title: Could two identical stars revolve around each other in a common orbit if we only account for Newtonian physics?

text: Both a parent star and its planet revolve around the center of mass of the system, the reason we see stellar wobble. But if we take this to be true, which it is, there can be a configuration in which two identical stars revolve around their center of mass, in a common orbit. What I find astonishing in this case is that they will be revolving around something having no mass at all, in a shared orbit, like two runners trying to catch each other but never quite being able to do so. In that case, in a purely Newtonian system, the centripetal force must be provided by the gravitational attraction between the stars. Now if I assume both the stars to have a mass $m$, at a separation of $d$ from each other, revolving in the common orbit diametrically opposite to each other, $$\frac{mv^2}{r} = \frac{Gm^2}{d^2}\; {\rm where}\; r = \frac{d}{2}$$ solving for $v$, we get a velocity where a stable orbit is formed, $v = \sqrt{\frac{Gm}{2d}}$ Note: I have failed to find such an expression, I am not certain about the math. I did find a system described in such a way that two stars orbit a common point in separate ellipses though. Is my conventional wisdom correct, or is my derivation of this expression somehow intrinsically flawed? Does this expression already exist? And I suppose the odds of observing such a system are 'astronomical', but has something like it ever been observed?


title: If the absolute horizon were exclusionary of matter, what supernova behaviors would that predict?

text: Kip S Thorne's "Black Holes & Time Warps", 1994 paperback, p.415, Box 12.1: > ... The absolute horizon is just a point when created, but it then expands > smoothly, like a balloon being blown up, and emerges through the star's > surface precisely when the surface shrinks through the critical > circumference ... It then stops expanding, and thereafter coincides with the > suddenly created apparent horizon. I would like to propose a amenable to both theoretical exploration and, through sufficient astrophysical observation, experimental verification: > What if the absolute horizon _excludes_ matter rather than allowing matter > to pass through? The most likely consequence would be that stars of sufficient mass for their iron cores to collapse into black holes would explode with a different signature, one likely with more explosive vigor, than they would in the standard model in which matter overtaken by the expanding absolute horizon falls into its interior to become a singularity. In an expulsion model, collapsing core matter would still seem to "disappear" as the absolute horizon overtakes it, but the disappearance would be the result of accretion and collapse onto the expanding absolute horizon, rather than from the matter simply falling through that horizon and continuing on to become a singularity at the center of the horizon. There is a good chance (but only the math would tell) that the resulting bottleneck in the "processing" of the infalling matter would push harder on the core layers just beyond the expanding absolute horizon. I do not know if expulsion is compatible with the original Hawking mathematical analysis that led to his concept of an expanding absolute horizon. I had been wondering if there was any self-consistent way to exclude matter from the interior of a black hole as it forms. With the exclusion tweak, the expanding-from-a-point absolute horizon concept provides exactly the dynamic structure needed. I was not expecting that. The theoretical implication of exclusionary absolute horizons would not be complicated, just different from standard theory: Singularities would simply not exist, at least not as a result of collapsing matter. Black holes would remain very real, but would be defined solely by their event horizons, and 100% of their mass would reside in the horizon. The interior of such a construct would be a region in which traditional spacetime simply does not exist. So: While it's too much to expect an answer to this large a question, does anyone know if the concept of exclusionary absolute horizons have been explored, and if so, what kinds of predictions it makes for supernova behaviors?


title: Are synthetically-produced diamonds as hard as natural diamonds?

text: I was having a discussion with my friend about the intrinsic worthlessness of diamonds (DeBeers and whatnot) and how synthetic diamonds haven't caught on, again because of the marketing/propoganda that natural diamonds are "better". My friend, partially to spite me because he doesn't like to lose an argument, claims that synthetic diamonds have only been produced with a hardness up to 9.8, and you can only get a hardness of 10 with a natural diamond. I call BS on that. A little research indicates that while natural diamonds may slightly vary in hardness (the impurities which alter the color also modify its crystal properties), synthetic diamonds have a more consistent hardness which is otherwise identical to a naturally-occurring diamond with the same chemical properties. This appears to make sense. Are chemically-identical synthetic and natural diamonds of equivalent hardness? Or is there a caveat to the synthetic diamond-making process that produces weaker diamonds?


title: How to relate compression ratio to work input for a compressor in a heat pump

text: So I need to calculate the COP (Coefficient of Performance) for a 6-ton air conditioner system in terms of the compression ratio. I understand that: $$COP=\frac{Q_L}{W_{net,in}}$$ I cannot find out how I can relate these energies to the compression ratio of the compressor. I realize this may not be the correct StackExchange website but Physics was the closest thing I could find. There was an Area 51 proposal for a Mechanical Engineering site but it did not make the cut.


title: What if we were to travel against the speed of our galaxy?

text: The Earth (and its inhabitants) is rotating and moving along with the Milkyway at approximately 600 000 m/h (not the of the Earth's rotation). Now let's say I'm in my spaceship in earth orbit watching the earth spin, I too would be traveling around 600 000 m/h (or 168 m/s) the same as the Earth, correct? Now let me compare myself to a ball on a conveyor belt. From my view point I am stationary, but to an outside observer I am moving with the conveyor belt. Now lets say I(the ball) have some method of producing motion in the opposite direction of the belt; in effect, I now look stationary to the outside observer (and as to make the motion unnoticeable, the ball is completely uniform). Back to my spaceship, what if I were to leave Earth's orbit and travel against the rotation of the galaxy enough to cancel out the effects of motion like the ball and conveyor belt and once at that point give no more means of motion. If all of this were to be correctly implemented, then:   1. Would I effectively have no motion?   2. Would I see the Galaxy traveling 600 000 m/h around me (assuming I would never hit something)?   3. Would I experience time differently?   4. If I would see the Galaxy moving, what would I see?


title: What physical forces allows for electromagnetic induction

text: In electromagnetic induction, what force is actually doing the work? what physical force actually drives the electrons around the circuit? Let's say we have a coil and an increasing magnetic field through the loop. The electrons start to flow by faraday's law. Is there a very real electric field that is induced around the wire which does the work? because magnetic fields can most certainly not do work. Also, why do the charges even start moving in the first place? At each instant in time, there is some instantaneous value for the magnetic field. The magnetic field cannot cause the electrons to start moving because it doesn't affect stationary charges


title: Angular Velocity expressed via Euler Angles

text: On the top of the fourth page from here, the author trivially derives the components of angular velocity, expressed via Euler angles of the system. I fail to understand how the components of angular velocity were derived. May you please enlighten me!


title: Accretion disk physics - Stellar formation

text: I was going through the Wikipedia page for Accretion disks, and I couldn't comprehend what the meaning of this is: > "If matter is to fall inwards it must lose not only gravitational energy but > also lose angular momentum." What does that mean? Also, why are these accretion disks planar? I learned that the reason is conservation of angular momentum, but what is the physics behind it?


title: How do I do error calculus right using gnuplot as an example?

text: Given is a set of measurements with their respective errors for example an energy spectrum. In gnuplot one is to fit a function $ f(x;\\{p_i\\})$ depending on a variable $x$ and on fit parameters $p_i$. When the fit is done one gets values for the $p_i$ with errors and a correlation matrix with values $ c_{ij}$. Now one has to calculate a value $v(\\{\text{some of the }p_i\\})$ that depends on some of the the $p_i$ and find its error $\Delta p_i$. How will a calculate the error $e$? Do I have to take correlations into account? Can I do it the way I attempted it in my solution attempt. Solution Attempt: $e^2=\sum_{i}\left(\frac{\partial v}{\partial p_i}\cdot \Delta p_i\right)^2+\sum_{ij}\frac{\partial v}{\partial p_i}\frac{\partial v}{\partial p_j}c_{ij}\Delta p_i \Delta p_j$ If this is right what happens if one of the $c_{ij}$ is negative?


title: Gauge fixing and equations of motion

text: Consider an action that is gauge invariant. Do we obtain the same information from the following:   1. Find the equations of motion, and then fix the gauge?   2. Fix the gauge in the action, and then find the equations of motion?


title: Why $\log \rho$ in the continuity equation?

text: I just saw the continuity equation, in a manuscript, written as $$\frac{\partial \log \rho }{\partial t} + \vec v \cdot \nabla \log \rho= - \nabla \cdot \vec v.$$ Now, just calculating the derivatives of $\log$, and multiplying by $\rho$, this comes back to the familiar $$\frac{\partial \rho }{\partial t} + \nabla \cdot (\rho \vec v)= 0.$$ But I am curious: what would be the reason to write it in that log-form? EDIT: The log-form also appears on page 53 (pdf page 69) in this manual:   http://www.nordita.org/pencil-code/doc/manual.pdf EDIT 2: page 2 here explains about the same as tpg2114's answer.   http://www.nordita.org/~brandenb/own/Bran_comp03.pdf


title: Action Reaction Pairs and Work

text: Say that a ball is sitting in front of a compressed spring launcher. The spring is then released. The spring applies a force on the ball for a certain distance. This force is accompanied by an equal and opposite force from the ball. Thus, the spring does some work $W$ on the ball, while the ball does exactly $-W$ work on the spring, correct? So, work is transfer of energy and always accompanied by a negative work on behalf of what is gaining kinetic energy, right?


title: Arduino Servo Torque Calculation for a Automated Telescope

text: I have a class project where I am using several servos to control a telescope. I'm trying to figure out the torque needed to rotate the tilt of the telescope with a servo from 0 to 90 degrees. I haven't come across any similar problems online that have examples. I know that the one I am currently using is close, but just isn't enough to lift the telescope up by itself. I currently have an HS-311 (51oz-in) that isn't working. The servo I would potentially purchase in place is the HS-645MG (133oz-in). From the experiments I've done, I've been able to rotate a ~6oz object almost perfectly, but the ~12oz telescope is too much. I assume that if I get the 133oz-in servo, it will be able to rotate 2.6 times the mass or 15.6oz in this case. Here is a modified diagram of the original setup. The circular disc on the servo tip is 1" in diameter. ![setup](http://i.imgur.com/AbQ3HGj.jpg) And here is what it looks like assembled (without telescope attached yet). ![assembled](http://i.imgur.com/qr6QXEq.jpg)


title: Generalizations of Bell's inequality to quantum field theory

text: Can anyone refer me to some sources on generalizations of Bell's inequalities to quantum field theory (as opposed to quantum mechanics)? Scalar fields would be enough.


title: How much runway does the space shuttle need to land?

text: A textbook question proposes the following scenario: NASA needs to plan emergency landing sites for the Space Shuttle (RIP). There are a number of things to consider. The shuttle touches down no sooner than $300$ meters into runway track, and upon touching down deploys a parachute, which takes 5 seconds to be fully deployed. Once the speed decreases to $60\ \mathrm{km/hr}$, the brakes can be applied and require $200$ meters further to arrive at a complete stop. Here are the rest of the specs (not necessarily accurate in real life): >   * Total Landing Mass: $104,328\ \mathrm{kg}$ > >   * Velocity: $350\ \mathrm{km/hr}$ > >   * Area of parachute: $68.0625 \pi$ > >   * $C_D$, the drag coefficient of the parachute: $1.4$ > >   * $\rho_\text{air}$, the density of air: $1.1774\ \mathrm{kg/m^3}$ > > The question is, asses NASA's decision to land the space decision at an airport with a runway of $11,000\ \mathrm{ft}$. > MY ATTEMPT I begun my assesment by assuming the Shuttle touches down at the $300$ meter mark, at which it is travelling at $350\ \mathrm{km/hr}$. Immediately the parachute deploys and takes five seconds to deploy. Once the parachute deploys, a negative Force is applied to the Shuttle in the form of $$m \frac{dV}{dt} = - \frac{1}{2} \rho_\text{air}C_DAV^2 = - \frac{1}{2}(1.1774)(1.4)(68.0625 \pi)(V^2) \approx-56.0958 \pi V^2$$ Out of this, I formed a function for _velocity_ based on _time_ by solving this as a differential equation and got: $$ \frac{1}{V^2}dV = .005377 \pi dt \longrightarrow - \frac{1}{V} = .0005377 \pi t +C$$ and as a function of $t$: $$v(t) = - 185.977 \frac{1}{ \pi t} + C $$ Since my initial condition is $350$ when $t=0$, my function finally is: $$ v(t) = - 185.977 \frac{1}{ \pi 0} + 350 $$ But I'm dividing by zero! My goal was to see how far the shuttle travels in the time it takes to reach $60\ \mathrm{km/hr}$ and determine if it has enough track to make it. I'm not sure if what I'm doing so far, or my strategy is correct. Obviously what I'm doing is somewhat flawed. I would really appreciate any help. Thanks!


title: vacuum level alignment

text: When two materials are connected like the case in heterojunctions, we always firstly align the so called vacuum level[1], and then decide the relative position of other energy levels like conductance and valence bands. But in the wikipedia page it says " The vacuum level alignment approach may or may not hold due to details of the interface", so my question today is, how to judge whether the vacuum levels align or not? [1]http://en.wikipedia.org/wiki/Vacuum_level


title: How to relate photon's higher frequency to time dilation?

text: The usual explanation for photon's higher frequency in lower altitudes (higher gravity), when the photon is going downward towards a massive body, is that gravitational potential energy is converted into photon's energy which, by its turn, through $E=hf$ implies higher frequency. Because, since $c$ is constant, it doesn't make sense to turn that potential energy into kinetic energy. That, I understand. My question is, why is it affirmed that this implies that time runs slower in the lower altitude? I mean why is it such that this higher frequency is equivalent to slower running clocks?


title: Is the universe we know just shaped due to time?

text: Call me a daydreamer, but I could not stop myself for asking this question.   I was idle when something ticked me and struck my head. I was thinking about universe, speed of light, and how time works. I imagined that what if there was another world where time was so slow that light travels at 1 m/s w.r.t. the present world. What I mean to say is that imagine the world which has time passing so slow that in that world, the speed of light is 1 m/s w.r.t. the present world. I was just wondering, would that be the possible 'speed limit' of that world? I think probably the answer is yes but if the answer is yes, does that mean that the universe is shaped the way we know is just due to how time behaves? I dont know but i do not think any scientist has said that universe is just shaped due to how time behaves but i think the time is not dependent on universe, the universe is dependent on time. Am i correct?


title: Identifying an optics equation in a code refactoring project

text: I apologise if this a somewhat naive post. My knowledge of optics is limited, and I am learning on the job, so to speak. I'm a computational biologist, dabbling in a bit of physics. I'm currently reading Sonke Johnsen's "The Optics of Life" to try and get a better picture. I am involved in an academic project to refactor some obsolete code from a program created over 15 years ago, which models the resolution and sensitivity in reflective superposition compound eyes. The problem is that a lot of the code uses non-descriptive variables that are difficult for me to interpret. I am stuck with working out what this is doing:               DA = sqrt((ER * ER) - (AR * AR))    /* DA??? */     AC = atan(AR / DA) / CONV           /* AC??? */      This is then used to adjust the aperture diameter:               AD = (AC / 360.0) * EC              /* adjust aperture diameter */      In this code the variables definitions are as follows: > ER = eye radius > > AR = aperture radius > > CONV = pi / 180.0 (converts radians to degrees) > > AD = aperture diameter > > EC = eye circumference What might `DA` and `AC` be? Any help should be greatly appreciated :)


title: Why can't batteries be charged quickly?

text: Charging of laptops, cell phones take so much time. Why can't we make such batteries easily/commercially which are charged more quickly? What's the thing behind this limiting?


title: Is it possible to levitate an object with piezoelectricity?

text: The delay for transmit pressure depend of speed of sound in the material. A lot of piezo slip-stick actuator can generate an asymetric wave. Like the differential pressure is not the same the force to up can cancel gravity ? In example, with 400 actuators, each actuator give force to down in 1/1000000s enough for cancel gravity while 1/1000000s, wait 10/1000000s and a force to up in (1/400 - 11/1000000)s. But next actuator action at time 1/1000000s. etc. ![actuactor](http://i.stack.imgur.com/cloel.jpg)


title: Stimulated Emission in Free-Electron Lasers

text: I've been reading about free-electron lasers these days. The basic principle is clear: In the undulator electrons spontaneousely emit synchrotron radiation. A self- attunement takes place, causing the electrons to be bunched into packages that are integer numbers of wavelengths apart. Thus the emitted radiation is coherent. I wonder if this is only spontaneous emission (and coherence is purely due to the self-attunement) or if stimulated emission actually does play a role.


title: A kind of Noether's theorem for the Hamiltonian

text: How can I (conveniently?) show that an invariance of the Lagrangian and Hamiltonian (i.e. the kinetic as well as the potential energy are independently invariant) will lead to a conservation law **using only the Hamiltonian formalism**? Am I correct, that the invariance of a Hamiltonian does not need to lead to a conservation law, if the corresponding Lagrangian is not invariant (i.e. only $T + V$ is invariant, but $T - V$ is not)?


title: Variational method applied to brownian motion

text: It's possible apply the variational method to the brownian motion ? I mean, one of requisites on $y(t)$ is that it must be continuous and $\partial_t{y(t)}$ too, and in this case, $\partial_t{y(t)}$ is not.


title: Charge distribution inside a parallel plate capacitor

text: I'm having difficulty deriving a known expression for the magnitude of the Electric Field within a parallel plate capacitor with applied potential difference $V$ and an immobile charge distribution within the capacitor with uniform density $\rho$. The expected result is: > $$\frac{\rho(d-2x)}{2\varepsilon_{0}}-\frac{V}{d},$$ Where $d$ is the > distance between the plates and $x$ is the distance from the positively > charged plate. My initial attempt involved using Gauss' law in a Gaussian surface $\Sigma$ projecting out of the plate a distance $x$: $$\iint_{\partial \Sigma}\vec{E}\cdot\mathrm{d}\vec{A}=\frac{\rho}{\varepsilon_{0}}\iiint_{\Sigma}\:\mathrm{d}V+\frac{Q}{\varepsilon_{0}},$$ Where $Q$ is the magnitude of the charge on each plate. We therefore get: $$EA=\frac{Ax\rho}{\varepsilon_{0}}+\frac{Q}{\varepsilon_{0}}\implies E=\frac{\rho x}{\varepsilon_{0}}+\frac{Q}{\varepsilon_{0}A}=\frac{\rho x}{\varepsilon_{0}}+\frac{V}{d}$$ As $V=\frac{Q}{\varepsilon_{0}A}d$. However, this is not what we should get, so I'd appreciate any pointers in the right direction.


title: Gravitational Time Dilation with big masses

text: From Wikipedia I read that the formula for calculating the time dilation caused by a mass is $t_0 = t_f{\sqrt{1-\frac{2GM}{rc^2}}}$ where:   * $t_0$ is the proper time between events A and B for a slow-ticking observer within the gravitational field   * $t_f$ is the coordinate time between events A and B for a fast-ticking observer at an arbitrarily large distance from the massive object   * $G$ is the gravitational constant   * $M$ is the mass of the object creating the gravitational field   * $r$ is the radial coordinate of the observer (which is analogous to the classical distance from the center of the object, but is actually a Schwarzschild coordinate),   * $c$ is the speed of light The question is, what if $\frac{2GM}{rc^2}$ is greater than 1? This can happen if the mass is too big or if the radius is too small.


title: How axiomatic is the symmetrization requirement (i.e. the Pauli principle)? (in QM)

text: I've so far always been told, that the symmetrization requirement is an axiom on the level of the Schrödinger equation and the statistical interpretation of the wave function (or it's absolute value). Some time ago, however, I found the following little calculation (which I modified a little bit, but it is hopefully still correct): Let $\Psi \left(\vec{n_1},\vec{n_2}\right)$ be the wave function of a two particle system and $\vec{n_1}$ and $\vec{n_2}$ be the quantum numbers of the particles. Now if the two particles are identical (i.e. indistinguishable), we shouldn't be able to observe any changes when exchanging their quantum numbers, which leaves us with: $$ {\left|\Psi \left(\vec{n_1},\vec{n_2}\right)\right|}^2={\left|\Psi \left(\vec{n_2},\vec{n_1}\right)\right|}^2 $$ Now we can conclude: $$ \Psi \left(\vec{n_1},\vec{n_2}\right)=\text{e}^{i\delta}\Psi \left(\vec{n_2},\vec{n_1}\right) $$ I.e. the wave function acquires a factor $\text{e}^{i\delta}$ when we exchange its arguments. Exchanging the arguments again, leaves us with: $$ \text{e}^{i 2\delta}=1\ \therefore\ \text{e}^{i\delta}=\pm1 $$ Which basically is what the Pauli principle states. If this calculation is correct, shouldn't the Pauli principle be regarded as a consequence of the indistinguishability of identical particles and the statistical interpretation, rather than an axiom?


title: What is the commutator?

text: $e$ and $f$ are unit vectors, $L_e$ is defined by $L_e=eL$, where $L$ is of course the angular momentum operator. A similar definition for $L_f=fL$ The commutator that I can't solve: \begin{equation} [L_e,L_f]=? \end{equation}


title: Equal and opposite reaction - forces coming in pairs - or just common sense?

text: I've never understood why the conservation of momentum law is taught in schools as "every force has an equal opposite reaction". To me a gun's recoil is self explanatory; the explosion sits in-between the bullet and the back of the barrel and the gun and bullet fly in opposite directions (as would be expected). The bullet has a much higher velocity than the gun because of its lower comparative mass. Am I wrong here? Same with stepping off a boat. You push against the boat and the mass differential between your body and the boat as well as the friction in the water enables you to jump off the boat onto the ground without falling on your face. The boat comparatively moves in the opposite direction minimally, but not because of some weird law, simply because you pushed it back! To tell kids in school "every force has an equal and opposite force" is misleading I think. It makes it seem as if an equal mysterious force manifests out of thin air to keep the universe happy. Just thinking - "direction of application" is only relevant to perspective, there really is no direction of application since the force and its counterpart occur simultaneously… so more accurately:   * The total force can never be propagated along a single vector?


title: Gravitational redshift of temperature and electrostatic potential

text: Consider a charged black hole in four-dimensional Minkowski spacetime, with charge $Q$, mass $M>Q$: $ds^2=-f(r)dt^2+\frac{1}{f(r)}dr^2+r^2d\Omega_2^2$, with $f(r)=1-\frac{2M}{r}+\frac{Q^2}{r^2}$. When an observer at radial coordinate $r_1$ emits a photon, an observer at radial coordinate $r_2>r_1$ will perceive the photon with a redshifted wavelength. This is easy to interpret. A similar thing happens for the temperature. If $\kappa$ is the surface gravity of the black hole, then the Hawking temperature is $T_H=\frac{\kappa}{2\pi}$. Due to gravitational redshift, the temperature measured by an observer at radial coordinate $r$ is $T_{loc}(r)=\frac{1}{\sqrt{f(r)}}T_H$. The redshift factor is the same as for a redshifted frequency, which I can understand by associating temperature with the inverse imaginary time period. I interpret this redshift as a consequence of the particles which constitute Hawking radiation experiencing gravitational redshift. **Is this correct?** Apparently, something similar also happens for the electrostatic potential. The electrostatic potential difference between the outer event horizon $r_+$ and infinity is given by $\Phi=\frac{Q}{r_+}$. However, the electrostatic potential between $r_+$ and some coordinate $r>r_+$, "blueshifted" from infinity to $r$, is given by $\phi(r)=\left(\frac{Q}{r_+}-\frac{Q}{r}\right)\frac{1}{\sqrt{f(r)}}$. ( _Source: Braden, Brown, Whiting and York, Charged black hole in a grand canonical ensemble, PRL Vol. 42 No. 10, 1990, equation 4.15._ ) This expression seems to tell me that $\frac{Q}{r_+}-\frac{Q}{r}$ is the electrostatic potential difference between $r$ and $r_+$ as measured by someone at infinity, and the above expression is this same potential difference as measured by someone at $r$. > Is there a simple interpretation why the measured electrostatic potential > should experience gravitational redshift as well, with the same redshift > factor as a frequency? What is the wavelength that is being redshifted in > this case, is it the one from the photons mediating the electromagnetic > force? (These photons are virtual though, so can they actually be > redshifted?)


title: What is the fringe separation in Young's double slit experiment?

text: In the double slit experiment, a pattern of dark and bright bands appears on the screen. Is the fringe separation the distance between two bright bands? Question: > In a double slit experiment using light from a helium-neon laser, a student > obtained the following results: > > width of 10 fringes 10x = 1.5 cm > > separation of of slits a = 1.0 mm > > slit-to-screen distance D = 2.40 m > > Determine the wavelength of light. How would I find the fringe separation here? If I divide 1.5 by 10, I would get the width of 1 fringe. Is that the same thing as the fringe separation? A diagram showing what the fringe separation is would be really helpful. The textbook that I'm currently reading defines the fringe separation as the distance between the centres of adjacent bright fringes. How would I find out the fringe separation here in this question? Thank you in advance.


title: It's possible to create a proton colliding a positron with enough energetic photons?

text: If a proton is supposed to decay in a positron and gamma ray photons is possible to obtain the opposite process colliding enough energetic photons with a positron and create a proton ?


title: What is the smallest distance between a decaying vertex and an interaction point a detector can measure?

text: A short lived particle is created at the interaction point and then decaying a distance $d$ away, in some detector. My question is what is the smallest distance $d$ that can be measured experimentally? Of course it will be different from a detector to another. Any links that explain how this analysis, of discovering a particle, is done experimentally would be appreciated.


title: Number of conditions for a two-particle state to be decomposable

text: Suppose we have a general two-particle state $ \Phi (x_1, x_2 ) = \sum_{n_1,n_2} \phi_{n_1,n_2}(x_1,x_2)|n_1,n_2> $, where $n_1$ can be any of $n$ possible states, and $n_2$ can be any of $m$ states. If the state is decomposable then the coefficients $\phi_{1,2}$ can be decomposed into $\phi_{n_1} * \phi_{n_2} $. It seems to me that for the decomposable state there are $n + m$ independent coefficients ($n$ coefficients describing the state of partice 1, and $m$ coefficients describing the state of particle 2), and if the system is not decomposable, i.e., if there is entanglement between the two particles, then the number of independent coefficients is $n * m$ ($n$ possibilities for $n_1$ times $ m$ possibilities for $n_2$ in $\phi_{n_1,n_2}$). If this logic is correct, then the number of conditions to be fulfilled by a decomposable state is $nm - n - m$. However, according to the book I am studying the number of conditions is $nm - n - m +1 $. I wonder why there is an extra condition. I am not considering normalization, because the states are seen as rays in projective space, and furthermore both the decomposable and entangled states would have to fulfill the same normalization requirements so I guess there would be no difference.


How accurate is the time on a digital clock versus the time on internet clocks? The internet clocks seem slower than internet clocks. So which is the correct time? What are the most accurate clocks?


title: "Regular" 20-sided die, vs "life counter" 20-sided die. Same probabilities?

text: Regular dice are made such that opposite sides of the die add to 1+the number of sides. For example, a 20-sided die has 14 and 7 opposite of each other, adding to 21. For certain types of games, "life counter" die are used. In these, sides are numbered sequentially, so that 1 is next to 2, which is next to 3, which is next to 4, and so on. You can see a picture here: http://www.coolstuffinc.com/main_supplies.php?fpid=Acc- QWSd20SpindownLifeCounterBluewhite Now the question: generally speaking, will these two kinds of die yield equal probabilities? (Assuming that both dice are well balanced). Particularly,   * Will the probability of getting a certain number be the same?   * Will the probability of getting a number above a threshold be the same? (i.e. rolling a 10 or more)


title: With the phone Vibration why the whole bed vibrates

text: While we are lying on the bed and there is a phone placed somewhere on the bed, when the phone vibrates we can feel the vibration all over the bed, but in the scenario when we are lying on the floor and our phone starts vibrating we couldn't feel the vibration as same as we feel on the bed. Why does it happen


title: What's the Significance of Color Changes Between EM-Spectrums?

text: As everybody knows that Black objects are black because almost all the light that falls on them is absorbed into the material. Little or no light is reflected back toward our eyes, therefore we see black. It is also known that, since black objects absorbs all the light they ultimately become heated-up..! If we see the snow in visible light, It appears to be White. But in the Infrared, it appears to be black. Does that mean that the Snow will feel warm in Infrared lights?


title: Causal reconnection

text: Can causally disconnected regions join up again? For example the universe is expanding faster than light creating cosmological horizon, but what if something causes the expansion to slow down and reverse, will everything just join up again or are things beyond the horizon forever disconnected no matter what?


title: Is there anything wrong with the cybernetic interpretation of quantum mechanics by Ross Rhodes?

text: The interpretation is outlined here It certainly gives a good logical explanation of most quantum oddities.


title: How to relate speed of sound with relative humidity?

text: I am exploring the idea of measuring the humidity of a space using sound waves, however I am having trouble finding a mathematical relationship between the speed of sound and the humidity level. $c_{air} = 331.3 \sqrt{1 + \frac{T}{273.15}}$ but this is for dry air (0%RH) How can I factor the effects of humidity into this relationship?


title: Shock Absorber - Bus - Not Uniform

text: Imagine a empty bus/lorry travelling at high speed over the "speed brake", why is that only rear seat is heaved relatively high compared to the fron seat?


title: When is the right ascension of the mean sun 0?

text: I understand that the right ascension of the mean sun changes (at least over a specified period) by a constant rate, but where is it zero? I had naively assumed that it would be zero at the most recent vernal equinox, but when I try to calculate the equation of time using this assumption and true sun positions, all my values are about 7.5 minutes larger than they should be. When (at what date and UT time) is the right ascension of the mean sun 0? And why?


title: few fermions in a harmonic trap — position density matrix from diagrammatics

text: I'm trying to calculate the momentum distribution of a 1D system of non- interacting identical fermions in a harmonic trap. Given Feynman's answer (from his Statistical Mechanics book) for the position density matrix of a single trapped particle at $T>0$, $ \rho_1 (x, x'; \beta) = \sqrt{\cfrac{m \omega}{2 \pi \hbar \sinh (\beta \hbar \omega) }} \exp \left\\{ \cfrac{-m \omega}{ 2 \hbar \sinh (\beta \hbar \omega) } \left[ (x^2 + x' ^2) \cosh (\beta \hbar \omega) - 2x x' \right] \right\\} $ , the translationally invariant distribution of $ (x'-x) $ is $ \tilde{\rho} (s; \beta) = \int_{-\infty} ^\infty \mathrm{d}x \mathrm{d}x' \delta (s - (x' - x) ) \rho_1 (x, x'; \beta) = \cfrac{\mathrm{e}^{\frac{-m \omega s^2}{4\hbar} \coth \frac{\beta \hbar \omega}{2}}}{2 \sinh \frac{\beta \hbar \omega}{2}} $ . Here, we define $\beta\equiv 1/(k_\text{B}T)$. The Fourier transform of $\tilde{\rho} (s ; \beta)$ is the momentum distribution of the system, which is also a Gaussian. **How would you construct a two-fermion version out of this?** What about 3 fermions? My first attempt was $ \tilde{\rho}_{2\text{F}} (s;\beta) = \frac{1}{2!} \left( 2 \tilde{\rho} (s; \beta) \tilde{\rho} (0; \beta) - 2 \tilde{\rho} (s; 2\beta) \right) $ using Feynman diagrams, keeping in mind the anti-periodicity $\rho_1 (x, x'; \beta + \beta) = -\rho_1 (x, x'; \beta)$ for fermions. Visualizing the path along the imaginary time on a hypercylinder of circumference $\beta$ would look something like ![enter image description here](http://i.stack.imgur.com/V19Nw.png) Note here that diagrams with fermion paths intersecting each other have zero statistical weight. I've written the full statement of my problem at http://mathb.in/1393 . Apparently my answer doesn't agree with two other numerical calculations... perhaps missing a normalizing factor in one of the terms. Any comments appreciated.


title: Maxwell's equations as the particular case of massive vector field equation

text: There was a discussion (please look to the comments on my answer) about getting Maxwell's equations for free spin-1 field by using massive spin-1 representation's equations. I'll start from the beginning. Massive spin-1 representation $\left( \frac{1}{2}, \frac{1}{2}\right)$ is realizing by vector field $A_{\mu}$ and conditions $$ (\partial^{2} + m^{2})A_{\mu} = 0, \quad \partial_{\mu}A^{\mu} = 0. \qquad (.0) $$ It's not hard to get the solution: $$ A_{\mu} = \int \sum_{\lambda = 0}^{3}e_{\mu}^{\lambda }(\mathbf p )\left( a_{\lambda }(\mathbf p)e^{-ipx} + a^{*}_{\lambda }(\mathbf p)e^{ipx}\right) \frac{d^{3}\mathbf p}{\sqrt{2(2 \pi)^{3}}}, \qquad (.1) $$ where $e_{\mu}^{\lambda }$ are polarization 4-vectors, for which we can choose conditions $$ e_{\mu}^{1} = (0, \mathbf e^{1} (\mathbf p)), \quad e_{\mu}^{2} = (0, \mathbf e_{2}(\mathbf p)), \quad e_{\mu}^{3} = \left( 0, \frac{\mathbf p}{\epsilon_{\mu}} \right), , \quad e_{0}^{\mu} = (1, 0), \quad (\mathbf e_{1,2} \cdot \mathbf p ) = 0. $$ and so $$ a_{0} = \frac{\mathbf p^{2}}{\mathbf p^{2} + m^{2}}a_{3}. \qquad (.2) $$ So field has three independent polarizations. Then let's have massless case. In my opinion, we may use solution $(.1)$ for it, but also we can analyze the conditions $(.0)$ in case $m = 0$, from which follows, that in some frame we may use $A_{0} = 0$. From this there is a result $e_{\mu}^{0} = 0$, and so $e_{\mu}^{3} = 0$. Massless spin-1 field has 2 independent polarization states, as it must be. Also, equations $(.0)$ can be derived independently of Proca equation, so it's not necessary $m \neq 0$ condition. So I have a few questions:   1. Are my statements correct?   2. If yes, can they be interpreted as transition from massive reps of Poincare group (with two non-trivial Casimir operators $W_{\mu}W^{\mu}, P_{\mu}P^{\mu}$) to massless rep (which is characterized by the helicity)?


title: Flow Rate and Orifice Diameter

text: I have a electrical/electronics background and have limited knowledge of Fluid Mechanics. So, i will try to be as clear as possible. I am currently working with brake fluid in an ABS system and to effectively construct my alternative control strategy,i have following questions: 1> Flow-rate: I just need to know how to measure the flow-rate across an orifice? I know of the formula **$$Q = CA\sqrt{\frac{2\Delta P}{d}}$$** $C$ - **discharge coefficient** $\Delta P$ - **Change in pressure** $d$ - **density of the fluid** $A$ - **Area of the orifice** Now, this formula should pretty much give me the answer, but my real question is; if i want to calculate simply the flow-rate across a circular orifice of different diameters. So, $Q1$ for $D1$,$Q2$ for $D2$ ... $Qn$ for $Dn$. I am simply interested in finding the flow rate without the consideration of pressure change or velocity etc. Just to clarify, FLOW RATE through a PIPE of varying diameters. I will be grateful for any and every suggestion!


title: Adding stuff to the path integral (Faddeev-Popov method)

text: I'm wondering about the Faddeev-Popov method described in Peskin Schroeder and also on page 7 in this link. What gives them the right to simply add the Gaussian $\omega$ and thus introduce the $\xi$ parameter? It seems so arbitrary to me. Are there any rigorous derivation of the Faddeev-Popov method?


title: Why England is so cloudy?

text: Why, physically, some regions of the globe are affected by a noticeable cloudiness respect to others? Is the reason why some places are also rainy the same?


title: Is it possible for the planets to align?

text: We've all heard the statement that on the 21st of December, the planets in the solar system will "align" from the point of view of the Earth. I assume this means that they would all be in the same spot in the sky if we looked from here. The theory says that the alignment of the planets will somehow exert some influence on the Earth which would bring varying levels of catastrophe, depending on who you ask. Now, it has been said many times that this will not actually happen, and that even if it happened there would be no effect on the Earth whatsoever. I know that, and that's not the question. What I'm wondering here if it is actually possible for the planets to align in this way, regardless of whether it'll actually happen. As far as I know, the planets' orbits aren't all in the same plane, so it doesn't seem even theoretically possible, i.e., there's no straight line passing through the orbits of all the planets. Am I right?


title: Why is information in a volume proportional to it's surface area?

text: I read the wiki article on the holographic principle, but it never answered this question. Can anyone explain the math that leads to this conclusion?


title: How many celestial bodies could be in stable orbit at roughly the same distance from a star?

text: **How many planet-like celestial bodies** of roughly the same mass (say within 50%) could orbit a star at roughly (say within 10%) the **same distance from the star** and be in stable orbits. By stable I mean they would not require tethers or significant energy to stay there. I feel sure the answer is at least 2, Earth plus another body at Lagrangian Point L3. I have heard it's not 3 -- that putting Earth-size bodies at Earth's L4 and L5 would destabilize. Is that true? Please explain concisely and cite reputably. I'm asking in the interests of Earthling living space within the solar system, in the far future of course. (Or other stellar systems in the uber far future.) Thanks to @dgh for steering the terminology away from "planet" which is defined by the IAU as having cleared its neighborhood (i.e. orbital radius) of other bodies. ## Research Possibilities   * Klemerer Rosette (theoretically unlimited, but practically no more stable than a ball on a hill)   * Iridium Satellites (66 are active in stable, Low Earth Orbit, in six planes)


title: Must neutron speed and diffusion speed be comparable for neutron scattering experiments to be probe atomic diffusion?

text: Neutron scattering experiments are sometimes used to study atomic diffusion in solid materials. My intuition is that the typical speeds of the atoms diffusing in the material--say, the diffusion constant (in cm^2/s) divided by the atomic spacing (in angstroms)--must be comparable to the speed of the incident neutrons for any information about diffusion to be gained. Much faster neutrons should see only a snapshot of the target atoms frozen in time, while much slower neutrons should see only average densities of the target atoms. Is this reasoning correct? (Can anyone point me to a good introduction?)


title: Why does electron-positron annihilation prefer to emit photons?

text: If gravitons are massless, and neutrinos nearly so, why aren't pairs of either of them normally expected products of electron-positron annihilations? Are they possible but simply unlikely, or is there actually some conserved quantity prohibiting their creation? Edit: I'm talking about the low energy limit, not in accelerator beam collisions.


title: How to calculate uncertainties of a natural exponential function?

text: (I apologize if this should be posted in mathematics, however I chose to post it here as it's technically about physics) I conducted an experiment in which position of items were shifted on an object, either on the ends of wings of it, or on the base (I'd rather not get too much into what it's about), and the effect on its fall rate over a certain distance was measured. The result was a decay model of the form: $T(N)=Ae^{-bN}+c$, where $A=1.44,b=0.132,c=0.303$ and $T =$ Time,$N =$ Number of items added to wings. However, for each of the times there is an uncertainty of between 0.08 and 0.09 seconds. So, I asked my teacher for assistance and he explained the following: First you remove the 0.303, and then you can rearrange it as follows: $T = 1.44*e^{-0.132N}$ $\ln{T} = \ln(1.44*e^{-0.132N})$ $\ln{T} = \ln{1.44} + \ln{e^{-0.132N}}$ $\ln{T} = 0.365 + -0.132N$ $\ln{T} = -0.132N + 0.365$ And thus you have a linear equation. Then I calculated $\ln{T}$ and $-0.132N + 0.365$ for each value of N, and graphed it in a graphic software, and made error bars of $±((\ln(T+\delta T)-\ln{(T-\delta T))/2})$, and thereby can get a best-fit gradient, a maximum possible gradient, and a minimum possible gradient, all in terms of $(ln(T)/(-0.132N + 0.365))$ if I'm not mistaken. But now for the questions:   1. Why could the (+0.303) simply be removed, and how can that be justified?   2. What do I do with my newly acquired values for the max. and min. gradients? I'd truly appreciate any help on this!


title: Pendulum system: how is derived the output as Energy?

text: Good day to everyone, I want to understand in which way the "Energy equation" is been implemented to this pendulum system. $x_1(t)$: The angular position of the mass $x_2(t)$: The angular velocity $k$: friction constant $M$: point mass The input $u(t)$ is a torque The output $y(t)$ is **Energy** So, we got: $\dot{x_1}(t) = x_2(t) $ $\dot{x_2}(t)= -(g/l) sin(x_1(t)) - (k/Ml^2)x_2(t) + (1/Ml^2)u(t)$ The last one, obviously, is derived by the fundamental law $M=Ix_2(t)$ The output: $y(t)=(1/2) Ml^2 x_2^2(t)-Mglcos(x_1(t))$ Considered the inertia $I=Ml^2$, and the friction in proportion to the velocity $x_2(t)$ The problem is the way that it uses to derive $y(t)$. In which way is implemented the Energy formula to obtain the result? Thank to all, in every case :D


title: Is measuring energy of photon released by electron enough to detect its previous state?

text: I am referring to an experiment mentioned in Quantum Entanglement lecture: https://www.youtube.com/watch?v=VtBRKw1Ab7E (Starting from: 14:20) In that lecture, as far as I understand, the description of the experiment is as follows: Assume: "UP state" means electron's (assumed as a bar magnet) north pole is vertically upwards   1. **Preparing the state:** Prepare electron to be in certain state (an arbitrary angle) by placing it in a magnetic field at certain angle.   2. Turn off that magnetic field.   3. **Detecting the state:** Turn on another magnetic field to detect the state of the electron. The north pole of external magnet is vertically downwards and the south pole is vertically upwards. Detect the state of the electron's state (up or down) by detecting if the electron releases any photon or not. If the electron releases photon, then it is "concluded" that previously, the electron had its north pole downwards i.e. electron was in "down" state. If electron does not emit any photon, then it is assumed that previously it was in "up" state. That "photon" released always has same energy (which is equal to energy released when transitioning from "down" to "up"). This is true even in cases where electron has been prepared in some other angle (i.e. neither down nor up but something in between). The argument made here is that, no matter what "state" (angle) the electron is prepared for (in step 1), while "detecting the state" we find that either electron releases no energy or some constant energy. So "detecting the state" results in either "up" or "down", but not any other state. But just because electron emits constant energy (equal to transition from down to up), why should it be "concluded" that it was previously "down"? May be the photon has quantized energy, so no matter what state it previously was in, it always releases that particular amount of energy to get to state "up". **My question is:** is it correct to assume electron was previously "down" just because it releases that particular amount of energy (equal to transitioning from down to up) to go to state "up"?


title: Using the monte carlo method to compute the magnetic field of a curent carrying loop

text: I have written a program in cpp that computes the magnetic field at a point from a current carrying loop. It uses the biot savart law and the monte carlo technique to carry out the integral. The program functions as expected for points that are not too close to the loop but from looking at a plot of the vectors you can see strange errors close to the loop. I checked my results against http://hyperphysics.phy- astr.gsu.edu/hbase/magnetic/curloo.html#c3 for a point on the axis and http://www.netdenizen.com/emagnet/offaxis/iloopcalculator.htm for a point off the axis. My program agrees with these sources for points not too close to the loop. This is my code. http://pastebin.com/4BH9qEZz My main question is if there is anything inherent in the monte carlo technique that would cause this error?


title: Does a random local unitary destroy entanglement

text: There is a massive debate raging about whether the wavefunction is an aspect of reality or just a way of handling the data you have about a system. This question is in that vein. A local unitary on a qubit is an operator, $O$ on a two dimensional hilbert space, for which there exists an adjoint, $O^{\dagger}$, such that $O O^{\dagger} = I$. Suppose you have an entagled pair of qubits. Next, suppose you perform a Uniformly random, local unitary, which you have no knowledge of, on one of the qubits. Does this destroy the entanglement? I would say that the entanglement is destroyed if the value $E= Tr(\rho_a ln(\rho_a))$ departs from its original value, of say 1, towards 0. $\rho_a$ is the density matrix of the $a$ qubit. Note: My first guess is that for states like this: $\psi = | 00 \rangle + |11\rangle$ the Von Neumann entropy, $E$ is 1. If you apply a random unitary to any one qubit, you are just randomizing an already completely random state, so there is no effect, so, no, you cannot destroy the entanglement with a random local unitary.


title: Why are momentum balance equations formulated in terms of net stress rather than net force?

text: In engineering, a steady state equilibrium force balance is typically expressed as $$\nabla\cdot \sigma=0$$ where $\sigma$ is the stress (force per unit area) applied at a point. It seems more intuitive to me to express the force balance as $$\sum F = 0$$. What does the first equation capture that the second equation doesn't? That is, what advantages does $\nabla\cdot \sigma=0$ have over $\sum F = 0$ to represent this steady state?


title: Proving that Measurement increases von Neumann entropy

text: Let $V$ be a finite dimensional complex inner product space. Let $\mathcal{M}$ be the classical sample space of measurement outcomes that may occur in a given experiment, and $M_\mu$, $\mu \in \mathcal{M}$ are a family of measurement operators acting on the pure states of $V$ that satisfy $\sum\limits_{\mu}M_\mu^\dagger M_\mu = Id_V$. The state of the system before the measurement is given by the density operator $\rho$. The probability that the outcome of the experiment is $\mu$ is $P_\rho (\mu) = \operatorname{Tr} ( M_\mu^\dagger M_\mu \rho)$. The system is now in the mixed state \begin{equation} \rho' = \sum\limits_{\mu} P_\rho(\mu) \frac{M_\mu \rho M_\mu^\dagger}{P_\rho(\mu)} = \sum\limits_{\mu}M_\mu \rho M_\mu^\dagger \end{equation} Now I am asked to prove that $$ S(\rho') \geq S(\rho)$$ where $S$ denotes the _von Neumann_ entropy. The hint says, that i should use that $$ - {\rm Tr} \left[ \rho \log \tau \right] \geq - {\rm Tr} \left[ \rho \log \rho \right] $$ which I have proven already in a previous exercise. Unfortunately I do not have any idea how to start. I'm not even sure that the definitions I took from the lecture notes are correct. All I know is that I have to get to $$ - {\rm Tr} \left[ \sum \limits_{\mu}M_\mu \rho M_\mu^\dagger \log \rho' \right] = - {\rm Tr} \left[ \rho \log \rho' \right]$$ in order to apply the hint. I tried to write it in a basis where either $\rho$ or $\rho'$ are diagonal but I found no way of achieving the above, I'm still feeling uncomfortable with density and measurement operators and appreciate any hint. PS: I'm not sure whether this is more of a math.SE question, but given the physical meaning of the calculation I decided to post it here. I'm happy to move it if it is considered unsuitable for physics.SE.


title: Spallation neutron generation and pure U-238 reactors

text: Main question: Is it possible to achieve net power generation based on linear proton accelerator and U-238 target? In the proposed reactor design there is a proton beam with energy ~10 GeV, and on U-238 target where these protons should generate cascade neutron generation & U-238 fission with high-energy neutrons. The problem though is that there is some disagreement on energy looses and neutron multiplicity. My estimations and understanding is the following:   1. 60% of incident proton energy goes into heat.    2. For high energy protons (>1GeV) there are significant loses on pi-0 decay, so using proton energy above 1-1.5GeV is inefficient. Is there other significant ways of loosing energy?   3. Estimated number of neutrons generated is 120-250 per incoming 10 GeV proton on Uranium target.   4. Only fraction of such neutrons have energy high enough to support fission of U-238   5. If only 25% of these neutrons would cause 1 fission, we'll only have ~180/4*0.2=9 GeV of fission + comparable amount of energy in neutrons and high-energy gamma (from pi-0 decay) Does that seems to be correct? Is there any papers investigating possibility of spallation-based net power generation on U-238/Thorium reactors? (i.e. without classical fissile materials like U-235, U-233 or Plutonium) Also, based on published specs of SINQ-2 spallation neutron generator, I calculated it's neutron generation efficiency to be 0,024% of proton beam power. Even if we replace spallation target with Uranium one, it's efficiency would be 0,1662% at best. Obviously, with such efficiency we cannot support net power generation. Does that mean that this idea is not workable?


title: Commutation relations of the generators of the conformal group

text: My question is from P.98 of the book by Di Francesco on Conformal Field theory. He gives the six non-vanishing commutation relations between the elements $P_{\mu}, D, L_{\mu \nu}$ and $K_{\mu}$ comprising the four dimensional space of generators of the conformal group. These are then rewritten by defining a set of four more generators: $$J_{\mu \nu} = L_{\mu \nu}\,\,\,,\,\,\,J_{-1, \mu} = \frac{1}{2}(P_{\mu} - K_{\mu})\,\,\,,\,\,\,J_{-1,0} = D\,\,\,,\,\,\,J_{0,\mu} = \frac{1}{2}(P_{\mu} + K_{\mu}),$$ and I believe the motivation for doing so is so that the six commutation relations mentioned above can be eloquently recast in a single one line commutation relation $$[J_{ab}, J_{cd}] = i(\eta_{ad}J_{bc} + \eta_{bc}J_{ad} - \eta_{ac}J_{bd} - \eta_{bd}J_{ac})$$ It also says that $a,b \in \left\\{-1,0,1,\dots,d\right\\}$ and that the new generators obey the $SO(d+1,1)$ commutation relations (which I think is the one-line equation above). My question is: What do the indices $a,b$ represent and what does the notation $SO(d+1,1)$ mean? I think there are $d$ spatial dimensions but I can't see what the significance of the $-1$ and $0$ elements are. Many thanks.


title: Beat like Interaction with electromagnetic waves

text: So I was reading earlier on this website and I came across some question similar to this but I want to lead it Into a different direction. I understand that two interacting superposition waves (that are light) will not combine to make a larger energy wave. This is due to some math that I just forgot. So my question was can u take two waves a high energy one and a low energy one could you possible get a area that has a visible spectrum wave in it where they constructively interfere. One problem to this i see is that with two wave with drastically different energy l evals and wavelengths you would have to use three waves due to the electric and magnetic parts of the wave will not line up with just two waves. I was wondering if there was a way around this. If there is I would love to get a mild answer then for someone to point me in the direction of the math behind it and not just give it to me because i am looking for a summer project lol. If that doesn't work i would love something for a very smart high school student to research and do during the summer. I understand it not your guys responsibility to entertain someone but hey I am bored.


title: Will a water-cooled system heat up a room less than an air-cooled system?

text: Recently - motivated by the recent heat where we live - an acquaintance argued that a water cooled computer will not heat up the room as much as an air cooled computer will. His reasoning was that the processors never reach temperatures as high as they would with air cooling, so the room will not absorb as much heat. This seems flawed to me; the processors have an energy consumption that is regulated by how much processing they do, and that doesn't depend on whether they are air cooled or water cooled. I figured the amount of energy that gets converted to heat should be the same in both cases, the water cooling just doesn't reach temperatures as high as the air cooling because water has a higher capacity for absorbing energy than air does. In the end, the same amount of heat energy would end up in my room in both cases, so I would expect the room to heat up the same, independent of whether we cool our computers with water or air cooling systems. Which one is right?


title: Sound pressure and velocity pressure concepts vs loudness

text: Here is my understanding and question: Atmospheric pressure is around 101325 Pa. Our ear drum is exposed to this pressure from both sides so total pressure is zero. But when we hear sound, ear drum starts vibrating and the effective pressure on ear drum is calculated for many scenarios. For example a jet engine produces 632 Pa, our threshold of pain is 63.2 Pa. In a wind-tunnel 15 m/s (54 km/h) wind speed corresponds to 140 Pa. When I enter a wind tunnel I don't feel any pain with 140 Pa, but why do I feel pain with 63.2 Pa when it comes to sound. It vibrates or not, isn't pain about the force on ear? Why does sound waves with less pressure results pain but not a constant 140 Pa wind directly blowing to my ear?


title: Friction force problem for a 4 wheel drive Car

text: > _The car is a 4 wheel drive in equilibrium, the car doesn't touch the ground > with the front wheel (A) but is in contact in B with the sidewalk (the > sidewalk has a hight equal to $h$), the friction in point B and D are both > equal to 0.8, and the weight of the car is equal to 18000 N, the question is > can the car go up the sidewalk without run up._ I am unable to understand the solution in this picture which says yes , as long as point I is between M and N, the car can go up the sidewalk. But according to my understanding, in order for the car to move with kinetic friction, the angle between vector B or D and the normal line should be equal to φ Can you please help me? Many thanks ![enter image description here](http://i.stack.imgur.com/DlGYr.png)


title: Types of photon qubit encoding

text: How many types of qubit encoding on photons exist nowadays? I know only two:   * Encoding on polarization: $$ \lvert \Psi \rangle = \alpha \lvert H \rangle + \beta \lvert V \rangle $$ $$ \lvert H \rangle = \int_{-\infty}^{\infty} d\mathbf{k}\ f(\mathbf{k}) e^{-iw_k t} \hat{a}^\dagger_{H}(\mathbf{k}) \lvert 0 \rangle_\text{Vacuum} $$ $$ \lvert V \rangle = \int_{-\infty}^{\infty} d\mathbf{k}\ f(\mathbf{k}) e^{-iw_k t} \hat{a}^\dagger_{V}(\mathbf{k}) \lvert 0 \rangle_\text{Vacuum} $$   * Time-bin: $$ \lvert \Psi \rangle = \alpha \lvert 0 \rangle + \beta \lvert 1 \rangle $$ $$ \lvert 0 \rangle = \int_{-\infty}^{\infty} dz\ f\left(\frac{t -z/c}{\delta t_{ph}}\right) e^{-i w_0 (t-z/c)} \hat{a}^\dagger(z) \lvert 0 \rangle_\text{Vacuum} $$ $$ \lvert 1 \rangle = \int_{-\infty}^{\infty} dz\ f\left(\frac{t -z/c+\tau}{\delta t_{ph}}\right) e^{-i w_0 (t-z/c+\tau)} \hat{a}^\dagger(z) \lvert 0 \rangle_\text{Vacuum} $$ Is there anything else?


title: Electromagnetic inertia due to advanced radiation?

text: The scalar potential $\phi$ and vector potential $A$ at a distance $r$ from a charge $q$ are given approximately by $$\phi = \frac{q}{r}$$ $$\mathbf{A} = \frac{q\mathbf v}{r}$$ where the constants have been suppressed. The corresponding electric and magnetic fields are given by $$\mathbf{E} = -\nabla \phi - \frac{\partial \mathbf A}{\partial t}$$ $$\mathbf{B} = \nabla \times \mathbf A$$ Now the gradient and curl terms fall off as $1/r^2$ so that at large distances from the charge $q$ we only have an electric field $E$ given approximately by $$\mathbf{E} = - \frac{\partial \mathbf A}{\partial t}\ \ \ \ \ \ \ \ \ (1)$$ where $$\mathbf{A} = \frac{q\mathbf v}{r}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $$ Thus at large distances an accelerating charge produces an electric field that is proportional to its acceleration. This is the standard (retarded) radiation field of an accelerated charge. But perhaps this is not the only solution to equation (1). Imagine that the charge is fixed with respect to an inertial frame and that the observer is moving with velocity $-v$. In the rest frame of the observer the charge is moving with velocity $v$. This implies that in the observer's rest frame there is an $A$-field. Now if the observer changes his velocity then in his comoving frame he experiences a changing $A$-field. My proposal is that Equation (1) still applies in this situation so that this "apparently" changing $A$-field induces an electric field in the observer's accelerating frame. This field causes any charges that are co-moving with the accelerating observer to feel a kind of electromagnetic inertial force. I realise that this is non-standard as Maxwell's equations are usually only applied in an inertial frame. My hypothesis is that the retarded solutions apply to an inertial frame whereas the previously unused advanced solutions apply to accelerated frames.


title: Plotting $\psi$ for finite square well potential

text: Lets say we have a finite square potential well like below: ![finite well](http://i.stack.imgur.com/3PGLK.png) This well has a $\psi$ which we can combine with $\psi_I$, $\psi_{II}$ and $\psi_{III}$. I have been playing around and got expressions for them, but they are not the same for ODD and EVEN solutions but lets do this only for ODD ones. * * * **ODD solutions:** $$ \boxed{\psi_{I}= Ae^{\mathcal{K} x}~~~~~~~~\psi_{II}= - \dfrac{A e^{-\mathcal{K}\tfrac{d}{2}}}{\sin\left( \mathcal{L} \tfrac{d}{2} \right)}\, \sin\left(\mathcal{L} x\right)~~~~~~~~ \psi_{III}=-Ae^{-\mathcal{K} x}} $$ When i applied boundary conditions to these equations i got transcendental equation which is: \begin{align} &\boxed{-\dfrac{\mathcal{L}}{\mathcal{K}} = \tan \left(\mathcal{L \dfrac{d}{2}}\right)} && \mathcal L \equiv \sqrt{\tfrac{2mW}{\hbar^2}} && \mathcal K \equiv \sqrt{\tfrac{2m(W_p-W)}{\hbar^2}} \\\ &{\scriptsize\text{transcendental eq.} }\\\ &\boxed{-\sqrt{\tfrac{1}{W_p/W-1}} = \tan\left(\tfrac{\sqrt{2mW}}{\hbar} \tfrac{d}{2} \right)}\\\ &{\scriptsize\text{transcendental eq. - used to graph} } \end{align} Transcendental equation can be solved graphically by separately plotting LHS and RHS and checking where crosssections are. $x$ coordinates of crossections represent possible energies $W$ in finite potential well. So i can theoreticaly get values for possible energies $W$ and when i get these i can calculate $\mathcal L$ and $\mathcal K$. But i am still missing constant $A$. * * * **ORIGINAL QUESTION:** I would like to plot $\psi_I$, $\psi_{II}$ and $\psi_{III}$ but my constant $A$ is still missing. How can i plot these functions so the normalisation will be applied? * * * **EDIT:** After all of your suggestions i decided to work on a speciffic case of an electron with mass $m_e$ which i put in a finite well. So the constants i know are: \begin{align} d &= 0.5nm\\\ m_e &= 9.109\cdot 10^{-31} kg\\\ W_p &= 25eV\\\ \hbar &= 1.055 \cdot 10^{-34} Js {\scriptsize~\dots\text{well known constant}}\\\ 1eV &= 1.602 \cdot 10^{-19} J {\scriptsize~\dots\text{need this to convert from eV to J}} \end{align} I first used constants above to again draw a graph of transcendental equation and i found 2 possible energies $W$ (i think those aren't quite accurate but should do). This looks like in any QM book (thanks to @Chris White): ![enter image description here](http://i.stack.imgur.com/KzudS.png) Lets chose only one of the possible energies and try to plot $\psi_I$, $\psi_{II}$ and $\psi_{III}$. I choose energy which is equal to $0.17\, W_p$ and calculate constants $\mathcal K$ and $\mathcal L$: \begin{align} \mathcal K &= 2.3325888\cdot 10^{10}\\\ \mathcal L &= 1.5573994\cdot 10^{10}\\\ \end{align} Now when the picture above looks like in a book i will try to use constants $\mathcal K$, $\mathcal L$ and $\boxed{A \\!=\\! 1}$ (like @Chris White sugested) to plot $\psi_I$, $\psi_{II}$ and $\psi_{III}$. Even now the boundary conditions at $-\tfrac{d}{2}$ and $\tfrac{d}{2}$ are not met: ![enter image description here](http://i.stack.imgur.com/KjZMf.png) **EXTENDED QUESTION:** It looks like boundary conditions are not met. I did calculate my constants quite accurately, but i really can't read the energies (which are graphicall solutions to the first graph) very accurately. Does anyone have any suggestions on how to meet the boundary conditions? Here is the GNUPLOT script used to draw 2nd graph:               set terminal epslatex color colortext size 9cm,5cm     set size 1.5,1.0     set output "potencialna_jama_6.tex"          set style line 1 linetype 1 linewidth 3 linecolor rgb "#FF0055"     set style line 2 linetype 2 linewidth 1 linecolor rgb "#FF0055"     set style line 3 linetype 1 linewidth 3 linecolor rgb "#2C397D"     set style line 4 linetype 2 linewidth 1 linecolor rgb "#2C397D"     set style line 5 linetype 1 linewidth 3 linecolor rgb "#793715"     set style line 6 linetype 2 linewidth 1 linecolor rgb "#793715"     set style line 7 linetype 1 linewidth 3 linecolor rgb "#b1b1b1"     set style line 8 linetype 3 linewidth 1 linecolor rgb "#b1b1b1"          set grid          set samples 7000          set key at graph .70, 0.4     set key samplen 2     set key spacing 0.8          m = 9.9109*10**(-31)     d = 0.5*10**(-9)     U = 25 * 1.602*10**(-19)     h = 1.055*10**(-34)          K = 2.3325888*10**10     L = 1.5573994*10**10     A = 1          f(x) = A*exp(K*x)     g(x) = -( A*exp(-L*(d/2)) )/( sin(L*(d/2)) )*sin(L*x)     h(x) = -A*exp(-K*x)          set xrange [-d:d]     set yrange [-8*10**(-2):8*10**(-2)]          set xtics ("$0$" 0, "$\\frac{d}{2}$" (d/2), "$-\\frac{d}{2}$" -(d/2))     set ytics ("$0$" 0)          set xlabel "$x$"          plot [-1.5*d:1.5*d] f(x) ls 1 title "$\\psi_{I}$", g(x) ls 3 title "$\\psi_{II}$", h(x) ls 5 title "$\\psi_{III}$"


title: How far will the stone travel in 2.0 s?

text: A stone is thrown vertically down from the top of a 40 story building with an initial speed of 1 m/s. How far will the stone travel in 2.0 s? the answer is 22 m and my answer is 17.6 m I used this equation to solve the problem $x=v_0t-(1/2)gt^2$ = (1)(2)-(0.5)(9.8)(4)=-17.6 = 17.6m So, where is the mistake?


title: Argument against computer consciousness

text: Imagine that we have a computer program that produces the conscious awareness of the present moment. Let us assume that every time the program is run a counter is incremented. Let us also assume that the program is encapsulated in an infinite loop. Each time the program is run the computer "wakes up" and asks itself "what is the present value of the loop counter?" The computer reasons that _a priori_ it is equally likely to find itself in any loop with any value of the loop counter. Therefore the prior probability that the loop counter has any particular value $n$ is $1/\infty=0$. But the computer will find that it has _some_ loop counter value. There seems to be a contradiction here. I think the problem is with the assumption that a computer program can produce conscious awareness in the first place. I think this argument shows that any deterministic machine cannot be conscious. Such determinism implies a single-valued time evolution whose (theoretically) infinite extent leads to this zero probability paradox. I believe to avoid this paradox one needs a branching concept of time in the manner of the many-worlds quantum theory in which each branch into an alternate future is only of finite extent. Maybe the brain itself avoids the paradox by exploiting this many-worlds ontology in its operation.


title: The Lagrangian in Scalar Field Theory

text: This is perhaps a naive question, but why do we write down the Lagrangian $$\mathcal{L}=\frac{1}{2}\eta^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi - \frac{1}{2}m^2\phi^2$$ as the simplest Lagrangian for a real scalar field? This is by no means obvious to me! After all it gives rise to a kinetic energy term (fine), some specific unmotivated potential energy (less fine) and a gradient energy (even less obvious). Is there some principle by which we know to study this Lagrangian? Is it just that $\mathcal{L}$ gives rise to a nice equation (the Klein-Gordon equation), which we can interpret in an appealing way? This reason seems somehow hollow! I've heard people mention causality as a motivation, but I can't see how that ties in. Could someone provide me with some intuition? Many thanks in advance!


title: Can Ohm's law break in metals?

text: I was rereading Purcell's Electricity and Magnetism as research for another question, and I found this passage: > In metals Ohm's law is obeyed exceedingly accurately up to current densities > far higher than any that can be long maintained. No deviation has ever been > clearly demonstrated experimentally. (Second edition, 1985, section 4.5, > p143.) Here all characteristics of the metal, including temperature, must be kept constant, and in such conditions Purcell states that no deviation from a linear relation between electric field and current density has been demonstrated. Now, this book is relatively outdated (two pages back it touches briefly on superconductivity and mentions that the highest Tc on record is 21 K), so I wonder whether the fact above, admittedly less technologically crucial than high-Tc superconductivity, is still held to be experimentally true, or whether deviations from linearity have in fact been observed. Purcell goes on to state that > According to one theoretical prediction, departures on the order of 1 > percent might be expected at a current density of 10$^9$ amps/cm$^2$ though I'm having some trouble guessing whether we have technology to reach such current densities. To put things on a more concrete footing, then: have deviations from Ohm's law in metals been observed? Is this an active question in any field of research? Do confirmations of this follow from other, related research? Or, alternatively: has the prediction above been verified or falsified? Do we have technology to test those regimes? (if not, what ranges are experimentally viable?) Additionally, can someone supply a reference to the theoretical prediction? * * * Brief edit: I just got my hands on a sample of the new (2013) edition of Purcell (now in SI units!), revised by Morin. They have updated the superconductivity bit to include high-Tc SCs, but the Ohm's law passage remains much the same. They do mention that the current density above is `over a million times stronger than normal`, but don't specify what the state of the art is in current density, and don't give references to the 'theoretical prediction'. It's real nice to have that book brought back to SI life, though!


title: Electrostatic notion of voltage as it applies to circuits

text: I have a question that's been bothering me about electric fields, voltage, and circuit analysis. Initially, I came to understand voltage as it was taught in the context of electrostatics - through energy analysis. The concept of the electric field as a conservative field makes a lot of sense. No matter what path a charge takes, if it travels from a positive charge to a negative charge in any way (in a dipole situation), it has the same difference in voltage, namely, the voltage between the positive and negative charge. This concept of voltage seems to take a huge leap when we jump to considering circuits. The terminals of a battery create an electric field, which tends to move charge carriers from the higher to the lower voltage. But how can the circuits be of absolutely any shape? If I have an extremely long, convoluted wire, how can we analyze voltage as simply as we can? Imagine a practical situation of a lamp plugged into a wall outlet far across the room. Circuit analysis says that the wall outlet is "at 120V" (AC) and the voltage drops through all resistance in the circuit on its way back to the ground wire. I just have trouble matching up this notion of voltage with the notion of voltage at a point in an electric field. On the other hand, the only way the charges in a circuit move is because of an electric field. Is the electric field just conveniently shaped such that it moves charges along the circuit at every point? What if the wire goes "uphill" and turns momentarily back toward the positive battery terminal but then turns around again. Wouldn't the charges get stuck in the local potential energy well? ![A simple circuit](http://i.stack.imgur.com/m9ATt.png)


title: Electron shells in atoms: What causes them to exist as they do?

text: I have seen similar posts, but I haven't seen what seems to be a clear and direct answer. Why do only a certain number of electrons occupy each shell? Why are the shells arranged in certain distances from the nucleus? Why don't electrons just collapse into the nucleus or fly away? It seems there are lots of equations and theories that describe HOW electrons behave (pauli exclusion principle), predictions about WHERE they may be located (Schrödinger equation, uncertainty principle), etc. But hard to find the WHY and/or causality behind these descriptive properties. What is it about the nucleus and the electrons that causes them to attract/repel in the form of these shells at regular intervals and numbers of electrons per shell? Thank you! Please be patient with me, new to this forum and just an amateur fan of physics.


title: Do we breathe air by creating a vacuum?

text: How do we suck air into our lungs, are we generating a vacuum?


title: Can a black hole be explained by newtonian gravity?

text: In the simple explanation that a black hole appears when a big star collapses under missing internal pressure and huge gravity, I can't see any need to invoke relativity. Is this correct?


title: I need to know how many pounds of force a 3 million pound object has moving at 1 mph

text: How many pounds of force does a 3 million pound object have moving at 0.1, 0.5, and 1 mph?


title: Jaumann rate, expansion in cartesian co-ordinates

text: I am stuck at implementing the Jaumann rate or the Hooke's law since i am unable to deduce the full cartesian form of the equations from the indices form. Is there some literature i can read where these equations are listed?


title: Curvilinear Coordinates and basis vectors

text: In these notes, $\frac{\partial \vec{r}} {\partial q_i}$ is stated to form a basis set for the vector space. How does this happen? Also, how does one justify this equation from Goldstein's Classical Mechanics using the above method? ![enter image description here](http://i.stack.imgur.com/gjuos.png)


title: Spontaneous breaking of Lorentz invariance in gauge theories

text: I was browsing through the hep-th arXiv and came across this article: > Spontaneous Lorentz Violation in Gauge Theories. A. P. Balachandran, S. > Vaidya. arXiv:1302.3406 [hep-th]. (Submitted on 14 Feb 2013) The authors say that Lorentz symmetry is spontaneously broken in QED and also in non-abelian gauge theories by infrared effects. However as far as I know the observed world is Lorentz invariant. Can someone kindly explain to me what is going on? Thanks for any help!


title: Is the assumption of space time to be a continuum is just a matter of mathematical taste?

text: Is the assumption of space time to be a continuum is just a matter of mathematical taste ? Isn't there any physical significance associated with it.


title: Thermionic emission and delayed emission

text: I want to understand the concepts behind the thermionic emission. In thermionic emission, the energy randomization occurs and the energy may be split to electronic or roto-vibrational states. If this energy randomization happens, I didn't understand how the electron will get enough energy to be detached from the molecule/cluster! The randomized energy could be dissipated via radiation (IR or visible). So, to make clear, I am wondering how this randomzed energy can cause electron emission. As far as I know (from readings), delayed emission of electrons is an indication for thermionic emission (Y Zhao et.al. J. Chem. Phys., 1996, 105(12), 4905). How the electron finally collect these randomized energy so that it eventually get detached?


title: A spinning bullet

text: I know the rifling in a gun or rifle puts a spin on the bullet along the axis of trajectory. Now I don’t understand exactly what does it make the trajectory more stable and the travel grater?


title: Puzzled by magnetized aluminum!

text: For a school project I bought a set of magnets and an aluminum strip. The magnets are places along both sides of the aluminum strip (all in the same direction). Now the aluminum strip is exhibiting magnetization. I was under the impression that aluminum cannot be magnetized. What is the explanation? The strip seems to have opposite polarity vs the magnets near it.


title: What happens to a single quark in lattice QCD simulations?

text: I understand that if a pair quark/antiquark, out of the vacuum, tries to separate then the energy increases, another pair is produced, and we finish with two mesons or generically two hadron jets. But in numerical simulations we can just put a single quark and calculate the field, can we? What does it happen, does the total energy diverge? Or does it increases up to, say, some "constituient" mass?


title: Particle physics conservation law checking tool

text: I'm just starting out with simple particle physics, and I'm doing a ton of exercises where I have to check if a certain reaction is allowed, from the point of lepton/baryon/energy conservation and such. Now in principle I am able to figure most of it out of course, as it's simply checking a few numbers and whatnot, but from time to time I'm asked to 'add' the missing particles to the equation, and I'm not always sure if my answer is sensible. Now, as I'm doing approximately 40 of these exercises posting them here and asking if they are sensible is neither time-efficient nor very useful for anyone other than myself, so I was wondering if there's something out there such as a simple tool that checks if the conservation laws are obeyed. Something where you'd say I begin with an antitau and I end up with a tau neutrino, an antimuon and a mu neutrino, and it'll tell you yes this is possible, or no this or that conservation law is not satisfied. I'm assuming something like this should be relatively simple to make (as it simply has to know a few particles and count some numbers), but I haven't been able to find one as of yet. Does anyone know if one exists?


title: Rotation from Goldstein's Classical Mechanics

text: I apologize for the ambiguity in my title. It was rather difficult to figure out what is the most appropriate title for my questions. My questions come from chapter 4 and chapter 5 of Goldstein, which are both about rotations. First regards to the paragraph following equation (4.84), which states $dG_i = a_{ji}dG_j' +da_{ji}G_j'$ where prime coordinates are the body coordinate, and unprimed represents the space coordinate. $a$ represents the transformation matrix from space to body. The paragraph states > It is no loss of generality to take the space and body axes as > instantaneously coincident at the time t. Components in the two systems will > then be the same instantaneously , but differentials will not be the same, > since the two systems are revolving relative to each other. Thus, $G'_j=G_j$ > but $a_{ji}dG_j' = dG_i'$. I get its reasoning, except the conclusion. I agree that the differentials are different respect to the two coordinate system, but why does that imply $a_{ji}dG_j' = dG_i'$? Second question I have comes from chapter 5, where it is stated (in section 1), > Any difference in the angular velocity vectors at two arbitrary points must > lie along the line joining the two points. Why is that true?


title: What determines the magnitude of force of an electromagnet?

text: Let's assume that there is an electromagnet with $B = 1 Tesla$ , what is the magnitude of the force it could generate on other dipoles equal to it's surface area? Can the magnitude of force (in newtons) be determined from $B$?


title: How do test lights (mains tester) work?

text: I was wondering how one-contact test lights work. Obviously there is a small current running from an energized pole through the human body somewhere that is still large enough to produce a visible light. German wikipedia talks about the current being in the range of µA. What I find puzzling is where the current actually goes. It seems to work on whatever material I stand on (eg. tables, chairs) and whatever thick rubber shoes I wear. I can't find a source that explains that.


title: BBC radiation: What is it?

text: I'm reading articles about blazars, and in a discussion about models concerning Blazar SEDs (Spectral Energy Densities) there are mentions of different processes: Sychrotron scattering, inverse Compton scattering, etc. Now concerning models for blazar emission, there are a lot of acronyms. I managed to figure out that SSC stands for "Self Synchrotron-Compton," but I have no idea what the acronyms "XC" and "BBC" stand for. And it isn't easy to google out "bbc + anything" without finding just references to... you know: http://en.wikipedia.org/wiki/BBC Does anyone know or have a good sense of what those acronyms might stand for?


title: Motivating the Legendre Transform Mathematically

text: If I begin with a functional of the form $$J[y] = \int_a^b f(x,y,y')dx$$ and find its Euler-Lagrange equations $$\frac{\partial f}{\partial y} - \frac{d}{dx}\frac{\partial f}{\partial y'} = 0 = \frac{d}{dx}\frac{\partial f}{\partial y'} - \frac{\partial f}{\partial y}.$$ I end up with a second order ODE $$\frac{d}{dx}\frac{\partial f}{\partial y'} - \frac{\partial f}{\partial y} = \left( \frac{\partial ^2 f}{\partial y' \partial y'} \right) \frac{d^2 y}{dx^2} \+ \left(\frac{\partial ^2 f}{\partial y \partial y'}\right) \frac{dy}{dx} \+ \frac{\partial ^2 f}{\partial x \partial y'} \- \frac{\partial f}{\partial y} = 0$$ Now every higher order ODE can be broken up into a system of first order ODEs in $y$ and the derivative $M = \frac{dy}{dx}$, giving $$ \frac{d}{dx}\frac{\partial f}{\partial y'} - \frac{\partial f}{\partial y} = \left( \frac{\partial ^2 f}{\partial y' \partial y'} \right) \frac{d M}{dx} \+ \left(\frac{\partial ^2 f}{\partial y \partial y'}\right)M \+ \frac{\partial ^2 f}{\partial x \partial y'} \- \frac{\partial f}{\partial y} = 0. $$ From this perspective, Hamilton's equations $$\left\\{ \begin{array} & \frac{dy}{dx} = \phantom{-}\frac{\partial \mathcal{H}}{\partial p}\\\ \frac{d p}{dx} = - \frac{\partial \mathcal{H}}{\partial y} \end{array}\right.$$ are merely a system of first order equations making my above system of first order ODEs look more symmetric, after a suitable change of variables. My question is, looking at $$\frac{d}{dx}\frac{\partial f}{\partial y'} \- \frac{\partial f}{\partial y} = \left( \frac{\partial ^2 f}{\partial y' \partial y'} \right) \frac{d^2 y}{dx^2} \+ \left(\frac{\partial ^2 f}{\partial y \partial y'}\right) \frac{dy}{dx} \+ \frac{\partial ^2 f}{\partial x \partial y'} \- \frac{\partial f}{\partial y} = 0$$ it should be possible to see why the Legendre transformation arises, first because its a transformation using derivatives to change variables but also because it should just make some terms go to zero in this second order ode so that everything looks nicer, but how do you see this explicitly? It'd be great if you could use my notation, ie. $J[y]$ etc... as you see I snuck in $\mathcal{H}$ in above which shouldn't really be there, I'd love to see how that comes about in my notation - thanks!


title: A confusion about notation in Goldstein

text: On treating systems of particles, Goldstein starts with the consideration that whenever there are $k$ particles on a system, the $i$-th one obeys the relation $$\dfrac{d}{dt}{\bf p}_i = {\bf F}_i^{(e)}+\sum_{j=1}^k {\bf F}_{ji}$$ Where ${\bf F}_{i}^{(e)}$ is the external force on the $i$-th particle and ${\bf F}_{ji}$ is the force that the $j$-th particle applies on the $i$-th one. Now if we compute the work on moving the system along a curve $\gamma$ we have: $$W=\sum_{i=1}^k\int_\gamma {\bf F}_i = \sum_{i=1}^k \int_{\gamma}{\bf F}_i^{(e)}+\sum_{i,j=1}^k\int_{\gamma}{\bf F}_{ji}$$ Now he says the following (on p. 10) > In the special case that the external forces are derivable in terms of the > gradient of a potential, the first term can be written as $$\sum_{i=1}^k > \int_{\gamma} {\bf F}_i^{(e)} = -\sum_{i=1}^k \int_{\gamma} {\bf > \nabla}_iV_i=-\sum_{i=1}^k V_i\bigg|_{a}^b$$ where the $i$ subscript on the > del symbol indicates that the derivatives are with respect to the > coordinates of $\mathbf{r}_i$. Now I can't understand why this $i$ on the del symbol. Basically we are considering the configuration manifold to be $\mathbb{R}^3$ so the natural coordinate functions are $x,y,z: \mathbb{R}^3\to \mathbb{R}$ which are just projections onto the axis and with those coordinates, there's just _one_ del operator, namely: $${\bf \nabla} = \dfrac{\partial}{\partial x} e_1 + \dfrac{\partial}{\partial y}e_2+\dfrac{\partial}{\partial z}e_3$$ So what really means "differentiate with respect to the coordinates of $\mathbf{r}_i$? And why do we need it? Does it have any relationship with the idea of taking the configuration space for a system of $k$ particles as $\mathbb{R}^{3k}$?


title: Chiral Anomaly in Massless QED

text: Classical massless QED has axial current conservation. When quantizing the theory, we expect that suddenly $\partial_\mu \hat{j}^{\mu5}\neq0$ (as an operator equality). I have two questions regarding this:   1. Why is it then, that when trying to prove indeed that $\partial_\mu \hat{j}^{\mu5}\neq0$ in the quantum version, we go on to try to prove that $$\partial_\mu\langle\Omega\left|\hat{j}^{5 mu} (x) \hat{j}^\nu (y) \hat{j}^\lambda (z)\right|\Omega\rangle\neq0$$? I understand that the above relation would imply the violation of a naive Ward-identity, but what's the connection between axial current conservation and Ward-identities? Is it the case that if the Ward identity is violated then _necessarily_ the current cannot be conserved?   2. When calculating $\langle\Omega\left|\hat{j}^{5 mu} (x) \hat{j}^\nu (y) \hat{j}^\lambda (z)\right|\Omega\rangle$, it appears that the $\hat{j}$'s are treated as external photon lines, as in this image: ![diagrams in the bottom](http://i.stack.imgur.com/sl64J.jpg) why is that? I would naively calculate such a transition amplitude by plugging in functional derivatives w.r.t. to the Fermion and anti-Fermion sources, for example, like so: $$\langle\Omega\left|\hat{j}^{5 mu} (z) \hat{j}^\nu (x) \hat{j}^\lambda (y)\right|\Omega\rangle = \gamma_{\mu\alpha_{1}\alpha_{2}}\gamma_{5\alpha_{2}\alpha_{3}}\gamma_{\nu\alpha_{4}\alpha_{5}}\gamma_{\lambda\alpha_{6}\alpha_{6}}\left.\left(\frac{\left(-\frac{1}{i}\frac{\delta}{\delta\eta_{\alpha_{1}}\left(z\right)}\right)\left(\frac{1}{i}\frac{\delta}{\delta\bar{\eta}_{\alpha_{3}}\left(z\right)}\right)\left(-\frac{1}{i}\frac{\delta}{\delta\eta_{\alpha_{4}}\left(x\right)}\right)\left(\frac{1}{i}\frac{\delta}{\delta\bar{\eta}_{\alpha_{5}}\left(x\right)}\right)\left(-\frac{1}{i}\frac{\delta}{\delta\eta_{\alpha_{6}}\left(y\right)}\right)\left(\frac{1}{i}\frac{\delta}{\delta\bar{\eta}_{\alpha_{7}}\left(y\right)}\right)\mathcal{Z}_{QED}}{\mathcal{Z}_{QED}}\right)\right|_{\mbox{sources}=0}$$


title: Wavelength and Frustum shaped resonant cavity

text: Given a right conical frustum resonator top radius r, bottom radius R and height h, what is the relationship of the wavelength w that we wish to create resonance with, to r, R, and h? Assume for now that the sides of the chamber are perfectly reflective and that the wave, acoustic or electromagnetic, is introduced on the sloping side of the frustum. Much more interested in the sources than the answer but both would be nice! Thank you!


title: On the Existence of Interstellar Meteoroids

text: A meteoroid in orbit around the Sun has a heliocentric velocity in the vicinity of Earth of about 42.1 km/s. So if a meteoroid has a heliocentric velocity exceeding that, it will have an open (hyperbolic) orbit and would not have originated in the solar system. As a meteoroid encounters Earth's atmosphere it encounters atmospheric atoms and molecules, generating heat that causes atoms to boil off and collide with those in the atmosphere. This produces ionized particles which surround the meteoroid with a glowing envelope leaving a column of plasma. The meteoroid is now seen as a meteor. Meteor velocities measured by radar and optical means have indicated the detection of many hyperbolic meteoroid orbits (heliocentric velocities exceeding 42.1 km/s), but velocity measurement errors which occur in the 10% range have cast doubt on their interstellar meteoroid status. However, it's clear that the solar system must be embedded in dusts and gasses originating from nearby star systems, so the existence of hyperbolic meteoroids should not be in doubt. > What is the technical difficulty in obtaining sufficiently accurate velocity > measurements to prove or disprove the existence of interstellar meteors?


title: Force Between Two Conducting Spheres

text: **Aakash PHYSICS JEE (Main & Advanced) Study Package - 5 & 6 (Class XII)** **Chapter -** Electric Charges and Field               Assignment (page 12)      _SECTION - A; Q.no - 1_ > The force of repulsion between two point charges is F, when they are d > distance apart. If the point charges are replaced by conducting spheres each > of radius r and the charge remains same. The separation between the center > of sphere is d, then force of repulsion between them is > > (1) Equal to F > > (2) Less than F > > (3) Greater than F > > (4) Cannot be said               Answer - (2) Less than F      What I expected the answer to be was (1). The electric field due to a conducting sphere of charge Q is equivalent to the same due to a point charge at the center of the sphere as total charge on it appears as concentrated at the center for the points outside the charged sphere. So, the force should have remained the same. Any suggestions?


title: Hairy ball theorem: references to applications

text: I'm looking for references to applications of the Hairy ball theorem. This is a result of mathematics (topology), but I am interested in applications. I already visited wikipedia and cited references, but I need a little more explanation in magnetic fields, meteorology and applications in computer graphics. If any other application would be excellent. I've also read some questions on this site and some articles uploaded (an article of John Milnor for example, in math.stackexchange), but none deals specifically with applications. Maybe it's a bit difficult because I seek not know much about magnetic fields or computer graphics, but I'm just interested I would like to know if this result is really effective to model these situations, it is mentioned briefly in some texts, but not how they actually work. Thanks for any help.


title: The two faces of $F = m*a$

text: As I have understood, $F(t)=m \cdot a(t)$ can have 2 different meanings:   * When applying an external force $F$ on a point mass of mass $m$, the resulting acceleration of that mass at time $t$ is $a$.   * If a point mass is going to be accelerated with $a$ (maybe by another force), the inertial force of the mass is $F$. (Pushing in the opposite direction of $a$). Which one did Newton mean? Are these two distinct expressions, or am I thinking the wrong way?


title: What should be used, $v=r \omega$ or radial and transverse components? What would be the next step?

text: An aircraft A is flying in a vertical plane containing two tracking stations P and Q which are 15km apart. At a cetain instant θ (measured anticlockqise from horizontal line PQ to the line PA) is 60° and the $ \dot \theta $ (angular velocity) is -0.025rad/s. At the same time, α (measured anticlockqise from extended horizontal line PQ to the line QA) is 150° and the $\dot \alpha$ is -0.02rad/s. Determine the magnitude and the direction of the velocity of the aircraft. I used the radial and transverse components of the velocities relative to P and Q and calculated the resultant. Relative to P, radial component = $$\frac{d(PA)}{dt}= \frac{d(15000Cos(\theta))}{dt} = -15000Sin(\theta) \dot \theta $$ The transverse component = $$r \dot \theta =-(PA)0.025$$ The components of velocity relative to Q could be obtained in the same way but I'm not sure about what to do next. Is taking the resultant of them correct? I took the resultant and got the following answers. Magnitude - 15133m/s Direction - 88.89° north of east


title: Toda equations and surface operator

text: I would like to know the reason why the equation (14) in the paper by Yamada is called the Toda equation. \begin{equation} \left[\frac12\sum_{i=1}^N\left(y_i\frac{\partial}{\partial y_i}-y_{i+1}\frac{\partial}{\partial y_{i+1}} \right)^2+\sum_{i=1}^N u_iy_i\frac{\partial}{\partial y_i}+{\rm const}\sum_{i=1}^N y_i\right]Z=0. \end{equation} This equation was obtained by taking the decoupling limit of the equation (12) which is the equation conformal blocks in $SL(N)$ WZNW theory satisfy.   1. Does a certain correlation function in Toda conformal field theory satisfy the equation (14)? If so, there should be a realization of a full surface operator by M2-branes in the context of the AGT relation.    2. What kind of correlation functions in Toda conformal field theory satisfy the equation (14)?    3. Is there any good reference which explains the Toda equations?


title: In quantum entanglement are the 2 sub atomic particles exhibiting all possible values before being measured

text: I have been reading a lot about quantum entanglement when I came up to a part which says that before measuring one of the particles there is a 50:50 chance that a particle will be one of the pair but also before measuring both subatomic particles can be both variations at the same time up and down I believe is the correct terminology. Is this correct?


title: flavor splitting of quark masses

text: It seems pretty clear that flavor symmetry splits the quark masses. It seems extremely odd that a global gauge symmetry of the Standard Model fields could or would produce such extreme splitting as is calculated in the typical models and confirmed in experiment. The range seems more suited to the magnitude of the color force itself, and yet the color force is flavor agnostic. Does this bother anyone else? Perhaps it's neatly explained somewhere that I've missed?


title: Why are continents wider at the north and tapering towards the south?

text: Why are the continents wider at the north and tapering towards the south?


title: If a ball spinning on a rod hits another ball, what is conserved linear or angular momentum?

text: Suppose a 1-kg ball A is fixed to a spoke 0.2 m long, which is attached to an axle so that the ball can rotate (v=10m/s, KE=50J, $\omega$=50 rps, L=2, p=0) Now, there is a second ball B (m=1kg), attached to a 0.1 m stick that can likewise rotate, but at rest. An elastic collision takes place at the point where the two circular paths intersect, ball A stops dead and ball B starts moving at 10 m/s (KE = 50J v, KE is always conserved), case a) - if ball B **is not** attached to a stick it acquires _linear momentum_ (v=p = 10 m/s), but ball A had no linear momentum (p = 0), case b) - if it **is** , it acquires angular momentum **L = 1** , ($\omega$= 100 rps), but ball A had L = 2. In this second case is angular momentum conserved? @JohnRennie, your answer covers the angular momentum of B in case a), what about its linear momentum (from 0 to 10)? can you also address the angular momentum of B in case b) ?


title: Bose Einstein condensation and macroscopic occupation

text: If have been thought, that Bose Einstein condensation occurs of the ground- state is occupied macroscopically, so $n_0\in \mathcal{O}(N)$ when performing the thermodynamic limit. So naively, this seems odd as of course, if I put in more particles without increasing system size this will always be fulfilled (when keeping the temperature constant). The problem with my imagination is now, that of course when performing the thermodynamic limit, I have to change the systems volume while I put in new particles and thus "create" more low energy states. How do I see, that above the critical temperature these effects cancel, so that due to the additional states $n_0$ does not scale linearly with $N$?


title: Do you need energy to exert a force?

text: Is force dependent or independent of energy?


title: Why can't light penetrate solid objects?

text: Light is combination of perpendicular electric and magnetic fields, since electric fields penetrate a conductor, why can't light travel in them? I think my argument does sound stupid, but I can't find the flaw in it. What do you think can be the reason?