Trajectory of a photon around a Schwarzschild black hole?

Consider a photon coming from the infinity in a unbounded orbit to a Schwarzschild black hole (Schwarzschild radius $r_{s}$) (see this for illustration). Its impact parameter is $b$ and its distance of closest approach is $r_{0}$ with $$b^2=\frac{r_{0}^{3}}{r_{0}-r_{s}}$$. Then its trajectory in polar coordinates is defined by : $$\frac{d\varphi}{dr} = \frac{1}{r^2\sqrt{\frac{1}{b^2}-\left(1-\frac{r_s}{r}\right)\frac{1}{r^2}}}$$ Consequently : $$\varphi\left(r\right) = \int_{r_{0}}^{r} \frac{dp}{p^2\sqrt{\frac{1}{b^2}-\left(1-\frac{r_s}{p}\right)\frac{1}{p^2}}}$$ and one can compute the total deviation using : $\Delta\varphi = 2\times\left(\lim_{r\to+\infty}\varphi\left(r\right)-\frac{\pi}{2}\right)$ But my question is : how can I plot/draw the trajectory using the integral expression of $\varphi\left(r\right)$ ? * * * Because if I compute : $$f\left(r\right) = 2\times\left(\int_{r_{0}}^{r} \frac{dp}{p^2\sqrt{\frac{1}{b^2}-\left(1-\frac{r_s}{p}\right)\frac{1}{p^2}}}-\frac{\pi}{2}\right)$$ I obtain $f\left(r_{0}\right) = -\pi$, and then $f$ increases up to zero, crosses zero, and tends to its positive value at infinity $\Delta\varphi$. It does not make sense for me and I do not understand how to compute the trajectory from that. * * * If I compute : $$g\left(r\right) = \int_{r_{0}}^{r} \frac{dp}{p^2\sqrt{\frac{1}{b^2}-\left(1-\frac{r_s}{p}\right)\frac{1}{p^2}}}$$ it starts from $0$, and increase up to $\frac{\pi}{2}+\frac{\Delta\varphi}{2}$. * * * I would like to compute the trajectory in the $\left(x, y\right)$ plane, so how to use the values of $f\left(r\right)$ or $g\left(r\right)$ to compute the function $y\left(x\right)$ ?


Dirac magnetic monopoles and quark fractional electric charge quantization

When applying the Dirac quantization rule for electric and magnetic charge, I assume one is considering unit electric charges such as electrons. How does the Dirac quantization rule apply for the fractional electric charges of quarks?


How many $fb^{-1}$ for the most likely $5\sigma$ 115 Gev Higgs at the 7 Tev LHC?

How many $fb^{-1}$ of integrated luminosity at the 7 Tev LHC do physicists expect are needed, to make a $5\sigma$ discovery of the most likey 115 Gev Higgs, if it exists?


Can a system that holds information about it's past ever be Markovian?

To my (basic) understanding a Markov process is a process wherein the future state of a system only depends on the current state, and not on the past states of the system. I was wondering on what the standard approach is on systems that inherently hold information about their past in them. One could image, in the discrete-time case, system carrying in it a representation of it's three previous states. There are two cases to distinguish here:   * The future state of the system is not a function of this memory and only of factors that don't have anything to do with it. This is perhaps a boring case (although there might be interesting examples I don't know of).   * The future state of the system is a function of this memory information, yet in a "Markovian" way. This is the case I am most interested in. Are these things that would be considered as Markovian systems (perhaps on a different scale)? Are there examples of these kinds of systems?


Does a cycle (in Simple Harmonic Motion) have to equal 2π?

So, I search for the definition of cycle and I get this in Wikipedia: > A turn is a unit of angle measurement equal to 360° or **_2π radians_** (or > ...). A turn is also referred to as a revolution or complete rotation or > full circle or **_cycle_** or rev or rot. When I look for motion in pendulum, I get:- > The time for **_one complete cycle_** , **_a left swing and a right swing_** > , is called the period. So, in the first quote, it says that one cycle has to be 2π radians, **_but_** in the second quote it says "one complete cycle" is "a left swing and a right swing" (i.e the full cycle doesn't have to be equal to "2π radians"). Can you please clarify what I see as contradicting statements?


How could $\textbf{S}^2$ not be a multiple of the identity?

I'm self-studying quantum mechanics with Sakurai's book ( _Modern Quantum Mechanics_ , 2nd edition) and came across the following in reference to the operator $\textbf{S}^2$: > As will be shown in Chapter 3, for spins higher than $\frac{1}{2}$, > $\textbf{S}^2$ is no longer a multiple of the identity operator; however, > $[\textbf{S}^2, S_i] = 0$ still holds (for $i = x, y, z$). (page 28) The square of the total spin commuting with the components, I'm comfortable with. But the first part just confuses me: for a system with spin $s$, is it not true that $$\textbf{S}^2|\cdot\rangle = \hbar^2s(s+1)|\cdot\rangle$$ whether or not $s = \frac{1}{2}$? Or do I have a fundamental misunderstanding of the situation? (I have read through Chapter 3, but apparently I kept missing the part where the book addresses this.)


How to calculate the buoyant force on a balloon at different altitudes

For a project into balloon simulation I'd like to know how the force on a balloon changes with altitude: I know that the Buoyant force on a balloon is: $F = (\rho_{air} - \rho_{helium})gV$ Using changing air density from the international standard atmosphere and changes in gravitational acceleration I can calculate the buoyant force and add this to the (negative) force of gravity _mg_. However I know that it isn't this simple, the balloon changes volume with altitude as the air pressure outside the balloon changes. I'd like to know how the balloon inflates - this would be a great help! I think it may be something to do with ideal gas laws - I can calculate air temperature and pressure from the international standard atmosphere but do I use pressure as the air pressure? - or air pressure minus balloon pressure?


Visual Fields testing: Decibels of sensitivity - meaning

Automated Visual Field test measures the patient light sensitivity in decibels. Questions: If one point has sensitivity of 30 decibels and another point has sensitivity of 27 decibels does it mean that point 1 is 2 times more sensitive than point 2? If point 3 has sensitivity of 31dB and point 4 has 21dB does it mean that point 3 sensitivity is 10 times greater than of point4? Third and most important question - How these decibels calculated mathematically? Is there way to reconstruct this (theoretical) calculation?


Ladder operator on momentum basis

Since in Quantum mechanics momentum operator can be written in terms of ladder operators $$\widehat{p}=-i\sqrt\frac{{\hbar m \omega}}{2}(\widehat{a}-\widehat{a}^\dagger)$$ these operators operate on energy eigen-state $|n\rangle$ of particle and we get $$\widehat{a}|n\rangle=\sqrt{n}|n-1\rangle$$ $$\widehat{a}^\dagger|n\rangle=\sqrt{n+1}|n+1\rangle$$ But what if these ladder operator or momentum operator operate on discrete momentum basis, for example $|k_x\rangle$, $|k_x-k_o\rangle$, $|k_x-2k_o\rangle$, $|k_x-3k_o\rangle$........etc. Will it looks like $$\widehat{a}|k_x-k_o\rangle=\sqrt{k_x- k_o}|k_x-2k_o\rangle$$ and $$\widehat{a}^\dagger|k_x- k_o\rangle=\sqrt{k_x}|k_x\rangle$$


Optical Bloch Oscillation

I have a doubt about how the optical Bloch oscillations happen in a 1D photonic crystal. I try to explain: in a photonic crystal with discrete translational symmetry in one direction I superimpose a second bigger periodicity by mean of microcavities. These reduce the amplitude of the first Brillouin zone. Now it is found that minibands are formed, that correspond to delocalized states. If now i introduce a gradient in the refractive index in the longitudinal direction, the minibands are tilted and a Wannier-Stark ladder of states is created. How can I explain the oscillation of an optical pulse from the Wannier-Stark states in the real space? (I ask this beacuse in many article the Bloch oscillations in real time are explain as the fourier transform of the Wannier-Stark ladder but I'm not able to appreciate this point of view). Thanks.


How fast did hydrogen atoms travel when they were first formed in the early universe?

I can't seem to find any data on this, is it a known value?


Non-destructive measurement of qbits

Yale news "New qubit control bodes well for future of quantum computing" (Original paper) says: > "The Yale physicists successfully devised a new, non-destructive measurement > system for observing, tracking and documenting all changes in a qubit’s > state, thus preserving the qubit’s informational value." Which would be the major applications of quantum theory that are impacted by this new technique? Wouldn't e.g. the security of quantum key distribution in cryptography be rendered questionable?


How can crystal structures be determined using X-ray diffraction?

You have the intensity peaks and the diffraction angles. Let's say you suspect the material is cubic, how do I find if it's simple cubic or BCC or FCC? I've googled and all my textbooks just state that xrd is really nifty for determining crystal structure but no further information is presented.


Is it possible to record the 'blue air" effect when a core goes critical?

I've been studying about radiation, and I saw something called the Demon Core. Apparently, it was a core that was under experimentation by the government in Los Alamos to see the exact point at which a mass of plutonium would go critical. It was composed of two hemispheres of beryllium partially enclosing a mass of plutonium and made to reflect the radiation back to the center. But the top dome slipped and shut off the opening and the whole structure briefly went critical. During this brief time, the surrounding are turned blue because the air was being ionized. I'd like to know if this effect can be recorded, or if perhaps the radiation would corrupt the recording system(s) at work. If anyone can give me an explanation, I would greatly appreciate it.


Why is there a linear relationship between charge and isospin?

So the title basically says it all. Something that's really bothering me is the fact that the Gell-Mann Nishijima relationship can be extended to provide a linear relationship between charge and all the flavor quantum numbers for baryons as follows: $Q = I_{3} + (A + S + C + B + T)/2$ Where Q is the charge, A is the baryon number, S is the strangeness number, B is the bottomness number and T is the topness number. What is the physical interpretation behind this relationship? Is charge really isospin or vice-versa? Are they manifestations of each other in the same sense as electricity and magnetism? Is it just a coincidence?


Why do excited states in $^4$He not decay by photon emission?

Here's a level scheme for the $^4$He nucleus (source; click image to see full size): ![Level scheme for he-4](http://i.stack.imgur.com/d6h4F.png) Notice that all of the confirmed decay modes are by disintegration — emission of a neutron, proton, or deuteron. There are a few states with the right spin- parity to decay to the ground state by an E1 (from $1^-$) or E2 (from $2^+$) electromagnetic transmission, but there seems to be no evidence that those internal transitions have been observed. Is there a simple reason why electromagnetic transitions are so strongly disfavored? If I were giving a talk and this question came up I would mumble something about the strong force winning out over electromagnetism in few-nucleon systems and get back on topic, but I'd like a slightly more sophisticated answer than that.


Uncertainly Principle in orthogonal directions

The Heisenberg Principle states that for each direction, $\Delta x\cdot \Delta p_x \ge \hbar , \Delta y\cdot \Delta p_y \ge \hbar$ and $\Delta z\cdot \Delta p_z \ge \hbar$. But, can anything be said about $\Delta x\cdot \Delta p_y$? Can I measure the position in one direction and the momentum in an orthogonal direction in any required precision?


Entanglement entropy for finite interval using heat kernel method

I am looking for some reference for calculation of entanglement entropy(EE) for a finite interval in 4 spacetime dimensions using heat kernel method. Till now I have only found references where the calculation has been done for a halfspace. The paper by Wilczek http://arxiv.org/abs/1007.0993v2 says that the heat kernel method works well only when the region of interest is a half space. So can anyone please let me know whether the heat kernel method is suited for finite size interval calculations of EE or suggest a reference for the same? Thanks in advance.


Why can the Euler beta function be interpreted as a scattering amplitude?

The Wikipedia article on the Veneziano Amplitude claims that the Euler beta function can be interpretted as a scattering amplitude. Why is this? In another word, when the Euler beta function is interpreted as a scattering amplitude, what features does it have that make it able to explain strong force of mesons? What properties of (or why) the Euler beta function (when interpreted as scattering amplitude) has string behavior?


Random bond Ising model and computational efficiency

If you want to find the ground state of the 2d random bond Ising model (no field), a computationally efficient algorithm exists to do it for you (based on minimum weight perfect matching). What about the thermal state for some finite temperature, T? Is there an efficient (i.e. time scales polynomially with system size) way to sample states of different spin configurations from this distribution?


Why is this radio telescope's reflector spherical and not parabolic?

This is the Arecibo Observatory in Arecibo, Puerto Rico. Its reflector is spherical, measuring 1,001 ft. in diameter. It is considered the most sensitive radio telescope on Earth, but the fact that its reflector is spherical and not parabolic makes me wonder how much more sensitive it could be if the reflector were parabolic. What are the pros and cons of a sphere vs. parabolic? ![Picture of the telescope](http://i.stack.imgur.com/0CmCC.jpg) ![Plaque at Arecibo](http://i.stack.imgur.com/9i9q1.jpg)


Video of Earth spinning?

If the Earth is spinning or rotating at a really fast speed, why haven't we seen any videos from space of it spinning when we get a lot of photos of it?


current operator in Hubbard model

How to derive the particle current operators for the non-interacting and interacting Hubbard model ? Hubbard Hamiltonian is given here with the interaction term: http://en.wikipedia.org/wiki/Hubbard_model Here's an introduction : http://quest.ucdavis.edu/tutorial/hubbard7.pdf


Time Reversal in a Black Hole

I had a lively discussion with a person about black holes recently, and was making the point about gravitational acceleration in GR being paralleled by speed in SR. One thing that I know people talk about with special relativity, is that if you could reach a speed faster than the speed of light then time would reverse (I don't get this as the Lorentz factor gives an imaginary number, and who knows what that means). This is impractical for things with mass due to mass increasing asymptotically with velocity. However, since the event horizon for black hole behaves exactly like asymptotic barrier to reaching the speed of light (we'll say for a stationary non-charged black hole to keep things simple), shouldn't the effects translate to a black hole? Ie once you're in a black hole (ie the mass that was there when the black hole formed), wouldn't the mass be essentially traveling backwards in time, fastest the further it was from the event horizon? For clarity sake, for matter, approaching the speed of light and approaching the event horizon of black hole are essentially the same with respect to redshift/blueshift, space and time dilation. There for approaching an event horizon from the inside of a black hole would be like de-accelerating toward the speed of light (v>c). So matter in a black hole should be falling outward toward the event horizon. All of this I am of course imagining from the reference frame of a distant observer. Just curious if I'm way off base on this.


Valley meaning explanation for foreigner

English is not my native language and I have some hard time translating this word. I was searching in couple dictionaries(both paper and online) and could not find it. Could anyone provide me definition of word "Valley". example usage: > Suggest using only the valley degree of freedom   > > > valley and spin degeneracy > > Valley relaxation    Thanks in advance!


Is TRUE invisibility possible?

First off, this question may seem highly stupid at first, but it arose from past cartoon memories. In the cartoon Tom & Jerry from the 1940s, there was already hypothesizing over cream that, when applied, makes you invisible. I have looked into invisibility, and I see that nothing surprising has been done. For example, only success in certain magnitudes, presets, lights, etc., but not enough for this to be any breakthrough yet. So my question is: Is an invisibility cloak that can make a completely visible object in visible light (me, for example) completely invisible (after I put it over me, my torso/body appears to magically vanish) to someone whilst wearing it possible, with just a cloak itself and nothing else? Truly invisible means: 1.It can't be camouflaged; you must literally appear invisible with no distortion, traces, or signaling that someone or something is/was there. 2.It must be only accomplished by a cloak itself, and no add-ons or special presets must be met; no light tricks, external items needed, etc. How about "vanishing cream"? Is It somewhat possible?


What is the speed for an object that travel close to the speed of light?

I have some questions regarding Einsteins theory of Relativity that should be fairly easy to answer. Lets say we make an experiment where we have a rocket (with an astronaut inside) that travels very close to the speed of light and it is heading away from Earth. Also back on earth I have an observer that makes measurements regarding the rocket that travels very close to the speed of light. **Question 1:** What is the speed of the rocket that the earth observer would measure? Lets change the above experiment and place the rocket very close to the event horizon of a black hole. According to what I have heard and read the rocket would seem to stand still as our observer see it back on earth. **Question 2:** What is the speed that the earth observer would make about the rocket in this situation? **Question 3:** Are the two experiments the same or are they somehow fundamentally different? Would an object seem standing still while traveling very close to the speed of light?


Gauge fixing choice for the gauge field $A_0$

In many situations, I have seen that the the author makes a gauge choice $A_0=0$, e.g. Manton in his paper on the force between the 't Hooft Polyakov monopole. Please can you provide me a mathematical justification of this? How can I always make a gauge transformation such that $A_0=0$? Under a gauge transformation $A_i$ transforms as $$A_i \to g A_i g^{-1} - \partial_i g g^{-1},$$ where $g$ is in the gauge group.


Why do we must initially assume that the wavefunction is complex?

The sound waves are real, and they can interfere, so corresponding apparat may be used in quantum mechanics. We also may use the time dependence in a form of orthogonal matrix multiplying the initial constant vector. So why do we must postulate that the wavefunction is initially complex? This question is very similar to the corresponding questions at the site, but there I didn't find the answer.


Why is Kirchoff's Current Rule valid?

The fact that Kirchoff's Current Rule is valid means that whatever current flows in flows out. But this will only be valid for a steady current circuit, that is, when there is no accumulation of charge. But there is no reason why there shouldn't be even small accumalation of charge and therefore inconsistencies in current, is there? Somehow it all just doesn't seem very intuitive. Am I looking at something the wrong way? Is there a proper explanation?


Why is there an $E^2$-term in the expression for the polarization density in nonlinear optics

In nonlinear optics the polarization density can be expressed as $$P=\epsilon_0(X(1)(E)+X(2)(E,E)+\dots)$$ As far as I know, $P$ arises from the dipole moment in materials, and $E$ is an applied electric field in this equation. Terms involving $E^2$ and $E^3$ cause the phenomenon of nonlinear optics. However, I don't understand why these $E^2, E^3 \dots$ terms appear in this equation? What's the physical meaning?


Effect of charges near a parallel plate capacitor

If I charged a parallel plate capacitor. And then, I insert a charged body near one of the plates. Will there be any interactions like attraction or repulsion?   What if I disconnected the battery?


Parabolic Motion

I was wondering if I was right about this simple problem: A projectile is shot at an angle of 37.0° with the horizontal. The velocity of the projectile at its peak height is 16 m/s. Then I was told to find the velocity in x direction. So I wrote: Given that at its peak, the velocity of the projectile in y direction is equal to 0, but stated that at its peak, the velocity of the projectile is 16 m/s, this must be the velocity in x direction. So my final answer was: 16 m/s. But then, my teacher just said it was wrong. That I had to find Vx using the formula Vx= Vcos(37°). I told her that 16 m/s could not be the initial velocity, since the problem says that 16 m/s is the velocity at its peak, therefore it must be Vx. Am I right? If so, can you tell me a good explanation besides the one I said to prove my teacher wrong. If not, what am I missing?


Why is the gas halo of the Milky Way so hot?

I have read on the webpage of NASA that there is a massive hot gas halo around our galaxy. Its temperature is between 100,000 and 1 million Kelvins or more. I do not understand why is it so hot. The temperature of the cosmic background radiation is only 2.7 Kelvins. The stars of our galaxy warm it up or one (or more) process in the gas? UPDATE: The question is about circumgalactic medium (CGM) not about the gases within our galaxy.


Are the physical laws scale-dependent?

If you read the article "More Is Different", by P.W. Anderson (Science, 4 August 1972), you will find a deep question: are the physical laws dependent of the size of the system under study? As an example, we can ask ourselves, are the description of a hundred of atoms more than simply one hundred times the description of one atom alone? Of course we have interactions, but, are these interactions dependent of the number of particles implied?


Principle of relativity - a second, equivalent form, using invariants

Most people state the principle of relativity like this: > "The rules of physics must take the same form in all inertial frames." Question: is this an equivalent way of saying the same thing: > "The rules of physics must be _expressible_ using only (Lorentz) > invariants." Note the word _expressible_. A given rule could be expressed in two different ways. A statement using action integrals with a Lagrangian would be an example of using only Lorentz invariants. A statement using covariant differential equation would not.


Is gravity a vector or tensor function and does gravity have velocity or is it instantaneous?

If gravity exists quantumly it must take up space, have frequency, velocity, etc. Perhaps gravity is too fine a frquency for detection.


Solving the 1D Schrodinger Equation for a Free Particle - Two Different Methods?

So starting from the time dependent schrodinger equation I perform separation of variables and obtain a time and spatial part. The spatial part is in effect the time independent schrodinger equation. Since we are dealing with a free particle I can take the time independent equation, set V = 0 and solve. I can do this successfully to obtain : $Ae^{+i\sqrt{{2mE}/{\hbar^{2}}}x}+Be^{-i\sqrt{{2mE}/{\hbar^{2}}}x}$ My lecturer has a small section titled : > Solving for the Free Schrodinger Equation > > $$V=0$$ > > $$\frac{\hbar^{2}}{2m}\frac{\partial^2\psi}{\partial x^2}+E\psi=0$$ > > $$E=\frac{p^2}{2m}$$ > > $$\psi=Ce^{-{iEt}/{\hbar}+{ipr}/{\hbar}}$$ > > This is the solution to the free TISE and TDSE. He seems to be doing the same thing as me initially but he's obtained a different result ? Also, the first section of his answer :$e^{-{iEt}/{\hbar}}$ is the solution to the time part of the equation (described above).


Comprehensive list of $\mathcal{N}=2$ supersymmetric topological quantum field theories in $d=4$

Could anyone provide a comprehensive list of $\mathcal{N}=2$ supersymmetric topological quantum field theories in $d=4$? I know of one - Kapustin-Witten TQFT, but I do not know of any more. Thanks!


Computing Curvature via Cartan Formalism

Given a metric $g_{\mu \nu}$, one can select an orthonormal basis $\omega^{\hat{a}}$ such that, $$ds^2= \omega^{\hat{t}}\otimes\omega^{\hat{t}} - \omega^{\hat{x}} \otimes \omega^{\hat{x}} - ...$$ By employing the Cartan structural equations, one can 'read off' the components of the Ricci tensor and Riemann curvature tensor **in the orthonormal basis**. My question is: **how do I take my curvature tensors in the orthonormal basis back to the coordinate basis after?**


Explaining Walter Lewin's "Complete Breakdown of Intuitiion"

This Walter Lewin lecture hinges on the solenoid producing a non-conservative field. I read in the Feynman Lectures that "all the fundamental forces in nature appear to be conservative" (Vol 1, 14.5) which comes decades after Faraday's Law of Induction so surely he would have known about it. 1) Who is right? Or, what did Feynman _really_ mean? 2) In Lewin's final diagram there are two voltmeters which, apart from a section of wire, attach in the same place. What happens as you bring the two attachment points together to meet? I can't see, in a physical on-the-table experiment, how one voltmeter is "measuring in the other direction".


Proof of the inverse square law for a uniformly bright sphere

On page 8 of "Radiative processes in Astrophysics" by Rybicki and Lightman they have a proof that the flux of a uniform sphere is inversely proportional to the distance of the sphere from the observer. I don't understand why they claim $\theta_c=\sin^{-1}R/r$, where the geometry is shown on pg 8, figure 1.6. It seems to me that its should rather be $\theta_c=\tan^{-1}R/r$. But then I don't think the proof would work. Any help in clarifying this for me would be greatly appreciated.


Euclidean quantum gravity and gravitational instantons

I have some questions concerning the calculations made by Hawking in the 80-90's using Euclidean (canonical) quantum gravity on gravitational instantons. Were those tunnelings only between identical closed FRW metrics (in the minisuperspace approximation where the only unfrozen mode is a(t) ) with different a (tunneling from $a_1$ to $a_2$) but with identical topology, or could those worm-holes represent topological changes or something more exotic? I tried to read this paper: http://siba.unipv.it/fisica/articoli/P/Physical%20review%20D_vol.28_no.12_1983_pp.2060-2975.pdf (Hawking Hartle 1983) but couldn't clearly find the answer to my question. Said in other words, what kind of transitions can be described by these instantons processes? and, if Euclidean gravity is a dead-end as it seems widely considered today, are those results totally irrelevant nowadays? Thank you.


How much force is required to propel a bicycle?

A friend of mine had an idea to attach a motor to a bicycle with the idea to be able to turn it on and have it maintain average bike speed, apparently 15-20 mph. Neither of us have taken physics classes before, though, so we aren't too sure how to calculate how much force (torque?) the motor has to put out. Any help would be appreciated, and I apologize for my general ignorance (and Imperial units).


Square of the Pauli matrices and the identity matrix

The square of any of the three Pauli Spin matrices is equal to the identity. Is there any physical meaning to this? Would you expect it? Maybe in the context of the $SU(2)$ group?


What conductance is measured for the quantum spin Hall state when the Hall conductance vanishes?

It's probably just a definition, but what did König et al. actually measure when he confirmed the existence of surface states in CdTe/HgTe/CdTe quantum wells (see http://arxiv.org/abs/0710.0582)? According to most reviews (e.g. http://www.annualreviews.org/doi/pdf/10.1146/annurev- conmatphys-062910-140538): > "However, because the Hall conductance of the QSH state vanishes, it is > clear that the TKNN or Chern number discussed above, which corresponds to > the the value of the Hall conductance in units of e2/h, cannot provide a > useful classification of the QSH state. This issue has been addressed within > both the topological band theory (23) and the topological field theory (23). > Accordingly, the proper topological invariant is valued in the Z2 group > containing only two elements, 0 or 1, with 1 corresponding to the > topologically nontrivial QSH insulator and 0 corresponding to a > topologically trivial insulator with no robust gapless edge states." and > "Because spin-orbit coupling destroys spin conservation, there is no such > thing as a quantized SH conductance in the QSH effect. This is another way > to understand why the correct topological invariant for the QSH effect is Z2 > and not Z. Finally, the BHZ Hamiltonian predicts a single helical edge state > per edge." I don't understand this. So there is no charge conductance, but we measure charge conductance? What is the difference between spin and charge conductance? I thought König did measure a charge conductance which was exactly twice the Hall conductance (e^2/h) (For me that's quantized...). Does this mean there actually are only two states that lead to twice the Hall conductance? Also: Why is there only a single helical edge state per edge? Why must we have at least one and why can't we have, let's say, two states per edge?


Ziolkowsky Rocket Equation

I've looked at multiple sources on the internet and just can't wrap my mind around it. If I reference the wikipedia page on it; http://en.wikipedia.org/wiki/Tsiolkovsky_rocket_equation I am getting stuck on certain places. $V_e=V-v_e $ Why does this occur? What integration limits do we use to derive this and why? $\Delta V\ = v_e \ln \frac {m_0} {m_1} $ (I am only concerned with this equation up to this part) If someone could explain it from the beginning in an easier fashion that would be great but solving these doubts for me would also help.


How does the proof of operator commutativity work with non-continuous operators?

In some books, a proof that if two self-adjoint operators $A$ and $B$ share a common eigenbasis $\\{\phi_n\\}$, then they commute is given as follows : > For any $\phi_n$, $$AB\ \phi_n = a_n\ b_n\ \phi_n = BA\ \phi_n,$$ and so $AB > - BA = 0$. > > Then, for any vector $\psi = \sum\limits_{n=1}^\infty c_n\ \phi_n$, > > $$(AB-BA)\ \psi = (AB-BA)\sum\limits_{n=1}^\infty c_n\ \phi_n= > \sum\limits_{n=1}^\infty \ (AB-BA)c_n \phi_n = 0$$ But how is the third step on the to last line valid, where the operator $C = (AB-BA)$ is moved across the summation sign? $C (\lim f_n) = \lim (C f_n)$ is only valid if $C$ is continuous. But here, while $C$ is linear and self- adjoint, it may not be continuous!


Is it possible for a material to have a polarization field greater than the applied field?

In the case of a dielectric (LIH at least, since that is all I've studied), the polarization field is always less than the applied field. In the case of a conductor, the polarization field is equal to the applied field, which is why there is no electric field inside conductors. Is there any sort of material which has a polarization field greater than the field applied to it? My initial thought is no, since you would be creating a more powerful electric field than you started out with, and that would be a way of creating energy, right?


Entropy of radiation emitted into space

In several papers I see something equivalent to the following expression for the entropy of radiation given by an astronomical object such as the Sun (assuming the object can be approximated as a black body): $$ s = \frac{4}{3}\frac{u}{T}, $$ where $u$ is the total flux of radiation energy through a spherical shell surrounding the object, $s$ is the entropy flux through the same imaginary surface, and $T$ is the black-body temperature of the object (and hence also of the radiation). It is also generally stated that no entropy is produced during the process of emitting radiation. In a much smaller number of papers I see a formula that corresponds to just $s=u/T$, which is what I would expect. So the short version of my question is, which of these is correct for an astronomical body? But please read the rest, so you can understand why I'm so confused about it. I understand the derivation of the 4/3 formula by considering a photon gas enclosed in a piston (see below), but in the context of a continuously emitting body like the Sun it doesn't seem to make sense. The issue is that if, over a given time period, the body loses an amount of heat $Q$ then this must balance the increase in energy of the radiation field $U$, i.e. $U=Q$. The body loses entropy at a rate $Q/T$, and if radiation is really a reversible process then this should equal the gain in entropy of the radiation, which should therefore be $U/T$. But according to the above formula it's actually $4U/3T$, meaning that the total entropy increases by $Q/3T$. The 4/3 formula above was derived by Planck (in his book "the Theory of Heat Radiation", of which I've read the relevant chapter), who considered a photon gas in a sealed cylinder of finite volume. At one end of the cylinder is a black body and at the other is a piston. The radiation comes into equilibrium with the black body and exerts a pressure on the piston. If one reversibly (i.e. slowly) allows the piston to move then this causes some heat to be lost from the black body. In this case it turns out that $U=3Q/4$, with the discrepancy being due to the fact that the radiation field loses energy when it does work on the piston. The entropy balance then requires that the entropy of the photon gas increases by $4U/3T$, as above. The point is, I can't see how the radiation being emitted by the Sun can be seen as doing work on anything. At first I thought it might be doing work on the outgoing radiation field. So let's draw an imaginary shell around the Sun again, but this time let the shell be expanding at the speed of light. Perhaps the radiation inside the shell is doing work on the radiation outside it, and that's what's "pushing" it away from the Sun? But it seems to me that for anything inside the shell to have an effect on anything outside it, some kind of influence would have to travel faster than light, so I don't think that can be right. In any case it's well known that for a normal gas (made of matter), expanding against a piston is quite different from just expanding into a vacuum. In the former case the temperature and internal energy decrease, because the molecules lose energy in pushing the piston, whereas in the latter case they both remain constant. I haven't found any source that addresses the question of why this would be different for a photon gas. So it seems like the emission of radiation from a body like the Sun into space is quite different from the emission of radiation into a sealed piston, and I'm puzzled as to how the same formula can apply. Below I've listed some possible resolutions to this which seem plausible to me. I haven't been able to find any sources that directly address this issue, or state any of these positions.   1. The emission of radiation into space is an irreversible process after all. $U=Q$, and the total entropy increases by $Q/3T$. (But then, what happens when radiation is absorbed by a body at a similar temperature? Surely the entropy doesn't decrease.)   2. There is some weird sense in which the outgoing radiation can be thought of as doing work on something, so $U\ne Q$. (If so, why does nobody ever explain this subtle and important point?)   3. $Q=U$, but the entropy of radiation emitted into space is actually different from the entropy of a photon gas in a sealed cylinder, such that its entropy is given by $U/T$, not $4U/3T$. (This one actually seems the most reasonable to me. The radiation in the closed cylinder has rays travelling in all directions, whereas the radiation coming off the Sun only has rays travelling in directions away from its surface, so it seems reasonable that they would have different entropies. This would mean that the 4/3 formula doesn't apply to the radiation emitted by astronomical bodies after all - but if that's the case then it's an extremely widespread mistake.) Any insights into which, if any, of these is correct would be greatly appreciated - as would any references to sources that directly address the relationship between Planck's photon-gas piston and the emission of radiation into empty space.


How can I determine density of a gas only given temperature?

I have a homework problem where exhaust is traveling through an exhaust system (assumed to be air for simplicity) from and engine and then released into the atmosphere. The exhaust is at a temperature of 250F. I need to find the pressure at the beginning of the exhaust system. I've been able to set this up and created an equation to solve for the pressure. I assumed that the pressure of the atmosphere where it it exits is 0 atm. But the problem is that the pressure depends on the density of the exhaust (air). I was only given temperature of the exhaust in the problem and I need a second property to determine density. Is there another way I can determine density? I am also given the volumetric flow rate and the area of the piping system


Gravitational waves as dark energy?

Is the energy carried by gravitational radiation a viable candidate for $\Lambda$ / dark energy?


What is a dynamical variable

I see the phrase 'dynamical variable' in several of my texts and they don't really define it. A google search doesn't yield much either. I have gathered that a dynamical variable can be represented in terms of position and momentum, and examples are velocity and kinetic energy, but what is the definition?


Is it possible to produce images of pair production in home-made cloud chamber?

There are some nice pictures on the web showing the counter-spiralling paths of an electron positron pair produced in a bubble chamber with a uniform magnetic field, for example:- ![e+ e- pair production](http://www.bigganblog.com/wp- content/uploads/2012/03/I15-70-positron.jpg) Would it be practical to produce a cloud chamber and Helmholz Coils capable of imaging such events in an ordinary domestic garage? What sort of source would I need? How often might such events be detected?


Reluctance of torus shaped iron core with embedded wire loop

**Imagine a circular wire loop** (r = 50mm), the wire has an assumed diameter of zero, **which is embedded in a torus shaped iron core with a circular cross-section** of R = 10mm. A current in that loop would cause a circular magnetic field around the wire. **Is there any possibility to calculate the reluctance of that core?** I'm looking for a solution for weeks now, without any success. A solution for harmonic currents is desired, but I would even be happy for a DC solution. ![enter image description here](http://i.stack.imgur.com/y65Cw.png) * * * # CONTEXT **to explain what this is all about.** My real geometry looks as following: ![enter image description here](http://i.stack.imgur.com/gVvQ5.png) A torodial coil surrounded by a core with a cross section of a rounded rectangle. So I'm interested in the reluctance of the greyish part (and the other corners). If you put all corners together you'd get the mentioned torus. The green lines are the magnetic flux, the rectangle in the middle the torodial coil. For high frequencies and/or high conductive and/or high permeable materials the influence of the corners is negligible, for my case unfortunately not. I'd guess there is no analytic solution, but any idea which could get me close to it, would help. Thank you! * * * * * * # Attempted solution # Preface If one wants to calculate the permeance $P$ of a rectangular bar: ![enter image description here](http://i.stack.imgur.com/YWrk8.png) it is an easy task: $$P = \frac{\mu a b}{L} ~~~~ \rightarrow ~~~~ P\propto ab ~~~~and~~~~ P\propto\frac{1}{L}$$ where $\mu$ is the material constant. (Permeability) But my geometry is a **torus with just a quarter of its circular cross section** and the field $V$ **passes through it parallel to the circumference** of the (full) cross section: ![enter image description here](http://i.stack.imgur.com/Uq1n1.png) How can I calculate the permeance of this geometry, when there are the same proportional relations as above? * * * # Attempted solution I divide my geometry in $N$ hollow toruses with constant wall thickness $\Delta R$ and medium length element $\Delta L$, so the field passes an area of $\Delta A$: ![enter image description here](http://i.stack.imgur.com/nXbbl.png) A little piece of the radius $R$ is $\Delta R = \frac{R}{N}$. Now one can calculate: $$\Delta P_{n} = \frac{\mu \Delta A_n}{\Delta L_n} $$ with $$ \Delta A_n = \pi \bigg( (r+(n+1) \Delta R)^2-(r+n \Delta R)^2\bigg) $$ (Consider the full torus circumference, not just a quarter as displayed) and $$ \Delta L_n = \frac{\pi}{2} (2n+1) \frac{\Delta R}{2} $$ (but quarter cross section!) follows: $$P = \sum^{N-1}_{n=0} \Delta P_{n} = \mu\sum^{N-1}_{n=0} \frac{\pi(2r\Delta R+(2n+1)(\Delta R)^2)}{\frac{\pi}{2}(2n+1)(\frac{\Delta R}{2})}~~~~~~~~~~~~~~~~~~~~~~~~~$$ $$= 4\mu\sum^{N-1}_{n=0} \frac{2r\Delta R+(2n+1)(\Delta R)^2}{(2n+1)(\Delta R)} $$ $$= 4\mu\sum^{N-1}_{n=0} \Bigg( \frac{2r}{(2n+1)} + \Delta R \Bigg)~~~~~~~~~~ $$ $$= 4\mu \Bigg( R + 2r \sum^{N-1}_{n=0} \frac{1}{(2n+1)} \Bigg)~~~~~~~~~~ $$ **And this series does not converge for $N\rightarrow\infty$.** Which is physically seen not possible, so there must be a problem with the math. Do you see what I'm missing?


On numerically solving the Schrödinger equation

I just read a paper 'A pocket calculator determination of energy eigenvalues' by J Killingbeck (1979). Link: http://iopscience.iop.org/0305-4470/10/6/001 I have some questions about section 2.   1. Why $R<0$ implies the $E$ estimate was too high and $R>1$ implies too low? And is it possible that $0<R<1$ remains?   2. Why "if the starting point is chosen at $r=h(l+1)$, then $R(hl)$ can be assigned any finite value without disturbing the calculation at larger $r$ values" ?   3. How to guess a trial energy $E$ (especially the ground state energy)? (Actually I'm interested in a later paper by Killingbeck (1982) 'Finite- difference methods for eigenvalues'(Link: http://iopscience.iop.org/0022-3700/15/6/009), in which the author applies the method in 'A pocket calculator determination of energy eigenvalues' to some potentials with highly singularity at origin) * * * _mod note: copying the text of whoplisp's post here since it isn't actually an answer to the question_ They convert the Schroedinger equation into a finite difference equation (instead of a variational approach): $-\frac{1}{2}\Delta\psi +V\psi=E\psi$ $V$ is a central potential Ansatz with solid harmonic $\xi_l$ of degree $l$ (e.g. $x$ or $xy$) $\psi=\xi_l\phi(r)$ leads to $2r(V-E)\phi=rD^2\phi+(2l+2)D\phi$ Notation: $D=\frac{d}{dr}$ finite difference approximation $h^2D^2\phi \rightarrow \phi(r+h)+\phi(r-h)-2\phi(r)$ $2hD\phi \rightarrow \phi(r+h)-\phi(r-h)$ crucial step: **introduce ratio variable** $R(r)$: $\phi(r+h)=R(r)\phi(r)$ convert Schroedinger equation in recursive equation to calculate $R(r)$: $2r[1+h^2(V-E)]=R(r)[r+h(l+1)]+[r-h(l+1)](R(r-h))^{-1}$ > the corresponding 1D Schroedinger equation is: > > $2[1+h^2(V-E)]=R(x)+(R(x-h))^{-1}$ if $r=h(l+1)$ is chosen then $R(hl)$ can be assigned any finite value without disturbing the calculation at large $r$ values to find the ground states with angular momentum $l$ start with an estimate for $E$, some non-zero $R(hl)$ and some small $h$. $R(r)$ as $r$ increases will either become negative or pass through unity from below. first thing happens when guessed $E$ was too high second thing indicates $E$ was too low (because $\psi$ increases with $r$) you just try various values and 'sandwhich' the solution example for $V=\lambda r-1/r$: $\lambda=1$, $h=.1$, state=2p: 1.9759


Fundamental Representation of $SU(3)$ is a complex representation

Let in a $D(R)$ dimensional representation of $SU(N)$ the generators, $T^a$s follow the following commutation rule:   $\qquad \qquad \qquad [T^a_R, T^b_R]=if^{abc}T^c_R$. Now if $-(T^a_R)^* = T^a_R $, the representation $R$ is real. Again if we can find a unitary matrix, $V(\neq I)$ such that $ \qquad \qquad \qquad -(T^a_R)^*=V^{-1} T^a_R V \quad \forall a $ then the representation $R$ is pseudoreal. If a representation is neither real nor pseudoreal, the representation $R$ is complex. **Claim: One way to show that a representation is complex is to show that at least one generator matrix $T^a_R$ has eigenvalues that do not come in plus- minus pairs.** Now let us consider $SU(3)$ group. The generators in the fundamental representation are given by $T^a =\lambda^a/2; \quad a=1,...8$,   where $\lambda^a$s are the Gell-Mann matrices. We see that $T^8$ has eigenvalues $(1/\sqrt{12}, 1/\sqrt{12}, -1/\sqrt{3} )$. My doubt is: **According to the claim, is the fundamental representation of $SU(3)$ a complex representation?**


Are there old aged particles?

To measure the lifetime of a specific particle one needs to look at very many such particles in order to calculate the average. It cannot matter when the experimentalist actually starts his stopwatch to measure the time it takes for the particles to decay. If he measures them now or in 5 minutes makes no difference, since he still needs to take an average. If he measures later there will be particles out of the picture already (those who have decayed in the last 5 min), which won't contribute and the ones his measuring now behave (statistically) the very same, of course. I have just read the following in _Introduction to Elementary Particles_ by Griffiths: > Now, elementary particles have no memories, so the probability of a given > muon decaying in the next microsecond is independent of how long ago that > muon was created. (It's quite different in biological systems: an 80-year- > old man is much more likely to die in the next year than is a 20-year-old, > and his body shows the signs of eight decades of wear and tear, But all > muons are identical, regardless of when they were produced; from an > actuarial point of view they’re all on an equal footing.) But this is not really the view I had. I was imagining, that a particle that has existed for a while is analogous to the 80 year old man, since it will probably die (decay) soon. It just doesn't matter because we are looking at a myriad of particles, so statistically there will be about as many old ones as babies. On the other hand it is true that I cannot see if a specific particle has already lived long or not; they are all indistinguishable. Still I am imagining particles as if they had an _inner age_ , but one just can't tell by their looks. So is the view presented in Griffiths truer than mine or are maybe both valid? How can one argue why my view is wrong?


Finding distance when the force is a function of time

I'm having trouble with this homework question > A mysterious rocket-propelled object of mass 49.0 kg is initially at rest in > the middle of the horizontal, frictionless surface of an ice-covered lake. > Then a force directed east and with magnitude $F(t) = (16.3\text{ N/s})t$ is > applied. > > How far does the object travel in the first 5.50s after the force is > applied? For some reason I'm not getting the correct answer. I think maybe I'm not understanding how to use the magnitude of the force they are giving me. I know how to use a constant force, but is this different because the force is a function of time? I tried starting it like this: $$\begin{align}F &= ma\\\ 16.3(t) &= (49)(a) \\\ 16.3(5.5) &= (49)(a) \\\ 89.65 &= (49)(a) \\\ a &= 1.82959\ \mathrm{m/s^2}\end{align}$$ So now we know that: $$\begin{align}t &= 5.5\text{ s} \\\ a &= 1.82959 m/s^2 \\\ V_o &= 0\end{align}$$ So I plug it into my equation: $$\begin{align}\Delta X &= V_o t + 1/2 a t^2\\\ \Delta X &= (0)(5.5) + (1/2)(1.82959)(5.5)^2\\\ \Delta X &= 27.7\text{ m}\end{align}$$ But that's not the right answer.


"Redshifting" of forces in stationary space - times

Here's the problem statement: Let $(M,g_{ab})$ be a stationary spacetime with timelike killing field $\xi ^{a}$. Let $V^{2} = -\xi _{a}\xi ^{a}$ ($V$ is called the redshift factor). (a) Show that the acceleration $a^{b} = u^{a}\triangledown _{a}u^{b}$ of a stationary observer is given by $a^{b} = \triangledown^{b}\ln V$. (b) Suppose in addition that $(M,g_{ab})$ is asymptotically flat. Then, the "energy as measured at infinity" of a particle of mass $m$ and 4 - velocity $u^{a}$ is $E = - m\xi _{a}u^{a}$. Suppose a particle of mass $m$ is held stationary by a (massless) string, with the other end of the string being held by a stationary observer at infinity. Let $F$ denote the magnitude of the local force exerted by the string on on the particle. According to part (a), we have $F = mV^{-1}(\triangledown ^{a}V\triangledown _{a}V)^{1/2}$. Use **conservation of energy** arguments to show that the magnitude of the force exerted on the **other end** of the string by the observer at infinity, is $F_{\infty } = VF$. My attempt: (a) A stationary observer's 4 - velocity must be proportional to the time - like killing vector and it must be normalized to -1 so we find that for a stationary observer, $$u^{a} = \frac{\xi ^{a}}{(-\xi ^{c}\xi _{c})^{1/2}}$$. Now we compute, $$\triangledown _{b}u^{a} = \frac{\triangledown _{b}\xi ^{a}}{(-\xi ^{c}\xi _{c})^{1/2}} + \frac{\xi ^{a}\xi ^{c}\triangledown _{b}\xi _{c}}{(-\xi ^{c}\xi _{c})^{3/2}}$$ so $$u^{b}\triangledown _{b}u^{a} = \frac{\xi^{b}\triangledown _{b}\xi ^{a}}{(-\xi ^{c}\xi _{c})} + \frac{\xi ^{a}\xi^{b}\xi ^{c}\triangledown _{b}\xi _{c}}{(-\xi ^{c}\xi _{c})^{5/2}} = -\frac{\xi^{b}\triangledown ^{a}\xi _{b}}{(-\xi ^{c}\xi _{c})}$$ where the second term vanishes because it is a contraction of a symmetric tensor with an anti - symmetric one and I have swapped the indices in the first expression using killing's equation. Therefore, $$a^{a} = u^{b}\triangledown _{b}u^{a} = \frac{1}{2}\frac{\triangledown ^{a}(-\xi^{b}\xi _{b})}{(-\xi ^{c}\xi _{c})} = \frac{1}{2}\triangledown ^{a}\ln V^{2} = \triangledown ^{a}\ln V$$ as desired. (b) This is where I'm totally stuck. As far as conservation of energy goes, we know that $E$, as defined above, is a conserved quantity **along** the worldline of the stationary particle; physically $E$ is the energy needed to bring in the particle from infinity to its orbit. Here we have a stationary observer at infinity holding this particle stationary by a long thread. There is a tension force at the end the observer holds and at the end the particle hangs by. Let's say the observer exerts the force $F_{\infty }$ at event $P_{1}$ and the particle feels the local force $F$ at event $P_{2}$. As far as I can tell, all we know is that in between these events, $E$ is constant. But how do I relate $E$ to $F_{\infty}$ and how do I do this using the conservation of energy explained above? I should note that I tried something on a whim and looked at $\triangledown _{b}E$. We know that for the stationary particle hanging from the string, to which this total energy is attributed, $$E = -m\xi _{a}u^{a} = -\frac{m\xi_{a}\xi^{a}}{(-\xi _{c}\xi^{c})^{1/2}} = m(-\xi^{c}\xi_{c})^{1/2}$$ so if we compute the derivative we get $\triangledown _{b}E = mV\triangledown _{b}\ln V$ thus $(\triangledown ^{b}E\triangledown _{b}E)^{1/2} = mV(\triangledown ^{b}\ln V\triangledown _{b}\ln V)^{1/2} = VF$ but I have NO idea how this quantity is related to $F_{\infty }$, if at all. If it is related somehow I have no idea how one uses conservation of energy to arrive at the relation. Thanks in advance!


How can enzyme/substrate reactions that adhere (largely) to quantum theory also require 'Newtonian' consideration of gravity?

I'd just like to ask for a little clarification here due to confusion from interdisciplinary studies. I'm currently reading the 1976 paper related to the recent 2013 Nobel Prize for Chemistry, by 2 of the laureates Warshel and Levitt — _Theoretical Studies of Enzymic Reactions: Dielectric, Electrostatic and Steric Stabilisation of the Carbonium Ion in the Reaction of Lysosyme_ — and have recently been reading into quantum chemistry, as well as studying biochemistry and have had further lectures in Physics to boot. I'm wondering if the points made here are comparable to points I've heard in mechanics lectures, about a cannonball not being modelled as a quantum object, since every subatomic particle would be too computationally intensive, so a coarse-grain model is used (i.e. Newton's "classical" laws). I wouldn't be so quick to draw parallels between mechanics lectures and enzyme biochemistry but it seems to be in the language they are using here (see highlights below). From what I understand (in mechanics) only 3 of the 4 'fundamental forces' (strong, weak, and EM) only strictly apply at the quantum/microscopic scale, and scientists are as it stands struggling to work gravity into the setup. Is this worth considering in this paper - if an enzyme is considered all of a sudden due to its size as a quantum object does this entail ignoring gravity QM now in favour of 'Newtonian', inertial frame-of-reference type mechanics, or considering it in some other 'Newtonian' manner? The introductory section is as follows (full paper linked above): > The conformation of the enzyme-substrate complex can be adequately studied > using empirical energy functions based on the “classical” contributions of > bond stretching, bond angle bending, bond twisting, non-bonded interactions, > etc. The mechanism and energetics of the enzymic reaction can only be > studied using **a quantum mechanical approach**. Previous quantum mechanical > calculations on enzymic reactions have been limited in several respects. In > the first place, they deal with an over-simplified model system consisting > of only small fractions of the atoms involved in the real enzyme-substrate > interaction. Second, all available quantum mechanical methods treat the > reaction as if in a vacuum and are, therefore, unable to include the > dielectric, which has a very important effect on the energy and charge > distribution of the system. This limitation does not depend on the actual > quantum mechanical scheme, being equally valid for the simplest Hiickel > treatment and most extensive _ab initio_ calculation. > > Because the **treatment of the whole enzyme-substrate complex quantum > mechanically is computationally impossible** , it is necessary to simplify > the problem and use a **hybrid classical/quantum mechanical approach**. The > method described here considers the whole enzyme-substrate complex: the > energy and charge distribution of those atoms that are directly involved in > the reaction are evaluated quantum mechanically, while the potential energy > surfac e of the rest of the system, including the urrounding solvent, is > evaluated classically. An important feature of the method is the treatment > of the “dielectric” effect due to both the protein atoms and the surrounding > water molecules. Our dielectric model is based on a direct calculation of > the electrostatic field due to the dipoles induced by polarizing the protein > atoms, and the dipoles induced by orienting the surrounding water molecules. > With the inclusion of the microscopic dielectric effect we feel that our > model for enzymic reactions includes all the important factors that may > contribute to the potential surface of the enzymic reaction. Because we use > analytical expressions to evaluate the energy and first derivative with > respect to the atomic Cartesian co-ordinates, it is possible to allow the > system to relax using a convergent energy minimization method. Here we use > the method to study the mechanism of the lysozyme reaction, with particular > emphasis on factors contributing to the stabilization of the carbonium ion > intermediate. We first outline our theoretical scheme, paying attention to > the **partition of the potential surface into classical and quantum > mechanical parts and to details of the microscopic dielectric model**. Then > we use the method to investigate factors that contribute to the > stabilization of the carbonium ion intermediate and deal with the relative > importance of each factor. The effect of steric strain is considered for > both the ground and the carbonium states and found to be of minor > importance. **Electrostatic stabilization** is treated at some length with > special emphasis on the balance between stabilization by induced dipoles > (polarizability) and stabilization by charge- charge interactions. The > results demonstrate the importance of electrostatic stabilization for this > reaction.


What is the Principle of Maximum Conformality?

I'm trying to understand this article about an advance in the theoretical understanding of QCD which centers on the Principal of Maximum Conformality. What is this Principle? In other words, what is being maximized and what does this tell us about the structure of QCD? Also, is this a new principle or a new application of an old principal? Here's the full paper on the principle's application to top physics.


How many atoms in the particle (AuNP)

Let's say I have to determine number of atoms of gold in ball-shaped nanoparticle with diameter of 75 nm. I need general method of solving such problems more than specific answer for only this question.


Why does a tire need to be filled with air?

The **Mini 4WD** 's tires aren't full of air, and it can run. Also, the tank doesn't have tires with air. So, the question is: why do real cars on the road need to be filled with air? What is the idea behind it? ![](http://i.stack.imgur.com/0ciAs.jpg)


Why is storage of spent nuclear fuel dangerous?

Just what the title states; there's a good deal of noise made about transport, and storage of spent nuclear fuel. Why all the hullabaloo when the fuel is all spent?


Bending a flat sheet

I am interested in the following problem: starting from a flat sheet, once can bend it into a (sector) of a cylinder isometrically. If further (orthogonal to the first plane) curvature is induced, the sheet will necessarily be stretched. The question is, how much stretch is locally caused by the increase in curvature? I could attempt to answer the point by classical continuum mechanics tecniques, but I am told differential geometry allows an answer to be given immediately. Could somebody please point me towards the answer, or suitable material to study? Many thanks


Cheap launching objects to orbit

Why slingatron hasn't been successful ? Same thing with gigantic nail clipper. Since we dont need great force when arm length increases, why couldn't we build gigantic nail clipper orbital launcher :) ? Provided of course that we had material able to sustain great compression stress. Disclaimer, no im not affiliated in any way with creators, dont know them, not even same country. Just thought that idea was cool.


Schrödinger's cat and the difficulty of macroscopic superposition state

The Schrödinger's cat was regarded as peculiar since we seldom encounter a superposition state in macroscopic scale: $$ | \mathrm{dead \,\,cat} \rangle + | \mathrm{alive \,\, cat}\rangle $$ We more often describe an unknown cat as $$ | \mathrm{dead \,\,cat} \rangle \langle \mathrm{dead \,\,cat} |+ | \mathrm{alive \,\, cat}\rangle \langle \mathrm{alive \,\, cat}| $$ without superposition. I often heard that it is difficult to prepare and maintain a large-scale superposition state. Similar difficulty also occurs in quantum computing. My question is, actually what is the reason for the difficulty to prepare and maintain a large-scale superposition state? If it is decoherence, why decoherence happens? Is that because of entropy?


Why is the $SO(4)$ symmetry of the Hydrogen atom called dynamical?

Why _dynamical?_ My previous quantum mechanics teacher could not answer it.


What is to be considered a "body" in physics?

Well, the question says it all; is there a definition of body in physics? What is to be considered a physical object and what it cannot?


Why a transistor is activated when I touch its base with my finger?

Is my body charged eletrically? If so, how much? Will it work with all types of transistors?


How to find maximum velocity

I stack a question about projectile question. The question was > _A projectile is being launched from ground level with no air resistance. > You want to avoid having it enter a temperature inversion layer in the > atmosphere a height $h$ above the ground (a) what us the maximum launch > speed you could give this projectile if you shot it straight up ? Express > your answer in terms of $h$ and $g$.(b)Suppose the launcher available shoots > projectiles at twice the maximum launch speed you found in part (a). At what > maximum angle above the horizontal should you launch the projectile ?_ I could solve the (a) part. How did was following (a) using the following formula to drive $V$ $\delta x$ = $\frac{V^{2}-Vi^{2}}{2g}$$ also we have $Vi = 0$, $\delta x = h$ I got $V = \sqrt{2gh}$ after that I think I need to use some kind of angle relative formula to create $arccosx$ or $arcsinx$ will be equal to some number then find the angle but I still don't have idea which formula do I need to use and find max angle. also Do I need to split $Vx$ and $Vy$ from $V$ ? One more question, I have seen some minimum spe


Force and Laws of Motion

While catching a fast moving cricket ball, why do fielders gradually pull back their hands backwards with the moving ball? Please advice which among the following is the correct answer. If both are incorrect kindly give the correct answer:      **A** : By pulling the hands backward the fielder increases the time during which the high velocity of the moving ball decreases to 0. Thus, the acceleration of the ball is decreased and therefore the impact of catching the fast moving ball is also reduced.      **B** : By pulling the hand backwards, the fielder gradually and continuously decreases the momentum of the moving ball and thereby reducing the velocity to zero.


Light bending surfaces and energy funneling properties

In the last year there has been fuzz regarding metamaterials studied for the purpose of cloaking, and it seem to me they are somewhat glorified 2D waveguides, but in any case it seems a reasonable application of this to funnel radiation captured over a wide area of incidence into smaller area, and i thought that this could be a cheap way to produce ignition radiation for a fusion reactor. Reasons why this wouldn't work, or not produce enough energy density? All stuff i've read about fusion ignition with lasers doesn't seem to depend too heavily on the coherence of the radiation


Possible colors of fire?

I have learnt that depending on the various gases those are involved in the reaction that produces fire, different colors (yellow, red or blue) of flames become visible. I have a question .. what are the possible colors that fire can possess? and do they (how, if yes) relate to the intensity? like I know blue is more intense than red, etc..


How does Delta baryon decay conserve angular momentum?

I'm a chemist so bear with me: I understand the Delta baryons $\Delta^{+}$ and $\Delta^{0}$ to be in some sense spin (and isospin) quartet states of the proton and neutron. These can decay straight to a proton or neutron through emission of a pion, which is spin-0. I've been puzzling about the mechanism by which this decay conserves spin. I note via Wikipedia that pions decay into either a lepton pair or a photon pair, for which it's easy to show spin is conserved, however pion production seems to induce a loss of $J=1$. This seems to suggest momenta of: $\Delta^{+} (J=\frac{3}{2}) \rightarrow{} p^{+} (J=\frac{1}{2}) + \pi^{0} (J=0)$ Am I missing a boson? Is my understanding of spin flawed? I assume that if this decay exists, similar things turn up everywhere else.


Electric field singularity

What's a general proof for the log-singularity of electric field near the edge of a uniformly charged surface?


How much better is salt water at letting through microwaves than pure water?

> **Radio Wave Attenuation** > > There are two general types of matter (substances) in the universe that > affect electromagnetic waves, conductors and insulators which are called > dielectrics by scientists. Most, but not all, conductors are metals, such as > copper, aluminum, silver and gold. However, salt water is also a rather poor > conductor! Most, but not all, dielectrics are non metals. Examples of > dielectrics are paper, plastic, Teflon, glass, ceramic and dry wood. Pure > water is a good dielectric substance! Source: http://www.madsci.org/posts/archives/2002-03/1015162213.Eg.r.html I read this paragraph, and was truly intrigued. Is it really true that salt water (NaCl) passes radio waves (specifically 3cm) through better than pure water? Why is this, and how large is the difference? Would it be noticeable in an experiment?


Why doesn't fusion contradict the 1st law of thermodynamics?

I was reading up on the 1st law of thermodynamics for my Chemistry exam and I was wondering why doesn't fusion contradict the 1st law of thermodynamics? The 1st law states that > The energy of an isolated system is constant or that whatever is put into the system, you get out of it, but in fusion you get more out of the initial reactions than you put in Hydrogen + Hydrogen = Helium etc. I am still a bit confused... Thanks for any help!


What is the definition of correlation length for the Ising model?

The correlation length $\xi$ is related to critical temperature $T_c$ as $$ \xi\sim|T-T_{c}|{}^{-\nu}, $$ where $\nu$ is the critical exponent.   1. Is this the **formal definition** of correlation length? If not, what is the formal definition of correlation length (for phase transition in the Ising model)?   2. Can you give a physical understanding of correlation length?


Lie algebra of axial charges

Starting from the lagrangian (linear sigma model without symmetry breaking, here $N$ is the nucleon doublet and $\tau_a$ are pauli matrices) $L=\bar Ni\gamma^\mu \partial_\mu N+ \frac{1}{2} \partial_\mu\sigma\partial^\mu\sigma+\frac{1}{2}\partial_\mu\pi_a\partial^\mu\pi_a+g\bar N(\sigma+i\gamma_5\pi_a \tau_a)N$ we can construct conserved currents using Noether's Theorem applied to $SU(2)_L\otimes SU(2)_R$ symmetry: we get three currents for every $SU(2)$. By adding and subtracting them, we obtain vector and axial currents.   We could have obtained vector charges quickly by observing that they are just isospin charges, so nucleons behave as an $SU(2)$ doublet (fundamental representation), pions as a triplet (adjoint representation) and sigma as a singlet (so basically it does not transform): $V_a=-i\int d^3x \,\,[iN^\dagger\frac{\tau_a}{2}N+\dot\pi_b(-i\epsilon_{abc})\pi_c]$ But if I wanted to do the same with axial charges, what Lie algebra/representation must I use for pions and sigma?   I mean, axial charges are $A_a=-i\int d^3x \,\,[iN^\dagger\frac{\tau_a}{2}\gamma_5N+i(\sigma\dot\pi_a-\dot\sigma\pi_a)]$ and I would like to reproduce the second term using a representation of Lie algebra generators of axial symmetry which act on $\sigma$ and $\pi$, but I don't know the algebra (I think it is $SU(2)$), neither the representation to use.   I tried to reproduce that form using the three matrices $T^1=\begin{bmatrix} 0&-i&0&0\\\i&0&0&0\\\0&0&0&0\\\0&0&0&0 \end{bmatrix}\quad T^2=\begin{bmatrix} 0&0&-i&0\\\0&0&0&0\\\i&0&0&0\\\0&0&0&0 \end{bmatrix}\quad T^3=\begin{bmatrix} 0&0&0&-i\\\0&0&0&0\\\0&0&0&0\\\i&0&0&0 \end{bmatrix}$ which should act on the vector $(\sigma,\pi_1,\pi_2,\pi_3)$, but I calculated their commutator and they don't form an algebra, so I think I'm getting wrong somewhere in my reasoning.


Why can movies get away with 24 frames per second, but games can't?

Movies look reasonably smooth at 24 frames per second; unless the screen is huge and you use your peripheral vision, you usually can't see each frame individually. This has lead to a large amount of textbooks claiming that the visual response time of the retina is slower than 1/24 seconds. However, games at 24 frames per second absolutely suck. The motion is so choppy it is unplayable. For a game, you seem to need at least 40 frames per second, and even at that rate you still see individual frames if the objects are moving quickly. Why is this so? Is this related to the fact that movies have motion blur due to a non-zero shutter speed? Is "jerkiness" mostly perceived in terms of "gap between object in frame 1 and frame 2" rather than visual response time (the gap is smaller in films due to blurred objects "connecting" across frames)? If so, doesn't this prove that our visual response time is much faster than 1/24 seconds? To make the question more precise: is this response determined by physical mechanisms, such as the relaxation times of pigments in the eye or the firing times of neurons? If so, which ones? Or is it mostly due some sort of postprocessing done by the brain, i.e. more to do with the cognitive than the physical side of perception?


Does Gödel preclude a workable ToE?

Gödel's incompleteness theorem prevents a universal axiomatic system for math. Is there any reason to believe that it also prevents a theory of everything for physics? * * * _Edit:_ I haven't before seen a formulation of Gödel that included time. The formulation I've seen is that any axiomatic systems capable of doing arithmetic can express statements that will be either 1) impossible to prove true or false or 2) possible to prove both true and false. This leads to the question: Are theories of (nearly) everything, axiomatic systems capable of doing arithmetic? (Given they are able to describe a digital computer, I think it's safe to say they are.) If so, it follows that such a theory will be able to describe something that the theory will be either unable to analyse or will result in an ambiguous result. (Might this be what forces things like the Heisenberg uncertainty principle?)


Covariance of rayleigh channel

**f** =[ _f1_ _f2_ _f3_ ]; **g** =[ _g1_ _g2_ _g3_ ]; _f1_ ,_f2_, _f3_ ,_g1_, _g2_ ,_g3_ are all independent identically distributed zero mean complex random variable. **h** = elementwise wise Schur-Hadamard product of **f** and **g**. What is the covariance matrix of **h**? Is that diagonal matrix?


The synthesis of $^{254}\text{No}$

How is $^{254}\text{No}$ synthesised? Could you explain the reaction where it is preceded by $^{208}\text{Pb}(^{48}\text{Ca}, 2\text{n})$? References to articles are well enough—I was somehow unable to find anything sufficiently detailed and informative.


electric-circuits

Why can't you connect the + of a device to one battery and the - to another battery (and have it work)?![enter image description here](http://i.stack.imgur.com/W9x5Q.png)


How can I find the motion equations of the 2-dim harmonic oscillator?

First of all: I am no physicist, so I am rather helpless. I need to find the moving equations of the 2-dim. harmonic oscillator. If it is possible it should be rather elementary, because, as I said, I am no physicist. I've heard that ONE possibility is Hamilton mechanics, but to be honest, I hardly know what that is! Maybe you can give me help? I would be very thankful. **Update** : For the general spherical pendulum I get $$ \ddot{\theta}=\sin\theta\cos\theta\dot{\varphi}^2+\frac{g}{l}\sin\theta,~~\ddot{\varphi}=\frac{-2\cos\theta\dot{\theta}\dot{\varphi}}{\sin\theta}. $$ Isn't the 2-dim. harmonic oscillator a special spherical pendulum so that I maybe can use this results?


Is it possible to prove that projectiles with drag will not travel in a parabola under the theory of classical mechanics?

Can you prove that projectiles will travel in a parabola, provided the acceleration due to gravity $g$ does not vary significantly along the trajectory? This should take air resistance into account. I know how to prove a parabolic trajectory without air resistance (see my user page on Wikipedia, for instance, where I have successfully proven this), but with air resistance, the equations get more complicated and difficult to solve analytically.


A question on the Bousso-Polchinski paper

In this famous paper by Bousso and Polchinski, Quantization of Four-form Fluxes and Dynamical Neutralization of the Cosmological Constant an example in M-theory compactification is given in section 2.2, page 7. Since I am new to string theory, I wonder how exactly (2.16) $$ S= V_72\pi M_{11}^9\int d^4x \sqrt{-g_4}\left(R - {1 \over 2 \cdot 4!} F_4^2\right) + S_{branes}$$ is obtained from (2.12) $$ S= 2\pi M_{11}^9\int d^{11}x\sqrt{-g_{11}}\left(R-{1 \over 2 \cdot 4!} F_4^2 \right) +S_{{branes}}$$ Is there any assumption made which is not stated in the paper? Especially, why "eleven dimensional $\bf F_4$ is directly reduced to a four dimensional $\bf F_4$" Since I am new to string theory, I really want to have a thorough understanding of this work. Detailed explanation is very much appreciated!! Thank you very much!


What's the point of NASA's impossible space engine? Wouldn't a mirror work better?

My understanding (which is very little) is that the point of NASA's recent virtual-particle engine is to convert solar energy into momentum. That's fine, but what's the point? If the spacecraft is already receiving solar energy, can't it just carry around a mirror to reflect light and accelerate via radiation pressure? Or is the quantum version somehow injecting "more" momentum than is actually being delivered via radiation pressure?


Leonard Susskind said that the small value of the cosmological constant could mean that the universe is much bigger than we thought. Why?

He said that near the beginning of this Youtube video that it could mean that the universe is much bigger than he had thought that it was, and that the cosmological constant wasn't even a constant. I don't have a clue what his reasoning was though.


Jet rapidity vs. Pseudorapidity

thanks to this post Why is pseudorapidity defined as $-\log \tan \theta/2$ I got a better understanding of what makes (pseudo) rapidity an interesting an often-used concept. Tangentially working with jet physics (the spray of hadrons kind of jet :), I am wondering why people work with rapidity (no pseudo here) when it comes to jets. What is the reason for this (doesn't pseudo rapidity offer the same convenient properties of Lorentz invariance wrt z boosts)? More importantly, how does one determine the rapidity (or rapidity difference) in a measurement at a hadron collider, where one does not know the the full four vector of any physics object? I suppose that the above question might be related to the topic of jet mass, which I have read in papers before. Any hints on the definition/usefullness of jet mass would also be appreciated. Thanks


Conservation of angular momentum tensor $L^{\mu\nu}$ in special relativity

I have edited this question because I don't think that the related post answers my question fully. It refers to Noether's theorem but I would like an explicit illustration in an easier fashion: The angular momentum tensor is defined: $$L^{\mu\nu}~=~x^\mu p^\nu-x^\nu p^\mu$$ I would like to show that if a particle is not acted on by an external force that the angular momentum is conserved. Despite it being a direct consequence of the theorem I want to compute it in an inferior way, would this be sufficient? Without loss of generality take the rest frame of the particle $$\frac{dx^\mu}{d\tau}=(c,\underline{0})$$ Since the particle is not acted on by an external force, we have $$\frac{dp^\mu}{d\tau}=\left(\frac{1}{c}\frac{dE}{ d\tau},\underline{0}\right)$$ $$L^{\mu\nu}L_{\mu\nu}=\left( 2 (x\cdot x) (p\cdot p) - 2 (x \cdot p)^2 \right)$$ Can I show the conservation by $$\frac{d(L^2)}{d\tau}=0$$


Do electrons collapse into nucleus, if electrons in the atom are constantly excited?

From the Bohr's atomic model, it is clear that electron can have only certain definite energy levels. When the electron is present as close to the nucleus as possible, the atom has the minimum possible energy and is said to be in the _ground state_. When energy from some outside source is supplied to it, it can absorb a definite amount of energy and jumps to higher energy state. Such a state of an atom in which the atom possesses more energy than possessed in the ground state is called _excited state_. These excited states are unstable and the electron tends to come back to lower energy level. This transition (change) from upper to lower energy level occurs with a jump and energy is emitted in the form of a quantum equal to difference in energies between the two levels. I have a doubt here, if electron absorbs energy from the outside source and jumps to the higher energy state, it will be now storing absorbed energy, as potential energy at the excited state. If electron emits all the absorbed energy (quanta-difference in energies between the two levels). By virtue of what energy does electron come down? I mean, in common situations, we say that an object at any height comes down converting it's potential energy into kinetic energy. Here in case of electron, it has already emitted absorbed energy as quanta. So, is it that electron losses some energy other than the energy absorbed from the source, to come down to ground state. I thought, if it was a possibility, then electron would constantly need to lose energy, whenever excited, at last, it would collapse into the nucleus. But, this not we really observe. I think there might be some misunderstanding by me or there might be any of the existing model like quantum mechanical model, which could account for this. If any were the case, please explain.


Irreversible heat engines strictly less efficient than reversible ones

I undertand how Canot's theorem implies that irreversible heat engines must be no more efficient than reversible one's, but it isless clear why they need to be less efficient, as I have seen stated in some places. If they could be equally efficient then an irreversible engine engine could be used to drive a reversible engine operating between the same heat reservoirs, without any net energy transfer between the reservoirs. It would then be unclear what is irreversible about the irreversible engine. Does that constitute an actual contradiction though? If so can the argument be stated more tightly? It feels a litle sloppy as is. It could also be a quation of how an irreversible engine is defined in Carnot's theorem. I understood it to mean one that cannot be run in reverse as a heat pump, which could presumably include a Carnot engine with a one way ratchet attached. If it really means an engine whose thermodynamic effects can not be undone, the implication would be trivial.


Spring Constant - Waves

I tried to prove that for the same length of spring , a one dimensional wave that moves faster in spring 2 than spring 1, its spring constant will be lower than the spring constant of spring 1. That's the begining of the proof I tried to make: $$w = \sqrt{\frac km} = 2π \cdot \nu$$ $$\nu = \frac{1}{2π} \cdot \sqrt{\frac km}$$ $$k = m \cdot \nu^ 2 \cdot 4π ^ 2$$ What am I missing?


Why don't you always use a high sampling rate?

I was just curious, as I was reading about aliasing. As far as I understand, aliasing comes from the fact, that you use a bad sampling rate, resulting in getting a wrong waveform compared to the one you had from the beginning. So why is it you just don't use a fast sampling rate all the time ? Or maybe you do that, and I just don't know ?


Does a black hole have enough time to actually form a singularity?

I am trying to wrap my head around black holes, singularties and hawking radiation. Physics.se contains many intresting questions and answers, but from none I could so far read about the interaction between formation of a singularity and hawking radiation (most seem to talk about the point at which matter plunges into a black hole that already has a singularity). From what I understand, from the perspective of a star collapsing into a singularity, the perceived time is just like in classical mechanics: the mass is accelerated towards the center of gravity, and when "everything" is there, it is a singularity. Now this takes a finite time, and the matter itself perceives just that. Part of the question is: how long is this time? I am assuming in the magnitude of milliseconds. Now as I understand hawking radiation, as soon as there is an event horizon, it will start emitting negative mass/energy into the direction of the singularity. Time dilation gets infinitely stronger as we approach the singularity, so I would assume that from the perspective of those negative particles, they would all the time reduce their perceived distance to the particles of positive energy/mass that are already on their way to the singularity. Since approaching the singularity, time dilation becomes kind of infinite, even the $10^{100}$ years or so it is supposed to take for hawking radiation to evaporate a black hole seem to be enough to send the necessary amount of negative energy/mass particles down to the singularity so that they can "chase" the positive energy/mass paticles very closely. But will they reach them before a singularity forms, so that they cancel out each other to the point there is not enough gravitational force? If this is just a matter of calculating "$n$s until matter reaches singularity" and "$m$s until negative mass/energy reaches singularity" and calculation shows that always $n<m$ then I would be intrested in the calculation (if possible presented in a way that doesn't require too exotic concepts). If this is a qualitative matter, in that some of my ideas are fundamentally wrong, please point those out (again if possible in a way that doesn't require too exotic concepts).


Do negative numbers have any physical meaning?

So, mindlessly wandering off into space, thinking about quantum and how cool physics is, I came to a realization that... well.. negative numbers to me make 0 sense. You have either something, or not something. It's always a yes or no answer, with everything And everything always has an opposite; it's nonexistence. I either have a red scarf, or not. If there are 3 doors, labeled 1 2 3. I have to choose one of them. The choice isn't me choosing which door, the choice is what door i'll pick. I'll either pick door 1, or not pick door 1. The opposite of door 1 is no door 1. Lets go really small. I have 2 grains of salt. I can't take 3 grains away. I believe negative numbers were invented by man to create debt. Because it doesn't make any sense in physics. Can someone please explain this to me!!


Balloon tethering cable tension model (catenary tether)

If a moving balloon (let's say it's a blimp system with some rotors for movement) is connected to the ground by a cable that describes a catenary curve, how can we model the tension being applied on the blimp by the cable? ![MSpaint draft](http://i.stack.imgur.com/mmAmG.png) This force has a vertical component that will be related to the mass of the cable and a horizontal component with the opposite direction of the movement of the blimp. Is there any model for this kind of behavior?


Cauchy's Second Law of Motion understanding

I'm told that Cauchy's Second law of motion is $$\tau_{ik} ~=~ \tau_{ki}.$$ Now my notes don't go into great details for simplicity sake, but they mention that the order of magnitude of body force $\rho F \delta V$ and inertia $\frac{D}{Dt}(\rho \delta V v)$ on a volume is smaller than that of stress $t(n) \delta S$ on a surface. What I'm not understanding is the order of magnitude stuff? I'm not sure why stress has a larger order of magnitude.


Force on impact in case of a free fall

In a free fall, on impact with the ground, the body experiences a massive force because of a huge acceleration since the velocity goes to 0 rather quickly. Since this force has to come as a result of reaction, can it be assumed that the body applied a massive force on impact far greater than its weight?


Are oscillations of electron chirality experimentally observable?

Is there any plausible experiment by which chirality oscillations in electrons could be observed experimentally, such as through some analogy to neutrino oscillation experiments?


Electromagnetism duality theorem

Concerning Electromagnetism, textbooks often refer to the Duality Theorem. Sometimes it is presented like this: «Consider the Maxwell's Equations (with phasors) and a known field $\mathbf{E}_1$, $\mathbf{H}_1$: $\nabla \times \mathbf{E}_1 = - j \omega \mu \mathbf{H}_1$ $\nabla \times \mathbf{H}_1 = j \omega \epsilon \mathbf{E}_1$ If $\mathbf{E}_1$ is replaced with $\mathbf{H}_2$ (the magnetic field of another electromagnetic field: $\mathbf{E}_2$, $\mathbf{H}_2$) and $\mathbf{H}_1$ is replaced with $-\mathbf{E}_2$, $\mu$ is replaced with $\epsilon$ and $\epsilon$ with $\mu$, then the above equations become respectively $\nabla \times \mathbf{H}_2 = j \omega \epsilon \mathbf{E}_2$ $\nabla \times \mathbf{E}_2 = - j \omega \mu \mathbf{H}_2$» They are valid Maxwell's Equations too. But now what does follow from this substitution? 1) Should be $\mathbf{E}_1 = \mathbf{H}_2$ and $\mathbf{H}_1 = -\mathbf{E}_2$? It is dimensionally incorrect. 2) I read also $\mathbf{E}_1 = \eta \mathbf{H}_2$ and $\eta \mathbf{H}_1 = -\mathbf{E}_2$. My question is double: what is the advantage of Duality Theorem and which is the correct form between the two just written? Thank you anyway!


Coulomb potential in atoms other than hydrogen

The energy of an electron on $H$ atom is given by the formula: $-13.6 \; \text{eV}/n^2$. The constant value is born from $H$ dielectric constant and efective mass of the electron. My question is: this formula can be used to estimate the energy of a electron on a $Si$ Coulomb potential? Is it sufficient to user the proper values of $\epsilon$ and electron effective mass?


Direction of future time cone inside Schwarzschild horizon

Following is the picture copied from enter link description here, illustrating the future time cone inside the Schwarzschild horizon. ![enter image description here](http://i.stack.imgur.com/ZP1w4.png) My question is, on the left side of $r=2M$, why must the time cone be left- oriented, but not right-oriented?


Does the Anti-Up, Anti-Strange, Anti-Strange Quark combination form a Baryon?

I was using this applet: http://www.lon-capa.org/~mmp/applist/q/q.htm And I saw that the up, strange, strange quark combination formed this Baryon: ![enter image description here](http://i.stack.imgur.com/1Oa2a.png) But the anti-up, anti-strange, anti-strange combination didn't form anything. So is that a fault of the applet or is there an actual reason? What would the reason be?


Why do those dolphin bubble rings I saw in a video behave the way they do?

This (see http://www.youtube.com/watch?v=TMCf7SNUb-Q&feature=related) is so cool, I wanted to show it. As a pro forma question for this forum, any idea what is going on with the breaking of larger, presumably vortex related bubble rings and reconnection making the smaller rings? The dolphins appear to be making smaller vortices by suddenly moving their snouts through a ring. On a simpler level, why do the rings get larger? Any other good questions come to mind?


Thermodynamics, PV diagrams?

My teacher told me that the total amount of work done on or by a gas can be represented by the area enclosed in the process in a PV diagram. This is only valid for non isothermic processes, right?


First law of thermodynamics?

The first law says that the change in internal energy is equal to the work done on the system (W) minus the work done by the system (Q). However, can $Q$ be any kind of work, such as mechanical work? For example, does a string heat up if it is attached to a block and a tension is applied to move the block? And what about the work done on the system by gravity when an object falls? Is there any internal energy change when that happens?


How do I find the Gain of this Transfer Function

I found the transfer function for the spring mass damper system to be $$G(s)=\frac{1}{ms^{2}+bs+k},$$ and now I need to find the gain of this transfer function. I know that the gain is $G=|G(j\omega)|$, but I'm not really sure how to go about finding the gain of a transfer function with a quadratic term in the denominator.


Abstract Algebra in Relativity and Cosmology?

Is Abstract Algebra useful in theoretical Relativity and/or Cosmology? If so can anyone give me some examples or point me towards a good book with that emphasis if it is one? Thanks in advance.


Calculation of the cross section

Why, when we calculate the total cross section, we make the average other initial states and the sum over final states?


Magnet spinning between two other magnets

Suppose, we have two magnets, MA, MB, and we have a third magnet MC in between the two magnets. Each magnets' north pole faces the other magnets south pole, and the magnets are placed horizontally side by side. We spin the magnet MC at a speed between super fast and slow. What will happen then?   I mean, once MC's north pole faces MA's south pole and MC's south pole faces MB's north pole, then the next moment MC's south pole faces MA's south pole and MC's north pole faces MB's north pole. So what will happen then?![the image](http://i.stack.imgur.com/E0yGZ.jpg)


Atom emitts a photon - what is the frequency of the energy of the photon?

Lets say I have an atom of known mass who goes from excited state to the ground state whose energy is $14.4 keV$ lower. I know that the emitted energy $14.4keV$ got converted into an energy of a photon and kinetic energy of the Atom - this means energy of the photon $E_\gamma < 14.4keV$. I tried to calculate it like this and got a nonsense: \begin{align} E_{1} &= E_{2}\\\ \sqrt{ {E_{0~Fe}}^2 + {p_1}^2c^2 } &=\sqrt{ \left(E_{0~Fe} - E_\gamma\right)^2 + {p_2}^2 c^2 } \longleftarrow \substack{\scriptsize \boxed{p_1 = 0}~\boxed{p_2 = E_\gamma /c}}\\\ \sqrt{ {E_{0~Fe}}^2 + 0 } &= \sqrt{ \left(E_{0~Fe} - E_\gamma\right)^2 + \frac{{E_\gamma}^2}{c^2} c^2 } \\\ E_{0~Fe} &= \sqrt{ \left(E_{0~Fe} - E_\gamma \right)^2 + {E_\gamma}^2 }\\\ {E_{0~Fe}}^2 &= {E_{0~Fe}}^2 - 2E_{0~Fe}E_\gamma +{E_\gamma}^2 + {E_\gamma}^2 \\\ {E_\gamma}^2 + (-E_{0~Fe})E_\gamma + 0 &= 0\\\ &\Downarrow\\\ E_\gamma &= 0\\\ E_\gamma &= E_{0~Fe} \end{align} None of the solutions to the quadratic equation make sense. Can anyone give me a hint where did i go wrong? * * * **EDIT:** After i read some of your comen I did rethink the situation and wrote equation like this. But still energy of a photon $E_\gamma$ is too large. Shouldnt it be smaller than $14.4keV$? \begin{align} E_1&=E_2\\\ E_{0~Fe} + \Delta E &= \sqrt{{E_{0~Fe}}^2 + {p_2}^2c^2}\\\ E_{0~Fe} + \Delta E &= \sqrt{{E_{0~Fe}}^2 + \frac{{E_\gamma}^2}{c^2}c^2}\\\ E_\gamma &= \sqrt{\left(E_{0~Fe} + \Delta E\right)^2-{E_{0~Fe}}^2}\longleftarrow{\substack{\text{Lets assume that we have a "Fe" atom}\\\\\text{whose rest energy is $E_{0~Fe}=53GeV$}}}\\\ E_\gamma &= \sqrt{\left(53\times10^9eV + 14.4\times10^3 eV\right)^2 - \left(53\times10^9 eV\right)^2}\\\ E_\gamma &\approx 3.91\times10^7 eV \end{align} * * * **EDIT:** By even improving the equation I got better result which I think must be the right one. \begin{align} E_1&=E_2\\\ E_{1~atom}&=E_{2~atom} + E_\gamma\\\ E_{0~Fe} + \Delta E &= \sqrt{{E_{0~Fe}}^2 + {p_2}^2c^2} + E_\gamma\\\ E_{0~Fe} + \Delta E &= \sqrt{{E_{0~Fe}}^2 + \frac{{E_\gamma}^2}{c^2}c^2} + E_\gamma\\\ E_{0~Fe} + \Delta E - E_\gamma &= \sqrt{{E_{0~Fe}}^2 + {E_\gamma}^2}\\\ \substack{\text{square of}\\\\\text{a trinomial}}\longrightarrow(E_{0~Fe} + \Delta E - E_\gamma)^2 &= {E_{0~Fe}}^2 + {E_\gamma}^2\\\ \big((E_{0~Fe} + \Delta E) - E_\gamma\big)^2 &= {E_{0~Fe}}^2 + {E_\gamma}^2\\\ (E_{0~Fe} + \Delta E)^2 - 2(E_{0~Fe} + \Delta E)E_\gamma + {E_\gamma}^2 &= {E_{0~Fe}}^2 + {E_\gamma}^2\\\ (E_{0~Fe} + \Delta E)^2 - 2(E_{0~Fe} + \Delta E)E_\gamma &= {E_{0~Fe}}^2\\\ E_\gamma &= \frac{(E_{0~Fe} + \Delta E)^2 - {E_{0~Fe}}^2}{2(E_{0~Fe} + \Delta E)}\\\ E_\gamma &= \frac{\left(53\times10^9eV + 14.4\times10^3eV\right)^2 - \left(53\times10^9eV\right)^2}{2\left(53\times10^9eV + 14.4\times10^3eV\right)}\\\ E_\gamma &= 14.399 keV \end{align} Is it possible that atom takes so little energy?


How clean will first generation fusion reactors be compared to fission reactors?

Googling the topic seems to indicate that fusion reactors will produce less waste and less toxic radioactivity, but this fact never seems to be mentioned during the current debate over nuclear power. Will fusion reactors be cleaner than fission reactors? If so, how much cleaner?


In which field theories with fermions do string- and fivebrane structures not come up?

A year ago, username @Greg Graviton asked in a thread here about the Spin group as covering of the spatial rotations. A subquestion was: _What other groups, even larger than SU(2) are there that (could) act on "physics" and are an extension of SO(3) Is it possible to classify all possibilities, in particular the ones that are not direct products?_ My answer to this was: _Of course, there are many more groups relevant for Quantum Mechanics and other physical theories (like gauge groups, as you mentioned). But concerning your second question, as far as "rotations only" go, SU(2) is already theuniversal covering group. These are the simple connected Lie-Groups, which lie on top of these upward chains you're searching for._ Now recently I came across a construction for structures lying above the spin structure, namely string stucture and then fivebrane structure. There are mainly the nLab references, and there is a thread on MathOverflow, relating the Whitehead tower to physics. Without knowing too much homotopy, I can still see the definitions seem geometrical and hence natural to me. The definitions seem to be quite general constructions over the manifold and are the discussed in the context of QFTs. I make some observations which surprise me a little and I want to ask these specific questions:   * The title of the thread tries to address this question. Their names kind of suggest it, but do these mathematical objects, the objects above the spin structure, necessarily have to do with string theory? Now this question is problematic in as far as string theory people will say QFT is generally the necessary conclusion when studying QFTs. I want to filter this thought out for a moment and ask: If you want to do generally non-stringly quantum field theory, do you for practical reasons want to spend your time learning about these objects? Do you come across them just by learning about Dirac operators?    * The presentation for field theories discussing this focus on a map $\Sigma\to X$, were $\Sigma$ gives the field configuration and $X$ is a manifold. I figure I have to view this just as generalized element of the manifold. Now I find it curious that the Spin, String and fivebrane structure seems to be something build over $X$ first, prior to considering physically propagating things in that world. This reverses my thinking of the spin being something associated with the particle, not with the space it lives in. When following this approach, to what extend to I think of the spin (bundle, etc.) to be a property of the world the object lives in, and then what is the object itself, if we say it only gets these feature when it's put into $X$?   * Why *five*brane? Why not some other number?


Why are the laws of thermodynamics "supreme among the laws of Nature"?

Eddington wrote > The law that entropy always increases holds, I think, the supreme position > among the laws of Nature. If someone points out to you that your pet theory > of the universe is in disagreement with Maxwell's equations — then so much > the worse for Maxwell's equations. If it is found to be contradicted by > observation — well, these experimentalists do bungle things sometimes. But > if your theory is found to be against the second law of thermodynamics I can > give you no hope; there is nothing for it but to collapse in deepest > humiliation. and Einstein wrote > [classical thermodynamics] is the only physical theory of universal content > which I am convinced will never be overthrown, within the framework of > applicability of its basic concepts. Why did they say that? Is it a very deep insight they had, or is it something one can be convinced of quite easily? Or even, is it trivial?


Detecting coherence

Is there a way to know if a particle is acting as a wave or a particle? Alternatively, if an entangled particle was already measured? A - Yes So any experiment over an entangled particle that let you know if they are coherent or decoherent, so would let we know that the distant particle is same "state" (coherence or decoherence), so independiently of any quantum state measured, having detected that "the wavicle has been measured", let you send information coded as decoherence time intervals. B - No So, does still have sense to talk about a coherence/decoherence "state" if there is no way to detect it? **EDIT** The wave has his meta-existence as a probability of a detectable particle interaction, but.. if there is no way to know about the system coherence or decoherence -before- it have "expressed" his behaviour (it's done always as a particle) then we could say that coherent state is never detectable?, I mean if you have measured a photon by a detector, you already know coherence is lost, but with engangled particles A, B, if you decoherence A, then what happend to B? only we could know that there is no "random"(umpredictable) value anymore.. then could we know if a particle is entangled with other? or explicitly could we know if a particle was measured or decoherenced from another entangled partner ? Fantastic way to say it: "Suposse you are employed to discover if our entire world is being spied from other distant world via entangled particles and the random we see (and its subsecuent evolution) was already known via an engangled model of our world, so.. there is a way to know if someone had got access from a distant engangled particle?, or there is no way to know if any measured value was already determined or is the current experiment where it's first known and fixed (and last years as a lab note in a yellow paper..) or was the experiment just redundant, because the value was already known in an entangled distant experiment (and by a distant world, with beings laughing at our surprise) it doesn't matter if the value is random, anyway is a known value, and they saw it first"