effective theory of graphene

This is a question about deriving effective mass theory for graphene. For the two sub-lattice atoms, the wave equation can be written as the massless Dirac equation: $ \displaystyle -i\hbar v_F \begin{pmatrix} 0 & \partial_x -i\partial_y \\\\\partial_x +i\partial_y & 0 \end{pmatrix} \left(\begin{array}{c} \Psi_A \\\ \Psi_B \end{array}\right)=E \left(\begin{array}{c} \Psi_A \\\ \Psi_B \end{array}\right) \ \ \ \ \ (1)$ where ${A,B}$ are two subatoms. The derivation of the equation went back to 1984, which is the paper I am trying to understand. In the article, they argued that at first order of ${\vec{\kappa}\cdot \vec{p}}$ expansion, the momentum matrix can be written in the form, under group-theoretic arguments (equation (3) in the paper): $ \displaystyle \bar{p}\begin{pmatrix} 0 & \hat{x} -i\hat{y} \\\ \hat{x} +i\hat{y} & 0 \end{pmatrix} . \ \ \ \ \ (2)$ What is the argument behind it? Is there anyone read the paper or know the answer?


Why is the sun so dense?

Looking at the radius and mass of stars on Wikipedia, I see that the Sun is the densest of all, often many times denser than other stars. Is that because only non-dense starts are easily seen from a distance? Are there any stars of comparable luminosity to the sun that can be seen with the naked eye, and do they have a similar density as the sun? If needed I can copy-paste the mass and radius of other stars here for reference.


Angular acceleration of stone disk

I have a conceputal question regarding the following problem: A round massive stone disk with diameter $0.600 m$ has a mass of $50.0 kg$. The stone rotates at an angular velocity of $115.2 rad/s$, around an axis located at the center of the disk. An axe pressed against the disk applies a tangential frictional force, $F_f$ at $50.0 N$. Assume that we turn off the rotating power of the disk, so that the frictional force is the only force acting on the disk. What is the magnitude of the angular acceleration, $\alpha$, of the disk? OK, so I have calculated the torque ($-15 Nm$), and the moment of inertia ($I = 2.25 kg \cdot m^2$). By using the formula: $$\tau = I \alpha$$ It is easy to show that $\alpha = -6.67 rad/s^2$, which is also the correct answer. However, say I want to try to solve this in another way, without using the torque and moment of inertia. Since the frictional force is the only force acting on the disk, I would assume that we would have: $$F_f = ma_T$$ $$-50 = ma_T$$ $$-50 = 50 \cdot a_T$$ $$a_T = - 1 m/s^2$$ And since we have: $$a_T = r \alpha$$ This gives: $$\alpha = \frac{-1}{0.3} = -3.33 rad/s^2$$ Which is not the same answer. So why doesn't this second approach work? If anyone can explain this to me, I would really appreciate it!


Is the Portal feasible in real life?

Firstly, I assume prior knowledge/experience with the game Portal. Should it be possible that two openings (circles/unidentified-constructions) would work as interchangeable gates? With one opening analyzing the molecular structure of the object passing through and the other reconstructing it accordingly? Maybe as a replica? The object doesn't need to be alive or biological at all.


What equations govern the formation of droplets on a surface?

When some smooth surface (like that of a steel or glass plate) is brought in contact with steam (over e.g. boiling milk) then water is usually seen to condense on that surface not uniformly but as droplets. What are the equations which govern the formation and growth of these droplets ? In particular what role does the geometry of the surface plays in it? Also it is possible to prepare experimental conditions where no droplets are formed but water condenses uniformly ?


Retarded (Advanced) GreenFunction

I need to show that Green Function: $\displaystyle G(\vec{r},t;\vec{r}^{\prime},t^{\prime}) = \frac{\delta(t-t^{\prime}\pm |\vec{r}-\vec{r}^{\prime}|\,/c)}{|\vec{r}-\vec{r}^{\prime}|}$ Obeys $\displaystyle \nabla^2 G -\frac{1}{c^2}\frac{\partial^2 G}{\partial t^2} = -4\pi \delta(\vec{r}-\vec{r}^{\prime})\delta(t-t^{\prime})$ I tried to use the identity for Laplacian : $\displaystyle \nabla^2 (\psi\phi) = \phi\nabla^2\psi + \psi\nabla^2\phi + 2\vec{\nabla}\phi\cdot\vec{\nabla}\psi $ As follows: ![My work](http://i.stack.imgur.com/9DJKa.jpg) But I couldn't get nice results. Any suggestions?


Rate of probability loss from absorbing boundary

The following is the solution to the 1D diffusion equation with diffusion coefficient D, initial particle position $x_0$, and a perfectly absorbing boundary at $x=0$ (s.t. $P(x=0)=0$). $$ P(x;t)=\frac{1}{\sqrt{4 \pi D t}}e^{-\frac{(x-x_0)^2}{4 D t}} - \frac{1}{\sqrt{4 \pi D t}}e^{-\frac{(x+x_0)^2}{4 D t}} $$ If I understand correctly, for an $x_0>0$. $$ P(\text{no collision at time $t$})=\int_0^\infty P(x;t) dx $$ In other words, the total probability in allowed space at time $t$ is exactly the probability that the particle _never_ contacted the absorbing boundary at $x=0$ up to time $t$. What I want to compute is the rate of probability loss $k(t)$. From above, it seems that would be: $$ k(t) = \frac{d}{dt} \int_0^\infty P(x;t) dx $$ evaluating with mathematica reveals: $$ k(t)=-\frac{D x_0}{2 \sqrt{\pi}} \left(\frac{1}{D t}\right)^{3/2} e^{-\frac{x_0^2}{4 D t}} $$ which seems reasonable. It seems that there should be a way to compute $k(t)$ _without_ computing the spatial integration across $x$, perhaps some computation involving only the boundary. I thought since all the loss occurs at $x=0$, the time derivative of $P(x;t)$ evaluated at $x=0$ should be k(t). However, the result of that calculation is 0. _Question: is there a way to compute k(t) without computing the spatial integral over the $x$ domain?_


Is there a deep reason why springs combine like capacitors?

I was solving a practice Physics GRE and there was a question about springs connected in series and parallel. I was too lazy to derive the way the spring constants add in each case. But I knew how capacitances and resistances add when they are connected in series/parallel. So I reasoned that spring constants should behave as capcitances because both springs and capacitors store energy. This line reasoning did give me the correct answer for how spring constants add, but I was just curious if this analogy makes sense, and if it does, how far one can take it. That is, knowing just that two things store energy, what all can you say will be similar for the two things.


Shrinking and Expanding objects in a CFT

In a conformal field theory, is it possible to construct a machine that shrinks or expands objects?


Including special relativistic effects in momentum in Heisenberg's Uncertainty Principle

I've been told that an electron is somewhere within the space of $10^{-10}m$ and am supposed to find the uncertainty in its velocity. Simply applying $m\Delta x \Delta v \geq \frac{h}{4\pi}$ results in $\Delta v \geq 5.79\times10^{10} m/s$. Now, this velocity is not approaching the speed of light, but I wanted to know if I could use the Lorentz factor to make this calculation more accurate (of course in this situation it would not make much of a difference. So I redefined momentum uncertainty as $\Delta p = \gamma m_0 \Delta v$ and solved the inequality... but my final answer states $\Delta v \leq 3\times10^8$... this relation is obvious... but why has it popped out of this? What if the space was even smaller when special relativistic effects actually would come into play, how would I then tackle the question?


What is the criterion of stability of thick-walled spherical shell?

Is there the formula (if someone already has discovered it) or what is the algorithm (if a particular formula was not deduced), to calculate the critical pressure of thick-walled spherical shell $−$ the pressure which provokes buckling?


Visualising scalar and vector fields

As I have been studying electromagnetic theory, I have always been lead to confusion when trying to visualise the fields. Fortunately, the electric and magnetic fields are vector fields and also along the propagation direction they provide a 3-Dimensional Coordinate system. However, in general to visualise a N-dimensional field (I mean field depending on N coordinates), N+1 dimensions are necessary. For instance, in the case of 1D field $\phi(x)$, we can plot a graph $x$ vs $\phi(x)$. But I am wondering if it is also possible to describe the fields using contours. To elaborate, if I have 2D field $\lambda(x,y)$ will it possible to visualise the field in 2D itself using contours (without actually plotting the variation of $\lambda(x,y)$ vs $x$ and $y$).


How can Excel be used to model the flight of a rocket?

Firstly, sorry for any silly mistakes or improper use of this website - this is my first time using it! I am trying to use Excel to model the flight of the Apollo 11 Saturn V rocket using the real values from the mission evaluation report. However, I can't seem to get the right acceleration. It is always too low. My initial acceleration at t=0.1s is 2.06 m/s^2 while the values in the evaluation report show the acceleration to be close to 10 m/s^2. My final goal is to calculate the acceleration at the S-IC OECO event (at t=161s). My spreadsheet showed 28m/s^2 while the real value is 39m/s^2. Obviously I have done something terribly wrong since the difference between the values is quite large, but I've been looking over my spreadsheet for days now and I just can't find my problem. What _should_ the columns be? This is what I have so far:   * Range time (s) - in increments of 0.1s down the column   * Mass (kg) - the mass at that time calculated by previous mass - mass flow rate*0.1s   * g (m/s^2) - the acceleration due to gravity at that altitude   * Weight (N) - mass*g   * Thrust (N) - the thrust at that time using real values from the graphs in the evaluation report   * Drag (N) - using the drag equation and estimated coefficients    * Total force (N) - weight + thrust + drag    * Acceleration (m/s^2) - using F=ma so I did total force/mass    * Velocity (m/s) - previous velocity + acceleration*0.1s    * Altitude (m) - previous altitude + velocity*0.1s I've taken into account changing thrust, mass flow rate, acceleration due to gravity and drag as well. The values in the report are given as 'inertial acceleration' but I haven't learnt this yet (still in high school). So is inertial acceleration different to the acceleration I have calculated using F=ma? Have I misused Newton's second law? Was I wrong to consider the weight of the propellants in this case? I'll appreciate any and all helpful feedback and I can provide more information if needed. Thank you so much! :)


Find drop-off rate of magnetic interference from a mass of pure iron on a magnetic compass

How can I find the magnetic interference a stationary 35000 kg block of 100% pure iron would have on a magnetic compass and what the drop off rate of the interference would be. So if said 35000 kg block of iron was 1 meter away from the compass, 100 meters away, or 1000 kilometers way I would like to calculate the rate of drop off of the interference. * * * This may seem absurd, but it is very important for a conceptual project I am working on. For the context of this question, assume everything is perfect, and that we are basically operating in a vacuum and there is no interference from anything else and that all instruments are 100% accurate and infinitely precise. And that I have only a very very basic understanding of physics, mathematics and magnetism.


What is the lifetime of an induced magnetization in a para/diamagnetic material?

To the best of my knowledge thermal fluctuations are responsible for washing out any effective magnetization, once the external field is switched off. Since thermal fluctuations need some time to drive the system back to random orientation, what would be the time scale for this process? Is it of the order of 200 ps?


Speed of a falling pencil

If you balance a pencil of length $d$ on its tip, and let it fall, how do you compute the final velocity of its other end just before it touches the ground? (Assume the pencil is a uniform one dimensional rod)


What are some interesting calculus of variation problems?

That I could create as a classical mechanics class project? Other than the classical examples that we see in textbooks (catenary, brachistochrone, Fermat, etc..)


How are Monte Carlo simulations used in experimental high energy physics? Particularly in studying detectors limitations (efficiencies?) and data analysis. I will appreciate giving a simple example to clarify how MC is used if the question is too technical to have a simplified answer


"String" infinity paradox

The smallest particles of the universe (or the smallest part of the smallest particles of the universe) must have infinity density. But if it have infinity density, you can calculate mass of one of these particles as infinity - so if any thing would composed of these particles, would it have infinity mass. How is it possible?


Should particles in Smoothed Particle Hydrodynamics (SPH) always move during a simulation?

Or can they just be used as an interpolation points and use some other "transported property" which are just evolved and propagated from boundary conditions like for eg. heat conduction through a solid metal bar. It seems to be possible, but isn't clearly elaborated. The force computation step is always used for fluid flow to move particles forward according to Newton's laws, but what if heat is transported purely by conduction in a solid? Here, nothing moves but still, heat flows and destroys temperature gradients. What equations are to be considered here (apart from the heat diffusion equation) to adapt this problem for this kind of discretization?


Does a diatomic molecule falling into a black hole dissociate?

I've just answered Dipping a Dyson Ring below the event horizon, and while I'm confident my answer is correct I'm less certain about the exact consequences. To simplify the situation consider a diatomic molecule falling into a Schwarzschild black hole with its long axis in a radial direction. ![Molecule](http://i.stack.imgur.com/KZea4.gif) The inner atom cannot interact with the outer atom because no radially outwards motion is possible, not even for electromagnetic fields travelling at the speed of light. However I find I'm uncertain exactly what the outer atom would experience. We know that the outer atom feels the gravitational force of the black hole even though gravitational waves cannot propagate outwards from the singularity. That's because it's experiencing the curvature _left behind_ as the black hole collapsed. Would the same be true for the interaction of the outer atom with the inner atom? Would it still feel an electromagnetic interaction with the inner atom because it's interacting with the field (or virtual photons if that's a better description) left behind by the inner atom? Or would the inner atom effectively have disappeared? If the latter, presumably the fanciful accounts of observers falling into the black hole (large enough to avoid tidal destruction) are indeed fancy since it's hard to see how any large scale organisation could persist inside the event horizon. **Later:** I realise I didn't ask what I originally intended to. In the above question my molecule is freely falling and the question arose from a situation where the object within the event horizon is attempting to resist the inwards motion. I'll have to go away and re-think my question, but thanks to Dmitry for answering what I asked even if it wasn't what I meant :-)


Spectrum of quantum fluctuations in a harmonic oscillator

If we have a harmonic oscillator and look at it on small scale the energy is quantized and we can calculate the different eigenstates. In general the energy eigenvalues are given by $$E_n = \left(\frac{1}{2}+n\right) \hbar \omega$$ Even if we can bring this system at $T=0$ into it's ground state, there will be zero point motion or quantum fluctuations remaining. Now if we heat a system, depending on its excitations we need Maxwell–Boltzmann, Bose-Einstein or Fermi statistics to calculate the occupation of each state. The resulting spectra are due to thermal excitations from the ground state. Now if we remain at about absolute zero the system still has quantum fluctuations. From Heisenberg's principle the energy is uncertain but that does not tell me which is the current energy or eigenstate. How can one calculate the 'quantum' spectrum of an harmonic oscillator, or bluntly, how can one calculate the probabilities for observing the system at the state $n=1,2,...$ at $T=0$ if we prepare the system to be in $n=0$?


Racing balls question

My question is related to simulation of racing ball demonstration. http://www.phy.ntnu.edu.tw/ntnujava/index.php?topic=142 One ball goes on a straight path, while another one goes on a curved path. On the simulation the second ball going on the blue part down hill which is parabola has a constant component of the velocity in the x direction. Why is that the case when the slope changes?


Speed of approach between image and object

A plane mirror image approaches the object at the same rate the object approaches the mirror. So the speed of approach is twice the speed at which the object approaches the mirror. If the object approaches the mirror at a speed greater than half the speed of light (c), does that mean that the speed of approach between object and image will exceed the speed of light?


Can I calculate the height of a cliff from weight of falling object and time taken?

I'm sure a simple question. I have a video of me jumping off a cliff into a river. I want to calculate how high it is. I know my weight, acceleration due to gravity of course, and I can get the time it took to hit the water. Can I calculate (albeit roughly) how high the cliff was? Cheers, Tim.


No photon interaction in free space

How can the Hanbury Brown and Twiss effect (photon bunching) be explained if photons don't interact in free space? To explain it with the influence of the two photons on the two detectors simultaneously would seem strange to me, because of the fact that "if the source consists of a single atom which can only emit one photon at a time, simultaneous detection in two closely spaced detectors is clearly impossible" (same Wiki under the "Quantum interpretation" heading). Comment: The question is not about uniting or dividing two photons, which is possible only on the atomic level. The question is about synchronizing two photons from different sources like the sun.


Water rocket physics

According to this page, the current record for greatest altitude achieved by a water and air propelled rocket is 2044 feet (623 meters).   In this connection, i had one question: How the greatest altitude depends of the density of the liquid used in the rocket?   Suppose that we have the opportunity to choose a liquid with an arbitrary density and with the same hydrodynamic properties as water. Then there must be a density at which the greatest altitude has its absolute maximum, i think. To be more specific let's consider a simpler problem: Let a two-liter soda bottle rocket, filled with some amount of a liquid mentioned above and air at the atmospheric pressure, is launched in deep space($g=0$)(say an astronaut is experimenting). What is the absolute maximum of the speed the rocket can achieve? For simplicity let's assume the expansion of the air to be isothermal (which it is not). ![enter image description here](http://i.stack.imgur.com/RVXX7.png)


Figuring out North and South Magnetic Poles for Earth

I currently make my living as an electrophysicist, so am a bit embarassed at having not thought this one through, before. Magnetically, the North pole is.... North. We get out our compass, and wait for the North-seeking needle to line up. I've always thought of this end of the needle as the north end. In a current carrying loop, if we use the right-hand rule and point our thumb in the direction of conventional current, the fingers show the direction of the mag field lines around the wire. The direction the mag field lines point "north" when they pass through the interior of the loop. By stacking many such loops we can create a cylindrical electromagnet, for which it is easy to see which end is the "north" end. I resume permanent physical magnets follow the same naming convention: that is, mag field lines emanate from the north end and cycle around to the south. We know that like poles of a magnet repel and opposites attract. So, now, hanging a magnet from a string (or going back and using our trusty compass), if the north pole of the magnet points towards teh magnetic north pole of Earth, doesn't that mean that the pole near the Arctic circle is really magnetically-speaking the south pole? Or conversely, when we say "north-seeking" pole of a magnet, is that really the south pole of the magnet? Which is it?


Is there an equivalent of a scalar potential for torques?

For a given scalar potential $V$, it is known that the corresponding force field $\mathbf{F}$ can be computed from $$ \mathbf{F} = -\nabla V $$ Suppose a rigid body is placed inside this potential. The torque on the body $\mathbf{T}$ exerted by the scalar field will be $$ \begin{align} \mathbf{T} &= \int_M \left( \mathbf{r} \times \mathbf{F}(\mathbf{r}) \right) dm \\\ &= \int_M \left( \mathbf{r} \times -\nabla V(\mathbf{r}) \right) dm \\\ &= \int_M \left( \nabla V(\mathbf{r}) \times \mathbf{r} \right) dm \end{align} $$ with $\mathbf{r}$ the position vector to the mass element $dm$, and the integration carried out over the entire body $M$. So, being not too familiar with rigid body dynamics, I was wondering -- does something like a (vector/scalar) potential $P$ exist, such that the local torque induced by the potential $V$ can be expressed as $$ \matrix{ \mathbf{T} = I \cdot \nabla P & & &\text{(or some similar form)} } $$ with $I$ the moment of inertia tensor of the rigid body? If such a thing exists:   * what is its name?   * where should I start reading?   * What is the proper expression for the torque $\mathbf{T}$?   * how does $P$ relate to $V$? If such a thing _doesn't_ exist:   * why not? :)


A mirror flips left and right, but not up and down

Why is it that when you look in the mirror left and right directions appear flipped, but not the up and down?


Axis of reflection in a mirror

When I look at my reflection in a planar mirror, the image I see is reflected about a vertical axis. Why is it this axis and not the horizontal axis?


How far would a radio signal propagate in free space as compared to Earth's atmosphere?

A curiosity question. Radio Propagation within Earth's atmosphere is via atmosphere. Broadly speaking a signal loses strength between bouncing off the ionized layer, and absorption by various earthy elements. If an identical signal were to be produced in free space, how far would it travel as compared to it's range within Earth's atmosphere? EDIT: A commercial FM transmitter on Earth probably has a range no more than 25 miles where the listener uses a broadcast receiver. For the purpose of elaboration assume the the receiver being placed in static orbit - how far out could the same signal be generated, and still be intelligible?


Propagator of the Klein-Gorden equation

## Does this integral converge? My question is related to this one: Free particle propagation amplitude calculation I am reading the book of Peskin and Schroesder. In the second page of their chapter 2, they estimated the propagator for the Klein-Gorden equation. The final step of the calculation is to calculate an integral of the form $$\int_0^\infty \mathrm{d}p\;p\;\sin(px)e^{-\mathrm{i}t\sqrt{p^2+m^2}}.$$ My question is very simple: does this integral converge in the first place? Note that the integrand is an oscillating function whose amplitude grows without bound. If we take $m=0$, then for sure the integral does not converge. In fact, I can reproduce their result very easily (namely, the $e^{-m\sqrt{x^2-t^2}}$ behavior). But I am not sure whether my calculation makes sense because I don't know whether the integral has a finite value. I know this integral can be converted into a loop integral in the complex plane, and then be estimated using stationary-phase method. But since the amplitude of the integrand does not go to zero uniformly at infinity, I feel such a procedure may not be valid. We may multiply a factor $e^{-\epsilon p}$ ($\epsilon>0$) to the integrand to make it converge, but then we are not calculating what we wanted to calculate in the beginning, right? Then, if the integral does not converge by itself, how should I make sense of the result in the Peskin and Schroesder book? I am quite confused by this. Thanks for any hints!


How much thrust would be needed to turn a hobbyist weather balloon into a deep space probe?

I was reading the article _Weather Balloon Space Probes_ that says you can put your own balloon probe at 65,000 ft temporarily. Is it even remotely possible to raise the probe high enough using balloons, and then put it in orbit or escape the field using an on-board propulsion system? Can it be done with easily purchased materials? How much mass can be added for instrumentation? How much fuel of is required? What kind? I know it's kind of a broad question, but I just want to know a rough estimate.


Magnetic moment with external magnetic field on a lattice?

Consider a system in which atoms are located in a regular lattice, each atom having a spin $1/2$ and an associated intrinsic magnetic moment $\mu_0>0$. Assume that each atom interacts only weakly with other atoms so that all the other atoms act as a heat reservior. If the system is placed in an external magnetic field $H_0 \hat{z}$, then the Hamiltonian is given by: $$\hat{H}=-\mu_0H_0\sum_i\sigma^z_i$$ where $\sigma_i^z$ are (the $z$ components of) the Pauli matrices, at the $i$th site. For the magnetic moment $\mu= \pm\mu_0$, the corresponding magnetic energy of the atom is $E=\mp\mu_0 H_0$. What would be the mean magnetic moment?


Why didn't electroweak symmetry breaking happen earlier than it did?

I've read that electroweak symmetry breaking occurred "at a picosecond or so" after the Big Bang (in a source I found). I can't help wondering why it took so long to get started. For instance, was something missing up until then, which was needed by the Higgs mechanism? What feature(s) of the Higgs mechanism, or perhaps what other theoretical consideration(s), was/were seemingly responsible for this very brief delay?


Maxwell's Demon Constant (Information-Energy equivalence)

New Scientist article: Summon a 'demon' to turn information into energy The speed of light c converts between space and time and also appears in e=mc^2. Maxwell's Demon can turn information supplied into a system into energy, which suggests there is a constant 'd' such that e=i*f(d) where i is information and f is a function of d. If d appears in the formula relating energy and information like c relates energy and mass, but c is a conversion factor between space and time, then what is d a conversion factor between? What dimensions does d have? Other than "Maxwell's demon constant", is there another name for d? Or is there not a 'd' after all, instead 'c' appears once again in the information-energy equivalence fomula e=i*f(c)? Or is there not a constant here at all and the situation is more complicated? Living things have evolved to exploit all kinds of opportunities - even quantum effects in the form of photosynthesis. Living things have senses to extract information from the environment, but do living cells utilise any demon effects to turn any of this information into energy ?


Distinguishing mechanical systems from general dynamical systems

[Remark: I admit that my first attempt on What makes a space a real space? was rather ill-posed and led to some confusion. Sorry for that, but please give me a second try. Part of the confusion arose from the ununsual term "real space". The far better term would have been "physical space" which I hope to be unambiguous.] In the following let a "mechanical system" be a system of $n$ spatial objects moving in physical space. Consider you are given a function $q:\mathbb{R} \rightarrow \mathcal{M}^n$ with $\mathcal{M}$ a mathematical space. **Question 1** > When would you assume that $q$ describes the evolution of a mechanical > system? I would, if I found or was given a function $\mathcal{L}(x,\dot{x},t)$ such that the Lagrange equations hold for all times $t$ when inserting $q(t)$ for $x$ and $\dot{q}(t)$ for $\dot{x}$, and which I can consider physically plausible. In this case I would interpret $\mathcal{M}$ as physical space, maybe distorted by generalized coordinates. I would _not_ , if there is provably no such function. In this case $q$ describes the evolution of a general dynamical system, but not a mechanical one. **Question 2** > Are there mathematically definable conditions on $\mathcal{L}$ for being > physically plausible, or possible, let's say? Besides, maybe, that $\mathcal{L}$ must be of the form $\mathcal{T} - \mathcal{V}$. **Question 3** > Can anyone give an example of a non-mechanical system that _behaves_ like a > mechanical one (in the narrow sense above)? One answer to this question comes immediately in mind: a system that is _designed_ to behave like a mechanical one, e.g. a computer running a simulation. I'd like to exclude such cases if I could, and restrict the question to "natural" systems. **Question 4** > What about possible worlds with a physical space like ours but in which the > Lagrange equations do not hold?


Does the image forced barrier lowering appear wnen the device is in Forward bias?

Because of the image forced barrier lowering the barrier potential reduces in reverse bias . What happens to barrier height (potential) in forward bias??


Why is supercritical fluid not considered a separate state of matter?

As given on this link, supercritical fluids are viewed more as a continuum which has both liquid and gas properties. This continuum is obtained when a gas is brought to a pressure and a temperature higher than its critical values. The intermolecular distances of a pure supercritical fluid are in between that of a liquid and gas. With so many qualitative differences between a supercritical fluid and a gas and a liquid. Why is it not considered a separate state of matter?


Is there some special cutoff density after which spacetime "collapses" and forms a black hole?

With crude calculations following densities can be approximated: Given that radius of proton is $1.75×10^{−15} m$ and it's mass is $1.67 × 10^{-27}kg$, this gives density of proton to be $\dfrac {1.67 × 10^{-27}kg} {\frac{4}{3}(1.75×10^{−15} m)^3}=\dfrac {3}{4} \dfrac{1.67 × 10^{-27}}{5.36×10^{−45}} \dfrac{kg}{m^3}=2.34 \times 10^{15} \dfrac{kg}{m^3}$ Density of the heaviest naturally occurring solid $\mathrm U^{238}$: 1 cubic meter of Uranium is about $2 \times 10^4 kg$, and since there are 238 protons+neutrons in Uranium, this gives the density $8 \times 10^{-3} \times 10^4 \dfrac{kg}{m^3}=8\times10 \dfrac{kg}{m^3}$ This means the protons are dispersed by a factor of $2.92\times10^{13}$, in other words there are $2.92\times10^{13}$ empty space units to each unit of space filled with proton. Density of water $=\frac{1}{20}\times10 \dfrac{kg}{m^3}=5\times10^{-1}\dfrac{kg}{m^3}$ The density in the core of neutron star is $8×10^{17} kg/m^3$, and the density of a black hole is supposedly infinite according to Wikipedia, but then again before transitioning to a black hole there must have been some finite mass distributed over a non zero volume of space therefore having a finite density. So far on logarithmic scale (base 10) followings look obvious for $\operatorname {log}_{10}(density \dfrac {kg}{m^3}) = \operatorname {log}_{10} \dfrac{kg}{m^3} + \operatorname {log}_{10}density$ Let $ d =\operatorname {log}_{10}density,\quad \operatorname {log}_{10} \dfrac{kg}{m^3}= \text{what is one to make of this}?$ Then following crude observations can be made: $-1\leq\mathcal O(d)\leq1$: Order of density of life as we know it. $\mathcal O(d)\approx 15$: Order of density of matter packed space $\mathcal O(d)\approx 17$:Order of density at the core of a Neutron star $\mathcal O(d)\geq x$:Order of density of black holes, (collapse of space to contain matter?) what is $x$? My question is: does these calculations make sense and is there anything more to the order of densities than these naïve observations?


Bending of light - photon's inertia instead of mass

Using classical mechanics, the formula for gravitational attraction is $$F = G\frac{m_1m_2}{r^2}.$$ This formula does not work for photons, and we need to use Einstein's theory of gravity to account for that, as the photon is massless. What if, instead of using the mass of the photon, we rather used the inertia of it? The momentum of a photon is $$p=\frac{h}{\lambda}.$$ Thus, using that, we can calculate its supposed 'mass' (inertia) as momentum is also equal to mass times velocity. So 'mass' of a photon is $$m=\frac{p}{v}=\frac{h}{\lambda c}.$$ If we use this mass as the mass of a photon, can we use Newton's equation for gravitational attraction?


Simple Parallel Electric Circuit

Ammeter (A1) and voltmeter (V) in series have parallel connection to another ammeter (A2). Currents in A1 and A2 are respectively 0.2 and 1.7 amp. Voltmeter's voltage is 6 volts. ![enter image description here](http://i.stack.imgur.com/w9oZl.png) How should I find resistances of all elements?


Can a disk like object (like UFO's) really fly?

UFOs as shown in movies are shown as disk like objects with raised centers that emit some sort of light from bottom. Can such a thing fly? My very limited knowledge in physics tell me that a disk like object may not be able to maneuver unless it has thrusters on sides and simple light can not be enough to make any object go up in the air. Is it possible?


Chain of balls on an inclined plane

Suppose we have some inclined plane, and there is some chain of balls of length $l$ and mass $m$ lying on it. No friction at all in the system. 1) What is $x_0$ (the vertical hanging part of the chain) needed for the chain to be in equilibrium? _Answer_ : $l/3$ 2) If we push the chain a little bit to the right, what is the acceleration of the chain, at instant when the vertical part of the chain is $x$ long ($x > x_0$)? _Answer_ : $a(x)=\frac{g}{l}(\frac{3}{2}x-\frac{l}{2})$ 3) What is the velocity of the chain at instant when all of it is vertical (e.g. all the left part on the inclined plane "became vertical")? ( _no answer_ ) ![enter image description here](http://i.stack.imgur.com/jydqH.png) 1) I do not understand, why the component of the weight of the left part should be equal to the weight of the right hanging part. Their vectors are not along the same line, so why should this condition be necessary for equilibrium? Also, the left part is being pulled by the right, so there should be another force along the incline. 2) Based on the first question, this one should be: $(1-\frac{x_0}{l})mg \sin 30^{\circ}-\frac{x_0}{l}mg=ma$ However, this leads to a wrong $a$, according to the answer. 3) Based on the book's answer of 2), is this right? From Torricelli's equation: $v^2=v_0^2+2a \Delta x$ $v(x)=\sqrt{v_0^2+2 \int_{l/3}^{x} a(x) dx}$ Therefore: $v(l)=\sqrt{\frac{20}{3}l}$


Why does a superfluid climb up the wall of container?

This has been asked  but I couldn't find a good answer why does super-fluid climb up the walls of container? What if the walls were too high? What forces are involved? ![enter image description here](http://i.stack.imgur.com/rgYNn.jpg)


Method of determining peculiar speed of the galaxy which moves on celestial sphere and emits the light

This question is the continuation of this one. I came up with solution, but I'm not sure that it is correct. Can someone check it? Let's introduce transverse (to the observer) proper speed $v_{\perp}$ of the galaxy. The galaxy travels distance $\Delta D$ in a time $\Delta t_{0}$: $$ \Delta D = v_{\perp}\Delta t = v_{\perp}\Delta t_{0}\frac{a(t)}{a_{0}}, $$ where $dt_{0}$ refers to the time interval of observation of the galaxy, $dt$ refers to the time interval of light emission. So $$ d\theta = \frac{dD}{a(t) r(t)} = \frac{v_{\perp}dt_{0}}{a_{0}r(t)}, $$ where $r(t)$ is the galaxy position, and finally $$ \frac{d \theta }{dt_{0}} = \frac{v_{\perp}}{a_{0}r(t)}. $$ Is this method correct? Also, can I identify the left side of this equation with the observing angular velocity? And how to find $r(t)$? To consider the FLRW interval for the light? $$ ds^2 = dt^2 - a(t)^2\frac{dr^2}{1 - kr^2} = 0 \Rightarrow f(r) = \int \limits_{0}^{z} \frac{dz}{H(z)} \Rightarrow r = f^{-1}\left[ \int \limits_{0}^{z} \frac{dz}{H(z)}\right]? $$


Maximum angular velocity to stop in one rotation with a known torque

I have an object I can rotate with a given torque. I would like to stop applying torque once I've reached a defined maximum rotational speed. The maximum rotational speed should be defined so that applying maximum torque will stop the rotation of the object within one rotation. If I know my torque and moment of inertia, how can I find the maximum rotational velocity to allow me to stop the object in one rotation? Time is whatever is needed. I've tried finding the angular acceleration required to stop the object, but that leaves me with the time variable. Of the equations I've tried, I'm left with a time variable as well as the maximum angular velocity.


How Long does it take for a roller coaster cart to reach point $x_1$?

Assume that we have a particle with weight m and initial speed v0, that follows a roller coaster path y(x). ![Roller Coaster Path](http://i.stack.imgur.com/kyUeA.png) Assuming no energy loss to friction or air resistance, the total energy in the system is $$E0 = m g y(x0) + 0.5 m vo^2$$ then the speed at each point $x_f$ is ![enter image description here](http://i.stack.imgur.com/jMCyQ.png) So, using the energy conservation principle, we can derive the speed at each point of the curve. It gets tricky though, to try to figure out at what time the particle reaches a specific point of the curve. Turning our attention to dynamics, should help us achieve the goal. $$dS/dt = v(x)$$ Is the approach followed correct? ![enter image description here](http://i.stack.imgur.com/hz3U7.png)


Does wind speed things up or accelerate things?

This question may seem odd, but I can't think of anything better. So I'll go straight to the point. Let's say there's a projectile in air going east, shot at a certain angle, with a certain speed. Then there's the wind blowing, for let's say 5 meters per second in the same direction as the projectile. So how will the wind affect the projectile? Will it accelerate for 5 meters per second every second it lasts in the air, or will the speed just increase by 5 meters per second all through out? And what if the wind goes exactly the opposite direction? Will it have a negative effect? Should I also take into consideration the drag the object makes, the mass, the terminal velocity, or if there are other things I've missed? I don't mind if you give formulas. It would also help. But I just wanted to know, in simple terms because I'm not really that inclined with terms used in physics, the effects of a blowing wind to a projectile motion. Thanks in advance and sorry if I made some errors or misunderstanding with this post or if I used the wrong tags. Feel free to edit it. Thanks again,   Vincent :)


"Dictionnary" between QFTs in D and D-1 dimensions?

Considering Einstein equations, suppose, for instance, that the RHS, the stress-energy tensor, is uniquely due to the electromagnetic field. Now, if we imagine a quantized version of these Einstein equations, we have a quantum field theory in $D$ dimensions (here the quantum electromagnetic field) as "source" of a "quantum gravity" theory in $D$ dimensions, so we may consider these equations as a kind of "dictionnary" between these two theories. On an other way, there are some holographic dualities, which relates some quantum theory of gravity in a $D$ dimensional space with boundary, with some quantum field theory on the boundary ($D-1$ dimensions) So, the question is, if, "skipping gravity", it is possible or not, and how, to directly relate the QFT in $D$ dimensions (source of gravity), and the other QFT (on the boundary) in $D-1$ dimensions.


Any FLOSS Monte-Carlo package for reactor physics? Are there any Free and Open Source software packages for nuclear reactor processes simulation? Maybe, something similar to MCNP?


No hair theorem and black hole entropy

The _no hair_ theorem says that black holes rapidly converge to a state that is completely described just by their mass, spin and charge. Black hole thermodynamics says that the black hole entropy is proportional to the surface area of the event horizon. As I understand it in information theoretic terms the black hole entropy is 1 bit per planck unit of area, which should then be the amount of information required to fully describe a black hole. These two statements seem to be incompatible. It seems that the thermodynamics says that if we fix the macroscopic no hair parameters there still remain all the surface bits to be fixed. Is there any real conflict here or is somehow illusory? Is it a quantum vs classical issue? Does the entropic information not count in some sense? If not why, can 2 equal mass Schwartzchild black holes be distinguished by their "surface information"? Is it that the surface information is just not observable because the surface is considered _in_ the black hole? Or is this apparent discrepancy part of the black hole information paradox? There is a very similar question with a detailed answer here, but I believe I am actually asking something different. The question and excellent answer there focus on the objects as they fall into a black hole. I am more interested in how to think about the total information in a steady state black hole and the current status of the no hair theorem wrt black hole entropy and the information paradox.


How do you represent an equation with four variables?

As the title asks, how do you represent an equation with four variables? Is there a practical way of representing them? (I've thought of one, but I don't think I can explain it clearly. This question has got me really curious.) Additionally, is there any practical way of solving these equations?


What is the fundamental probabilistic interpretation of Quantum Fields?

In quantum mechanics, particles are described by wave functions, which describe probability amplitudes. In quantum field theory, particles are described by excitations of quantum fields. What is the analog of the quantum mechanical wave function? Is it a spectrum of field configurations (in analogy with QM wave functions' spectrum of particle observables), where each field configuration can be associated with a probability amplitude? Or is the field just essentially a superposition of infinitely many wave functions for each point along the field (as if you quantized a continuous mattress of infinitesimal coupled particles)?


Depalma Free Energy fields

Few years ago I read some papers about Free Energy, written by Bruce de Palma, a physicist who is said to be the inventor of the N-Machine, which is an device that works with free energy latent in the space around us. There are several informal references to him and his work. However, I tried to find papers of him several times in Scielo, Science Direct, IEEE, Physical Letters, Nature and several other scientific databases and found no references to Dr. Bruce de Palma. Personally I find quite strange that a research of such importance do not have a formal paper in an journal of big impact. Does his work make sense?


Synthesizing elements (Nuclear Physics/Alchemy?)

Based on my limited knowledge of nuclear physics, it seems that one day it may/will be possible to synthesize whatever elements we may need, given enough energy. Is this accurate? Is there a table that lists the most efficient ways to create certain elements from more abundant ones via fusion or fission? I daresay this has rather large implications for precious metals such as gold/silver which are used as stores of wealth, provided the energy requirements are not exorbitant.


How to bend 3d space embededd in a 4d enviornment?

> **Possible Duplicate:**   >  Calculating position in space assuming general relativity I recently started to program a 4 dimensional visualization program. I have the 4 dimension space perspectively projected in the 3d space. Now I describe 3d sphere inside this space. I want to describe the movement of those 3d spheres. What I thought it will be cool is to apply the general relativity theory to it. As far as I know from the general relativity theory, is that masses bends the space around it. What I think it means: let's imagine a 2d plane. if I put a mass on this 2d plane, the plane would be actually pinched in the 3d space. Parctically what I have to do is to find a function that describe this pinch, in function of the mass. like a Gauss distribution under the mass. Now this transformation will be applied to the moving objects, which I think would bend the trajectory of the objet to a curve. What I want to do is to apply this to the 3d space: if you put a mass in the 3d space it will cause a pinch in the 4th dimension. So what I want is to find the transformation function which describe this pinch. I think that what Zaslavsky is the actual answer, I din't got it at the beginning because I didn't tought it was actually what I was looking for but anyway: $$\frac{\mathrm{d}^2x^\lambda}{\mathrm{d}t^2} + \Gamma^{\lambda}_{\mu\nu}\frac{\mathrm{d}x^\mu}{\mathrm{d}t}\frac{\mathrm{d}x^\nu}{\mathrm{d}t} = 0$$ The solution to this differential equation should give the pinch I mean, am I right? PS: found by surfing random on the internet: Lorentz transformations are just hyperbolic rotations. edit: I've edited the question, I hope, in a way is comprehensible.


Could gravity hold electron charge together?

Could the gravitational force be what holds the charge of the electron together? It seems to be the only obvious possibility; what other ideas have been proposed besides side-stepping the issue and assuming a "point charge"? How would this affect the electron "self-energy" problem? The question is related to the idea of geons.


Ginzburg-Landau model for superconductivity

Could someone kindly elaborate more on the Simple Interpretation section from this Wikipedia Article? I refer to the part on the natures of $\alpha , \beta$. Why can one assume that $\alpha(T)=\alpha_0(T-T_c)$, for example?


Pull Force of an electromagnet

How do I calculate the pull force of a cylindrical electromagnet to Iron as a function of distance? Is there any difference betweeen magnets and electromagnets?


Kinetic energy absorbing in order to avoid damages?

is this possible? To absorb kinetic energy and disable inertia force? http://www.youtube.com/watch?v=KnKPbAbJI0w http://www.youtube.com/watch?v=4_5oseSVUc4


Net work done on the body when we lift it and put it on the table is zero?

I'm little confused here. Work done on the body when we lift it and put it on the table is zero, because according to work energy theorem, change in kinetic energy of the body is zero. So, the net work done is zero. Fine, but now the object has $mgh$ (where $m=$mass of body; $h=$height of body) amount of potential energy stored in it. If the net work done is zero then who has increased its energy? I'm confused and unable to think it through kindly help.


Is glass a liquid?

I was told by a condensed matter physicist that glass is a liquid with a very high viscosity (it would be more precise to say that it is a supercooled liquid). The example given was that in cathedral windows the glass flows very, very slowly, so they are thicker at the bottom. I recently read this article in Scientific American, which contradicts the previous statement. Are there any serious studies about this aspect of glass? Is it only a problem of definition?


> **Possible Duplicate:**   >  Protons' repulsion within a nucleus How positively charged protons remain glued to each other while they should repel each other out of nucleus?


Order of phase transitions

I got to read things like > _In case of a first order phase transition, the volume and temperature > change in a discontinuous manner. However for phase transitions of higher > order the change in volume and temperature can be continuous._ I could not exactly understand the statement and thats why I thought what exactly it means by the order of a phase transition. May I know what exactly it means by the order of phase transition?


What are the differences between indistinguishable and identical?

What is the difference between indistinguishable particles and identical particles?


Chemical Potential of Ideal Fermi Gas

In Wikipedia's article on Fermi Gases, they have the following equation for the chemical potential: > $$\mu = E_0 + E_F \left[ 1- \frac{\pi ^2}{12} \left(\frac{kT}{E_F}\right) ^2 > - \frac{\pi^4}{80} \left(\frac{kT}{E_F}\right)^4 + \cdots \right]$$ where > $E_0$ is the potential energy per particle, $k$ is the Boltzmann constant > and $T$ is the temperature. I don't understand how they get the third term, particularly the 1/80 factor. I've often seen this equation expressed just to the tau^2 term, and I understand how to get the 1/12 factor from a binomial expansion of $$\left( 1 + \frac{\pi^{2}}{8} \left(\frac{\tau}{E}\right)^{2}\right)^{-2/3} $$ However, I've tried continuing the binomial expansion and cannot figure out why the factor is 1/80.


How long it will take for a upright rigid body to fall on a ground

Let's suppose there is a straight rigid bar with height $h$ and center of mass at the middle of height $h/2$. Now if the bar is vertically upright from ground, how long will it take to fall on the ground and what is the equation of motion of the center of mass (Lagrangian)?


Cerenkov light - a practical calculation

I need to calculate the approximate total amount of energy radiated, via Cerenkov, by a muon as it traverses $10\:\rm{cm}$ of quartz glass. Unfortunately I've spent so much time fiddling with electronics in the lab recently that I've completely forgotten how to do maths. The formula to use is the Frank–Tamm Equation: $$\frac{dE}{dx} = \frac{q^2}{4\pi} \int_{v>c/n(\omega)} \mu(\omega)\omega\biggl(1-\frac{c^2}{v^2n^2(\omega)}\biggr)d\omega$$ Where $\mu(\omega)$ and $n(\omega)$ are the permeability and refractive index of the medium being traversed by the charged particle. For a $5\:\rm{keV}$ muon travelling through quartz glass I have looked up the following numbers (but I'm really not sure over what frequency range to consider the last two): $$x=0.1\:\rm{m},$$ $$q=1.6\times10^{-19}\:\rm{C},$$ $$v\approx c,$$ $$n(\omega)=1.54-1.46 (\lambda=200-700\:\rm{nm})\approx1.5,$$ $$\mu(\omega)\approx1NA^2.$$


Efficiently calculating the gravitational force from spherical shell

I am reading Kolenkow and Kleppner's _Classical Mechanics_ and they have tried to calculate the gravitational force between a uniform thin spherical shell of mass $M$ and a particle of mass $m$ located at a distance $r$ from the center. The shell has been divided into narrow rings.$R$ has been assumed to be the radius of the shell with thickness $t$ ($t<<R$). The ring at angle $\theta$ which subtends angle $d\theta$ has circumference $2\pi R\sin\theta$. The volume is $dV=2\pi R^2t\sin \theta d\theta$ and its mass is $pdV=2\pi R^2t\rho\sin\theta d\theta$. If $\alpha$ be the angle between the force vector and the line of centers, $$dF=\frac{Gm\rho dV}{r'^2}\cos\alpha$$ where $r'$ is the distance of each part of the ring from $m$. Next, an integration has been carried out using $\cos\alpha=\frac{r-R\cos\theta}{r'}$ and $r'=\sqrt{r'^2+R^2-2\pi R\cos\theta}$. Question: I would like to avoid these calculations and I was wondering if there exists a better solution.


The Concepts of Path Integral in Quantitative Finance

I realize that path integral techniques can be applied to quantitative finance such as option value calculation. But I don't quite understand how this is done. Is it possible to explain this to me in qualitative and quantitative terms?


Derivation of Lagrangian density for an infinite classical dielectric in interaction with the EM field

I am tasked with reading and reproducing all the steps in J.J. Hopfield's 1958 paper "Theory of the Contribution of Excitons to the Complex Dielectric Constant of Crystals". Embarrassingly I am stuck on equation (3). Hopfield takes the Lagrangian _density_ for an infinite classical dielectric in interaction with the electromagnetic field to be ![enter image description here](http://i.stack.imgur.com/0byQD.png) Instead of taking this equation at face value, I would like to fully derive this Lagrangian density, but am unsure how to approach the problem. I have found many articles and sites tackling a similar problem, but never ending up with an expression as here. Any pointers on how to approach the problem or links to material that could be useful would be very much appreciated.


Where is a particle bound in a delta potential?

I can picture a bound state in a harmonic oscillator, or in an infinite square well, but where is a particle bound in a delta potential?


How do I measure the total weight of a container with scales on four corners?

I have a very large movable trash container, and I have mounted four scales, one on each corner. How do I get the **total weight** of the entire container? Is it simply a sum of the four weights, or is it more complex than that? The contents of the container are not evenly distributed, one corner will be heavier than the others. Also, there is a possibility that one corner will not be completely resting on the ground, and thus the scale on that corner will read zero, but the other three corners will bear it's weight.


Sunrise/Sunset on a planet with an A-Class or B-Class Star

I'm a sci-fi writer, and I have a question regarding a hypothetical planet that orbits an A-class or B-class Star. **Assuming Rayleigh Scattering occurs roughly the same way on this planet as it does on Earth, what colors would we see at Sunrise and Sunset?** I understand that what colours we see from our sun comes from the Rayleigh Scattering, but seeing as it's sci-fi I'm writing, I need a plausible colour scheme to use for my new planet. The Planet has a breathable atmosphere not too different to Earth, so for the sake of the argument, I'm assuming that the Rayleigh Scattering would occur roughly the same way, but my physics knowledge is sadly lacking in this area, and the Wikipedia article only refers to our own G-class Sun. On that note, would the Sky still appear Blue, from an A-class or B-class Star? Or would the Rayleigh scattering trigger a completely different colour of the sky? Is there no 'right answer' when it comes to what sunrise/sunset/sky would look like on a distant planet?


What path will an object in a cockpit take?

Situation: A fly is in the cockpit of a fighter jet, and flying directly in front of the pilot. The pilot becomes annoyed and decides to get rid of the fly by executing a sudden, high speed turn. The cockpit is so dark the fly cannot see any of the cockpit. will the fly continue to be in front of the pilot, or will it hit the side of the cockpit? If the pilot executes a slow turn will the result be different?


Can a long pipe stop the flow of a liquid from a pump?

If a pump is discharging a fluid into a pipe, is it possible to add a long enough pipe such that the flow of the liquid will be stopped and none will come out of the open end of the pipe?


Concerning drag on a flow past a cylinder

I am wondering about the drag coefficient for a flow past a cylinder. I am reading this article. I understand why the drag is high to begin with (point 2), when the boundary layer separates and the recirculations appear, the pressure behind the cylinder (where the wakes are located) will drop, creating a drag. This is similar to what happens when vortices appear at the end of an airplane wing, the result is a drag force. However, I am wondering about the following:   1) Why does the drag decrease as the Re goes up?   2) Why does the drag coefficient take a dive at point 5   3) Why does it rise after point 5? The text doesn't really explain why the drag coefficient is increasing or decreasing.


Singularity in Newton's gravitational law

If $r=0$ in the well know equation $F= G\dfrac{m_1\cdot m_2}{r^2}$, it will not follow that the force will be infinite? May someone please clarify it to me?


Separation of variables in various PDEs, physical meaning

The method of separation of variables produces an undetermined separation constant and a family of solutions indexed by the values of this constant. For instance, in the case of an infinitely long rod along the positive x axis (with suitable boundary conditions, like a fixed temperature at the beginning, and a given temp. distribution at initial time), the separation constant, $-k^2$, can vary continuously, and the solutions are of the form $$T_k(x,t)=C(k)\cdot \sin{kx} \cdot e^{-k^2\alpha t}$$ One is then supposed to integrate over $k$ to obtain the answer. In the case of a finitely long rod, the values of the constant are quantized and can be summed, rather than integrated. But when we're dealing with Schroinger's equation for a particle in a box, we are only interested in the eigenfunctions corresponding to the values of the separation constant, which we associate with energy. But the general solution would be a superposition of these wavefunctions. Suppose we're dealing with a 3D box of length $L$, then the eigenfunctions are: $$\psi_{k,\ell,m}(x,y,z)=\sqrt{\frac{8}{L^3}} \sin{\frac{\pi}{L}kx}\cdot\sin{\frac{\pi}{L}\ell y}\cdot\sin{\frac{\pi}{L}mz}, \quad k,\ell,m\in\mathbb{N}$$ My question: is there any physical significance to the function: $$\psi=\sum_k\sum_\ell\sum_m c_{k,\ell,m}\psi_{k,\ell,m}, $$ where $c_{k,\ell,m}$ are constant coefficients. Clearly this function is a solution to the Schrodinger equation, by the superposition principle. But does this function describe anything at all? Or is it that, unlike in the case of heat transfer, only the eigenfunctions matter?


Mass,Energy,Matter

In $E=mc^2$ what type of energy is $E$? For instance, you can find the kinetic energy of an electron by using $\frac{1}{2}mv^2$ but you can also find the energy from $E=mc^2$. How are those two different besides the values?


What's the total Cosmic Microwave Background energy?

This is a part of my wider attempts at accounting matter in the observable universe. The total light ever emitted by stars and other space objects is capped around 1% of baryonic mass (in energy equivalent). How much light is there in space and how heavy is it? How does the CMB compare to that? How would we calculate it?


Can relativistic momentum (photons) be used as propulsion for 'free' after the initial generation?

In discussing this question about propelling a spacecraft with photons and their relativistic momentum, the author asked that I restate my comment as another question. If photons can really be used for propulsion, could two mirrors be affixed between two objects, focused well and continually adjusting their respective 'where they will be at the time of light arrival', and use the same photons bouncing back and forth between the two objects for perpetual thrust after the initial photon generation? The initial question was on an assignment problem: "A rocket of mass $m_0$ is propelled by a giant monochromatic laser mounted on the back of the rocket. The laser emits a beam with a power of $P_0$ watts and a frequency $f_0$, both measured in the rest frame of the rocket. When the beam is turned on, the rocket is driven in the opposite direction by the recoil."


What happens to light and mass in the center of a black hole?

I know that black holes are "black" because nothing can escape it due to the massive gravity, but I am wondering if there are any theories as to what happens to the light or mass that enters a black hole and cannot escape.


Whis is the difference between charge fractionalization in 1D and 2D?

Both 1D Polyacetelene and 2D fractional quantum Hall state can support fractional excitations. But as I can see, there are some differences: the ground state of Polyacetelene breaks translational invariance but FQHE does not. In general a symmetry breaking state in 2D are considered as conventional state and does not support fractional excitations. I wonder why is there such a difference between 1D and 2D fractionalizations?   Is there any 1D model can support fractional excitations from a symmetry preserved ground state ?


Can two spools shield you effectively against radiation

a friend of mine thinks that by buying a product called rayguard he can effectively shield against electrosmog. On the website, there are multiple evidences that this might work. However the product seems to be just created of one plastic chasi with two spools inside. The functionality of the product is described as following: > With the RayGuards, the solution therefore is not based upon electronics but > on physics. In a radius depending on the model of device used, all > frequencies are absorbed. > > This spectrum of frequencies is being put through a dividing and analysing > system. There the compen- sation of the harmful waveparts is made. > > To explain it better, we will use the example of a camera. Compare the > effect of the RayGuard to the polarising filter. Disturbing effects of > reflection are being cleared by the filter and the foto looks clear and > colourful again. I do not claim that I am a physicist , however after some research my personal opinion on this product is that it is not so simple to shield against radiation. My questions to you are:   * Is it possible with the rayguard to shield effectively against radiation? If it is a scam? Why "not"   * From a physical standpoint, how to shield properly against radiation? What device would be needed to shield effectively against for example mobile phone radiation   * Is the everyday radiation from a physical standpoint "unhealthy"? I really appreciate for your reply! PS.: I know that this question is not the typical physician question which is asked here, however it would be great if you could give me some hard facts/theories to prevent my friend from wasting a lot of money!


Electric flux through nonspherical surphace

Here's an explanation from a book that I read: ![enter image description here](http://i.stack.imgur.com/9q7I7.png) Now, I have a few problems with this explanation as well as with the "cone method". I guess that all my problems boil down to the concept of infinitesimal.   1. The book says "a normal to $dA$ makes an angle $\phi$ with a **radial** line from $q$". Now, the problem is that I can take another radial line intersecting $dA$ which will make a different angle with the area.   2. The strength of the electric field passing through the projected area ($dA \cos \phi$) is different from $\vec{E}$ because the projected area is closer to the charge, yet the book seems to ignore that. The same goes for the "cone method": ![enter image description here](http://i.stack.imgur.com/mXMH1.png) Books calculate proportion between the areas and the radii using the same radius for $dA \cos \phi$ and $dA$ although the distance of the projected area from the charge is different than from the original area to the charge. So how can I solve these contradictions? Is there a strict mathematical proof that an electric flux through any area inside a cone (regardless of its orientation) is the same?


What's wrong with this reactionless drive?

I think this is analogues for the Woodward effect, but macroscopic: We assume a spacecraft consisting of a broomstick, a donut and lots of gear for storing and transfering mechanical energy. Take the donut, spin it till the outer edge moves at relativistic speed (We assume a tough glazing), and push it along the broomstick, brake it's spin and then the donut, move donut forward along broomstick, repeat. Since the donut is heavier when it's being accelerated backwards then when it's braked/accelerated forward, there should be a net momentum. Since nothing leaves the broomstick, this has to be wrong. But where?


What will happen if we place salt water on a induction cooker?

As we know that induction cooker works on the principal of induction of current in a conducting plate. So I just wanted to know what will happen if we place salt water in a plastic container on the induction cooker will it get hot as it is also a conductor or nothing will happen?


Is the molecule of hot water heavier than that of cold water?

We know that the molecule of hot water($H_2O$) has more energy than that of cold water (temperature = energy) and according to Einstein relation $E=mc^2$ ,this extra energy of the hot molecule has a mass. Does that make the hot molecule heavier?


Correcting for bad eyesight on display monitors

I think it would be neat if one could configure one's eyesight parameters (astigmatism and myopia in my case), viewing distance, and perhaps age into a special display driver, such that a computer would present its user interface distorted (from the view of a normal-sighted person) in such a way that the bad eye, by applying its own "distortion", would essentially un-distort it so that the brain would receive a sharp view without the need for physical glasses or lenses. I suspect though that this is not possible partly on grounds of limited information. If the error in eyesight would be linear (e.g. shift to the left or enlargement) an appropriate distortion seems trivial. But a realistic error (hence a realistic distortion) would lead to a situation where several image points get mixed into the same "pixel" (in the physical eye this would mean: hit the same receptor), and disentangling those would require a-priori knowledge about original image, which is not always available (could be available for regular window shapes, or moving images -- this may lead to headache :) It's strange though, because my glasses are simple physical devices that a computer should be able to simulate, but perhaps not in the limited confines of the 2D surface of its display. Now all of this is just a hunch and my question is this: is there a straightforward answer for what kinds of "invertible" distortions (in a mathematical sense) the imagined apparatus could work, and has the problem ever been formalized in a better way?


A problematic integral in calculating the entanglement entropy in 1+1 D free massive bosonic field theory

I encountered a curious integration identity when I was reading the paper by Pasquale Calabrese and John Cardy on the entanglement entropy of 1+1D quantum field theory (arXiv). The identity is given below: $$ \int^\infty_0 x I_{\alpha}(x)K_{\alpha}(x) dx \,\, ``=" \frac{\alpha}{2}. $$ Here $I_\alpha(x)$ and $K_{\alpha}(x)$ are the modified Bessel functions of the fist and second kind. The above identity is implicitly used in Eq.4.19 and Eq.4.22 of the said paper. However, the above identity looks false to me as the left hand side is apparently divergent. Recall that we have the following asymptotic expansion: $$ I_\alpha (x) \sim \frac{e^{x}}{\sqrt{2\pi x}};\\\ K_\alpha (x) \sim \sqrt{\frac{\pi}{2 x}}e^{-x}, $$ from which we obtain $$ xI_\alpha(x)K_\alpha(x) \sim \frac{1}{2}. $$ I believe that some sort of regularization has been done on the integral or I simply miss something. I hope someone here could help me understanding this identity. EDIT: I think the following might be a useful hint. The above integral rises from the expression of $\int^\infty_0 rG(r,r)dr$, where $r$ is the radial coordinate and $G$ is the usual Green's function (see Eq.4.18). Now let us consider $\int^\infty_0 G((1+\epsilon)r,r)rdr$, motiviated by the short distance expansion. The corresponding integral is: $$ \int^\infty_0 x I_{\alpha}((1+\epsilon)x)K_{\alpha}(x) dx = \frac{(1+\epsilon)^{-\alpha}} {(1+\epsilon)^2-1} = \frac{1}{2\epsilon}-\frac{1}{2}(\alpha+\frac{1}{2})+O(\epsilon). $$ The above identity can be found in Gradshteyn & Ryzhik's table, 7th Edition, entries (6.521.2), (6.521.8), and (6.521.9). Now we take the limit $\epsilon\to 0$. The first term of the above is divergent, which could be interpreted as the usual UV divergence part. The second term, however, is finite. If somehow we can argue that it is meaningful to drop the first term and take the $\alpha$-dependent as the "physical" answer of the integral, then we are done.


The states of the adjoint representation correspond to the generators

![enter image description here](http://i.stack.imgur.com/3pHzS.png) I understand the definition of the adjoint representation. It uses structure constants as matrix components of generators, but I can't understand meaning of the states $|X_{a}\rangle$. What does "correspond" mean? What is the exact definition of $|X_{a}\rangle$? Thanks in advance for any help.


Is this a valid derivation of the Legendre transformation from the Euler-Lagrange condition

E-L condition: $$\frac{d p}{dt}=\frac{\partial L}{\partial q}$$ Where $p=\frac{\partial L}{\partial \dot{q}}$ Are the following steps valid: $$\frac{\partial q}{dt} dp=\partial L$$ $$\dot{q} \: dp = \partial L$$ $$ \int \dot{q} \: dp = L+C $$ By integration by parts the LHS becomes: $$ \int \dot{q} \: dp = \dot{q} p-\int p \: d \dot{q} = \dot{q} p- \int \frac{\partial L}{\partial \dot{q}} \:d \dot{q}=\dot{q} p-L+C_1$$ Substituting this back into the LHS: $$\dot{q} p-L=L+C_2$$ If steps are valid, then this indicates that the Legendre transformation of the Lagrangian is just the Lagrangian plus some constant $C_2$, and that the Hamiltonian is thus $H=L+C_2$. Seems pretty fishy. If it is not fishy (if above steps are valid), then this question: Since it is derived from the E-L condition, does this result imply that the action of all functions that are their own Legendre transformation (plus a constant) is stationary?


How to understand "the speed of propagation of the condition of constant phase"?

I still can't understand that the phase can be a constant until now. If the phase is constant, from the $$y(x,t) = a \times \sin(phase)$$ the shape of wave will be a line parallel to x-axis.But I know that I am wrong,how to explain it?


At what point is the spin determined in a Stern-Gerlach Apparatus

Consider a particle with spin that travels through a Stern Gerlach box (SGB), which projects the particle’s spin onto one of the eigenstates in the $z$-direction. The SGB defines separate trajectories for the particles that travel through it depending on their spin. My Question: At what point is the spin determined when it is in superposition? When the particle starts to feel the magnetic field? Or only when the trajectory is measured in the detector? This is a similar question, however it does not answer my question.


Why don’t photons interact with the Higgs field?

Why don’t photons interact with the Higgs field and hence remain massless?


Why do electrons around nucleus radiate light according to classical physics

As I navigate through physics stackexchange, I noticed Electron model under Maxwell's theory. Electrons radiate light when revolving around nucleus? Why is it so obvious? Note that I do not know anything about quantum mechanics - and I just want classical stories.


Power radiated by the sun at different locations

_I am wondering can someone help to solve second part which extends first part;_ The power radiated by the sun is ${3.9*10^{26}}_{watt}$. The earth orbits the sun in a nearly circular orbit of radius $1.5 *10^{11}m$. The earth’s axis of rotation is tilted by $27^o$ relative to the plane of the orbit , so sunlight does not strike the equator perpendicularly. What power strikes a $0.75$ $m^2$patch of flat land at the **_equator_**? $P_0 = I*4*π*r^2$ $P = I*A*cos(\alpha)$ then $P=\frac{P_0 * A *cos(\alpha)}{ 4*π *r^2}$ then $P=\frac{(3.9*10^{26})(0.75)(cos(27))}{(4*π*(1.5*10^{11})^2)}=920_{watt}$ ~ How much power strikes a patch of the same area located at latitude (at the same time of year) of 43° S (like Christchurch, New Zealand) or 43° N?


How do I quantize a classical field theory

I have not been able to find any information about this on the Internet. I am a middle-schooler, 14, who self-studies physics, and I know up to and including ODEs, and some of the calculus of variations, enough so that I can take a variation of the action. However, I have absolutely zero budget, and no ability to get textbooks. So, I was wondering how a physicist would quantize a classical field theory, such as Gauss's law for gravity, which I'm currently trying to create a quantum field theory for, as part of my plan to create a possible theory of quantum gravity, by adding in the relativistic corrections later.


Why the color motion picture film appears as black?

Why the color motion picture film, or a color photographic slide, appears as black? The film appears black but when the light from the projector lamp hit it then you can see all the colors.


An Ideal Transformer

In a transformer assumed to be transformer, power in the primary is equal to power in the secondary. So in a sense, the power in the secondary is 'fixed'. Output voltage in the secondary is also fixed by the turns ratio and the voltage in the primary. Then, The current tapped from the secondary coil should also be fixed $I=\frac{P}{V}$. Doesn't the resistance in the circuit connected to the secondary coil also affect the current tapped from the secondary coil? So if a load resistance in the secondary coil circuit is varied, would the current still be fixed the same $I=\frac{P}{V}$ or would it change accordingly with $I=\frac{V}{R}$


Can the apparent velocity of an approaching spacecraft be faster than light?

Consider watching an alien space ship at Alpha Centauri (at 4.5 light years away) through a telescope from earth. This space ship turns towards us, and starts travelling toward us at what appears to be a velocity of exactly 2 million kilometers per second, when measuring the distance every hour. The space ship will arrive here in about 8 months. How fast is the space ship really travelling?


Why is the Heisenberg uncertainty principle stated the way it is?

I spent a long time being confused by the Heisenberg uncertainty principle in my quantum chemistry class. It is frequently stated that the "position and momentum of a particle cannot be simultaneously known to arbitrary precision" (or any other observables $[A, B] \neq 0$). This made no sense to me -- why can't you measure both of these? Is my instrument just going to stop working at a certain length scale? The Internet was of little help; Wikipedia describes it this way as well and gets into philosophical arguments on what "position" and "momentum" mean and whether they really exist (in my opinion, irrelevant nonsense that has no effect on our ability to predict things). Eventually it was the equation itself that gave me the most insight: $$\sigma_x \sigma_p \geq \frac{\hbar}{2}$$ Look at that – there's two standard deviations in there! It is _impossible_ by definition to have a standard deviation of one measurement. It requires multiple measurements to have any meaning at all. After some probing and asking around I figured out what this really means: Multiple repeated measurements of identically prepared systems don't give identical results. The distribution of these results is limited by that formula. Wow! So much clearer. Thus $\hat{r}(t)$ and $\hat{p}(t)$ _can_ be known for the same values of $t$ to as much precision as your measuring equipment will allow. But if you repeat the experiment, you won't get identical data. Why doesn't everyone just state it that way? I feel like that would eliminate many a student's confusion. (Unless, of course, I'm still missing something – feel free to enlighten me should that be the case). EDIT: This post was at +1. Who downvoted me? I took a while to write out my question clearly and made sure it followed the guidelines on here.


Why are galactic centers always brighter than the edges?

As you can see this image http://i.stack.imgur.com/YOt8C.jpg and other galaxy images, the centers generally much brighter. Why is that? Is there a very big star? A very big gravitational field?


Read-off particle from (projected) Dynkin labels

In the review of Slanksy "Group theory for unified model building" in chapter 6: How do one relate the projected Dynkin diagrams from for example $\overline{5}+10$ of $su(5)$ to the corresponding representation in of the subgroup $su(2)\times su(3)$? An example is $P( 0 0 1)=(0)(01)$, how do one see that this must be a charge $1/3$ antiquark singlet? Reference: http://cds.cern.ch/record/134739/files/198109187.pdf?origin=publication_detail


Confused about Impulse

Encountered a problem that involves impulse while studying for my exam and I'm not sure how to even approach it. I know that momentum is conserved, but I'm not sure how to relate that to avg force. Maybe someone can help point me in the right direction? I know that it's in quadrant III, through intuition, but I can't come up with a provable explanation Relevant equation: $J=F_{avg}\Delta T$ ![Couldn't copy and paste the problem, description in link](http://i.stack.imgur.com/dDlRS.png)


Why is no EMF being induced in this ring passing through a magnetic field?

![enter image description here](http://i.stack.imgur.com/B1KNO.png) The book's logic is that there is no induced EMF because flux is constant as it passes through the magnetic field. Which makes sense, but this seems counter-intuitive to what I previously learned. If i'm not mistaken, if an electron is relatively moving perpendicular to a magnetic field, it WILL experience a force. The electrons in the ring are moving perpendicularly relative to the stationary field, so why aren't they experiencing a force?


How would one begin to try to identify the properties of the mathematical structure corresponding to physical reality?

I was thinking about the Mathematical universe hypothesis and a natural question popped into my mind: _Assuming that the universe (by universe I mean the complete physical reality) is really isomorphic to some conceivable, mathematically constructible structure, how would one begin to narrow down the possibilities? How would one identify its properties without necessarily finding the structure itself?_ My first guess is that one should look at QFT and GR and assume that the mathematical structure we want would have to be **consistent** with those two theories at least in the appropriate approximations/limits and that we could somehow find all the familiar symmetries in some form on that structure. But those are just words, I don't understand what would one have to do rigorously to rule out some of the structures, I would really appreciate some non-handwavy guidelines if it's possible. **P.S. I'll gladly elaborate further and edit my question if something is particularly unclear.** **EDIT 1:** I'm not asking how theoretical physics should proceed in general and how should the scientific method be used in order to understand the world. I'm interested in how much can we say about the **"final, true and complete"** theory (assuming it exists) without actually having it. I'm interested in the mathematical structure associated with that theory, what are the most general statements about it that are almost certainly true?


ATLAS Higgs Interpretation

I came across this abstract, and I am curious as to what the ATLAS Team has actually discovered: > Abstract Motivated by the result of the Higgs boson candidates at LEP with a > mass of about 115~GeV/c2, the observation given in ATLAS note ATL-COM- > PHYS-2010-935 (November 18, 2010) and the publication “Production of > isolated Higgs particle at the Large Hadron Collider Physics” (Letters B 683 > 2010 354-357), we studied the γγ invariant mass distribution over the range > of 80 to 150 GeV/c2. With 37.5~pb−1 data from 2010 and 26.0~pb−1 from 2011, > we observe a γγ resonance around 115~GeV/c2 with a significance of 4σ. The > event rate for this resonance is about thirty times larger than the > expectation from Higgs to γγ in the standard model. This channel H→γγ is of > great importance because the presence of new heavy particles can enhance > strongly both the Higgs production cross section and the decay branching > ratio. This large enhancement over the standard model rate implies that the > present result is the first definitive observation of physics beyond the > standard model. Exciting new physics, including new particles, may be > expected to be found in the very near future. The abstract seems to be from a restricted web site (CERN Log-in required), however I have been able to track down a PDF that seems to be discussing the same phenomenon, however the abstracts are not the same. Currently, as I understand it, there is a lot of skepticism about the initial Higgs candidate. If this isn't the Higgs, then what part of the standard model is actually being represented? Or is this an entirely new phenomenon?


Sampling typical clusters between distant points in subcritical percolation

I have on several occasions wondered how one might proceed in order to sample large subcritical Bernoulli bond-percolation clusters, say on the square lattice. More precisely, let's consider the lattice $\mathbb{Z}^2$, and open each edge independently with probability $p<p_c=0.5$. I am interested in the event that the vertices $(0,0)$ and $(N,0)$ are connected by a path of open edges, when $N$ some large number ("large" meaning here "many times the correlation length"). It is well-known (rigorously) that this probability decays exponentially fast with $N$, and one has a lot of information about the geometry of the corresponding cluster (e.g., the corresponding cluster converges to a brownian bridge as $N\to\infty$ under diffusive scaling, it has a maximal "width" of order $\log N$, etc.). **Question:** Are there (not too inefficient) algorithms that would sample typical clusters contributing to this event? **Edit:** googling a bit, I have stumbled upon this paper. I have yet to read it (and to decide whether it is relevant to my question). What the author seems to be doing is growing his objects in a biased way, which helps create the very unlikely configurations he's after, but would result in the wrong final distribution. So he's at the same time correcting the bias (in a way I haven't yet understood by just browsing through the paper). As an example, he's sampling 18 disjoint crossing clusters for critical percolation on $\mathbb{Z}^3$ in a box of size $128\times128\times2000$, an event of probability of order $10^{-300}$. So: **Alternative question:** does anybody know how this works? (I can also simply read the paper, but it might be useful for me and others if a nice self- contained description was given here by someone knowledgeable). Apparently, looking more thoroughly, there is a lot of material that could be useful. It seems that relevant keywords here are: rare events simulation, importance sampling, splitting, etc.


Word for the star around which an exoplanet orbits:

Is there accepted nomenclature for the star around which a particular exoplanet orbits? Meaning, if I were to say "The exoplanet blah blah blah's (noun)" what noun would I put there? Sun? Star? Designated Gravity Buddy? Something else? As I understand it, this would not come up often currently in the real world (because Exoplanets are named by their star), and it more in the realm of science fiction and fantasy at the moment. However, just because something seems currently useless (to me) from a scientific standpoint doesn't mean that someone hasn't thought up a name. Would this name change if you're actually on the planet or in the system? (ie, you hear about Tatooine having "two suns" not "two stars").


Some questions on a version of the O'Raifeartaigh model

This form is taken from a talk by Seiberg to which I was listening to, Take the Kahler potential ($K$) and the supersymmetric potential ($W$) as, $K = \vert X\vert ^2 + \vert \phi _1 \vert ^2 + \vert \phi_2\vert ^2 $ $W = fX + m\phi_1 \phi_2 + \frac{h}{2}X\phi_1 ^2 $   * This notation looks a bit confusing to me. Are the fields $X$, $\phi_1$ and $\phi_2$ real or complex? The form of $K$ seems to suggest that they are complex - since I would be inclined to read $\vert \psi \vert ^2$ as $\psi ^* \psi$ - but then the form of $W$ looks misleading - it seems that $W$ could be complex. Is that okay?  Now he looks at the potential $V$ defined as $V = \frac{\partial ^2 K}{\partial \psi_m \partial \psi_n} \left ( \frac {\partial W}{\partial \psi_m} \right )^* \frac {\partial W}{\partial \psi_n}$ (..where $\psi_n$ and $\psi_m$ sums over all fields in the theory..) For this case this will give, $V = \vert \frac{h}{2}\phi_1^2 + f\vert ^2 + \vert m\phi_1 \vert ^2 + \vert hX\phi_1 + m\phi_2 \vert ^2 $   * Though for the last term Seiberg seemed to have a "-" sign as $\vert hX\phi_1 - m\phi_2 \vert ^2 $ - which I could not understand.  I think the first point he was making is that it is clear by looking at the above expression for $V$ that it can't go to $0$ anywhere and hence supersymmetry is not broken at any value of the fields.   * I would like to hear of some discussion as to why this particular function $V$ is important for the analysis - after all this is one among several terms that will appear in the Lagrangian with this Kahler potential and the supersymmetry potential.    * He seemed to say that if *``$\phi_1$ and $\phi_2$ are integrated out then in terms of the massless field $X$ the potential is just $f^2$"* - I would be glad if someone can elaborate the calculation that he is referring to - I would naively think that in the limit of $h$ and $m$ going to $0$ the potential is looking like just $f^2$.    * With reference to the above case when the potential is just $f^2$ he seemed to be referring to the case when $\phi_2 = -\frac{hX\phi_1}{m}$. I could not get the significance of this. The equations of motion from this $V$ are clearly much more complicated.    * He said that one can work out the spectrum of the field theory by _"diagonalizing the small fluctuations"_ \- what did he mean? Was he meaning to drop all terms cubic or higher in the fields $\phi_1, \phi_2, X$ ? In this what would the "mass matrix" be defined as?  The confusion arises because of the initial doubt about whether the fields are real or complex. It seems that $V$ will have terms like $\phi^*\phi^*$ and $\phi \phi$ and also a constant term $f^2$ - these features are confusing me as to what diagonalizing will mean. Normally with complex fields say $\psi_i$ the "mass-matrix" would be defined the $M$ in the terms $\psi_i ^* M_{ij}\psi_j$ But here I can't see that structure!   * The point he wanted to make is that once the mass-matrix is diagonalized it will have the same number of bosonic and fermionic masses and also the super-trace of its square will be $0$ - I can't see from where will fermionic masses come here!    * If the mass-matrix is $M$ then he seemed to claim - almost magically out of the top of his hat! - that the 1-loop effective action is $\frac{1}{64\pi^2} STr \left ( M^4 log \frac{M^2}{M_{cut_off}^2} \right ) $ - he seemed to be saying that it follows from something else and he didn't need to do any loop calculation for that!  I would be glad if someone can help with these.


Does decoherence explain all instances of wave function collapse?

Specifically, how can decoherence explain the appearance of flecks of metallic silver on a photographic plate when exposed to the very weak light of a distant star? EDIT: Perhaps the advocates of decoherence need some context for this question. There is a certain definite quantity of energy on the order of one or two eV to drive the chemical reaction 2AgBr -> 2Ag + Br2 This is the reaction responsible for the fleck of silver on the photographic plate. The amount of energy is far greater than can be accounted for in any realistic time frame by the classical e-m wave energy of the light of a distant star. Any explanation must explain where this energy comes from. How does "decoherence" claim to do this? I have heard over and over again that there is a matrix which is diagonalized, but no one has so much as volunteered to say just what matrix they are talking about. Is it, for example, the matrix of position states of the photon? Or perhaps it is the oxidation states of the silver atom? And I would really like a better explanation of how the matrix is "diagonalized" than to simply repeat that it is in "thermal contact with the environment." EDIT: I have reviewed the comments again and I find that no one has come close to dealing with the question. I cannot find anything wrong in the way I have asked it so far, so I am posting this edit as my only means to prompt people to attempt an answer.


Imaginary time in quantum and thermodynamics

The following question is about chapter 2 of Sakurai's _Modern Quantum Mechanics_. I wish I could link to the Google book, but it doesn't seem to have a satisfactory preview to be able to read the section I'm talking about, so I'll do my best to write out the part I'm talking about... In the section about propagators and Feynman path integrals (p. 113 in my edition) he gives the following example: $G(t) \equiv \int d^3 x' K(x', t; x',0) $ $=\int d^3 x' \sum_{a'} |\langle x'|a'\rangle|^2 \textrm{exp} \left(\frac{-iE_{a'}t}{\hbar}\right)$ $=\sum_{a'}\textrm{exp} \left( \frac{-iE_{a'}t}{\hbar} \right)$ He goes on to say that this is equivalent to taking the trace of the time evolution operator in the $\\{|a'\rangle\\}$ basis, or a "sum over states", reminiscent of the partition function in statistical mechanics. He then writes $\beta$ defined by $\beta=\frac{it}{\hbar}$ real and positive, but with with $t$ purely imaginary, rewriting the last line of the previous example as the partition function itself: $Z=\sum_{a'} \textrm{exp} \left( -\beta E_{a'} \right)$. So my question is this: What is the physical significance (if any) of representing time as purely imaginary? What does this say about the connection between thermodynamics and quantum? The fact that you get the partition function exactly, save for the imaginary time, here seems too perfect to be just a trick. Can someone explain this to me?


Writing Schrodinger equation with central potential in Atomic unit

I'm struggling to write Schrodinger equation with a central potential in Atomic unit. $$ \frac{\hbar^2}{2\mu}\Big(\frac{d^2}{dr^2}+\frac{2}{r}\frac{d}{dr}-\frac{l(l+1)}{r^2}\Big)R(r)+\Big(E-V-\frac{e^2}{4\pi \epsilon_0 \epsilon r}\Big)R(r)=0 $$ Taking Effective Bohr radius $a_0$ as length scale and effective Rydberg as energy scale $$Ry=\frac{\mu}{m_e \epsilon^2}\big(\frac{hc}{e}\big)\frac{m_e e^4}{8\epsilon_0^2 \hbar^3c}$$ in MKSA or $Ry=\frac{\mu e^4}{2\epsilon^2 \hbar^2}$ in CGS. What I tried is: By substitution of $r'=r/a_0$ : $$ \Big(\frac{d^2}{dr'^2}+\frac{2}{r'}\frac{d}{dr'}-\frac{l(l+1)}{r'^2}\Big)R(r')+\frac{2\mu a_0^2}{\hbar^2}\Big(E-V-\frac{e^2}{4\pi \epsilon_0 \epsilon a_0 r'}\Big)R(r)=0 $$ where $\mu$ is effective mass $\mu=.06m_e$. Then we can write $m_e=\hbar=e=1$. The potential is for example $V=4 \ Ry$. Know I don't know what can I write instead of   $$ \frac{2\mu a_0^2}{\hbar^2}\Big(-\frac{e^2}{4\pi \epsilon_0 \epsilon a_0 r'}\Big) $$ Any help appreciated.


Uniqueness of eigenvector representation in a complete set of compatible observables

Sakurai states that if we have a complete, maximal set of compatible observables, say $A,B,C...$ Then, an eigenvector represented by $|a,b,c....>$, where $a,b,c...$ are respective eigenvalues, is unique. Why is it so? Why can't there be two eigenvectors with same eigenvalues for each observable? Does maximality of the set has some role to play in it?


What is the connection between extra dimensions in Kaluza-Klein type theories and those in string theories?

This follows to some extent from a question I asked previously about the flaws of Kaluza-Klein theories. It appears to me that Kaluza-Klein theories attach additional dimensions to spacetime that are related to the gauge freedoms of field theories. I believe the original model was to attach a $U(1)$ dimension to the usual 4-dimensional spacetime to reproduce electromagnetism. But, as explained in the answers to my previous question, these extra dimensions have all sorts of problems. String theories also (famously?) require extra dimensions. So, is there a connection between the higher-dimensional descriptions? What do they have in common and how do they differ? For example, the $U(1)\times SU(2)\times SU(3)$ group is 7-dimensional, which, when attached to the 4 dimensions of spacetime, gives 11 dimensions. I hear the same number is bandied about in string theory although there's no obvious reason they should be related at all.


What size aperature would I need to find and see the supernova in M51?

On June 2, 2011, a new supernova was identified by an amateur French astronomer in M51. What size telescope aperture would one need to have any chance of seeing this? Given the right sized scope, how would I best identify which point of light is the supernova?


Stress in a thick-walled pressure vessel

I can find many references that give the stress in the walls of a pressure vessel for spheres and tubes, but they all seem to be limited to a thin-wall approximation. I'll limit my writing here to spherical vessels (I don't think it should be very different to find the answer for any structure with symmetry). My main trouble is in setting up the integral. $$ \sigma = \frac{P R }{2 t}$$ So my question is: how do you find material stress for a pressure vessel with a wall that goes from $R_i$ to $R_o$, and you want a solution that is still perfectly valid when $R_i \ll R_o$? My guess is that you would take the equation for the thin wall case, and recast it into an integral. $$ P = \frac{2 \sigma t }{ R }$$ In the more general case, would we write something like this? $$ P = \int_{R_i}^{R_o} \frac{2 \sigma }{r} dr$$ This would only be valid if the stress was constant for all differential shells. I'm wondering if that's a bad assumption, but if the material was highly elastic I think it would be decent assumption. My intuition is that the material would have to be much _more_ elastic for it to hold with the inner radius many times smaller than the outer radius. But regardless of that detail, is the above integral correct? Is there some good intuitive logic to justify it? And is there any other reference that gives the expression for stress in this case? It should be a logarithm if the above is correct, but I haven't seen this before.


Why does a remote car key work when held to your head/body?

I was trying to unlock my car, but I was out of range. A friend of mine said that I have to hold the transmitter next to my head. It worked, so I tried the following later that day:   * Walked away from the car until I was out of range   * Put key next to my head (it worked)   * Put key on my chest (it worked)   * Put key on my leg (didn't work) So first I thought it has to do with height. But I am out of range if I use the key at the same height as my head but not next to it, and normally my key is at the same height as my chest. So it has nothing to do with height. Or my whole body is acting like an antenna, but how is that possible if I am holding the key? Why would it be an antenna if I hold it against my head and not in my hand? Here's a vid of Top Gear demonstrating it.


What would happen if a hydrogen bomb were to explode in Saturn's atmosphere?

Purely hypothetical since any kind of testing in atmosphere/space is banned by international legislation/agreement. The humans have already bombed Luna so ... what could be expected to happen on Saturn if a hydrogen bomb were to explode in it's atmosphere? Would the explosion set the planet's atmosphere ablaze?


How can the microstates be measured with zero energy expenditure?

_James P. Sethna. Statistical Mechanics. Exercise 5.2:_ > What prevents a Maxwellian demon from using an atom in an unknown state to > extract work? The demon must first measure which side of the box the atom is > on. Early workers suggested that there must be a minimum energy cost to take > this measurement, equal to the energy gain extractable from the bit. Bennett > showed that no energy need be expended in the measurement process. Why does > this not violate the second law of thermodynamics? Reference to Bennett's paper didn't help me much. Here the relevant model is a tape consisting of single atoms in pistons, where knowing which side an atom is in a piston counts 1 bit of information, which can be used to extract useful work by expanding the piston, as shown below: ![enter image description here](http://i.stack.imgur.com/aKzX4.jpg) ![enter image description here](http://i.stack.imgur.com/2MbfM.jpg) **My understanding is that after the measurement uncertainty of position is reduced by half and entropy decreases. But without energy expenditure, it seems that this decrease comes free. How can the second law hold if there's no corresponding increase of entropy elsewhere (which I can't identify)?** Something like an explanation is given at the end of the exercise: > The demon can extract an unlimited amount of useful work from a tape with an > unknown bit sequence if it has enough internal states to store the > sequence—basically it can copy the information onto a second, internal tape. > But the same work must be expended to re-zero this internal tape, preparing > it to be used again. **Does this mean after measurement, the reduced entropy in the first tape goes to the second "internal tape" which stores the information? How can such measurement take place?**


Questions (intuitive) about circuits and the flow of electrons

Why don't electrons accelerate in a circuit? If there is a potential difference and the electron moves in the circuit a force is exerted on it. Why does that force produce no acceleration, keeping the current constant? If we add a resistor to a circuit why is the resistance 'spread' over the circuit? Is it because electrons moving faster in the areas of lower resistance would create charge imbalances at places in the conductor?