title: Are the results from models considered "data"?

text: At the AGU, I presented a poster on vocabulary for discussing data systems, and someone left a note at my poster stating: > You have a bias here towards observational data. Need to recognize that a > lot of data comes from models and analyses. And I completely agree; I didn't mention models and the values that come from them at all, and I had defined 'data' in such a way as it only covered observational data: > Values collected as part of a scientific investigation; may be qualified as > ‘science data’. This includes uncalibrated values (raw data), derived values > (calibrated data), and other transformations of the values (processed data). ... but do scientists consider those resultant values to be 'data'? I've talked to a few scientists over this last week (all in solar or space physics), and everyone of them including those who deal with modeling, were of the option that it was 'model output', but not 'data', (although one made the distinction between the philosphical concept of 'data' being multiple values, but said he didn't consider it to be 'science data' and commented that some of the earth science folks considered the values from their models to be data) So, the question -- is there a better term to use other than 'model output', and are there fields where 'model output' is considered to be 'data'? **note** : there's also the issue of the definitions of raw vs. derived vs. processed data, as there's different definitions used by instrument operators, but just ignore that issue in the definition of 'data' for the time being. **note 2** : I originally asked this on meta, and there were suggetions I move it to the main site; I've edited the text slightly due to a response I had yesterday from my surveying the scientists I work with. **note 3** : to set the context, the original intent was to identify clear or ambiguous terms across science disciplines, but the intended users were the data informatics community (those building data systems for the most part, not necessarily the scientists), but part of the issue is using language that won't offend or confuse the discipline scientists


title: Importance of Kohn anomaly?

text: What is practical importance of Kohn anomaly for experimentalists and/or theorists?


title: GUT that includes all 3 particle families into a large group?

text: Explaining SU(5) GUTs (using the first particle family as an example) in the last SUSY lecture 10, Lenny Susskind mentioned that there are at present no ideas how to combine simultaneously all 3 particle families into a large GUT theory. I somehow dont believe him, suspecting that he just didnt want to talk about this :-P... So, are there any ideas around how to incorporate all 3 families into a larger structer? If so, I would appreciate explanations about how it works at a "Demystified" level :-)


title: compact and non-compact gauge theory

text: From the answer to this question, I see that the gauge theory has to be compact if the charges need to be quantized. I am not sure in what sense these are necessary and sufficient. So here is a question I had: In electromagnetism, the electric charges are quantized in units of electronic charge, so shouldn't the gauge fields $A$ have values on a compact manifold rather than be a real number ?


title: why monochromatic light is necessary for interference?

text: One of the conditions stated for interference to occur was source should be monochromatic. why this condition needs to be satisfied for interference to occur? Is it because all mathematical equations use wavelength and single color means single wavelength?


title: Degeneracy, spherical harmonics

text: In a 3D oscillator, the energy levels are known to be $(n_x + n_y + n_z + \frac{3}{2})\hbar \omega = (n + \frac{3}{2})\hbar \omega$. Say for $n = 1$, any of the $n$'s can be $1$ and the rest are $0$. For $n = 2$ and $3$, same concept applies. How do I use this information about degeneracy to suggest values of $\ell$ in spherical harmonics $Y_\ell^m$?


title: How to find the equillibrium points using Jacobian and Hessian?

text: Given that I have Jacobian and Hessian matrices of three particles interacting with each other in a harmonic trap through Coulomb's law in a 2D plane, how do I find the equilibrium points of them (I know it should be a triangle)?


title: Distance between interference and diffraction fringes

text: In problems related to Interference and Diffraction fringes - I often have encountered phrases like "bright fringes 1.5 mm apart" or "distance between first bright fringe and 4th dark fringe". Fringe is some sort of rectangle not a point. we can talk about distance between two points not distance between two say, bands. I think both interference and diffraction fringes are equal width - determined by wavelength, distance between slits and screen and distance between slits or width of one slit. in one text book - it is mentioned that in diffraction fringe widths are not same where it is same in interference. Is it right?


title: What happens to orbits at small radii in general relativity?

text: I know that (most) elliptic orbits precess due to the math of general relativity, like this: ![precession](http://i.stack.imgur.com/H8l5b.png) source: http://en.wikipedia.org/wiki/Two-body_problem_in_general_relativity I also know that something is different for orbits at radii below a certain value. Wikipedia explains this as follows, I'm confused by this and just want to ask for clarification: > _If the particle slips slightly inwards from $r_{inner}$ (where all three > forces are in balance), the third force dominates the other two and draws > the particle inexorably inwards to $r=0$._ What does this mean? If you drew a graph of the path a particle takes in this unstable regime, what would it look like? Why is the stability transition point further than the Schwarzschild radius? Why does this graph show an unstable point beyond the Schwarzschild radius? For elliptical orbits that come very close to the event horizon, is there some kind of orbital decay? How is energy conserved? In short, do all orbits (with GR effects) look like the precession shown above, or is there another shape that we see if it gets closer to the Schwarzschild radius?


title: Stability of balanced masses with different surface areas

text: ![enter image description here](http://i.stack.imgur.com/Eze3s.png) Say I have this setup. The two round objects have equal mass and their centers of gravity at the same distance from the shaft. The objects only differ in that they have different surface areas (assume they are both flat-faced). If this apparatus is spun, will it fall over? I believe it won't. The centrifugal forces are balanced since they only depend on mass, and the aerodynamic drag will be in the direction of motion, so it won't pull the shaft in any outward direction.


title: Is it possible that the Big Bang was caused by virtual particle creation?

text: As far as I understand, it is understood that throughout the universe there exists, what is known as, a quantum field from which, due to its fluctuations, temporary (pairs of) virtual particles continuously appear in a random, unpredictable (or should I say probabilistic?) fashion. For my idea to have any viability, I am assuming that this quantum field is an intrinsic part of reality, that existed "before" or rather _at_ the time of the Big Bang already. Is this a correct assumption? Or is it believed that this quantum field was "formed" at the time, or perhaps even later than, the moment of the Big Bang? I hope not. Assuming the former, and also understanding that those pairs of virtual particles typically annihilate each other almost instantaneously, _but_ sometimes actually create real particles — for instance in Hawking radiation — is it a far stretch to think that the Big Bang was possibly started by the highly improbable (but in an "infinite" timespan of underlying reality, likely to occur), cataclysmic event of a huge amount of virtual particle all appearing, either "at the same time", or in such a sequential manner, that they couldn't annihilate each other anymore and were destined to actually form large quantities of real particles?


title: A simple question about matrix product with spinor indices

text: I have a big problem with dotted and undotted spinor indices. For example, suppose we have two convolutions: $$ \sigma^{\dot {a} a}F_{ab}, \quad \sigma^{\dot {a} a}F_{\dot {a} \dot {b}}, \quad F_{ab} = F_{ba}, \quad F_{\dot {a} \dot {b}} = F_{\dot {b} \dot {a}}. $$ Is it right that without index notation they are reduced to $$ \sigma^{\dot {a} a}F_{ab} = \hat {\sigma}\hat {F}, \quad \sigma^{\dot {a} a}F_{\dot {a} \dot {b}} = \hat {F}\hat {\sigma}? $$ Is it right that I can change the indices' order, $F_{\dot {b} \dot {a}} = F_{\dot {a} \dot {b}}$, so I will get $F_{\dot {b}\dot {a}}\sigma^{\dot {a} a} = \hat {F}\hat {\sigma}$? What if $F_{ \dot {a} \dot {b}} \neq F_{\dot {b} \dot {a}}$? Should I write something like $\hat {F}^{T}\hat {\sigma}$?


title: What starts the movement of a photon

text: Although a photon has no (rest) mass, it does have a measurable speed. Its movement can be altered by gravity. A photon "travels". If I turn on a flashlight, seen by someone at a distance, the photons from that flashlight traveled, at the speed of light, from the flashlight to the observer. But what "force" instantly starts the movement of a photon at the speed of light?


title: Quantum field theory, particle interpretations and path integrals?

text: I am trying to find some names or models of a particle interpretation of quantum field theory which isn't a literal path integral approach? Are there any particle interpretations of quantum field theory which don't use path integrals?


title: Generalised Kepler's III law?

text: I have derived the following equation for the time-derivative of the angle that an orbiting particle subtends with one of the coordinate axes, with the other particle at the origin (this is the focus of the ellipse) \begin{equation} \dot{\theta} = \left(\hspace{-0.2mm}\frac{GM}{a^{3}}\hspace{-0.2mm}\right)^{\\!\\!1/2} \frac{(1+e\cos\theta)^{2}}{{(1-e^{2})^{3/2}}}. \end{equation} Note that $\dot{\theta}=\omega$ and that this equaton is correct, according to a physics textbook. I then noticed that this is nothing other than Kepler's III law, but with some additional factor relating to angle and ellipticity. Why does this factor appear? What does it mean?


title: Computer Science Modeling of Physical World

text: I am curious what efforts have been made to date to define virtual computer worlds based on the physics we know in the real world? I think it would be awesome to say start off with an atom defining a class atom with characteristics atomic number, weight, etc. then model how these atoms interact and try to build up Helium, Hydrogen, Oxygen, etc. Tie this into a computer graphics engine and voila. ; ) I know this is major over simplification and you could go down to quarks or something else but you get the idea. I am curious about exploring discussion about what the bottlenecks are here. Maybe modeling the interactions between virtual atoms is impossible (too large a number to compute on today's hardware). Maybe quantum computing has advantages here? Maybe there are efforts out there to model interacts between larger molecular constructs then... or even an abstraction the size of a human cell? Would be interesting to discuss what is theoretically possible here. What got me thinking about this was in the context of Artificial Intelligence. If you could take a snapshot of the physical world and load it into this hypothetical model... what would that mean, i.e. a brain of sorts?


title: What is a physical example of a Saddle-Node Bifurcation?

text: I am doing a presentation on bifurcations and would like **_physical_** examples to go along with each type of bifurcation but I am unable to find or think of any good example of a simple Saddle Node Bifurcation. The most basic saddle node-bifurcation can be described as: Any type of system that has no equilibrium or steady state solutions initially but bifurcates as a parameter is varied into having 2 solutions, 1 stable, another unstable. or vice versa, any system that has 2 solutions that vanish as a parameter is varied.


title: Is the time reversed laser really a laser?

text: I'm looking at the time reversed laser and I was having trouble understanding why we call this device a laser. To me this device is more like the absorbers found in FDTD codes, something like a CPML. I am having trouble finding the analogs to the following processes in traditional lasers.   1. What is the mechanism for feedback. In a laser we have stimulated emission producing an exponential increase in output intensity, until the system saturates. In the reverse laser I only see an attenuation constant, whose constant value to me seems exactly like an absorber and not a laser.    2. Does this device have modes, does the feedback select an absorption width that narrows? The claims of an absorb band corresponding to the visible range (200nm-800nm) is a rather broad range that is characteristic of mostly everything?   3. What does population inversion and threshold mean for the reverse laser? If it is an absorber doesn't that mean that even a single photon knocks it out stability. In one paper the authors described a 2 state reverse laser system, we know that a 2 state laser is not possible.... Some images of time reversed lasers. ![enter image description here](http://i.stack.imgur.com/awqPw.png) ![This also looks nice, maybe](http://i.stack.imgur.com/7NoKK.gif) * * * Where is the lasing?


title: Wilson lines, boundary conditions, surface defects of TQFTs

text: I asked the following question in mathematics stack exchange but I'd like to have answers from physicists too; I have been studying (extended) topological quantum field theories (in short TQFTs) from the mathematical point of view and I have no background of the physics point of view. Sometimes I encountered papers talking about Wilson lines, boundary conditions, surface defects and so on. I looked up these terminology but I couldn't find a good explanation how these are related to TQFTs. Could you suggest me references (papers, websites, books, vidoes etc) that explains the relationship between these physical jargon and TQFTs from mathematical point of view? Or could you explain it here?


title: Moving from point A to point B in zero-gravity

text: Here's an proposal on how to get from point A to point B in zero-gravity without using any propellant and the question why it wouldn't work: A closed tube, filled with water and a round (solid) object. If you need equations, the volume of enclosed water is the same as the round object, but the round object is 10 times lighter. (imagine a glass with water plus a ping- pong ball). here's a picture: ![alt text](http://img210.imageshack.us/img210/6640/sprite143.jpg) On earth the round object will float on the water inside the tube (subject to one G). In zero-gravity the round object has no preferred position. If we accelerate the tube in zero-gravity by one G, the situation is the same as on earth, the round object "floats". In this example we are accelerating the tube from the left side to the right side. The round object will consider floating to the right AS LONG AS THERE IS acceleration. Now consider adding a pipe to the bottom of the tube and connecting it to the top, a loop. Inside the pipe is a small water pump. If we give this apparatus a push, say one G in a zero-g environment, the round object will move "up", but now we start the water pump and spray the water on the round object, we try to submerge it. It will resist and impart a impulse on the water. like trying to hose down a air balloon floating on the pool. The pump will feel a resistance and hence the whole apparatus will move. Just running the pump at constant speed, same volume of water per second, will do nothing. but if we run the pump faster and faster the whole apparatus will start moving: the amount needs to be geometrical. The point is that we need to keep the apparatus feeling an acceleration, since only then will the round-object "float" and resist the incoming water at the top, hence we have something to "push on". Before you blow the "foul whistle": consider the situation if there were no round floating ball in the apparatus. (the pump runs on solar power or pre-charged battery) (disclaimer: i know standing on a sailboat and blowing into the sail will not get me anywhere, action<->reaction) thanks a bunch Sklivvz for the edit. sometimes the idea just needs to get out, never the mind how it looks like : P


title: Math of eigenvalue problem in quantum mechanics

text: I learned the eigenvalue problem in linear algebra before and I just find that the quantum mechanics happen to associate the Schrodinger equation with the eigenvalue problem. In linear algebra, we always gives the matrix of certain size (e.g. 4x4) so to find the eigenvalues is identical to solve the related secular equation. In quantum mechanics, in most case, it seems that the dimension of the matrix is not known or very large so to how to find the eigenvalue in those case? I just saw a problem in a book, if we know $A|a\rangle=a|a\rangle$ and $A|b\rangle=b|b\rangle$ for a Hermitian operator $A$. What's the eigenvalue of a Hamilton operator $H=|a\rangle\langle b| + |b\rangle\langle a|$. It is so confusing to me because if we don't know the matrix and elements, how can we write the secular equation and find the eigenvalues? Sorry I didn't take any class on quantum mechanics before, I learn it all my myself so I might be wrong in some statement above.


title: Valid theory in all dimensions for solitary waves

text: I'm studying soliton (solitary waves). They are many theory which explain the phenomenon, like sine-Gordon model. But sine-Gordon model has limitations when it applies to 4 dimension because it is valid for only two dimensions only. So which topics/theory will cover to understands solitons fully?


title: Understanding the math behind a falling object attached to a parachute

text: I'm trying to understand the mathematics behind calculating the total speed and force of a $40 kg_f$ object hooked up to a parachute, falling to Earth. From what I understand, the formula for this is $$m \frac{dv}{dt} = (mass \times gravity) -Force_{deceleration}= 0 $$ Here's where I'm confused: I understand that $m \frac{dv}{dt} = 0$ because once the parachute is deployed, the object reaches a constant speed at which it falls for the rest of the way down. However I also learned that $m \frac{dv}{dt} = F$, which is the total force of any moving mass. The object _is_ falling to Earth and therefore it _is_ moving, so how could the force, $F$, be equal to zero?? If you could help me get things straight, I'd really appreciate it. Thanks!


title: A question on lewis model

text: Electron affinity of chlorine = -349 kJ per mole. And, first ionization energy of sodium = 496 kJ per mole while energy required for combining= 147 kJ per mole. Isn't it a contradiction against genereal physics and G.N. Lewis' "electronic theory of valency"?


title: Rotation Group and Lorentz Group

text: It is often stated that rotations in the 3 spatial dimensions are examples of Lorentz transformations. But Lorentz transformations form a group named the Lorentz Group, $O(1,3)$ which is a group a $4 \times 4$ matrices, $\Lambda$ having the following property: $$ \Lambda^T g \Lambda = g$$ where $g$ is the metric tensor. Now rotations matrices for the 3 spatial dimensions are $3 \times 3$ matrices and form $SO(3)$. How can they be in the $O(1,3)$ ?


title: Matrix element approximation

text: In the formula for the decay width of $\Upsilon(4S)$ to B-mesons from $\text{e}^+\text{e}^-$ collisions: $$\Gamma_{\Upsilon(4S)\to B\bar{B}}=\frac{\left|\underline{P}_B \right|}{8\pi M_{\Upsilon(4S)}^2}\left| \mathcal{M}_{\Upsilon(4S)\to B\bar{B}} \right|^2$$ Isn't the units of the LHS energy and the RHS inverse energy? To make the $\Upsilon(4S)$ which has a mass of $10.58 \text{ GeV}$, in the centre of mass frame, $\sqrt{s}=10.58 \text{ GeV}$ at a minimum? Say the B-mesons all have the same mass of approximately 5.28 GeV, and $\Gamma_{\Upsilon(4S)\to B\bar{B}}\approx20 \text{ MeV}$, doesn't this make the matrix element larger than one, which, as a probability squared is not allowed? Maybe I have calculated $\underline{P}_B$ wrong? In the comments section: $$\Gamma=\frac{S \left|\underline{p} \right|}{8\pi \hbar cm_1^2}\left|\mathcal{M}\right|^2$$ I think there is an energy term that goes missing by setting $c=\hbar=1$ $\left|\underline{p}\right| = \frac{c}{2m_1}\sqrt{m_1^4+m^4_2+m_3^4-2m_1^2 m_2^2-2m_1^2m_3^2-2m_2^2m_3^2}$ And $m_2=m_3=m_B$, $m_1=M_{\Upsilon(4S)}$ and $S=1$. Any ideas?


title: Is conservation of statistics logically independent of spin?

text: If the number of fermions is $n$, we expect the quantity $(-1)^n$ to be conserved, i.e., $n$ never changes between even and odd. This is known as conservation of statistics. In the normal context of particles with the properties traditionally expected for fermions and bosons, it can be derived from conservation of angular momentum. If $n$ were to change from, say, even to odd, then the total angular momentum of the system would have to change from an integer to a half-integer. In the context of tachyons, there's an odd twist, however. The spin-statistics theorem is based on the assumption that the field has to commute at points that are spacelike in relation to one another. This commutation relation wouldn't hold for tachyons, so they get a free pass on spin-statistics. In fact Feinberg, in the classic paper that introduced the term "tachyon," found that the most plausible possibility was that tachyons would be spin-0 fermions (Feinberg 1967). Feinberg says on p. 1099, discussing selection rules, > [...]there are restrictions following from the conservation of statistics. I > shall simply assume here that if we assign a number $+1$ for bosons and $-1$ > to fermions and multiply these numbers for a multiparticle system that these > products are conserved in any transition.[16] Footnote 16 says, > See, for example, Greenberg and Messiah (Ref. 10) where this is proven, > however, under assumptions that may be invalid for tachyon theories. Given Feinberg's assumption, we have two separate conservation laws, conservation of statistics and conservation of angular momentum, and the result is that if tachyons are unique in combining integer spin with Fermi statistics, they have a special conservation law that only applies to them: if $t$ is the number of tachyons, then $(-1)^t$ is conserved. This means, for example, that you can't produce just one tachyon, and you also can't have a process that produces a tachyon plus an electron, even though in normal field theory it's perfectly OK to produce two unlike fermions. Does conservation of statistics really have some independent logical status, or is Feinberg just making an assumption that could be false, but that might make the field theory he's constructing easier to work with or more familiar? The understanding of tachyons in QFT has progressed a lot since 1967, so I wonder if this issue is better understood today than it was then. > G. Feinberg, "Possibility of Faster-Than-light Particles". _Phys. Rev._ > **159** no. 5 (1967) 1089. Copy available here (may be illegal, or may fall > under fair use, depending on your interpretation of your country's laws).


title: Derivation of the volume element (which uses the metric tensor)?

text: I have often seen $\sqrt{-g}$ in integrals, especially actions, where $g=\mathrm{det}(g_{\mu \nu})$. Does anyone know of a derivation that shows that this is indeed the volume element which must be utilized? I have searched online, but all the literature I have encountered assumes you accept it as the volume element.


title: Collision between a rod and a bullet

text: There lies a homogeneous rigid rod of mass $M$ and of length $H$ on a frictionless table at rest. A small bullet of mass $m$ moves toward the rod with velocity $v_0$, perpendicular to the rod and collides absolutely elastically with the rod at a point, located at a distance $x$ from the center of the rod. Before they finally move away from each other, they may still collide each other a few times. What is the maximum number of possible collisions of the bullet with the rod before they leave each other forever?


title: When should angles be expressed in degrees vs. radians?

text: I am trying to calculate the albedo of a given latitude by following the methods of Brutsaert (1982), I have copied the formula below: > # 3.6 Shortwave and long-wave radiative fluxes > > Albedo $\alpha$, the reflectance of solar radiation, increases as the solar > angle decreases. To estimate $\alpha$, a function of solar altitude > (Brutsaery 1982) was used: > > $$\alpha = 1.18\Lambda^{-0.77}\tag{37}$$ > > Solar altitude $\Lambda$ (in degrees) was calculated from latitude, the time > of the day and Julian day number ($J$ = day of the year) following Martin > and McCutcheon (1999) as > > $$\Lambda = \frac{180\lambda}{\pi}\tag{38}$$ > > where $\lambda$ is the solar altitude in radians > > $$\lambda = \tan^{-1}\biggl[\frac{\Omega}{\sqrt{1 - > \Omega^2}}\biggr]\tag{39}$$ > > where > > $$\Omega = \sin\biggl(\frac{\pi\theta}{180}\biggr)\sin(\delta) + > \cos\biggl(\frac{\pi\theta}{180}\biggr)\cos(\delta)\cos(\omega)\tag{40}$$ > > where $\theta$ is the latitude (in degrees, negative for southern > hemisphere), $\omega$ is the solar hour angle (radians) and $\delta$ is the > declination of the sun > > $$\delta = > \frac{23.45\pi}{180}\cos\biggl(\frac{2\pi}{365}(172-J)\biggr)\tag{41}$$ where the hour angle, expressed as $\Omega$ here is given by: > ...and $H$ is the hour angle. The hour angle is > > $$H = (\pi/12)(t_\text{noon} - t),\tag{7}$$ > > where $t_\text{noon}$ is local solar noon ($\sim 12.5$ in Oklahoma) and $t$ > is the local solar time (e.g. $t = 12.5$ and $H = 0$ at local solar noon). I have written a Matlab script to calculate the albedo for Oklahoma during mid summer but I'm not sure that the code is accurate. It gives an albedo of 0.0386 where it is expected that the albedo will be in the region of 0.07. I guess I'm a bit confused about when I should use degrees and when I should use radians in the equations e.g. for $\Omega$ and $\lambda$, thus I do not know if I should trust my result here. Could someone please clarify this for me?


title: Are electrons moving? If so relative to whom? and why is there no magnetic field?

text: electrons on earth **moving** with us, due to rotation of earth, revolution of earth, sun and our galaxy right?   Then, why is there **no** magnetic field around a **piece** of copper wire?


title: How to apply Andreev reflection formalism to ferromagnet ,normal metal interface?

text: The traditional formalism for andreev reflection deals with what happens at normal metal, super conductor interface.http://en.wikipedia.org/wiki/Andreev_reflection (i.e when an electron from normal metal is incident on the super conductor interface, a hole is generated in the normal metal which lets the formation of a cooper pair in the SC which propagates in the appropriate direction) How do we apply a similar formalism when an electron of say up-spin is incident from a normal metal to half-metal (magnet) ? what happens when this electron comes spin polarized at an angle to the interface ? note that unlike in superconductivity, there is a direction associated with the half metal (which is basically a magnet with an orientation)


title: What happens where an electron is annihilated by a spontaneously generated positron-electron pair?

text: I was visiting the Australian Synchrotron earlier today as part of a tour group; as the guide was going over the booster and storage rings I was reminded of something I learnt of quantum. If I know my quantum well enough, every so often, there are spontaneous pairs of electrons and positrons created and destroyed everywhere. Now I know that they don't last very long, but I got to thinking, what would happen if a pair was created at just the right instant for the positron to collide with one of the super-high-energy electrons in the booster or storage ring of a synchrotron? When I asked our guide he said that the probability was pretty low since there were only 1x10^6 electrons in there and naturally the pair would have to generate at just the right spot at just the right time. Ultimately he wasn't quite sure what would happen if the circumstances were correct for it though. I have been doing some further thinking, if I remember correctly, these pairs form as an electron spiralling in towards a central positron. I think that therefore at ordinary energies any nearby electrons would be repelled by the electron around the positron (which is a little reminiscent of the atomic stuff of first year chemistry). Question is, could an electron, at the sorts of energy levels of the order of giga-electron-volts, get past the spiralling electron and collide with the positron instead of its paired electron doing so? As this energy could not simply vanish, would you then get a lonely (now unpaired) electron and an explosion? or would the energy somehow transfer itself to the other electron? My interest here is towards what would actually happen if all circumstances were just right; also appreciated would be anyone who might be able to tell me the theoretical frequency of electron-positron pairs being spontaneously generated, as well as helping me to think up a way of determining the probability of finding an electron within one of the rings (for instance, perhaps if electrons can be considered to have an effective 'volume' within which they are most likely to appear and comparing that to the volume of the ring?) Thanks for your time, I really appreciate it.


title: Deriving the conservation of mass in a perfect fluid

text: I have a problem on a homework assignment. I will write the question and then what I have so far. I just want some guidance on what I am doing wrong (or right). Any help would be greatly appreciated: In these exercises we study the equations of motion of a perfect fluid. These are the conservation laws: \begin{eqnarray} \nabla_{\mu}T^{\mu\nu} = 0 \space (1) \\\ \nabla_{\mu}I^{\mu} = 0 \space (2) \end{eqnarray} Where $T^{\mu\nu}$ is the energy-momentum-stress tensor, and $I^{\mu}$ the particle current: \begin{eqnarray} T^{\mu\nu} = \rho{u}^{\mu}u^{\nu} + p((g^{-1})^{\mu\nu} + u^{\mu}u^{\nu}) \space (3) \\\ I^{\mu} = nu^{\mu} \space (4) \end{eqnarray} Here $\rho$ is the energy density, $u^{\mu}$ the material velocity (a future- directed unit time-like vectorfield), and p denotes the pressure while n is the [article density. Exercise 1 [Nonrelatvisitic limit of the conservation laws]: Let us first study the quasi-Newtonian limit of the conservation laws. We consider (1) and (2) in Minkowski space and assume that with respect to a system of stationary observers: \begin{eqnarray} u^{0} = \frac{1}{\sqrt{1-\frac{|v|^2}{c^2}}}, u^i = \frac{\frac{v^i}{c}}{\sqrt{1-\frac{|v|^2}{c^2}}} \space (5) \end{eqnarray} Moreover, we assume that \begin{eqnarray} \rho = \mu{c^2} + h \space (6) \end{eqnarray} where $\mu$ is the mass density and $h$ is the internal energy density. (i) Derive in the nonrelativistic limit $c\rightarrow\infty$ the conservation of mass law, \begin{eqnarray} \frac{\partial\mu}{\partial{t}} + \nabla\cdot(\mu{v}) = 0. \space (7) \end{eqnarray} Hints: Express the coponents of (3) explicitly in orders of c using (5) and (6), and derive the conservation law from the 0-component of (1). So I have done a little work on this but I don't see how I can derive (7). Maybe there is some math trick I need to use or maybe this is a lot easier than I am making it out to be. So, I put (5) and (6) into (3) and I get the following: \begin{eqnarray} T^{\mu\nu} = (p + \mu{c^2} + h)\frac{v^{\mu}v^{\nu}}{c^2-|v|^2} + p(g^{-1})^{\mu\nu} \end{eqnarray} I am not sure what I am supposed to do from this point. I am not sure if I derived this properly and am I supposed to take the derivative of this in terms of $t$ and what do they mean by expressing things as orders of $c$. Thank you and have a great day! Well after doing some work on my own in this problem I did get somewhere but not this question. I will show only the result of my derivation for at least one part and maybe more detail on the derivation of another part that is giving me trouble, but just to recall what I want to derive: \begin{eqnarray} &&\nabla_{\mu}T^{\mu0} = \partial_0T^{00}+\partial_iT^{i0} = 0 \end{eqnarray} I got the first term just fine (I think): \begin{eqnarray} \partial_0T^{00} = \frac{d\mu}{dt}c^2 \end{eqnarray} but I am not getting something nice for the second term, so I will show my derivation in more detail: \begin{eqnarray} &&\partial_iT^{i0} = \partial_i((p+\rho)u^{0}u^{i}+p(g^{-1})^{i0} \\\ &&\partial_iT^{i0} = \partial_i((p+\rho)u^{0}u^{i}) \\\ &&\partial_iT^{i0} = \partial_i((p+\mu{c^2}+h)u^{0}u^{i}) \\\ &&\partial_iT^{i0} = \partial_i((p+\rho)(\frac{\frac{v^i}{c}}{1-\frac{|v|^2}{c^2}})) \\\ \end{eqnarray} If I play around with this I somehow get: \begin{eqnarray} \partial_iT^{i0} = \nabla\cdot(\mu{v})c \end{eqnarray} Which I don't think is quite right... any help would be nice :)


title: Why it is said that Universe is an isolated system?

text: I have read in many places > "Entropy of an isolated systems never decreases" And as a corollary: > "As **Universe is an isolated system** (I) then its entropy is constantly > **increasing** (II)" I) If there are boundaries, how can we know about what happens there with entropy? and if there aren't any boundaries, what does it mean to be isolated? II) Why it couldn't remain constant? What are experimental evidences of **I** and/or **II** ? Thanks


title: How could a particle be isolated to avoid decoherence?

text: The question aims to this issue : if there is some technological arrangement (or action) to take over the particle/system in order to keep it in a coherent state, then the field, (force or whatever) keeping it away from interacting with "an external system" isn't it itself an interaction? I mean, supposing you reach enough isolation to avoid decoherence. How do you know the particle is still there? Thanks


title: Why is $R^2$ gravity not unitary?

text: I have often heard that $R^2$ gravity (as studied by Stelle) is renormalisable but not unitary. My question is: what is it that causes the theory to suffer from problems with unitarity? My naive understanding is that if the the Hamiltonian is hermitian then the $S$-matrix $$ \langle \text{out} \mid S \mid \text{in} \rangle = \lim_{T\to\infty} \langle \text{out} \mid e^{-iH(2T)} \mid \text{in} \rangle $$ must be unitary by definition. So why is this not the case for $R^2$ gravity? I see that Luboš Motl has a nice discussion related to such things here, but I am not sure which, if any, of the reasons he mentions relate to $R^2$ gravity. Are there other well known theories that have similar problems?


title: How much force is required to hold an umbrella?

text: Assuming a small umbrella (e.g. http://www.amazon.com/ShedRain-WindPro- Umbrella-Close-Size/dp/B001DL5WN0). What are the factors that need to be considered? I assume where I hold the umbrella might dictate part of the calculation of torque which in turn is dependent on the weight of the umbrella. But what about effects like (a light) wind that can turn the umbrella into a sail and make it seemingly harder to hold? How would I calculate for example total torque required to hold the umbrella accounting for light wind as well?


title: How does my hand amplify an FM radio signal?

text: I have a gizmo that broadcasts an FM radio signal from my phone's headphone socket to my car's radio antenna. Sometimes, there's another station on the same frequency with a stronger signal and my car plays that signal instead. However I've noticed that if I hold my hand up near the transmitter, my signal takes over again. What's going on when this happens? Am I somehow boosting the signal with my hand?


title: AC passed to a DC motor

text: What would happen when an alternating current (AC) is passed to a DC motor? And a direct current (DC) passed to an AC motor? Will it work or not? What will happen to it?


title: Simulating how comets travel with Newtonian physics

text: Imagine that a comet enters the solar system from Oort cloud, what concepts and laws does one need to take into consideration to simulate how it travels? I'm about to start researching how to write such a simulation. Of course there is the force of gravity between the comet and the planets. Ideas for what other things I need to take into consideration? If you know any introductory books that deal with the physics of how comets travel through the solar system that would be very welcome as well.


title: (Iso-baric / iso-choric) Thermodynamic cycle question

text: A very simply question: Is something that even somewhat resemble a thermodynamic cycle consisting of two iso-baric and two iso-choric cycles possible (such that you get a simple rectangle representing the work done by the process). I can understand and think of examples were a contained gas is heated iso- barically and then expanded is-chorically, but then the reverse - isobaric contraction followed directly by is-choric contraction - I can't imagine happening. Maybe I am just uncreative - you tell me! However, I believe, the system could exist at least to some degree, if a iso- choric and a isobaric process were replaced by adiabatic process (forming a quarter circle/ellipsoid kinda thing). btw, perhaps more importantly, does anyone know any free websites on which one can review and learn, some what more rigorously than google answers, about thermodynamic cycles (e.g. carnot cycle, dies cycle etc.).


title: Beta-decay of the neutron

text: Could you please tell me, where can I find a straightforward calculation with all details of beta-decay of neutron? To find the amplitude and to square it is difficult, but possible. The main problem is to take the integral over phase space (momentums of proton, neutrino and electron) and get final answer.


title: How do derive this result in stat-mech style

text: I'm going through (well, at least I'm planning to) Rief's book about statistical mechanic (I want to improve my knowledge). I want to be serious about this so I'm trying to solve as much problem as I can (and understand them). I came upon this one (1.5): > In a game of a Russian roulette ( _not_ recommended by the author), one > inserts a single cartridge into the drum and of a revolver, leaving the > other five chambers of the drum empty. One then spins the drum, aims at > one's head, and pulls the trigger.   >  (a) What it the probability of being still alive after playing the game $N$ > times?   >  (b) What it the probability of surviving ($N-1$) turns in this game and > then being shot the $N$th time one pulls the trigger?   >  (c) What is the mean number of times a player gets the opportunity of > pulling the trigger in this macabre game? I believe that the starting point for (c) is: $\left \langle n\right \rangle = \sum_{n=1}^\infty n \left(\frac{5}{6}\right)^{(n-1)} \frac{1}{6}$ Wolfram alpha tells me this is an exact sum which equals to 6 (logical). I want however, to try and imply the standard derivative trick (and this is why I'm asking this here and not in the math thread): $\left \langle n\right \rangle = \frac{1}{6}\sum_{n=1}^\infty \frac{\partial}{\partial n} \left(\frac{5}{6}\right)^{n} = \frac{1}{6} \frac{\partial}{\partial n} \sum_{n=1}^\infty \left(\frac{5}{6}\right)^{n}$ But this is an exact sum (that equals to 1) and I can't take it's derivative. I know I could use some table to solve this, the question is if there some way to solve this in a way that a student can do this in the exam? I hope any of what I wrote is clear and approved by the site policy.


title: How can you calculate (or convert) the Wh of a capacitor whose energy is given in Farads

text: When trying to compare the energy in a battery to the energy in a capacitor, the units don't match up. How can one compare a battery whose Ah are 10 and Voltage is 3 (for a total of 30 Wh) to a capacitor whose Farads is X and voltage is Y?


title: Canonical momentum in different coordinate system

text: The canonical momentum is defined as $p_{i} = \frac {\partial L}{\partial \dot{q_{i}}} $, where $L$ is the Lagrangian. So actually how does $p_{i}$ transform in one coordinate system $\textbf{q}$ to another coordinate system $\textbf{Q}$ ? http://en.wikipedia.org/wiki/Hamiltonian_mechanics#Charged_particle_in_an_electromagnetic_field When dealing with the Hamiltonian of the electromagnetic field, the derivation of $p_{j} = m \dot{x_j} + eA_j$ on the above link is usually written as $\textbf{p} = m \textbf{v} + e\textbf{A}$ but the derivation is based on using Cartesian coordinates, does it mean that $\textbf{p}$ is really a vector? If we are using another general coordinates, say, spherical coordinates, can we still have $\textbf{p} = m \textbf{v} + e\textbf{A}$ ? If no, I think the form of Hamiltonian in electromagnetic field $H = \frac{(\textbf{p} - e\textbf{A})^2}{2m} + e\phi$ will only be valid in Cartesian coordinates. In any other coordinates, $H$ carries a different form! Any comments are appreciated.


title: How can one calculate the distance a particular sound will travel?

text: What do you need and how to calculate a distance traveled by sound? For example if you hit a bell with a specific amount of power how far will it travel?


title: Angle between acceleration and velocity

text: Problem: > _A particle is constrained to move in a circle with a 10-meter radius. At > one instant, the particle's speed is 10 meters per second and is increasing > at a rate of 10 meters per second squared._ The book's answer: > The angle between the particle's velocity and acceleration vectors is given > $45^0$. My answer: > But I got $90^0$ by using this formula, $$v= r \omega \sin\theta$$ where > $\omega= \sqrt{\frac{a}{r}}$ Where am I wrong?


title: Position of two bodies as a function of time, special case

text: I'm looking for a closed equation like $x=f(t)$ (if there is) for a simplified case of the two bodies problem: Suppose that we have two point-like objects with the same mass, nothing more than their own gravity is effected on them. The initial distance between them is $D$. As the gravity force acts on them, they slowly speed up, reaching each other (let's say that they cannot collide), then begin to move away from each other. The following facts can be considered:   1. They and their velocity vector are always in a line with their mutual barycenter point.   2. The barycenter is always exactly halfway between them.   3. The force and the acceleration is increasing to infinity until they reach each other at the barycenter.   4. The total energy of the system is constant, the given potential energy transforms to kinetic energy, and vice versa.   5. The maximum speed is not infinite, can be calculated easily.   6. If there is an $x=f(t)$ function, then $f$ is a periodic function. If $x$ represents the current distance between the two objects, then $0\le x\le D$ Please help me to find the correct formula to calculate the distance $(x)$ in function of time. If there is no closed-form expression for this, then please suggest another simple way to calculate it. Thank you!


title: Is Sound Considered a Subtopic of Physics? What are the SI Units of Sound?

text: Does physics cover sound? If so, why does sound not have units in the SI system, or how would we measure sounds and frequency? I guess it's debatable, but why isn't it standard practice in physics?


title: Irradiance (or intensity) of an electromagnetic wave

text: Is the irradiance (or intensity) of an electromagnetic wave given by \begin{equation} I = \left<\|\mathbf{S}\|\right>_{T} \end{equation} or by \begin{equation} I = \|\left<\mathbf{S}\right>_{T}\| \; , \end{equation} with $\mathbf{S} = \mathbf{E} \times \mathbf{H}$ being the Poynting vector and $\left<\right>_{T}$ denoting the time average? In other words, is it the time average of the Poynting vector's norm or the norm of the time-averaged Poynting vector? **First update:**   Also textbooks seem to have a diverging view on this question. "Optics" from E. Hecht teaches the first variant, while "Fundamentals of Photonics" from B.E.A. Saleh and M.C. Teich presents the second one referring to "Principles of Optics" from M. Born and E. Wolf. In addition, Born and Wolf emphasize that $\mathbf{S} \cdot \mathbf{n}$ with $\mathbf{n}$ being the normal of the considered surface (irradiance is power per area) is the relevant quantity. So maybe the best description is \begin{equation} I = \left<\mathbf{S} \cdot \mathbf{n}\right>_{T} \; ? \end{equation}


title: Efficiency increase via ducting a fan - why is this more in viscous media?

text: I am looking at mixers for liquids like water or slurries. This brochure claims the following: > With the optional jet ring, it is possible to increase efficiency by up 15% > in water - even higher in viscous media I do not soubt that a jet ring, or ducting, can increase the efficieny of a propeller (So don't answer that). But I don not understand why this effect is stronger in viscous media. In fact, I would guess the opposite, that in a viscous media the drag of fluid - jet ring plays a bigger role, and that the centrifugal forces that propel fluid tangentially out of the propeller are less relevant in a viscous fluid. But my intuition is probably wrong, as the people selling the mixers measured something else. **Why is the effect of the jet ring stronger with a more viscous fluid?**


title: Topological disorder in condensed matter?

text: What is meant by topological disorder in condensed matter (both crystalline and amorphous)? For example, please see the following two papers from arxiv.org http://arxiv.org/pdf/0906.3848.pdf http://arxiv.org/pdf/cond-mat/0308612v1.pdf Thanks


title: Has anyone checked whether the speed of light varies according to gravitation

text: My physics is fairly basic, but I hope someone can answer without being too rude. A transparent medium such as water or glass refracts light and also reduces its speed, so I was wondering whether since light can be bent by gravity as in gravitational lensing, its speed would also be affected.


title: Is there any proof for the 2nd law of thermodynamics?

text: Are there **any analytical proofs for the2nd law of thermodynamics**? Or is it based **entirely on empirical evidence**?


title: Angular velocity $\omega$ by $v$

text: We have two girls, with mass ($M$). They become close to each other in speed of $V$. The distance between them is $3L$. ![](http://img2.timg.co.il/forums/1_159160419.jpg) I was asked to calculate the Angular velocity ($\omega$) of the two girls. So I set the rotation axis in the middle (where the distance between the two girls and the axis is $1.5L$, and I calculate the Angular velocity using this equation: $\omega=v/r$, where $r=1.5L$, and I got that $\omega={2v\over 3L}$. As I understood, the answer is correct, but this is not the correct way. What is my mistake? (sorry about my english)


title: Physically what happens during Avalanche breakdown to the pn junction?

text: What does breakdown mean physically? I saw this in wikipedia: > The avalanche process occurs when the carriers in the transition region are > accelerated by the electric field to energies sufficient to free electron- > hole pairs via collisions with bound electrons. Say I have a lightly doped Germanium pn junction, will I be able to use it again or throw it away once Avalanche breakdown take place? And also what happens in case of Zener breakdown? I know it requires highly doped pn junction and not so much pot diff..


title: What are the theoretical explanations behind the "no-boundary condition" in cosmology?

text: In his book "The Grand Design" on page 167-173 Stephen Hawking explains how one can get rid of the so called "time zero issue", meaning by this the very special initial state needed for inflation to work (?), by applying the so called "no-boundery condition" which says that time behaves as a closed space dimension for times small enough such that the universe is still governed by quantum gravity. To see better what Hawking is talking or writing about I`d like to understand this a bit more "technically" and detailed, so my questions are: 1) How can situations where QG kicks in lead to a transition of the time dimension to a space dimension? I mean what is the physical reason for this "ambiguity" of the signature of the correspondng spacetime metric? 2) What are the triggering causes of the "symmetry breaking" between the 4 space dimensions when starting from the no-boundary condition such that one of the space dimensions becomes an open time dimension at the beginning of time? 3) Are there (quantum gravity) theories or models that can explain 1) and 2) and if so, how? Since I've just read about these issues recently in the "Grand Design" (and nothing else) I`m probably confused and hope that somebody can enlighten me by explaining some details ...


title: Space and time translation of quantum field

text: In book I am studying, the authors want to calculate the spectral representation of the g-lesser (and g-greater) functions and find a general expression for these. In order to do so they start with the expression for the fields in Wigner coordnates (translated in time and space) $\hat \psi \left( \mathbf R + \frac{\mathbf r}{2} , t + \frac{\tau}{2} \right) $. They then claim that this may be written like $ \hat \psi \left( \mathbf R + \frac{\mathbf r}{2} , t + \frac{\tau}{2} \right) = \exp \left[ \frac{i}{2} \left( \hat H \tau + \hat{\mathbf P} \cdot \mathbf r \right) \right] \hat \psi \left( \mathbf R , t \right) \exp \left[ - \frac{i}{2} \left( \hat H \tau + \hat{\mathbf P} \cdot \mathbf r \right) \right] $ where $H$ is the Hamiltonian and $\hat{\mathbf P}$ is the total momentum operator. What are the underlying assumptions for the equation above to be true? One obvious thing is that the Hamiltonian must not be explicitly time dependent (since the time translation operator then gets a more complicated expression involving the time ordering operator). Is it for example understood that the total momentum operator and the Hamiltonian are commuting? If the Hamiltonian and the momentum operator do not commute then since $\exp \left[ \frac{i}{2} \left( \hat H \tau + \hat{\mathbf P} \cdot \mathbf r \right) \right] \neq \exp \left[ \frac{i}{2} \hat H \tau \right] \exp \left[ \frac{i}{2} \hat{\mathbf P} \cdot \mathbf r \right] $ the order of which the time and space translations are made matter. Perhaps someone could also help me clarify what the meaning of this is...


title: Centre of instantaneous rotation problem

text: Is there a point of Centre of Instantaneous Rotation (CIR) for every type of motion or only for cases of rolling?


title: Does the Sun orbit a much larger nearby star?

text: EDIT: I got thumbed down, so I removed the details. The question is already crystal clear, IMO.


title: Why do most of the motor with blades rotate anti-clockwise when viewed from the front facing the blade?

text: I have noticed that most of the motor with blades and engines rotate anti- clockwise direction when viewed from front facing blade. Is there any specific reason for this? Is it because of any kind of force acting? Or simply coincidence?


title: Is there just one fundamental frequency?

text: I read simple definitions of the terms _frequency_ , and _fundamental frequency_ , which defined them thus,   * _Frequency_ : the number of occurrences of a periodic wave during a second   * _Fundamental Frequency_ : the lowest frequency of a periodic wave.  By replacing _frequency_ in the definition of _fundamental frequency_ with the definition of _frequency_ renders the definition 'the fewest occurrences of a periodic wave during a second'. It seems to me that that the fewest possible occurrences of something must be one. Is the fundamental frequency one cycle per second?


title: Trying to understand infinite gravitational energy

text: Google has not been helpful because so many derivations of gravitational potential energy discuss $r$ at infinity. My understanding of this https://www.youtube.com/watch?v=IcxptIJS7kQ is: Dark energy increases with volume The volume of the universe is increasing. Therefore the total amount of dark energy is increasing. If the universe is flat , it can expand forever at an increasing rate. So the amount of dark energy $\to\infty$. The conservation of energy is not violated because gravitational energy is infinite. We know that $$U=-G\frac{mM}{r}$$ so when $r\to0$ $U \to\infty$. But no particle has zero radius so $r$ cannot be 0 so $U$ must be finite. So how can dark energy $\to\infty$?


title: Deducing instability growth rates from the Hamiltonian for the nonlinear schrodinger equation

text: Consider the following nonlinear Schrodinger equation (NLSE): $$A_t+iA_{xx}+i|A|^2A = 0$$ where $A$ is a complex valued function of $(x,t)$. A solution to this equation is $A=a_oe^{-ia_o^2t}$. We investigate the stability of these solutions by considering a perturbation to the above solution of the form $a_oe^{-it} (1+\alpha_+e^{i(kx-\Omega t)}+\alpha_-e^{-i(kx-\Omega t)})$, for constants $(\alpha_+,\alpha_-, k,\Omega)$. Putting this back into the NLSE and collecting terms of order $\alpha_{\pm}$, we find a system of equations $ \left( \begin{array}{cc} a_o^2-k^2-\Omega & a_o^2 \\\ a_o^2 & a_o^2-k^2+\Omega \end{array} \right) \left( \begin{array}{c} \alpha_+ \\\ \alpha_- \end{array} \right) = 0$. This has non-trivial solution when the determinant of the matrix is zero, which is the condition that $\Omega^2 = k^2(k^2-a_o^2) $, which takes on imaginary values (ie leads to unstable growth) when $k < \sqrt{2} a_o$. (Note: The way the problem has been non-dimensionalized, all of the parameters (eg $k,a_o$) are unitless. ) This has maximal growth rate $Im(\Omega_*) = a_o^2$ for $k_* = a_o$. Now, this is equivalent to the so called Benjamin-Feir (or modulation instability) inherent to a variety of physical systems (eg water waves, lasers). Next, recall that the NLSE can be derived from a Hamiltonian density H, where $H = |A_x|^2 - \frac{1}{2} |A|^4$, and Hamilton's equations take the form $i\frac{\partial A}{\partial t} = \frac{\partial H}{\partial A}$ and the complex conjugate of this. My question is, is there anything in the structure of this Hamiltonian that can give us the results of the spectral stability analysis in a different manner? Naively, I want to know if I can deduce the instability criteria and the growth rate directly from the structure of the Hamiltonian. I am vaguely aware of certain stability criterion used in the study of the stability of Hamiltonian systems (eg Krein signatures etc), but am not familiar with how it works in practice (and in particular in this relatively simple example) and would greatly appreciate any tips, or references to relevant resources. Thanks! Nick


title: Is there Bremsstrahlung radiation for a charged massless particle?

text: This is a follow up question from: Massless charged particles Since by definition such a particle would interact with photons- resulting in some change in momentum- would the particle emit Bremsstrahlung during this interaction? If it does, it seems that there is a problem as the particle cannot decelerate, yet the Bremsstrahlung would necessarily carry away some of the energy. Or perhaps the fact the particle is massless precludes such emissions. Is this a real problem? What would the physical effects of Bremsstrahlung radiation be for this particle? Note: for some reference- this question came out of a discussion in the comments after my answer to Massless charged particles


title: What the rate of heat transfer between my thumb and a surface of known dimensions and properties?

text: I am trying to calibrate a high speed thermocouple for measuring heat transfer rates. One of the calibration techniques I am using is sticking my thumb to the thermocouple (which is embedded flush into a solid steel surface onto which I am pressing my finger) and measuring the result. The time taken for my thumb to fully compress against the surface appears to be trivial judging from the temperature trace we are recording; let's assume this is the case anyway. We have used the measurements to derive time-accurate heat transfer rates (peaking around 22kW/m^2 for anyone interested - seems awfully high), but now I wish to compare that peak starting value (before my thumb cools and the surface heats up) to the expected result based on what I think should be simple theory. How would I go about predicting this? Specifically - **what is a good _analytical_ approximation of the heat transfer over a short _but finite_ period of time immediately following contact between my thumb and the metallic surface?** (By short I mean a period sufficiently short that a) temperature changes do not propagate to the end of either the thermocouple or my finger and, hopefully b) a numerical solution is not necessary to provide a solution). Some background information: The thermocouple is constantan/chromel with known (and assumed to be uniform) thermal properties. The thermocouple is insulated from the rest of the metal surface and any cross-conduction can be ignored. The temperature of both my thumb and the surface prior to contact are known. Let's assume my body temperature is also uniform throughout my thumb, and that the thermal properties can be approximated by those of water (is this a good assumption?). My thumb and the thermocouple are semi-infinite over the time period.


title: Is it possible to derive fermi-dirac or bose-einstein statistics using quantum operator formulations?

text: I've been looking through theory on identical particles to get a better grasp of the uncertainty principle but it would be very interesting if these results could be extracted from the formalism as well. Does anyone have any leads on this?


title: Motion of elevating airplane

text: I wonder how the motion of an airplane which will not try stop to elevate should look like. Of course, at some point the plane will no longer be able to go to higher altitudes but is it going to stay on a circular trajectory or things are more involved?


title: Quantum Hamiltonian commuting with the Pauli-Runge vector

text: I have to prove that $[A_j, H] = 0$, with; $$\vec{A} = \frac{1}{2Ze^{2}m}(\vec{L} \times \vec{P} - \vec{p} \times \vec{L}) + \frac{\vec{r}}{r}$$ $$H = \frac{p^2}{2m} - \frac{Ze^2}{r}$$ And, $Z, e, m$ are all constants. Now, after expanding out the commutator bracket, I'm left with six terms, which I need to prove are all equal to zero. Now, I can do this for $[\vec{L} \times \vec{p}, p^2]$, $[\vec{p} \times \vec{L}, p^2]$, and $[\frac{\vec{r}}{r}, \frac{1}{r}]$, but I'm having trouble with $[\vec{L} \times \vec{p}, \frac{1}{r}]$, $[\vec{p} \times \vec{L}, \frac{1}{r}]$, and $[\frac{\vec{r}}{r}, p^2]$. I'm sure once I've $[\vec{L} \times \vec{p}, \frac{1}{r}]$, I can prove $[\vec{p} \times \vec{L}, \frac{1}{r}]$, but, I'm not really sure what to do. With $[\vec{L} \times \vec{p}, \frac{1}{r}]$, I've done the following; $$[\vec{L} \times \vec{p}, \frac{1}{r}] = \varepsilon_{ijk}[L_{j}p_{k}, \frac{1}{r}]$$ $$ = \varepsilon_{ijk}(L_j[p_k, \frac{1}{r}] + [L_j, \frac{1}{r}]p_k)$$ $$ = \varepsilon_{ijk}L_{j}[p_k, \frac{1}{r}]$$ $$ = \frac{i \varepsilon_{ijk} L_j q_k}{r^3}$$ But from here, I'm not sure where to go. I was hoping to use the cyclic nature of the Levi-Civita symbol, but I can't really see how that helps here. Any input would be fantastic!!


How to prove that $i\hbar\frac{\partial}{\partial \mathbf{p}}$ is the operator of $\mathbf{x}$ in momentum space?


title: Specific Gravity vs. Pore Space in Chromatography Column

text: Wondering if we can apply molecular specific gravity to elucidate average time required to pass through a column given porespace, distance and viscosity? This would eliminate the need for special dyes or fluorescence to create exacted captures of required material out of solution.


title: self-antiparticles and broken symmetries

text: certain particles (i.e: certain bosons like the photon) do not have an anti- particle, or rather, they are they own anti-particles. lets assume that such symmetry is only approximate and these particles are actually different than their antiparticles, just that the broken symmetry is very small and hard to detect **Question** what experimental facts would we expect to change if such symmetries were only approximate? what physical consequences would they show? * * * **EDIT** The answers have focused on the argument that any symmetry breaking of this kind would be an all-or-nothing proposition. I would like to explore that argument a bit more with a more precise notion of a soft-broken symmetry, which would hopefully, elicit answers that will expose better the reason why symmetry breaking has to be an all-or-nothing: what about photons hypothetically having a slightly higher coupling constant to matter than to antimatter, and a symmetric situation with their dagger counterparts, which would couple slightly stronger to anti-matter? what argument exists to discard that possibility theoretically? maybe we have bounds on the delta of such couplings? to what fraction of $\alpha$ are such bounds known experimentally?


title: holographic principle and Wheeler's bag of gold

text: How is it possible to explain "bag of gold" spacetimes (see Marlof) such that the ideas are compatible with AdS/CFT and the holographic principle?


title: Why is ski jumping not suicidal?

text: At least on television, ski jumpers seem to fall great vertical distances before they hit the ground - at least a few dozen meters, though I couldn't find exact distances via a quick search. And yet they almost always land on their feet as if they just fell two or three meters. (Here's a whole lot of footage from the Vancouver Olympics if you need to refresh your memory.) Without going into the level of equations (which I wouldn't understand), why are ski jumpers able to fall such great heights without seriously injuring themselves?


title: Bose-Einstein-statistics out of fermionic many body system

text: ok, let me try this again. How do I get to atoms obeying Bose Einstein statistics from considering the fermionic many body problem of a bunch of electrons, protons and neutrons forming these (bosonic) atoms. [ORIGINAL QUESTION] this keeps bugging me: Whenever there is a system of bosons (lets not consider photons/phonons) then we use BE statistics to describe it. But shouldn't we also be able to describe it (more accurately) by FD statistics with regard to its constituent electrons, protons and neutrons? How is this consistent with the symmetry requirements on the wave function and phenomenas as Bose Einstein condensation?


title: How to derive the Karman-Howarth-Monin relation for anisotropic turbulence?

text: I find the derivation of the Karman-Howarth-Monin relation in the book Turbulence from Frisch (1995) a bit to short. Can someone point me to a more detailed derivation of that relation, if possible in a publicly available source (lecture notes, thesis, arXiv paper)? Or can someone explain to me how Frisch obtains the equation that follows the sentence on page 78: > Starting from the Navier-Stokes equation (6.6), we obtain $$\begin{align} > \partial_t \frac{1}{2}\langle v_i v'_i \rangle =& \- \frac{1}{2} \partial_j > \langle v_i v_j v'_i\rangle \- \frac{1}{2} \partial'_j \langle v'_i v'_j > v_i\rangle \\\ & \- \frac{1}{2} \langle v'_i \partial_i p \rangle \- > \frac{1}{2} \langle v_i \partial'_i p' \rangle\\\ &\+ \frac{1}{2} \langle > v'_i f_i \rangle \+ \frac{1}{2} \langle v_i f'_i \rangle \\\ & \+ > \frac{1}{2} \nu \left( \partial_{jj} + \partial'_{jj} \right) \langle > v_iv'_i\rangle. \qquad (6.11) \end{align} $$


title: Calculation of a bending moment

text: I'd like to calculate the bending moment of a cantilever, fixed at its base, and submitted to a certain stress on a specific spot, but I can't find the proper definition of this bending moment (first time I'm encountering this name, and google didn't not help me so far).


title: About the gauge formalism in statistical quantum field theory

text: I would like to understand a bit more the aspects of the gauge theory in statistical field theory. In particular, I would like to understand how the replacement $\tau \rightarrow it/\hbar$ is performed in a mathematically proper way, when $\tau$ is an inverse temperature. This replacement comes from the resemblance between the evolution operator $e^{-iHt/\hbar}$ in quantum field theory, and the statistical weight $e^{-H\tau}$ in statistical physics (one sees then that $\tau=\left(k_{B}T\right)^{-1}$ in case you wonder :-). In principle it leads to a compact momentum space, when the frequencies become discrete, and called Matsubara frequencies $\omega_{n}=2\pi k_B T \left(n+1/2\right)$ and $\omega_{n}=2\pi n k_{B}T$ with $n$ an integer for fermions and bosons. I'm perfectly aware of the classic book > _Methods of quantum field theory in statistical physics_ by Abrikosov, > Gor'kov and Dzyalochinski - Dover Books on Physics but I'm stuck on the gauge formalism. Can we do a gauge transformation in _imaginary time_ -$\tau$ as we do with real time-$t$ ? Does the _imaginary- time_ -covariant derivative $\partial_{\tau}+A_{\tau}$ make any sense? Are there some precautions to take? Any comment, answer, indications or even good reference (or even just keywords) about this topic is warm welcome. I precise I'm a condensed matter physicist, so if you could adapt your parlance to me (for instance, please talk slowly and loudly), I would greatly appreciate :-) **EDIT:** Clearly the keyword is _thermal quantum field theory_ and there is an associated Wikipedia page with plenty good references. Anyway, any comment are still welcome, since I progress really slowly understanding this, especially what a gauge choice mean? Thanks in advance.


title: How can I interpret negative values of potential evapotranspiration?

text: If I extract Potential Evaporation (PET, W/m$^2$) from the National Centers for Environmental Prediction (NCEP), climate reanalysis data (downloadable as netCDF files here), there are some negative values, e.g. for summer in southern Manitoba (50N, 100W), I get the attached distribution of PET values.: ![enter image description here](http://i.stack.imgur.com/G6Gct.png) Is this a real physical process, such as condensation (dew formation), or is it an error (e.g. should I average over these values or set them to zero)? * * * **update** requested metadata: The metadata (contained in the files) says that the valid range is -800, 5200; here is further description of the 'pevpr' variable (provided in the data header):   * long_name = "Monthly Mean of Potential Evaporation Rate"   * valid_range = -800,5200 ;   * units = "W/m^2" ;   * add_offset = 0 ;   * scale_factor = 1 ;   * missing_value = -9.96921e+36 ;   * precision = 1 ;   * least_significant_digit = 0 ;   * var_desc = "Potential Evaporation Rate" ;   * dataset = "CDC Derived NCEP Reanalysis Products" ;   * level_desc = "Surface" ;   * statistic = "Mean" ;   * parent_stat = "Individual Obs" ;


title: Could Space Travel be achieved using a very strong magnetic field to warp space?

text: An Alcubierre Warp Drive could theoretically achieve faster than light space travel but requires exotic matter or negative energy to compress space in front of the spaceship and expand space behind it. Could a similar effect be achieved by only compressing space in front of the spaceship? As the spaceship fell into the gravity well and moved forward, space would automatically expand back to normal behind the ship as it rode the distortion wave. If this propulsion method worked, it would require only a fraction of the energy needed by the Alcubierre Warp Drive. Also, if space could be compressed ahead of the spaceship using a very, strong magnetic field, no exotic matter or negative energy would be needed, just the electromagnetic force. Even if faster than light travel could not be achieved this way, a small constant acceleration, maintained over a long period of time would achieve a high near light velocity.


title: Holomorphic coupling as a source for gaugino condensation

text: On the top of page 23 of hep-th/03061119, it is pointed out that treating the holomorphic gauge coupling $\tau$ as a background (spurion) superfield allows one to think of its $F$-term, $F_\tau$ as the source for the gaugino condensate, $\lambda^a\lambda^a$. This is because in the supersymmetric gauge kinetic term, $\mathcal L \supset \int \tau \mathcal{W}^a\mathcal{W}_a$, $F_\tau$ is the coupling for the $\lambda^2$ term. I think I understand how this acts as a constant source for $\lambda^2$ 'tadpoles', but I don't undertand why the $\theta^0$ term of $\tau$ (i.e. the lowest component of the background superfield, the gauge coupling itself) shouldn't also be thought of as a source for $F^2$. It makes sense to me that $F^2$ is just the usual gauge kinetic term and its coefficient is related to the gauge coupling. However, why is this term any different from the $\lambda^2$ term: why should $F_\tau$ be thought of as a source for a gaugino condensate whereas $\tau$ shouldn't be thought of as a source for a gluon condensate? Thanks!


title: Is there a relation between supersymmetry and entropy?

text: Considering that entropy denotes the level of order/disorder in a system, would it be possible for entropy and supersymmetry to exist at the same time? Or, are they entirely unrelated?


title: Excitation probability for 2 Level Atom

text: I currently started learning Quantum Optics using the book "Atom Physics" of Christopher Foot. Currently I am working through Chapter 7 and a question occured concerning the intuitively physical understanding: Assume a two level atom with energy levels $E_{1}$ and $E_{2}$ (with $E_{1}$<$E_{2}$ is given). The eigenstates of the system are given by $|1\rangle$ and $|2\rangle$ respectively. Now assume that we perturbe the system by using a time dependent Electric field of the form \begin{equation} \vec{E} = \vec E_{0} \cos\left(kz - \omega t\right) \end{equation} and expand the solution of the Schroedinger Equation in terms of the unperturbed eigenstates \begin{equation} \Psi(\vec r,t) = c_{1}(t) \ e^{-i (E_{1}/\hbar) t} \ |1\rangle \+ c_{2}(t) \ e^{-i (E_{2}/\hbar) t} \ |2\rangle \end{equation} Let's assume that $c_{1}(0)=1$ and $c_{2}(0)=0$. Using the dipole approximation as well as the rotating wave approximation as discussed for example in www.physics.ox.ac.uk/Users/lucas/BIII/QuantumNotes.pdf, we obtain that \begin{equation} |c_{2}(t)| \sim t^{2} \ \frac{\sin^{2}(x)}{x^{2}} \end{equation} with $x=(\omega-\omega_{0})t/2$ and $\hbar \omega_{0}:=E_{2}-E_{1}$. Plotting this as a function of $\omega-\omega_{0}$ we obtain a "Fraunhofer-diffraction"-like graph. My Question: 1) As the interaction time increases the width of the maxima is decreasing. What is from a physical / intuitive perspective the reason for this. 2) There are also certain minima occuring for certain frequencies of the light. What is the physical interpretation (intuitively) for this? I am looking forward to your responses. Thanks in advance.


title: Mnemonic for creation and anhiliation operators

text: I am not sure if this is question is withing the scope of this page. I have an exam in quantum mechanics in a few weeks and I always confuse anhilation and creation operator. $$ a^\dagger |n\rangle = \sqrt{n+1} |n+1\rangle $$ $$ a|n\rangle = \sqrt{n} |n-1\rangle $$ Does someone have a mnemonic for these relations?


title: To what extent is theoretical physics done in a non-univeristy setting?

text: Maybe it's just me, but in the last weeks or so I noticed a small increas of the number of more theoretical question on this site. I consider myself on this side and I wondered how many, if this is quantifiable, of the physical models are eventually "put to use" (under quotation marks, see below). Of course, the aim is to eventually find a physical theory, maybe _the_ physical theory, but in reality, nobody in high energy physics can expect to land the big hit in less than quite some years. The thing is that whenever I concern myself with some set of models, privately, then I don't really care if what I do would be labeled "physics" or "the study of some sections over some manifold and it's associated algebraic structures." All these things are beautiful to work with. And so often I'm not sure if people really _want_ to make physics or if that's just what they have to say to be able to keep doing it. I guess this paragraph basically seeks for a theoretical physicists apology. So I think especially of the different field and lattice models in this case. Clearly, theories like electromagnetism, semi-conductor models or microscopic theories for reaction rates are transformed to money in the industy. Considering only models which involve, say, some Lagrangian or Hamiltonian, can it be estimated how many of them see the day of light in the above sense? I'm fairly young and I don't have any overview. Are these kind of theories always only born in universities? (I guess maybe goverment funded theory departments are somewhere in between.)


title: Finding the mass of 7cm string

text: If a string has $120$ g and length $40$ cm. I found that the linear mass density is given by $$120g/40cm=\ linear\ mass\ density$$ How do I use this to find the mass of a $7$cm of the same kind of string? * * * What I have done so far: $$120g/40cm = m/7cm$$ $$m=7cm*120g/40cm$$ $$m = 21$$


title: Lorentz spinors of $SO(n,1)$ and conformal spinors of $SO(n,2)$

text: * It would be great if someone can give me a reference (short enough!) which explains the (spinor) representation theory of the groups $SO(n,1)$ and $SO(n,2)$. I have searched through a few standard representation theory books and I couldn't find any.   * More specifically I would like to know how a Lorentz spinor of $SO(n-1,1)$ (say $Q$) is "completed" to a conformal spinor of $SO(n,2)$ (say $V$) by saying,  $V = (Q, C\bar{S})$ where $C$ is a "charge conjugation operator" and $S$ is probably another $SO(n-1,1)$ spinor. Is there some natural Clifford algebra representation ($\Gamma$) lurking around here with respect to which I can define the "charge conjugation operator" as $C$ such that $C^{-1}\Gamma C = - \Gamma ^T$? (...in general a representation of the Clifford algebra also gives a representation of $SO(n,1)$..I would like to know as to how this general idea might be working here...)   * Some of the other aspects of this group theory that I want to know are an explanation for facts like,      * $Sp(4)$ is the same as $SO(3,2)$, and the fundamental of $Sp(4)$ is the spinor of $SO(3,2)$     * $SU(2,2)$ is the same as SO(4,2), and the fundamental of $SU(2,2)$ is the spinor of $SO(4,2)$ (...just two "facts" hoping that people can point me to some literature (hopefully short!) which explains the systematics of which the above are probably two examples...)


title: simulate a quantum computer on a normal one

text: Would it be possible to simulate a quantum computer on a normal computer. I was thinking something like have a bunch of GPU's running a set of entangled quantum particles and then produce the results. Would this be feasible or one of those take the lifetime of the universe to run problems?


title: Do we always ignore zero energy solutions to the (one dimensional) Schrödinger equation?

text: When we solve the Schrödinger equation on an infinite domain with a given potential $U$, much of the time the lowest possible energy for a solution corresponds to a non-zero energy. For example, for the simple harmonic oscillator with frequency $\omega$, the possible energies are $\hbar\omega(n+\frac12)$ for $n =0,1,\dots$ . Some of the time, solutions with zero energy are possibly mathematically, but boundary conditions would mean that such solutions would be everywhere zero, and hence the probability of finding a particle anywhere would be zero. (For example, an infinite potential well). However, when solving the Schrödinger equation for a particle moving freely on a circle of lenfth $2\pi$ with periodic boundary conditions $\Psi(0,t)=\Psi(2\pi,t)$ and $\frac{\partial\Psi}{\partial x}(0,t)=\frac{\partial\Psi}{\partial x}(2\pi,t)$, I have found a (normalised) solution $\Psi(x,t)=\frac1{2\pi}$ with corresponding energy $0$. I can't find a way to discount this mathematically, and it seems to make sense physically as well. Is this a valid solution, and so is it sometimes allowable to have solutions with $0$ energy? Or is there something I'm missing?


title: Conical spacetime of cosmic string

text: Inspired by: Angular deficit   The 2+1 spacetime is easier for me to visualize, so let's use that here. (so I guess the cosmic string is now just a 'point' in space, but a 'line' in spacetime) Edward says it is possible to cut a wedge out of a flat spacetime and glue the edges together. So in my mind this looks like a paper cone. I'm having trouble understanding why this is flat everywhere except at the 'tip' of the cone. Imagine a triangle on the original paper, and now after piecing it together, wouldn't an observer think the lines are now curved? And in piecing it together, wouldn't there now be another angle, so it is a 4 sided polygon, and the exterior angles won't correctly add up to 360 degrees anymore? I'm just very confused because Edward and Lubos say the spacetime is flat everywhere except at the center, so the Riemann curvature tensor is zero everywhere except at the center, but Lubos says a parallel transported vector around a path on this flat spacetime can change angle!? Does this mean we can't describe the parallel transport of a vector with the local Riemann curvature? Hopefully I've said enough that someone knowledgeable can see what is confusing me and can help me understand. If a clear question is needed then let it be this:   **How can we explicitly calculate the curvature, and the effect this has on angles of paths or vectors, in conical spacetime?** The 'paste flat spacetime pieces together' process seems very fishy to me. **UPDATE:**   Okay, thanks to Ted and Edward I got most of it figured out (although my attempt couldn't say anything about the curvature 'spike' at the center)', but still can't figure out how to see the parallel transport of vector in a closed loop ala Lubos's comment. It would be neat to see this last part worked out for an arbitrary loop. In particular Ted's comment _"that (in some appropriately-defined sense) the average curvature inside the triangle is nonzero. In this particular case, that average comes entirely from a curvature "spike" at the origin."_ sounds like there may be an easy way to transfer the integral around a path to an integral over the area bounded by the path, ala Gauss-Bonnet but the integral I'm getting doesn't even look like a normal integral and I don't really understand what Gauss-Bonnet is saying physically. Can someone work out this last little piece explicitly, and if you use something like Gauss-Bonnet maybe help explain what the math is telling us about the physics here?


title: The most stable metal for radioactive environment

text: Could you suggest which metal/alloy shows the least activation & better long- term stability for space (LEO, inside Van allen belts, deep space) applications? Or stability of construction elements isn't an issue comparing to stability of electronics?


title: Effective Hamiltonian / Perturbation theory for non-degenerate case

text: Trying to wrap my head around the following situation. Consider first a case that I understand well: Let's assume a three level system where the lowest two levels are degenerate and individually couple to a third level, higher in energy by some large $\Delta$. The Hamiltonian may be written as $$\begin{pmatrix}0 & 0 &t \\\0 & 0 & t \\\ t & t & \Delta\end{pmatrix}$$ If we want to integrate out the higher energy level, we can use the partition method / downfolding to get an effective coupling $t' = -t^2/\Delta$ to arrive at an effective Hamiltonian for the two-level system, $$\begin{pmatrix}0 & t' \\\ t' & 0\end{pmatrix}$$ The denominator $\Delta$ comes from a term $E - H_1$ in the perturbation expansion where $H_1$ is the Hamiltonian for the higher-energy sector. So far, so good, because both states in the low-energy sector have the same energy. **But what if the energies aren't the same?** Let's consider the Hamiltonian $$\begin{pmatrix} 0 & 0 & t \\\ 0 & \epsilon & t \\\ t & t & \Delta \end{pmatrix}$$ with a small $\epsilon$. However, then I cannot simply set $E = 0$ in the denominator for the perturbation expansion. I know that I can still use Rayleigh-Schroedinger perturbation theory to calculate the energy corrections, but what if I don't want to do that and rather want an effective Hamiltonian for those two low-energy states? Just to be clear: I don't want to "solve" the Hamiltonian per se, and rather want to arrive at an effective Hamiltonian for the low-energy subspace, which will look something like $$\begin{pmatrix} 0 & t_{\rm eff} \\\ t_{\rm eff} & \epsilon\end{pmatrix}$$


title: Why does string not wick down?

text: I regularly drink tea at work and I often reuse the tea bags (yes I know I'm a cheapskate). Yesterday afternoon I used a tea bag once and kept it in the cup in case I wanted another cup before I left. As it turned out, I didn't and the tea bag was still there when I came into the office this morning with a very curious pattern of discoloration. I believe I understand most of the process; over night, the bag was drying out with the upper portion drying most quickly and the liquid from the leaves wicking up the bag and into the string. As the process continued, progressively more suspended solids were carried up the system and deposited near the top where the rate of drying was most rapid. As a result, there is a dark line of tea deposits around the upper edge of the bag and on the string. Oddly, though, the deposits very abruptly stop at the edge of the cup where the string drops to the dangling tag. Close examination reveals that this discoloration stops not exactly at the edge of the cup (as might be expected if contact with the cup were the cause) but slightly beyond it where the string descends. The only reason I can think of as to why this should be so is that the wicking action responsible for the discoloration of part of the string, stopped at that point. But why should this be? After all, a siphon ONLY works when the receiving end is below the apex. Am I wrong in my assumptions? Or does anyone know: Why does a string not wick down? ![A photo of the situation.](http://i.stack.imgur.com/y9t9w.jpg)


title: Flat space metrics

text: This question concerns the metric of a flat space: $$ds^2=dr^2+cr^2\,\,d\theta^2$$ where $c$ is a constant. Why is it necessary to set $c=1$ to avoid singularities and to restrict $r\ge 0$? Thanks.


title: canonical and microcanonical ensemble

text: What does one mean by **canonical** and **micro canonical ensemble** in statistical mechanics? Can one elaborate on this in a very simple way with examples? Pardon me, if it is a very simple thing; I am a math student.


title: Why do objects accelerate as they fall?

text: Most importantly, what must change in order for the falling object to change its speed? Is it the distance to the centre of the planet? If you pull the earth away from the object as the object falls, will the object slow down or will it keep accelerating?


title: What determines the sign of an image distance?

text: > A lens placed at the origin with its axis pointing along the x axis produces > a real inverted image at $x = - 24 cm$ that is twice as tall as the object. > > What is the image distance? Why is the image distance $s' = 24\text{ cm}$ not $-24\text{ cm}$ (actually already given in the question)?


title: the temperature of photon and its energy

text: Do photons have temperature? If not, does it mean that photon lose energy while travelling through space? As the planets farther away from the sun are comparatively cooler than the one that are closer to it, does it imply that photon also lose energy?


title: Do EM waves transmit spin polarization?

text: Suppose you have a normal dipole antennae (transmitter and receiver) . Spin polarized current (as opposed to normal current) is sent into the transmitter, it emits an EM wave and the Receiver receives it. Will the charge carriers in the receiver become spin polarized as well? In other words, will the spin polarization of the transmitter current have some effect on the receiver, like for example imposing the spin polarization on the receiver carriers by means of making EM radiation circularly polarized? I am aware that this effect is possible using certain semiconductors. But I am talking about a normal metal chunk used as the antennae. I am wondering whether the spin polarization of the transmitter current will have any effect on the receiver on a deeper level: using principles of Quantum Field Theory and Quantum Electrodynamics? (I don't know anything about QFT and QED btw)


title: Tips on teaching Dimensional Analysis?

text: What's a good way to explain dimensional analysis to a student? Here's a simple question which this method would be useful: Let's say a truck is moving with a speed of 18 m/s to a new speed of 13 m/s over a distance of 48 meters. How long did it take for the truck to break to its new speed, and what was the acceleration of the truck? Edit @TimGoodman and @Laar, Hmm. I was under the impression that you could solve such questions by analyzing their dimensional properties. For example, based on V1 and V0 in the example above I know the person decelerated 5m/s over a distance of 48 meters. If I want the time it took (first question) I can simply look at the units and see if I can eliminate all units except s (seconds). 48m/5m/s allows the meters to cancel out leaving 9.6 seconds. Now I have a new peice of information and I can now solve the second question "What was the acceleration of the truck?" Since I'm solving for acceleration I know my final units will be m/s^2. I know the velocity of the deceleration was 5m/s and it took him 9.6s. Dividing 5m/s/9.6s = .52 m/s^2. My final units match and so by using dimensional analysis I've solved the problem. Maybe I have a misunderstanding of dimensional analysis. Any suggestions?


title: When does centripetal force cause constant circular motion?

text: As far as I know, if an object has a speed $s$ and a force is applied to it which generates an acceleration of $s^2/r$, then the object will start moving in a circle of radius $r$. Also, as far as I understand, this circular motion is constant (the radius will remain constant at all times). However, I've seen physics problems that go something along the lines of _a person stands on a merry-go-round, going at a certain tangential speed, find the friction needed for the person to not slip off_ , and the correct answer would be $$\mu N m = m \frac{s^2}{r} \implies \mu = \frac{s^2}{N r}$$ Given what I've said in the first paragraph, I don't understand why the person would slip off. If the acceleration is equal to $s^2/r$, why is friction needed?


title: How is gravitational lensing possible?

text: How is it possible that a force that affects mass (gravity) can affect massless particles like photons?


title: Classical static equilibrium in atomic physics

text: Consider a collection of $n+1$ mass weighted points in $\mathbb{R}^3$. Suppose we have one mass located at the point (0,0,0) with mass $m\in\mathbb{N}$ and further suppose we have $n$ masses arranged arbitrarily each with mass -1. Now we want to assign to each mass weighted coordinate a collection of vectors in the following way:   1. There will be a total of $n$ vectors emanating from each weighted point.   2. The assigned vectors $\vec v$ will be of two varieties: $$\vec v\in\\{\frac{\hat r_{ij}}{||r_{ij}||^2},\frac{-m\hat r_{i0}}{||\vec r_{i0}||^2}\\}$$ where $||r_{ij}||$ is the norm of the vector between points $i$ and $j$, $\hat r_{ij}$ denotes a direction away from both points $i,j$ and along the line between $i$ and $j$. $||r_{i0}||$ is the norm of the vector between a point $i$ and the origin and $-\hat r_{i0}$ denotes a direction from point $i$ to the origin. I am interested in whether there exists a 3 dimensional configuration of the points where the sum of all the vectors is zero for each point. If this is not possible I hope someone would give some hints on how to prove it or point me towards a proof in the literature. In essence what I am seeking is whether there exist three dimensional equilibrium configurations for Coulomb's law of electrostatics as you come across the periodic table of the elements. I suspect not but who was the first to prove it?


title: What prevents a natural fibers rope from untwisting when it is elongated?

text: I don't understand why a rope doesn't untwist when an attempt is made to increase its length. It seems counter-intuitive to me, because untwisting would allow the rope to increase its length, but in practice it does not untwist. Do the braids spiral-like shape and the friction between the braids prevents this from happening ?


title: Dimensional analysis for gravitational radiation expression

text: on this paper, please refer to equation 2.117 for the power emitted for a rotating mass system: $$ P = - \frac{128}{5} G M^2 R^4 \Omega^6 $$ power in cgs should be (g is grams, m is meter, s is seconds): $$ g m^2 s^{-3} $$ now, $G$ is in $m^3 g^{-1} s^{-2}$ and $\Omega$ is in $s^{-1}$ so 2.117 right hand side is $$m^7 g s^{-8} $$ so, i'm going to infer that the right hand side is missing a factor of $\frac{1}{c^5}$, so the dimensionally accurate expression for power (without weird normalized units) is; $$ P = - \frac{128}{5 c^5} G M^2 R^4 \Omega^6 $$ is that dimensional analysis accurate?


title: Calculating probability current for scattering problem

text: I'm trying to calculate the probability current for a scattering problem. The potential is $V = V_0 > 0$ in $x>0$, with $E>V_0$ So I have in the region $x \le 0$: $$\psi = \exp(ikx) + R \exp(-ikx)$$ And in $x>0$ $$\psi = T \exp(i \kappa x)$$ I am trying to calculate the probability current, $j = \frac{-ih}{2m} (\bar{\psi}\psi' - \bar{\psi'}\psi)$, in each region and show that it is equal. However when I calculate the probability current in $x<0$, I get: $$\bar{\psi} = \exp(-ikx) + \bar{R}\exp(ikx)$$ $$\psi' = ik\exp(ikx) -ikR\exp(-ikx)$$ $$\bar{\psi'} = -ik\exp(-ikx) + ik\bar{R}\exp(ikx)$$ $$\psi = \exp(ikx) + R\exp(-ikx)$$ So: $$\bar{\psi} \psi' = ik -ikR\exp(-2ikx) + \bar{R}ik\exp(2ikx) - ikR\bar{R}$$ $$\bar{\psi}' \psi = -ik -ikR\exp(-2ikx) +ik\bar{R}exp(2ikx) +ik R\bar{R}$$ Hence, $$j = \frac{hk}{2m} (1 + R \exp(-2ikx) - \bar{R} \exp(2ikx) - R\bar{R})$$ And in the region $x>0$: $$j = \frac{\kappa h}{2m}(T\bar{T})$$. I can show that (by imposing continuity conditions at the boundary): $$k(1-|R|^2) = \kappa |T|^2$$ So I would expect the first probability current to just be: $\frac{kh}{2m}(1-|R|^2)$. Any help with this issue is much appreciated! I'm pretty sure I'm making some stupid mistake somewhere, but it's very frustrating as I cannot find it! Thanks


title: Advanced atomic physics: From Liouville Equations to the Bloch equations

text: I'm trying to derive the Bloch equations from the Liouville equation. This should be possible according to this paper, where it discusses higher order Bloch equations (second order spherical tensors). I'm trying to derive the same for the simple vector Bloch equations. The way I'm doing this is the Following. The Lioville equation is this: $$\frac{d}{dt}\rho=\frac{1}{i\hbar}\left[H,\rho\right]$$ The Hamiltonian for an interaction with the magnetic field can be written as $$H=-\vec{\mu}\cdot\vec{B}=-\frac{g\mu_{B}}{\hbar}\vec{B}\cdot\vec{F}=-\vec{\omega}\cdot\vec{F},$$ where $$\left\\{ \omega_{x},\omega_{y},\omega_{z}\right\\} =\frac{g\mu_{B}}{\hbar}\left\\{ B_{x},B_{y},B_{z}\right\\} $$ And I expand the dot product using spherical basis dot product: $$\vec{\omega}\cdot\vec{F}=\sum_{q=-k}^{k}\left(-1\right)^{q}\omega_{q}F_{-q}=-\omega_{-1}F_{1}+\omega_{0}F_{0}-\omega_{1}F_{-1}$$ The density matrix can be expanded in terms of spherical operators as: $$\rho=\sum_{k=0}^{2F}\sum_{q=-k}^{k}m_{k,q}T_{q}^{\left(k\right)},$$ Now I try to compute the commutator using all the information I calculated: $$\left[H,\rho\right]=\sum_{q=-k}^{k}\left[\omega_{-1}F_{1}-\omega_{0}F_{0}+\omega_{1}F_{-1},m_{k,q}T_{q}^{\left(k\right)}\right]$$ $$=\sum_{q=-k}^{k}m_{k,q}\left[\omega_{-1}F_{1}-\omega_{0}F_{0}+\omega_{1}F_{-1},T_{q}^{\left(k\right)}\right]$$ $$=\sum_{q=-k}^{k}m_{k,q}\omega_{-1}\left[F_{1},T_{q}^{\left(k\right)}\right]-\sum_{q=-k}^{k}m_{k2,q}\omega_{0}\left[F_{0},T_{q}^{\left(k\right)}\right]+\sum_{q=-k}^{k}m_{k,q}\omega_{1}\left[F_{-1},T_{q}^{\left(k\right)}\right]$$ Now we use the commutation relations with tensor operators: $$\left[F_{\pm1},T_{q}^{k}\right]=\hbar\sqrt{\left(k\mp q\right)\left(k\pm q+1\right)}T_{q\pm1}^{k}$$ $$\left[F_{0},T_{q}^{k}\right]=\hbar qT_{q}^{k}$$ And we get: $$\left[H,\rho\right]=\sum_{q=-k}^{k}m_{k,q}\omega_{-1}\hbar\sqrt{\left(k-q\right)\left(k+q+1\right)}T_{q+1}^{k}-\sum_{q=-k}^{k}m_{k,q}\hbar q\omega_{0}T_{q}^{k}+\sum_{q=-k}^{k}m_{k,q}\omega_{+1}\hbar\sqrt{\left(k+q\right)\left(k-q+1\right)}T_{q-1}^{k}$$ Now we substitute this in the Liouville equation $$i\sum_{q}\frac{d}{dt}m_{k,q}T_{q}^{\left(k\right)}=\sum_{q=-k}^{k}m_{k,q}\omega_{-1}\sqrt{\left(k-q\right)\left(k+q+1\right)}T_{q+1}^{k}-\sum_{q=-k}^{k}m_{k,q}q\omega_{0}T_{q}^{k}+\sum_{q=-k}^{k}m_{k,q}\omega_{+1}\sqrt{\left(k+q\right)\left(k-q+1\right)}T_{q-1}^{k}$$ Then we multiply this equation from the left by $T_{q^{\prime}}^{\left(k^{\prime}\right)\dagger}$ and use the orthogonality relation $\text{Tr}\left(T_{q}^{\left(k\right)\dagger}T_{q^{\prime}}^{\left(k^{\prime}\right)}\right)=\delta_{qq^{\prime}}\delta_{kk^{\prime}}$ and we finally get: $$i\frac{d}{dt}m_{k,q}=m_{k,q+1}\omega_{-1}\sqrt{\left(k-q\right)\left(k+q+1\right)}-m_{k,q}q\omega_{0}+m_{k,q-1}\omega_{+1}\sqrt{\left(k+q\right)\left(k-q+1\right)},$$ **My question starts here:** How do we get the standard Bloch equations from this? I'm gonna try to do this. So use the vector version of this by saying that $k=1$ and $q={-1,0,1}$, which gives me the set of equations: $$i\frac{d}{dt}m_{1,-1}=m_{1,0}\omega_{-1}\sqrt{2}+m_{1,-1}\omega_{0}$$ $$i\frac{d}{dt}m_{1,0}=m_{1,1}\omega_{-1}\sqrt{2}+m_{1,-1}\omega_{+1}\sqrt{2}$$ $$i\frac{d}{dt}m_{1,1}=-m_{1,1}\omega_{0}+m_{1,0}\omega_{+1}\sqrt{2}$$ Now to go to Cartesian coordinates I use the conversion of spherical basis, which are: $$m_{1,+1}=\frac{m_{x}-im_{y}}{\sqrt{2}}$$ $$m_{1,0}=m_{z}$$ $$m_{1,-1}=-\frac{m_{x}+im_{y}}{\sqrt{2}}$$ **But this doesn't give theBloch equations! When I substitute those in the last result I get (by also adding and subtracting the first and last equations to separate mx and my equations)** $$\frac{d}{dt}m_{x}=-\sqrt{2}m_{z}\left(i\omega_{x}\right)-m_{y}\omega_{z}$$ $$\frac{d}{dt}m_{y}=-\sqrt{2}m_{z}\left(i\omega_{y}\right)-m_{x}\omega_{z}$$ $$\frac{d}{dt}m_{z}=\sqrt{2}m_{x}\left(i\omega_{x}\right)-\sqrt{2}m_{y}\left(i\omega_{y}\right)$$ Which are weird and I don't understand... could someone please explain where I did a mistake? Why am I not getting the Bloch equations?


title: Microscopic source of pressure in an incompressible fluid

text: The pressure of a fluid can be explained microscopically in terms of molecules bouncing off the walls of a container. The molecules have a certain mass and speed, so when they bounce they transfer a certain momentum to the wall; and since a certain number of bounces occur per unit area and time, the averaged- out result of all the bounces is pressure. So the pressure depends on the speed (temperature) and density of the molecules. If we consider an incompressible fluid, however, we can _apparently_ alter the pressure without changing the temperature or density. Suppose we have a container of fluid and apply force to it using a piston. Since it's incompressible, the piston doesn't move, so the density doesn't change; no work is done on the fluid, so we can't heat it up. Nevertheless the force is transmitted through the fluid "somehow" and results in an increase in pressure. A large force will cause a large increase in pressure. Evidently the approximation of a perfectly incompressible fluid breaks down here. Pushing on it with a piston must _slightly_ change the density, temperature, or both. How does this work? How do _slight_ changes in these properties result in a _large_ change in pressure, microscopically?


title: Vibration as a means of passing through solid objects?

text: First of all, I think this is a weird question but I still want to know whether it's possible. There is a character in a cartoon network series who is really really fast. Sometimes he lets his own body vibrate really hard so he can go through for example a wall. My question is: Is it possible for materials to go through other materials when they are vibrated fast enough?


title: Why are there equations in physics with factors of 2, 3 and 5, but there aren't any with factors of 7 or 11?

text: I noticed that there are a lot of equations in physics with factors of 2, 3 and 5 (either in the numerator or in the denominator), but there aren't any with factors of 7 or any prime number greater than 7. It's like the first three prime numbers (2, 3 and 5) are "more important" than the other prime numbers in physics. What is the reason for this?


title: The relation between electric field and magnetic potential

text: In every electrodynamics book there is one chapter on special relativity which includes one section about" covariant formulation of electrodynamics" which uses tensor to describe the two fields and the four potential and lorentz force etc. and unifies maxwell's equation in a simple way. The problem encountered is the relation between E field and A field:some book gives $\vec{E}=\frac{\partial \vec A}{∂t} - \nabla{V}$, while some $\vec{E}=-\frac{∂ \vec A}{∂t} - \nabla{V}$. (c.f. Jackson or Landau for the first choice, Goldstein's CM or Dubrovin, Fomenko, Novikov's modern geometry for the second choice) I do understand the second one, for Faraday's law(here particularly Lenz's law) needs to be satisfied, and I do understand the first one as well, for the four-d Lorentz force becomes clear in such way. E field, however, is not like A field which can be given a gauge; E field can be measured directly (in an inertial reference frame), so the choice mustn't be arbitrary, and one of them must be wrong. This is my question.