 2 particles in a gas](http://i.stack.imgur.com/JCZO0.png) We have the relation $\frac{sin\theta_1}{l_0} = \frac{sin\theta_0}{2r_0} \approx \frac{\theta_0}{2r_0}$ where $l_0$ is the mean free path and $r_0$ is the radius of the particle knowing the uncertainty $\delta\theta_0$ , we can estimate the uncertainty on $\delta\theta_1 \approx \delta\theta_0\frac{d\theta_1}{d \theta_0} \approx \delta\theta_0(\frac{l_0}{2r_0}) \approx \delta\theta_0 * 10^3$. After n collisions, the error is $\delta\theta_0*10^{3n}$ which rapidly explode. Now see how important this effect was, I tried to apply it to the smallest angle I could find. The observable universe being about $10^{25}m$ in radius and the planck length $10^{-35}m$, the smallest angle one can observe is about $10^{-60} rad = 10^{-3*20} rad$. To get an error of $\pi$, it would therefor take 20 collisions. This doesn't seem possible, is this result correct? If not, where is the problem? I can't find it but I can't believe that 20 collisions are enough to lose all information about the particle.


Sign of acceleration

I'm developing an application using accelerometer sensor. I'm not good at physics so forgive me if the question is trivial. If I have 3 values of acceleration: $x$, $y$, $z$, I find acceleration magnitude by taking square root of $x^2+y^2+z^2$, but how do I find it's sign? Example reading: x: -0.010020584 y: 0.010257386 z: -0.04910469 The magnitude will be around 0.05115, but how do I know if it is deceleration or acceleration?