title: Is my simulation result for unpolarized light correct?

text: This is a follow-up of this question. After that, I picked up some knowledge of FDTD (an algorithm for solving Maxwell's equations) and simulated following scene: **_Pic 1_** ![enter image description here](http://i.stack.imgur.com/6hJ0C.jpg) As the picture shows, a silicon cone ($r=5 μm, θ=30°$) is placed in vacuum (truncated with PML). Incident light ($λ=532 nm$) is perpendicular to the bottom of the cone. In my simulation the incident light is modeled as a electric field hard source placed at the bottom of the cone and its electric field intensity obeys Gaussian distribution i.e. $$E\left(x,y,z_0\right)=A_0 \sin (2 \pi f t) e^{-\frac{x^2+y^2}{w^2}}$$ Here $E$ can be either $E_x$, $E_y$, $E_z$, $z=z_0$ is the plane at which the bottom surface of the cone is. The whole domain is discretized with 200×200×400 uniform grids ($\Delta x=\Delta y=\Delta z=50 nm$), along with 20-grid-thick PML layer enclosing it. Here's my simulation result of the light intensity ($|E|^2$) at the plane $100 nm$ away from the tip of the cone for z-polarized light _(notice that the z-axis is parallel to the central axis of the cone in my coordinates)_ : **_Pic 2_** ![dd](http://i.stack.imgur.com/llHvj.jpg) and y-polarized light: **_Pic 3_** ![enter image description here](http://i.stack.imgur.com/OcKlL.jpg) So here comes the confusion. As Steve B suggest in his answer, "the intensity for unpolarized light is the average of the intensity in the two simulations (for polarized light)", but since this two result is… not independent of the rotational orientation of polarization analyzer (if there exists one), a simple average of them won't be, too: **_Pic 4_** ![enter image description here](http://i.stack.imgur.com/V1Yvo.jpg) _This is the average of z-polarized and y-polarized result, it's almost the same as that of the y-polarized light because that of z-polarized one is in fact very weak._ So the unpolarized light is no longer unpolarized after it passes the cone? Moreover, it's not hard to realize that simulation result for x-polarized light is just the same as the y-polarized one, only a 90° rotation needed, so the average of x-polarized and y-polarized light is: **_Pic 5_** ![enter image description here](http://i.stack.imgur.com/5A8Ld.jpg) Still, the picture don't have full rotational symmetry, what's more, it's different from the average of the z-polarized and y-polarized one! So I'll get different simulation result for unpolarized light with different average method? Is my way to "take the average" wrong? Or it's just the truth? My code isn't contained in this post for its lengthiness. I think it's trivial since it has been checked for times and nothing seems to be wrong. Any help would be appreciated. * * * **~~Wrong Guess~~ Update 1:** In the comments below, DumpsterDoofus and Ruslan guessed that the PML I used in the model maybe not work as expected i.e. the lack of full rotational symmetry of my image may be caused by the reflection of the boundary I use. If this guess is correct, the image I get will be different when I change the size of simulation domain, so I tried two more simulation, and the result, in a word, proves negative. To make the simulation faster, I've chose a smaller cone ($r=\frac{5}{2} μm, h=\frac{5 \sqrt{3}}{4}μm $) this time. Two domains with different sizes (200×200×100 grids and 120×120×70 grids) are used in the two simulations separately. Another tiny change is that this time I've turned to additive source instead of the hard source used in the simulation above. Grid size and the thickness of the PML layer (for this part, see my edit above) remains the same as that of the simulation above. Here's how the whole domain looks like in the two simulations (the sizes of the cones are same). Domain formed with 200×200×100 grids: **_Pic 6_** ![enter image description here](http://i.stack.imgur.com/VA67I.jpg) Domain formed with 120×120×70 grids: **_Pic 7_** ![enter image description here](http://i.stack.imgur.com/Vl2Up.jpg) And here's the result (the picture is still taken at the plane $100 nm$ away from the tip): y-polarized light (simulation result of the larger domain): **_Pic 8_** ![enter image description here](http://i.stack.imgur.com/WKQ2h.jpg) y-polarized light (simulation result of the smaller domain): **_Pic 9_** ![enter image description here](http://i.stack.imgur.com/kgo8s.jpg) Supposed unpolarized light (simulation result of the larger domain): **_Pic 10_** ![enter image description here](http://i.stack.imgur.com/whf9U.jpg) Supposed unpolarized light (simulation result of the smaller domain): **_Pic 11_** ![enter image description here](http://i.stack.imgur.com/ax4re.jpg) Apparently there is no significant difference except the spot size( **Note:** I forgot to adjust the `DataRange` of `ListDensityPlot` 囧, and I decide not to modify it in the following part of this post), and we still don't have full rotational symmetry. * * * **~~Wrong Guess~~ Update 2:** Ruslan then suggested that the round source I chose may generate unequally widening wave in x and y direction depending on polarization i.e. it maybe not work as expected. If this guess is correct, the simulation result won't show perfect symmetry even if I remove the cone i.e do the simulation in an empty space. So I simulated the y-polarized light in vacuum without the cone, and the result shows perfect symmetry, so the source seems to work as expected. The following picture shows how the whole domain looks like in this simulation (formed with 120×120×70 grids): **_Pic 12_** ![enter image description here](http://i.stack.imgur.com/r7tke.jpg) Here's the result (taken at same place as _**pic 9**_ ): **_Pic 13_** ![enter image description here](http://i.stack.imgur.com/9no6x.jpg)


title: How are anti-matter atoms created?

text: What is the reaction, or reactions that make anti-matter? I don't understand how anti-matter is created by CERN if interaction with normal matter causes annihilation.


title: Time dilation confusion

text: I'm just starting to learn about special relativity, and I'm a little bit confused about something. Take the example of an observer in $S$ on the ground observing a train move at constant velocity $v$ relative to $S$, an observer in $S'$ is on the train, and this observer in $S'$ flashes a light that reflects from the ceiling and returns to him in a time, $t'$ he measures. ![enter image description here](http://i.stack.imgur.com/DUabI.jpg) I know that in general, if two frames $S$ and $S'$ are in relative uniform motion with respect to each other, and an observer in $S$ can see the clock of an observer $S'$ and the observer in $S'$ can see the clock of the observer $S$, then the observer in $S$ will see the clock of the observer in $S'$ run slower than his own, and vice versa. But I also know that $S'$ time is proper time, and so $t'\leq t$. This seems very strange to me. Observer $S$ sees his clock running faster, so intuitively, observer $S$ expects to measure less time for the event, but the time he measures for the light to return is no less than the time that observer $S'$ measures. Am I understanding this correctly?


title: Moseley's Law Conceptual Question

text: From Young & Freedman, _Univ. Phys._ p. 1394 about Moseley's law: > _Suppose one electron is knocked out of the K shell. This process leaves a > vacancy, which we’ll call a hole. (One electron remains in the K shell.) The > hole can then be filled by an electron falling in from one of the outer > shells, such as the L, M, N, ... shell. This transition is accompanied by a > decrease in the energy of the atom (because less energy would be needed to > remove an electron from an L, M, N, Á shell)._ I don't quite get why there is a decrease in the energy of the atom. Can someone explain to me please?


title: Techniques to cool gas to -100 celsius degrees

text: I am planning to design a scrubber system to separate CO2 and some other gases from a gas mixture. For this purpose, I thought to cool the gas below CO2's boiling point which is -57 Celsius. There are some other gases which is below this and going to about -90 Celsius. By doing so, the actual gas I want which has much more lower boiling point will stay as gas, and others will turn into liquid. As a result of my research, Peltier modules look like a suitable solution to do this cooling, though as far as I read, they reach to about 70 Celsius difference at maximum. What other and not-expensive techniques are available to reach about -90 Celsius?


title: How do you calculate the air and water pressure inside the bell siphon during and after the vacuum is formed?

text: I am interested in the math/physics behind the simple bell siphon. In this YouTube video, the presenter says that the space in the top of the bell becomes a vacuum when the water flow eventually fills the entire downpipe. Right after the vacuum has been formed, am I right to guess that the water pressure in the bell has lowered significantly to allow more water to flow? Is the remaining air trapped in the bell during the formation of the vacuum exit out the downpipe along with the water? What are the pressures and forces in play inside the bell during and after the formation of the vacuum? How would you go about calculating the air and water pressure inside the bell during and after the formation of the vacuum?


title: Derivation of existence of energy band gap in semiconductor (solid State)

text: I am looking for both a mathematical and a physical reason for energy band gap in metals. For Physical reason, I was told that at each reciprocal lattice, you could have Bragg scattering, that would cause the band gap, but this does not really make full sense to me. For a mathematical reason, do we have to solve the Schrodinger equation? Before we can see why? Similarly these kind of bang gap have been found in a diatomic chain and I have seen the derivation which makes perfect sense.


title: What is a tensor?

text: I have a pretty good knowledge of physics but couldn't understand what a tensor is. I just couldn't understand it, and the wiki page is very hard to understand as well. Can someone refer me to a good tutorial about it?


title: Tensors in special relativity

text: I'm trying to understand tensors, but I've come across the following question: >   1. Let $T^{\mu\nu}$ by a $(2,0)$ tensor. Give the definitions of > $T_\mu^{\,\nu}$, $T_{\mu\nu}$, and $T^{\mu}_{\,\nu}$. Then give explicit > formulas for the following components in terms of components of > $T^{\mu\nu}$: > $T_{00},\,T_{10},\,T_{12},\,T_0^{\,1},\,T_1^{\,1},\,T_0^{\,0},\,T^1_{\,1}$   >  $$T_\mu^{\,\nu}\equiv\eta_{\mu\sigma}T^{\sigma\nu} \\\ > T_{\mu\nu}\equiv\eta_{\sigma\mu}\eta_{\rho\nu}T^{\sigma\rho}\\\ > T^\mu_{\,\nu}\equiv\eta_{\sigma\nu}T^{\mu\sigma}$$ > I'm finding it hard to understand these definitions. I really don't know what these tensors represent or what it means when both the indices are raised? Also are these standard definitions in special relativity or is it more likely that they are just made up by my professor for this assignment? I've always thought that things with two indices are matrices in general.


title: Interpretations of (r,s) tensors

text: A tensor of type (r,s) on a vector space V is a C-valued function T on V×V×...×V×W×W×...×W (there are r V's and s W's in which W is dual space of V) which is linear in each argument. We take (0, 0) tensors to be scalars, as a matter of convention. The interpretations of (r,0) tensors are trivial, since they are definitions of multilinear functionals (as a special case (1,0) tensor interpreted as covector (elements of dual space)). We can interpret (1,1) tensors as follows: A(v,f ) ≡ f (Av). Say we have a linear operator R; then we can turn R into a second rank tensor T by T(v,w) ≡ v · Rw where · denotes the usual dot product of vectors. If we compute the components of T we find that the components of the tensor T are the same as the components of the linear operator R. Ok. Everything is good. But I cant understand interpretations of other (r,s) tensors. For example I found in Wikipedia (0,1) tensor intepreted as a vector or (0,2) as a bivector and in general (0,s) tensor as n-vector tensor; or (2,1) tensor as cross product and so on. I want you to show how the tensors in general interpreted. Is it possible for you to show these interpretations like as I did for (1,1) tensor ? Please dont close this post. My answer is not in "What is tensor ?"


title: How do magnetic objects exhibit attraction/repulsion across empty space?

text: Magnets will attract or repel over a distance before physically touching each other. What makes this effect possible? My best guess is that the forces generated by the angular momentum of the electrons in the outer shell produce a "wind" of force that is similar to the draft you feel when a fast moving object passes you. I'm sure there's an explanation for this, but why is this effect only produced by magnetic fields at a close proximity? Also, do all atoms have this property but at a scale too small to perceive?


title: Torque and angular acceleration with bicycle wheel

text: This might be a simple problem for many of you. However, please help me understand it too. I have been looking trough a lot of materials online, and I still have the following questions, that would help me a lot. I saw this video on youtube: see wheel in action If we consider the following model: ![enter image description here](http://i.stack.imgur.com/bGYsu.gif) Please clarify for me the following questions:   1. I believe the gravity is not pulling the wheel down because the torque resulting from multiplying gravity by the l vector is pointing sideways. In other word, the down-pointing gravity vector is transformed into a vector pointing sideways and spinning the wheel around the string. (Is this how force multiplication works? do the multiplied factors transform into the resulting force?)   2. If the wheel would keep a constant angular velocity(assuming no friction) the wheel would spin indefinitely without rising or falling. What would happen if the angular velocity would be increased while spinning? would the wheel rise (ie the free floating end would rise)?   3. Mathematically what causes the wheel to fall when the angular velocity decreases? The only changing quantity is L, which depends on the speed of the spinning wheel. But the direction of L is not changing. And from (1) above, the gravity should not be pointing downwards. So why does the wheel fall when L gets smaller(by wheel falling I mean the free floating end goes down, until it is under the string)


title: Work done (by gravity) on paired blocks

text: ![pulley](http://i.stack.imgur.com/3Dwfj.png) Why does gravity do more work on the block going down? The total work done (by gravity) on the $8kg$ block is more than the work done (by gravity) on the lighter block. How?


title: Energy fluctuations in quantum canonical ensemble

text: How would you go about showing that in the quantum canonical ensemble (that is, in the density matrix and operator formulation), the energy fluctuations, namely $\langle H^2\rangle - \langle H\rangle^2$ is the same as in the classical case, i.e. $k_b T^2 C_V$? I've seen a lot of books asserting this, but can't find a formal derivation similar to the classical one, as given here, for example.


title: How can we move an object with zero velocity?

text: Consider there is a box of mass $m$ at rest on the floor. Most books give an example that we need to do a work of $mgh$ to lift the box $h$ upward. If we analyze this work done, the external force acting on the box by us should be equal to the weight of the box. Therefore the net force is zero which in turn there is no acceleration. If there is no acceleration and the initial velocity of the box is also zero, how can the box move upward?


title: How to pronounce $\textrm{eV}\!/c^2$

text: It seems that $\textrm{eV}\\!/c^2$ (and its multiples) is commonly used as _the_ unit of mass in particle physics. For example, David Griffiths uses it quite naturally in _Introduction to Elementary Particles_, so does Wolfram Alpha. However, a problem occurs when one is to pronounce a quantity expressed in this unit. My question is: What is the common, or most widely used pronunciation of this unit? Is it even ever pronounced? Video or audio recordings to back up individual cases are welcome.


title: What's the wavelength of an electron after hitting a potential barrier?

text: I have this question: > An electron with Energy $E = 40 eV$ hits a potential barrier with $E_0 = 30 > eV$. What is the wavelength of the electron after hitting the potential > barrier? I worked from the energy $E = p\cdot v \Rightarrow m_e \cdot E = p^2$ and combined it with the DeBroglie Wavelength $\lambda \cdot p = h$ which yields $$ \lambda = \frac{h}{\sqrt{m_e \cdot (E-E_0)}} $$ However, the sample solution says the wave number is $$ k = \frac{2\pi}{\lambda} = \frac{\sqrt{2\cdot m_e \cdot (E-E_0)}}{\hbar} $$ Which is exactly what I got, except for the $2$ inside the root. **Where does that factor come from? Why is my lacking of it wrong?** * * * _Note: I tagged this homework although it's not really homework, but homework- like head-scratching._


title: What do I need to know to calculate the price per km for plug-in hybrid or electric cars?

text: Knowing the per kWh price of electricity, what other factors are needed to calculate how much would one pay per km traveled on electric power compared to gasoline power?   EDIT: For example, the Chevrolet Volt reports a 35 kWh per 100 miles. Supposing the price for electricity is 11 cents per kWh, so this mean one can drive 100 miles for $3.85, or a mile for 3.8 cents?


title: Why does the fluid inside a cup not spin when the cup is spinning, but starts to spin when the cup stops spinning?

text: How come when I spin a cup with water in it, the water does not spin, but the moment I stop spinning the cup, the water starts spinning the other way?


title: Dark matter detection

text: In the detection of weakly interacting massive particles (WIMPs), which is the basis of dark matter, what is the use of the tank filled with liquid xenon? I mean, how does the releasing of photons contribute in the detection of WIMPs?   To relate to what I'm talking about, you can follow this link:   http://www.universetoday.com/105943/new-dark-matter-detector-draws-a-blank-in- first-test-round/


title: How can contact force and gravitational pull be a Newton Pair?

text: Just a little background before I present my question - I am a Psychology graduate looking to move into Neuroscience so I'm going through some Physics to learn more about this topic as it is relevant for some Neuroscience courses. As such I lack some background knowledge of Physics and have last studied it at GCSE about 5 years ago, so forgive me if my question is a little basic! I'm going through a section on Newton's Laws at the moment and my textbook gives an example of a Newton Pair, identifying the gravitational pull of a person on the Earth and the contact force of the person on the Earth as the Newton Pair. Right underneath it says that to help identify Newton Pairs, you should remember they have to be the same type, e.g. both gravitational forces. In the above example, it does not seem that the forces are the same type, but I have seen other similar examples, so feel I am missing something. I'd really appreciate it if somebody could explain this to me a little better, including why those two are a Newton Pair if they are different types of force and what I am missing in this example. Thanks!


title: Does coffee cool faster than tap water?

text: A friend of mine, based on casual observation, believes that coffee will cool faster than ordinary hot tap water. Being curious about this I have tried to investigate it myself, but I'm not well versed in physics and thermodnamics. Also, I know that there are many variables to this, so I will attempt to ask the question as specifically as possible: I have 8oz of water (we can assume that it is impure, but we can not know what those impurities are) that is heated to 200 degrees F in a microwave (let's say we have a really precise microwave that can do that easily) and that is then placed in a standard, throw-away styrofoam coffee cup. I have an exact duplicate scenario, except that this time we will also brew coffee in the water for several minutes, filter out the coffee using a standard coffee filter (we can assume some coffee particles remain in the water, but we can not know anything about those coffee particles) and then re- heat the water to 200 degrees F before adding it to the foam coffee cup. If each of these coffee cups are then placed on separate tables in a room- temperature room (let's say 70 degrees F) and allowed to sit, undisturbed, will the liquids cool at roughly equivalent speeds (i.e., will the water and the coffee reach 180 degrees F at roughly the same time) or will one cool faster than the other? For either scenario, why? If there's any clarification or wording choices that should be changed, please let me know as I am genuinely curious. My guess is that there's something to do with the specific heats begin different, and that the surface tension of the coffee mixture will be disrupted due to the coffee particles. But, again, I don't really know :-)


title: A meaningful distinction between determinism and causality

text: Causality is generally accepted to be a fundamental physical principle. But quantum mechanics is acausal (e.g. there is no 'why' as to the result of a measurement of the position of a particle in an infinite 1D square potential well). In which case what is a distinction between causality and determinism, that has a meaningful non-acausal definition of causality. It seems to me that "causality = determinism + quantum mechanics". It seems that it doesn't provide any new concepts, it's just a flawed attempt to reconcile determinism with the arbitrary probabilistic nature of QM, which are ultimately unreconcilable. Does anyone agree with this idea? If not, I would like someone to try and convince me that I'm wrong. I am fully aware that a lot of threads barking up this tree, and I have tried reading many of them, but none seem to crystallize my problem very well.


title: Room temperature and fan orientation

text: So I'm in a tiny dorm room and I normally point my fan blowing outside the window to cool my room off. I've been in some debates on blowing air out or in is more effective, so I'm hoping to get some empirical data to back up my claim. Things I have   * A Heater with only low, medium, high settings.   * The Fan.   * I do _not_ have a thermometer. So the best I could think of was melting ice cubes. I don't care about the exact temperature, just which method is more effective.   I've outline the basis of my experiment: Shut the windows and turn the heat on high, 15 minutes Open the windows, turn off the heat, turn on the fan on high, 15 minutes Leave $n$ ice cubes out and take the average of how long it takes them to melt. All the ice cubes are from the same tray. Can anyone devise a better experiment with no tools...


title: Ground velocity of satellite

text: I was wondering whether there is a simple formula in order to calculate the ground velocity (the velocity with which the satellite is moving along the planet's surface) on basis of the orbital velocity. So for instance, a GEO satellite at Earth has an orbital velocity of around 3.07 km/s, while its ground velocity is 0. The reason I would like to calculate the ground velocity, is to find the time to traverse one pixel on the planet's surface.


title: Speeds and frames of reference

text: When reports state that Voyager is traveling at ~17kmps or that a comet is traveling at 600kmps, what are these numbers relative to? Given that the Earth travels at ~30kmps and orbits the Sun which is traveling at ~25kmps (and so on), is there an implicit qualifier attached to them? Similarly, when a spacecraft attains escape velocity and leaves Earth (as it were), what would observers, say on Earth, Mars, and Alpha Centauri note as its speed? How about a stationary point between the Milky Way and the Andromeda galaxies?


title: Is there a Tungsten alloy more dense than gold?

text: This thought came from researching a previous question. In the spirit of our get-rich-quick scheming I looked up element densities and prices near gold.   * Element (g/cm^3) (dollars/kg)   * Uranium 18.95 40   * **Tungsten 19.25 24**   * _Gold 19.282 53,000_   * Roentgenium 19.282 no market   * Plutonium 19.84 4,000,000   * Neptunium 20.25 660,000   * **Rhenium 21.02 5,000**   * Platinum 21.46 54,000   * Darmstadtium 21.46 no market   * Osmium 22.61 77,000   * Iridium 22.65 42,000 It would be so easy if we could just substitute some of the gold for a cheaper metal, but it seems that gold is well-protected from this kind of forgery (which is perhaps part of the reason its so special to begin with). Almost all of the elements more dense are vastly more expensive, or only producible in small quantities in a lab. There is a very thin market for Rhenium, so we could buy some at a price lower than gold, but any more and we'll risk either bumping up the Rh market price, or being foiled by the authorities. As thrifty criminals we would like to go with Tungsten instead. Can an allow increase the density of a metal? Qualitatively, I imagine that a lighter element could take up interstitial sites, and allow a higher density. But is there any known Tungsten compound that fits this, or any known method of alloying that would increase its density by that ever-so-slight amount?


title: Can peridynamics give accurate quantitative results?

text: I'm applying for a position that is interested in studying crack growth using peridynamics. I have experience with MD on polymeric systems, but I had never knew of this method before. It seems very well suited for at least qualitative description of cracks. My concern is: can it also provide quantitative results? Crack growth and propagation is not a trivial task to model!


title: Where is the missing gram unit in this expression for energy loss?

text: In the discussion of "Landau Straggling", the following expression comes up for the energy loss $\xi$ in Landau's original paper: $$\xi = x\frac{2\pi N e^4\varrho\sum Z}{mv^2\sum A}$$ Here $N$ is avogadro's number, $e$ is the electron charge, $\varrho$ is the density, $m$ the electron's mass and $v$ a velocity. Moreover $x$ is a measure of length and $Z$ and $A$ are the usual atomic and mass numbers. The point here is not to get into the physics, but rather to look at the dimensional analysis: $$\rm [\xi]=[MeV]$$ But the units of the RHS are: $$ \rm[cm][statC^4][gm\cdot cm^{-3}][gm^{-1}][s^2\cdot cm^{-2}]=[MeV\cdot gm] \neq[MeV]\quad\bf!!$$ I'd ignore this as a typo, but all of the literature on the topic seems to follow the same discrepancy in units, so there must be something I'm missing. I'd be tempted to say that $\varrho$ might be a number density, but Landau (and others) explicitly states this is mass per volume. **What am I missing? Where is the missing gram unit?** **Edit:** This snippet from Leroy (2009) on the topic has the same exact problem and may be simpler to understand: ![enter image description here][1] So where is the missing gram? > The value of $n_A$, in units of $\rm cm^{-3},$ is given by $$ > n_A=\dfrac{N_\rho}{A}, \tag{1.39} $$ where $N$ is the _Avagadro constant_ > (see Appendix $\text{A.2}$), $\rho$ is the absorber density, in $\rm > g/cm^3$, and $A$ is the _atomic weight_ [also known as _relative atomic > mass_ ].


title: Can somebody help me find the tension in this string?

text: My friends and I are debating over the correct answer for this particular physics problem. There's a pendulum with a $2kg.$ mass and a $5m.$ string attached to a ceiling at $53^{\circ}$ to the vertical. It's released. At point B, $37^{\circ}$ from the vertical, we are asked to find the tension in the string. Both of us agree that $v = 2\sqrt{5}$. My equation was $T\cos 37^{\circ} = 20 + F_c\cos{37^{\circ}}$, which gives me $T = 33N$ I got the upward component of $T$ and set it equal to the weight of the block + the downward component of the centripetal force. My friend's equation was $T = 20\cos{37^{\circ}} + F_c$ which makes $T = 24N$. She got the component of the weight at $37^{\circ}$. Conceptually, I think we're both correct, but we get different answers. Could somebody answer who's right and why? Note: we assume $\cos 37^{\circ} = 0.6, \cos 53^{\circ} = 0.8, g = 10m^2/s$


title: The topology of a "closed" universe - is it really closed?

text: The spatial part of the positive curvature FRW metric has the form \begin{equation} ds^2=\frac{dr^2}{1-(r/R)^2}+r^2d\Omega^2 \end{equation} or \begin{equation} ds^2=R^2(d\chi^2+\sin{\chi}^2d\Omega^2) \end{equation} This is described as "closed", as it has the metric of a three-sphere, but I want to know what this actually means. Is $\chi$ not limited to the interval $[0,\pi/2)$, since beyond that we are simply reproducing the same $r$? In which case, is the universe not really one "hemi-threesphere" and not closed in the same way as a full sphere is? (Edit: I note that Carroll states that the "only possible global structure is the complete three-sphere", but he doesn't go into why.)


title: Mathematical Universe Hypothesis

text: What is the current "consensus" on Max Tegmark's Mathematical Universe Hypothesis (MUH) which claims every concievable mathematical structure exists, including infinite different Universes etc. I realize it's more metaphysics than physics and that it is not falsifiable, yet a lot of people seem to be taking a liking to it, so is it something that is plausible ? I've yet to hear any very good objections to it other than "it's crazy", but are there any real technical problems with it?


title: Batteries and fields?

text: Batteries generate fields in wires that essentially cause the movement of electrons. I think of batteries as two charged plates that essentially contain a mechanism between them to move electrons from the positive to negative plate. If you have two charged plates they would essentially create a field everywhere. Now consider a circuit that consists of two batteries and a resistor. The first battery would generate a field everywhere. The second battery would also generate a field everywhere. Now say the first battery is 6V and the second battery is 3V. That means inside the first battery they're is a field that would result in 1C of charge gaining 6J of potential energy if moved from the negative to positive terminal. But in a circuit, shouldn't the field from the second battery affect the field inside the first battery? In other words, shouldn't the field from the 3V battery affect the field inside the 6V battery? Since fields can be added, shouldn't the field inside the 6V battery change if a 3V battery is brought near it? So why do we assume the a n-V battery remains constant in voltage regardless of what other batteries are near it?


title: Current when the slide wire attains a constant velocity

text: In a slide wire generator, when the slide wire attains a final constant velocity, is the current in the circuit 0? It does not make sense if it is 0 as the area keeps increasing, allowing more flux to pass, meaning more current would need to be generated to counteract the increasing flux.


title: Does the cell phone make dipole particles in our body dance the same way Microwave oven does in food

text: I am a non-physics person trying to wrap my head around some EM radiation facts. Please help me with this. I have read about Dielectric Heating on Wikipedia. And watched quite a few videos on internet on how Microwave oven works. It all boils down to the following facts. Please correct me if I am wrong on any of them.   1. The Microwaves are absorbed by Water molecule and the Negative charged Hydrogen atoms want to align with negative radiation and Positive Oxygen atom wants to face positive radiation which makes the water molecule dance.    2. Microwave oven Magnetron changes the negative-positive polarity in charge as often as 2.4 billion times in a second which makes Water molecules dance really really fast.   3. This dance makes the water molecules heated. Now coming to my question, I learned that Cell Phones use UHF frequencies such as 800 MHz and 400 MHz. Does that mean water molecules in our body are dancing at the rate of 800/400 steps per seconds ? Why don't they heat up then ?


title: Would electromagnetic radiation impart a pressure on a surface of neutrons?

text: In my physics textbook, it says that a qualitative way to envision pressure from EM waves is as follows: the electric field drives charges in the $x$ direction, and the magnetic field then exerts on them a force $q\vec{v}\times\vec{B}$ in the $z$ direction. The net force on all the charges in the surface produces the pressure. But if this is the case, then does that mean that EM waves wouldn't impart any pressure on a surface of neutrons, given there are no charges to move? Let me know if this is just an analogy that doesn't hold... but in this case, is there another intuitive way to understand why there's a pressure associated with the waves? I understand mathematically, but not physically.


![Chemical potential of Bose-Einstein condensation](http://i.stack.imgur.com/Vy9SB.jpg) This is a problem in An Introduction to Thermal Physics. (a)(b) is easy to work out. In (c), I use the formula $N_{excited}=(\frac{T}{T_c})^{3/2}N, \ (T<T_c)$ to calculate how many atoms are there in the ground state. But I am completely lost about the difference between chemical potential and ground-state energy, because the book always assumes $\mu \approx 0$ when $T<T_c$. I don't think the answer should simply be $\epsilon_0$ but don't know to how to calculate. I know this question seems to be silly but please provide some hints about it.


title: Does the gain medium in a laser need to be replenished?

text: Just what the title states. The gain medium in a gas/fluid laser works by shedding stimulant energy. My elementary knowledge of physics tells me a photon is mass-less. Ergo, photon emission would continue ad infinitum in the presence of stimulant energy - no need to replenish the medium. Is this correct? Does the medium eventually need to be replenished?


title: Beer bottle leftovers pour quickly only after waiting?

text: Why is it that after pouring a delicious beer from a bottle, I can hold it upside down for several seconds without reward, but if I wait a bit, the remainder presumably settles at the bottom and immediately pours out on demand?


title: Bragg condition for transmission: Why is the full diffracted angle Two times Theta? Or isn't it?

text: On a Bragg _reflection_ with incomming angle Theta the total diffraction angle of the incomming wave is 2*Theta, of course. But I have Bragg _transmission_ with electrons on a graphite crystal (experiment about de-Broglie-wavelength and wave-particle-duality). The Bragg interference condition is still the same. But do the diffracted patterns behind the crystal appear under Theta or under 2*Theta? And why? All I found was the pure statement "They appear under 2*Theta", but completly without explanation why this should be the case for transmission. What I think here: I can't apply Snell's law (incoming = outgoing angle) here, because I have transmission and not reflection. So I'd assume that the diffracted electrons will appear also under Theta (and not 2*Theta). Because they enter the crystal under Theta, do their interfering and exit in the same direction as they came in (no reflection, but transmission).


title: Dark matter: degrees of freedom

text: I'm afraid this question could sound a little too vague. I don't even know if dark matter (DM) can be genuinely described by quantum field theory, or if quantum field theory should be somehow "modified" in order to include dark matter. Assuming that ordinary QFT describes DM, what can be said (or what is known) about the number of degrees of freedom dark matter should have?


title: Classical vs qubits: Superposition

text: Since a quantum information lecture today I have been wondering what does it really mean for a state to be in superposition? Is this something that is answerable? This is what we learnt (or what I gathered :) ) A classical bit is always in a state 0 or 1. Sometimes there exists a degree of uncertainty and so probabilities are assigned to either state but in reality it still is 0 or 1 right. However a qubit can be in a state 0, 1 or a superposition whereby this superposition is fundamentally different from the probability mixture for a classical bit. But how can this be so? Surely at any given time a system can in reality either be in the 0 or 1? Does it have something to do with the interference properties of the system? If you could answer my questions and explain the fundamental differences between the qubit and the classical bit I would really appreciate it. Thanks!


title: Can cos(x) or sin(x) be the function of stationary action?

text: Is there a way to express $\cos(x(t))$ (or $\sin(x(t))$) as the solution to the Euler-Lagrange equation, in other words is there a sense in which this function is the path of stationary action?


title: Shouldn't the change in kinetic energy be more in a moving elevator from a stationary frame of reference?

text: Consider an elevator moving down with uniform velocity. A person standing inside watches an object fall from the ceiling of the elevator to the floor. Say the height of the elevator is $h$. Then the work done by gravity in that frame of reference should be $mgh$. But consider this same event being watched by someone else in the stationary frame of reference. In his reference frame, the object travels a larger distance as it falls from the ceiling to the floor of the elevator because the floor itself is moving downwards (one can calculate this extra distance covered to be $u \sqrt{\frac{2h}{g}} $) and hence the change in kinetic energy should be more in that frame than in the moving frame! I just can't seem to figure out where I'm going wrong here. I'm probably missing something very obvious. So I would be very grateful if anyone could explain this to me. **Edit:** Okay, let's say the object is a clay ball and it collides with the floor inelastically such that it's kinetic energy is converted into heat. In the moving frame of reference the heat would be simply equal to $\frac{1}{2}mv^2$ which is equal to $mgh$. In the stationary frame of reference it would be equal to $\frac{1}{2}mv^2-\frac{1}{2}mu^2$ since the ball after colliding is moving with speed $u$. This can be calculated to be equal to $mgu\sqrt{\frac{2h}{g}} + mgh$ which is clearly greater than the heat produced in the frame attached to the elevator and this is a contradiction because the heat measured in any frame should be the same.


title: Does black hole evaporation respect quantum superpositions?

text: I've confused myself about the following scenario: Suppose you make a black hole out of states with spin aligned into one direction, say the positive x-direction, and let's call this "up". Then the so-created black hole has an non-zero angular momentum because the expectation value of the spin is nonzero. (Let's assume there's no other net contribution to the angular momentum, just for simplicity. I don't think this matters.) The so-created black hole evaporates in the semi-classical limit with an energy spectrum that depends on this expectation value of the total angular momentum. (Forget about what happens in the final stage where the semi-classical approximation breaks down, I don't think this matters.) Next you make a black hole out of the same initial state with the only difference that all spins are down. Same thing: the emission spectrum - the distribution of outgoing particles over energy - depends on the expectation value of the angular momentum, at least in the semi-classical limit we're pretty sure of that. Finally, you make a black hole out of a superposition of the two initial states. The expectation value of the total angular momentum is now zero, correspondingly the angular momentum in the semi-classical Kerr-background is zero. What is the decay spectrum (in the early radiation) in this case? Is the outgoing radiation a) a superposition of the radiation that one obtained from the purely up and purely down case? But then it would depend on the angular momentum which the black hole doesn't have. Or does it b) not depend on the angular momentum of the superposed states (note that what is and what isn't a superposition is ambiguous anyway) but then the evolution doesn't seem to be linear? (In the sense that |psi_in> + |psi'_in> doesn't result in |psi_out> + |psi'_out> when |psi_in> resulted in |psi_out> and same for the dashed case).


title: Linear combination of anomalous dimensions in effective potential on pseudomoduli space

text: In the paper of Intriligator, Seiberg, and Shih from 2007, they give an expression for the effective potential on the pseudo-moduli space $X$, estimated at large $X$ (equation 1.3). In this equation, "$\gamma$ is the anomalous dimension of a particular linear combination of them". Later on in the paper they give examples for this (eq. 4.9). What is the formula for this $\gamma$ and how do we reach this expression?


title: The asymptotic behavior of the propagator of a field

text: In Steven Weinberg's book "The Quantum Theory of Fields" vol. I, Section 12.1, page 500, it writes: _"We will write the asymptotic behavior of the propagator $\Delta_f(k)$ of a field of type $f$ in the form_ $$\Delta_f(k)\sim k^{-2+2s_f}$$ _Looking back at Chapter 6, we see that $s_f=0$ for scalar fields, $s_f=\frac{1}{2}$ for Dirac fields, ans $s_f=1$ for massive vector fields. More generally, it can be shown that for massive fields of Lorentz transformation type $(A,B)$, we have $s_f=A+B$. Speaking loosely, we may call $s_f$ the 'spin'."_ How can we show that $s_f=A+B$ holds for massive fields of type $(A,B)$? Does anyone have some ideas of the proof? Thanks a lot in advance!


title: The $U(1)$ charge of a representation

text: My question is about the reduction of a representation of a group $SU(5)$ to irreps of the subgroup $SU(3)\times SU(2) \times U(1)$. For example the weights of the 10 dimensional representation of SU(5) are ![enter image description here](http://i.stack.imgur.com/SlEAK.png) One can identify the irreps of the subgroup by regrouping the dynkin labels into $((a_3 a_4) ,(a_1), a_2)$ such that (denoting $-1$ by $\bar{1}$): $$ (1,1)_{Y} \rightarrow \left\\{ \begin{array}{l l} (0 0,0,1 ) \end{array} \right. $$ $$ (\overline{3},1)_{Y} \rightarrow \left\\{ \begin{array}{l l} (0 1,(0),\bar{1}) \\\ (1 \bar{1},(0),\bar{1})\\\ (\bar{1}0,(0),0) \end{array} \right. $$ $$ (3,2)_{Y} \rightarrow \left\\{ \begin{array}{l l} (1 0,1,\bar{1}) \\\ (\bar{1} 1,\bar{1},1)\\\ (0\bar{1},\bar{1},1)\\\ (1 0,\bar{1},0)\\\ (\bar{1}1,1,0)\\\ (0\bar{1},1,0) \end{array} \right. $$ My problem is: how can I derive the $Y$ charge of the $U(1)$ factor for each of these from the Dynkin labels? * * * **Edit** The metrictensor for SU(5) is thus $$G= \frac{1}{5}\left( \begin{array}{cccc} 4 & 3 & 2 & 1 \\\ 3 & 6 & 4 & 2 \\\ 2 & 4 & 6 & 3 \\\ 1 & 2 & 3 & 4 \end{array} \right). $$ However in the reference, Slansky, on page 84 the same exercise is done but the axis have negative values... $$\tilde{Y}^W = \frac{1}{3} [-2 \;1\, -1\; 2]. $$ How come they do not agree?


title: Equivalent Representations of Clifford Algebra

text: I'm reviewing David Tong's excellent QFT lecture notes here and am a little confused by something he writes on page 94. We've considered the standard chiral representation of the Clifford Algebra, and he is now generalising to a different representation. He writes that this will involve $$\gamma^{\mu}\rightarrow U\gamma^{\mu}U^{-1} \ \textrm{and} \ \psi \rightarrow U\psi$$ What does the second transformation mean? I don't see how it has anything to do with a representation of the Clifford algebra (which is an assignment of a matrix to every element of the algebra, as far as I know). Does he mean that in the resulting projective representation of the Lorentz group we should transform $$\psi\rightarrow S[\Lambda]U\psi$$ or am I barking up the wrong tree? This is all very reminiscent of changing pictures in QM, but I've never talked about representation theory in that context! Is there a rigorous link? Many thanks!


title: How do I measure the spin along an arbitrary direction?

text: I need to measure the atoms spin along a direction perpendicular to the z-axis but at an angle $$ \phi $$ with the x-axis. They then get the output of the spin measured is + with probability $$ \cos^2(\phi/2) $$ Is there a general formula I can use to compute the + spin state and then form that determine this probability?


title: Why must the final state be stationary?

text: I faced the following sentences: > We consider a gravitational collapse taking place in this spacetime. The > singularity theorems assure us that a singularity will form. The assumption > that the cosmic censorship conjecture is valid will then imply that the > collapse must produce a black hole which will settle down to a stationary > final state. Can anyone explain to me why **The assumption that the cosmic censorship conjecture is valid will then imply that the collapse must produce a black hole which will settle down to a stationary final state**? Why must the final state be stationary?


title: QM perturbation theory : When do I have to use degenerate/non-degenerate perturbation theory?

text: I am considering a perturbation theory problem in quantum mechanics. The unperturbed hamiltonian is $$H_0 = A_1 \boldsymbol{B} S_{1z} + A_2 \boldsymbol{B} S_{2z}.$$ The eigenstates of the unperturbed hamiltonien are $ \mid M_1 M_2 \rangle$, where $-S_{1,2} \leq M_{1,2} \leq S_{1,2}$ ($S$'s are spin operators). Here, $A_1 \ll A_2 $. I want to know if this system is degenerate to apply the perturbation theory. The energy of each state is : $$ E_{M_1 M_2} = \hbar(A_1 B M_1 + A_2 B M_2).$$ (Am I correct ?) But because $A_1$ is way smaller than $A_2$, it means that the energy will have some kind of structure : separated energy levels of order $A_2 B M_2$, and then close to that, small variations around this value owing to $A_1BM_1$. In my problem, it would be simpler to do non-degenerate perturbation theory. But I have the feeling that it may be incorrect because of the small difference of energy added by the first operator (like : $\mid M_1, M_2\rangle $ and $\mid M_1+1, M_2\rangle$ have almost the same energy, but not $\mid M_1 M_2\rangle$ and $\mid M_1 M_2+1\rangle$), even if technically each state has a different energy. What should I do ? Thank you for any comment.


title: 4 element super conductor without oxygen and iron

text: I am trying to find a list with super conductors. In particular I am searching for a superconductor, that consists of 4 elements none of which is Oxygen. The reason I rejected Oxygen, is because(from the kinematic factor) the maximum energy from scattering with the Oxygen is about $730 keV$. Therefore-unfortunately-I have to reject _YBCO_ superconductors. In particular I have to analyse an RBS spectrum of a supposed 4-element well known super conductor. I can't find any 4-element-superconductor without oxygen. Is there anyone, or it might be the case where the sample is bad described? Just for reference the RBS spectrum is ![enter image description here](http://i.stack.imgur.com/eEndF.png) **EDIT** I also tested iron based superconductors, but it seems that I need a heavier element. For instance a superconductor containing $Ba, K, Fe, As$ has the following RBS spectrum ![enter image description here](http://i.stack.imgur.com/P5ssM.png) Please ignore the concentrations... The analysis indicates that the superconductors consists of 4 elements which are   * Bi(28.5%)   * Sr(28.4%)   * Cu(28.2%)   * K(14.8%) Does a superconductor like that really exists? I mean OK about Bismuth or Copper or even Strontium...How about Potassium? The combination of those doesn't seem right to me...


title: Radio vs optical telescope imaging

text: As I understand, the visible light from an optical telescope is focused on a sensor which correlates light exposure to an electrical voltage, which is then converted to an image. A single antenna radio telescope's signal is focused on some sort a sensor but this signal represents only a single 'pixel.' To generate a composite image of an object, a time intensive scan pattern needs to be carried out. Why does this difference exist? Why does a radio telescope see a single pixel, while optical telescopes see an entire image? Are radio telescope sensors only a single pixel, or is a radio telescope itself, regardless of the sensor, only capable of seeing a single pixel?


title: How do I calculate the power consumed by a lightbulb?

text: I'm studying a lightbulb and its variable resistance, given by the expression: $R(T) = Ro[1 + α(T-T_0)]$, where $R_0$ is the resistance of the lamp at $T_0$. In this case, $R$ is not given by Ohm's law ($V=Ri$). So, which expression can I use to calculate the power consumed by a lightbulb?   * $P = R\cdot i^2$   * $P = \frac{V^2}{R}$   * $P = V\cdot i$ (I don't think I can use this one) I have measured the current and the e.p.d. at the bulb using instruments.


title: What is the voltage difference between the Moon and the Earth?

text: The Moon and Earth are approximately spherical conductive balls and the Earth has a self-capacitance of around 710uF. Is there ever a significant potential difference in the Earth-Moon system? Is there even a way to tell if there is? Presumably the Apollo astronauts didn't expect to be electrocuted on landing. I was thinking that a huge charge would not build up because they are both bathed in a flow of charged particles from the Sun which I would have thought would equalize the voltage. Is this true? Also what voltage difference would have to build up for a massive bolt of lightening to jump through space between them?


title: Gauge fermions versus gauge bosons

text: Why are all the interactions particle of a gauge theory bosons. Are fermionic gauge particle fields somehow forbidden by the theory ?


title: Why is the divergence of a magnetic field equal to zero?

text: We know due to Maxwell's equations that: $$\vec{\nabla} \cdot \vec{B}=0$$ But if we get far from the magnetic field, shouldn't it be weaker? Shouldn't the divergence of the field be positive? If we define the vector field as a function of distance, then if the distance increases then the magnitude of the vector applied to a distant point of the "source" should be weaker. Is my reasoning correct?


title: Speed of sound in astrophysics

text: Why is the speed of sound given so much importance in Astrophysics? For example in gas outflow (and accretion) problems, we often calculate the sonic point (the point at which the outflow speed becomes equal to the speed of sound). Is it simply because speed of sound is a convenient number that can be used to scale down the variables, or is something deep involved?


title: How use the Higgs branching ratio plot to extract information about the Higgs mass compared to experiment?

text: What does the plot of higgs branching ratio (see figure below) say about the higgs mass anyway? How can one use it as a guide to find the higgs mass experimentally? If we e.g. go to $M_H=126$ GeV (in figure below) and then read off the different branching ratio: then if experiment agrees with this, it would mean that the higgs mass is $126$ GeV? ![Higgs branching ratio.](http://i.stack.imgur.com/KcvRz.jpg) In other words, what's so special about this plot?


title: Physical meaning of Legendre transformation

text: I would like to know the physical meaning of the Legendre transformation, if there is any? I've used it in thermodynamics and classical mechanics and it seemed only a change of coordinates?


title: How are complex sound waves combined?

text: Audio is often explained by single frequencies. Typically _this is a sound wave_ :               plot sin(x) * 2 from 0 to 10      ![plot sin\(x\) * 2 from 0 to 10](http://i.stack.imgur.com/FzIdQ.gif) However we usually deal with more complex sounds, more specific various frequencies and amplitudes. Varying amplitudes within same frequency etc. in the same time frame. How are such sound patterns visualized? Say we have the previous tone + this one:               plot sin(5x) * 2 from 0 to 10      ![plot sin\(5x\) * 2 from 0 to 10](http://i.stack.imgur.com/prW59.gif) Would this be the end-result? Would a wave like this travel trough the air?               plot (sin(5x) + sin(x)) * 2 from 0 to 10      ![plot \(sin\(5x\) + sin\(x\)\) * 2 from 0 to 10](http://i.stack.imgur.com/lhC4Y.gif) Even a mono track can have a rather detailed sound picture of say a philharmonic playing some composition. Would such a recording (or live observation for that matter – but that would give a different picture I guess as one would have multiple sound sources) be a _complexified_ × ten folds version of the two sine waves? * * * My end goal is to better understand PCM audio and how digital audio works. A starting point is to better understand the physics behind audio. Then again a most of what I find deals with _one frequency samples_ and the like. (I'm likely missing some terminology.)


title: In KK theory, is proper time defined using the 5 dimensional or the 4 dimensional line element?

text: Let's consider five dimensional KK theory. This is Klein's metric $\hat{g}_{AB}= \begin{pmatrix} g_{00}+A_{0}A_{0}&g_{01}+A_{0}A_{1}&g_{02}+A_{0}A_{2}&g_{03}+A_{0}A_{3}&A_ 0\\\ g_{10}+A_{1}A_{0}&g_{11}+A_{1}A_{1}&g_{12}+A_{1}A_{2}&g_{13}+A_{1}A_{3}&A_ 1\\\ g_{20}+A_{2}A_{0}&g_{21}+A_{2}A_{1}&g_{22}+A_{2}A_{2}&g_{23}+A_{2}A_{3}&A_ 2\\\ g_{30}+A_{3}A_{0}&g_{31}+A_{3}A_{1}&g_{32}+A_{3}A_{2}&g_{33}+A_{3}A_{3}&A_3\\\ A_{0}&A_{1}&A_{2}&A_{3}&1 \end{pmatrix}$ my question is the following, is proper time defined using the five dimensional line element? $d\tau^2=\sqrt{-\hat{g}_{AB}dx^{A}dx^{B}}$ or is it defined as usual using the 4 dimensional line element? $d\tau^2=\sqrt{-g_{\mu\nu}dx^{\mu}dx^{\nu}}$ in either case, why?


title: A paradox with spin: Is spin a physical degree of freedom?

text: Suppose I want to calculate the state associated with a spin particle under a magnetic field. I suppose the particle interacts via the Zeeman effect, and only through it. Then I want to resolve the Schrödinger equation: $$\mathbf{i}\hbar\dfrac{\partial\Psi}{\partial t}+\dfrac{\hbar^{2}}{2m}\dfrac{\partial^{2}\Psi}{\partial x^{2}}=h_{Z}\sigma_{z}\Psi\left(x,t\right)$$ with $h_{Z}$ the Zeeman term. All other terms are usual. To resolve, I apply the transformation $$\Psi=e^{-\mathbf{i}h_{Z}\sigma_{z}t/\hbar}e^{-\mathbf{i}Et/\hbar}e^{\mathbf{i}kx}\Psi_{0}$$ with $\Psi_{0}$ a constant taken to be real for commodity, and the dispersion relation reads $$E=\dfrac{\left(\hbar k\right)^{2}}{2m}$$ as usual for a **free** particle. There is no mark of the spin splitting we are used to, like the energy doublet $E_{\pm}=E_{0}\pm h_{Z}$ due to the Zeeman term. Of course this is just a redefinition of the energy origin, done in two different ways for spin up and spin down (or something like that). If I calculate observable like the density $\rho\propto\left|\Psi\right|^{2}=\Psi_{0}^{2}$ or the current $j\propto\Im\left\\{ \Psi\partial_{x}\Psi^{\ast}\right\\} =k\Psi_{0}^{2}$ again I find no _measurable_ signature of the spin degree of freedom... So where is the spin degree of freedom gone ? Do we need to discuss only space and/or time dependent spin degree of freedom to make any sense of it ? And then the spin is only a _relative_ degree of freedom (in other word, I know what is a spin _mismatch_ , but not what is a spin, a bit like a phase: only phase difference matters). Or anyone gets a better idea of what a spin is? Thanks in advance.


title: Superconformal theories

text: * Can anyone tell me where can I read about the notion of "short" and "long" representations? Like what they are etc.    * From where can I learn the arguments which show that the bosonic subalgebra of $\cal{N}=2$ and $\cal{N}=3$ Lie superalgebra are given by $SO(3,2)\times SO(2)$ and $SO(3,2)\times SO(3)$ respectively?    * I have often seen the following statement being made about the above which I do not understand, that primary states of these algebras are labeled by $(\Delta,j,h)$ where $\Delta$ is the scaling dimension of the primaries, $j$ is its spin and $h$ is its $R-$charge (or R is its charge highest weight) Can someone explain the above labeling scheme. I don't understand much of the above. Somehow the above doesn't seem to fit with my elementary understanding of what are primary operators in 1+1 CFT which are defined using particular forms of the OPE with stress-energy tensor.   * Then one argues that for $j\neq 0$ unitarity forces $\Delta \geq \vert h\vert + j+1$ and for $j=0$ unitary representations occur when $\Delta = h$ or $\Delta \geq \vert h \vert + 1$ I haven't been able to trace the above argument in any reference book or exposition.   * It is said that the isolated representations which saturate the above unitarity bound for $j=0$ are all short.    * Apparently the Witten Index in this context is defined as $Tr(-1)^Fx^{\Delta +j}$ and it vanishes on all long representations but is nonzero on all short representations.    * Further to understand the state content of all unitary representations in superconformal algebras people defined two different kinds of Witten indexes, $\cal{I}^+$ and $\cal{I}^-$ as, $${\cal I}^+ = Tr(-1)^Fx^{\Delta+j}e^{-\beta(\Delta-j-h)}$$ and $${\cal I}^+ = Tr(-1)^Fx^{\Delta+j}e^{-\beta(\Delta-j+h)}$$ The above are apparently independent of $\beta$. I would like to know why. Apparently the first index above receives contributions only from states with $\Delta = j+h$ and all such states probably are annihilated in some sense and have something to do with the `cohomology of the supercharge with charges $(\frac{1}{2},-\frac{1}{2},1)$ Similarly somehow the second index above receives contributions only from states with $\Delta = j-h$ and all such states probably are also annihilated in some sense and have something to do with the`cohomology of the supercharge with charges $(\frac{1}{2},\frac{1}{2},1)$   * Eventually I see these indices to be writable as products (sometimes infinite) of rational polynomials in $x$. I don't know how it happens. This way of writing has something to do with "single/multitrace operators/primaries" (another concept which I don't understand!)  I would be grateful to hear of explanations of the above arguments and constructions and also if detailed references/expositions exist for the above.


title: Help understanding the potential outcome of solving the Unified Theory in modern physics

text: I find myself greatly fascinated by physics and astronomy, though I really do not understand most of it. For example, I avidly watch programs and read books that discuss our current search for a Unified Theory which will unite the major forces (Strong Interactive, Weak Interactive, Electromagnetism on the one side, with Gravitation on the other). I also try to understand a little bit about our work with subatomic particles, the search for the Higgs particle, and our use of tools like the Large Hadron Collider. Despite this enthusiasm, I find myself asking one major question: What might come of definitively finding a Unified Theory (Theory of Everything)? Most of the material I absorb seem to imply that there are major technological advances from such an accomplishment, but rarely provide examples. Obviously we cannot predict accurately all of the potential for the outcome of this scientific progression, but I get the impression that physicists and astronomers have a great deal of pending problems that await the solution. For a layperson, what kinds of advancements and breakthroughs might one realistically expect?


title: Existence and uniqueness of solutions for Einstein equations

text: Now that an equivalence of Navier Stokes and Einstein equations has been established, and it is known solutions to Einstein-Maxwell-Boltzmann exist and are unique, and it is known that Einstein equations in a wave guage with electromagnetic source with a lorentz gauge also have unique solutions that exist, what are the next steps? What are the good references on research for existence and uniqueness for einsten solutions without electromagnetic sources? Also, what does it mean physically that there are unique solutions to Einstein Field equations when electromagnetic sources are included? For those struggling with the concept of existance and uniqueness, I think this post provides some good insight.


title: Mixing of fluid in a rotating barrel

text: A barrel/drum with a diameter of 60cm is rotating at 20RPM to get a good mixing of the fluid contained (type thick oil). At what RPM should a barrel of 30cm rotate to get the same mixing efficiency with the same fluid? To get the same peripheral velocity, the RPM should be 40RPM, but will this give the same mixing of the fluid? I guess gravitational and centripetal forces play a role her... EDIT: More information to my otherwise incomplete question :) The barrel is rotating around the length axis and the degree of fill is 90%.


title: Accelerating towards the center of your orbit

text: If you are in a perfectly circular orbit and accelerate perpendicular to your velocity (directly towards the center of the orbit), what would happen to your orbit? Would it just shift in the direction of acceleration? Is there a different result if the orbit is not perfectly round? The basis for this question comes from me playing _Kerbal Space Program_. I never really considered orbital mechanics before.


title: Euclidean derivation of the black hole temperature; conical singularities

text: I am studying the derivation of the black hole temperature by means of the Euclidean approach, i.e. by Wick rotating, compactifying the Euclidean time and identifying the period with the inverse temperature. Consider the Schwarzschild case as an example. The Euclidean Schwarzschild metric is of course $ds^2=\left(1-\frac{2M}{r}\right)d\tau^2+\left(1-\frac{2M}{r}\right)^{-1}dr^2$ where $\tau=it$ the Euclidean time. Here $\tau\in[0,\beta]$ with $\beta=T^{-1}$ the inverse temperature, where the points $\tau=0$ and $\tau=\beta$ are equivalent. (I ignore the 2-sphere part of the metric.) Outside but close to the event horizon $r=2M$, we can (after simple some steps) write this as $ds^2=\frac{\sigma^2}{16M^2}d\tau^2+d\sigma^2$. Here $\sigma^2 \equiv 8M(r-2M) $, though this is not really relevant for my question. Now the next step in all the literature is to require that $\tau/4M$ is periodic with period $2\pi$ to prevent conical singularities. I have trouble understanding this.   * A conical singularity basically means that the point at $\sigma=0$ looks like the tip of a cone, right? So we have a singularity at $\sigma=0$. But don't we still have a singularity there for polar coordinates $(r,\theta)$, since the coordinate chart $\theta$ is not continuous there?    * If so, removing the point $\sigma=0$ would erase the conical singularity, right? Then why do we want to get rid of the conical singularity, why is it any worse than the polar coordinate singularity? Of course polar coordinates just describe flat space without the origin, is this not the case for the conical coordinates? Clearly I have not understood the concept of a conical singularity... Last question: suppose I did understand it, and continued the derivation to get the temperature $T=1/8\pi M$. Apparently this is the temperature as measured by an observer at infinity, how can I see this? I know the temperature gets redshifted, like frequency, but I don't see where the derivation identifies $T$ with the one measured at infinity.


title: Quantum circuit equivalent of quantum pseudo-telepathy game

text: I'm trying to understand the wikipedia article on quantum pseudotelepathy. I've been trying to figure out the quantum circuits the players can use to win the game from the wiki article. (Level of knowledge: Everything I know about quantum physics is from the computer science side. I can explain how Grover's algorithm works and understand quantum logic circuits, but I have no idea how those map to the underlying physics or what an observable is or how eigen values relate to observables or how wavefunctions come into things or etc.) My main stumbling point is what the heck is going on in the center column and row. For example, here's my current terrible guess at a circuit (using this online simulator) for the center column: ![Quantum Circuit taking one entangled qubit from each pair and outputting cell choices](http://i.stack.imgur.com/gLz55.png) In the above circuit q1 is understood to be entangled with an unseen q1' to be used as part of a corresponding circuit for one of the rows, and the same for q2 and q2'. The circuit is xoring together the X, Y, and Z rotations of the input qubits and using that output to determine the values to place in the cells of the column. This circuit doesn't work. I know it doesn't work because it's never mixing anything; never taking advantage of superpositions. It could be simulated classically, and the game can't be won with certainty classically. So... I'm lost. A link to a more introductory explanation would be great. I'm pretty sure I'm missing something related to the pauli XYZ matrices satisfying XYZ = -iI, and rotations in 3d being order-dependent, but I don't know where to apply those facts.


title: What are the current experimental limits on the polarizability of the vacuum?

text: Many (all?) materials are polarizable at some level; meaning that when we apply an electric field across the material, there is an induced dipole moment $$ \vec{p}=\alpha(\omega)\vec{E}, $$ where $\alpha(\omega)$ is the polarizability of the material at frequency $\omega$, $\vec{E}$ is the applied electric field, and $\vec{p}$ is the induced polarization. If we assume that the electrons in the material are bound to their nuclei as simple harmonic oscillators (the standard physics assumption), then we find that the polarizability of the material is given by $$ \alpha(\omega)=\frac{e^2/m}{\omega_0^2-\omega^2}, $$ where $\omega_0$ contains the information about the stiffness of the binding. This leads to interesting optical phenomena in materials, one example of which is that it can be related to the index of refraction by $$ n^2(\omega)=1+\frac{N}{\epsilon_0}\alpha(\omega), $$ where $N$ is the number density of the atoms and $\epsilon_0$ is the permittivity of free space. All of this is just background information. **My actual questions are:** What are the experimental limits on the polarizibility of the vacuum, and which experiments set these limits? Is anyone actively trying to improve the limit? ~~It is obviously expected to be identically zero, but we have been surprised by measurements in the past.~~


title: Do we have a fundamental Hamiltonian for the system of H$_2$O molecules?

text: From the quantum mechanics(QM) viewpoint, does there exist a Hamiltonian $H$ for the system of H$_2$O molecules? Assume that the number of H$_2$O molecules is fixed. Imagine that by calculating the free energy $F(T)$ from the Hamiltonian $H$, one can reproduce the three common phases **ice** , **water** , and **steam** at different temperatures $T$(Here do we need _order parameters_ to characterize these phases?). On the other hand, the Hamiltonian $H$ may depend on some parameters, i.e., $H=H(\lambda )$, and at $T=273K$, varying the parameter $\lambda $ causes the phase transition between **ice** and **water** without temperature changing. So can we write a microscopic Hamiltonian $H$ of this kind in terms of the QM language?


title: Does thermal energy change within a chamber depending on what's in it, where it is, what it is?

text: I have noticed that some people will say that, when you put more mass in an oven, it takes longer for both to cook than if just were one. Some others will say that timings are the same. Assume we have an oven that's heated to 350 degrees fahrenheit, and it's 10 cubic feet(it should hold about 340 liters of liquid, volume-wise). If we stick a 1 lb, 6 ounce cake in the oven in a 9x9 aluminum pan, let's say that it cooks until succinct in about 25 minutes. If we were to add the exact same mass, but in another cake and equivalent pan (with the exact same ingredients to the nearest degree possible on the atomic scale), and we place that pan right next to it in the center of the one-rack oven, will cooking times differ? Basically, does thermal energy become affected when you place more mass over an area of heat within a chamber? It takes different cooking times for different things(molecule-wise, etc.), and mass is always a factor(e.g. a chicken drumstick will cook at 350F in less than 15 minutes, but a whole 5 lb chicken, say, will most definitely take quite longer). So my question is, in an enclosure of 10 cubic feet (or the like), if you add more mass/volume to the chamber, will thermal energy change, will cooking times change with varying mass and volume, distributions and placement inside the chamber, and why or why not? PS: I know this is broad, but we can narrow it down by just taking my aforementioned examples _as just examples_ , and provide answers relevant to the idea within any scope of measurements desired.


title: Force applied to wheel in pure rolling motion at contact point with road

text: Suppose a wheel with radius $R$ is resting on a non-inclined surface. A torque $\tau$ is applied to the wheel center. In an attempt to prevent wheel from spinning, the ground applies a static friction force to the wheel at the contact point (parallel to the surface), then the wheel starts rolling without spinning. The same friction force also acts as a torque on the wheel around its axis. This scenario is depicted below: ![Wheel in pure rotation motion](http://i.imgur.com/hCv4D32.png) I'm trying to find the magnitude of the force the ground applies to the wheel - that is, the force which drives the wheel forward (which should be the same in magnitude as the force the wheel applies to the ground at the contact point). That's how I'm doing it: The relation between linear acceleration $a$ and angular acceleration $\alpha$ for a pure rolling movement is given by \begin{equation} \tag{1} a = \alpha \centerdot R, \end{equation} The relation between torque $\tau$ and angular acceleration $\alpha$ is \begin{equation} \tau = I \centerdot \alpha \end{equation} where $I$ is the moment of inertia of the wheel around its axis. The relation between torque $\tau$, force $f$ and lever arm $R$ is: $$\tau = f \centerdot R$$ Being the engine torque $\tau_e$, the friction force $f$, the counter torque due to friction force $\tau_f$ and the moment of inertia of the wheel $I$ around its axis given by $\frac{1}{2}mR^2$: The linear acceleration of the wheel is due to the friction force only: $$f = ma$$ $$a = \frac{f}{m}$$ This is the counter torque the ground applies on the wheel's edge (negative, because it pointing in the opposite direction of $\tau_e$): $$\tau_f = -f \centerdot R$$ The net torque causes angular acceleration on the wheel: $$\tau = \tau_e + \tau_f$$ $$\tau = \tau_e - f \centerdot R$$ $$\tau = I \centerdot \alpha$$ $$\alpha = \frac{\tau}{I}$$ $$\alpha = \frac{\tau_e - f \centerdot R}{I}$$ $$\alpha = \frac{\tau_e - f \centerdot R}{\frac{1}{2}mR^2}$$ Substituting $\alpha$ and $a$ in $(1)$ gives: $$\frac{f}{m} = \frac{\tau_e - f \centerdot R}{\frac{1}{2}mR^2}R$$ Rearranging: $$f = \frac{2}{3}\frac{\tau_e}{R}$$ And that is the force $f$ from static friction which pulls the wheel forwards without making it spin or slip, and consequently, the force the wheel applies to the road surface at the contact point. But I found this link: http://www.asawicki.info/Mirror/Car%20Physics%20for%20Games/Car%20Physics%20for%20Games.html And it says: "The torque on the rear axle can be converted to a force of the wheel on the road surface by dividing by the wheel radius. (Force is torque divided by distance)." That statement doesn't match the approach I used above. If the force of the wheel on the ground was simply engine torque divided by radius (negative, since it pointing in the opposing direction): $$f = -\frac{\tau_e}{R}$$ then the counter torque applied to the wheel would be $$\tau_f = -f \centerdot R$$ that implies that $\tau_f = -\tau_e$. That means the net torque would be zero, and the wheel would just slip without rotating at all. Am I doing something wrong on my calculations?


title: Why don't fogbows appear on clouds?

text: As far as I know clouds are lot of small droplets condensed in air. If droplets are large enough we see a rainbow. If they are small we see a fogbow. Although the size of the droplets are big enough for fogbows to form on cloud according to this link, I haven never seen a fogbow on clouds nor found a picture of it appearing on clouds only on fog. Why?


title: Yet another question on the Lindhard function

text: Here's another question concerning the Lindhard function as used in the physical description of metals. First we define the general Lindhard function in the Random Phase approximation as $\chi(q,\omega)=\frac{\chi_{0}}{1-\frac{4\pi e^2}{q^2}\chi_{0}}$ where $\chi_{0}$ denotes the "bare" Lindhard function" excluding any feedback effects of electrons. A reference is Kittel's book "Quantum Theory of solids" Chapter 6, however he is using the dielectric function instead of the Lindhard function. By writing $\chi_{0}(q,\omega)=\chi_{01}+i\chi_{02}$ as a complex number and using the relation $\frac{1}{z-i\eta} = P(1/z) + i\pi\delta(z)$ in the limit that $\eta$ goes to zero from above and P(..) denotes the principal value it is straight forward to see that the imaginary part of the lindhard function is zero except of the case when we have electron-hole excitations of the form $\hbar\omega=\epsilon_{k+q}-\epsilon_{k}$. Now let's assume we study plasma excitations using the actual Lindhard function (not the bare one). These plasma excitations appear for small q outside of the electron hole continuum. Therefor we have a vanishing imaginary part of $\chi_{0}$ in this case and consequently since also a vanishing part of $\chi$ itself, since its a function of $\chi_{0}$ which is "real" except of the "$\chi_{0}$"-part. However when deriving the plasma excitations using a small q expansion we obtain a none vanishing imaginary part of $\chi$ as given in equation (3.48) of the following document: http://www.itp.phys.ethz.ch/education/lectures_fs10/Solid/Notes.pdf I don't see why this is consistent. I'd be more than happy if you could help me. Thanks in advance.


title: Opposite of Cryogenics

text: Cryogenics is related to very low temperatures, so, what is the term when referring to very high temperatures?


title: Path integral formulation understanding

text: I have done basic quantum mechanics and now I want to do the path integral formulation. I find Feynman's book _Path Integrals in Quantum Mechanics_ difficult. Is there an easier alternative?


title: Relativistic Doppler effect on gamma rays

text: I'm trying to solve the following problem : > An electron-positron pair annihilates, creating two photons. At what speed > must an observer move along the line of the photons in order that the > wavelength of one photon be twice that of the other? So far, I've found using $E=m_0c^2$ that the energy of each photon is 511 keV, and using $E = \frac{hc}{\lambda}$ that the corresponding wavelength is $\lambda = 2.429\cdot10^{-12}$m. The observed wavelength for the photon coming _towards_ the observer is then given by the relativistic Doppler formula : $\lambda_o = \lambda_s \sqrt{\frac{1+\beta}{1-\beta}}$ where $\beta = \frac{v^2}{c^2}$. What I don't understand is how to calcutate the wavelength for the photon propagating _away_ from the observer. It seems to me that it'd be impossible to even observe a photon moving away from you... Also, since the two photons are emitted from the same source, and a photon's speed relative to the observer is always $c$ in all IRFs, shouldn't both photons experience the same redshift/blueshift anyway? Thanks, \--hlouis Edit : Thanks to Mr. dmckee, I was able to solve the problem. If the observer carries a long pole with detectors at both ends, and the emission of both photons occurs when both ends are on opposite sides of the source, then one detector is moving away from the source, while the other is moving towards it. Then the answer is the speed needed to shift down the wavelength by 25% for one detector, and up 25% for the other.


title: Fundamental Commutation Relations in Quantum Mechanics

text: I am trying to compile a list of fundamental commutation relations involving position, linear momentum, total angular momentum, orbital angular momentum, and spin angular momentum. Here is what I have so far: $$ [x_i,x_j]=0 \\\ [p_i,p_j]=0 \\\ [x_i,p_j]=i\hbar\delta_{ij} \\\ [J_i,J_j]=i\hbar\epsilon_{ijk}J_k \\\ [L_i,L_j]=i\hbar\epsilon_{ijk}L_k \\\ [S_i,S_j]=i\hbar\epsilon_{ijk}S_k \\\ [J^2,J_i]=[L^2,L_i]=[S^2,S_i]=0, where \ X^2=X_i^2+X_j^2+X_k^2 $$ I also know that $L$ and $S$ commute, but I am unsure why. I've heard that it is simply because they act on different variables, but I don't understand exactly what this means. Is there a way to show this explicitly? What are the remaining commutation relations between $J$, $J^2$, $L$, $L^2$, $S$, and $S^2$?


title: Shortcut to find $\hat{p}^2$ expectation value

text: I have been going through several calculations where I am asked to calculate $\langle p^2 \rangle$ and the task is proving to be pretty tedious. Does anyone know of a shortcut for this? Such as with $\langle p \rangle$ where: $$ \langle p \rangle = m\frac{d\langle x\rangle}{dt} $$ I have seen a few specific examples where it can be done knowing the energy eigenvalues and the potential... $$ \frac{\langle p^2 \rangle}{2m} + \langle V \rangle = E_n $$ But I was hoping for something more fruitful than this. I guess another way would be to utilize the Hermicity of $\hat{p}$.


title: Humour: Noble Gases, Music, Feynman diagrams

text: Google found these lyrics:               I love the morn, to see the first Radon,     When the stars Argon,     As the night Krypton,     And, if it’s cloudy, you won’t Xenon.      Years ago, I saw approximately the same lyrics with sheet music. The staffs of the music turned into Feynman diagrams. I can't remember what other visual gags were involved, and based on what I didn't know then, and still don't know now, about music and physics, I'm sure there were gags that I didn't catch. Does anyone have a copy? It dates back to before we all had internet access and drawing software, so if it survives at all it might be as one of those photocopies of a photocopy of a photocopy that used to circulate. Edit: I was tempted to add a character named Liam. He (Liam) would scrape his knee on a rock as he stumbled around in the dark.


title: What would Rutherford scattering be like off a "plum pudding model" atom?

text: Qualitatively, for a plum pudding model atom (where the positive charge of each atom is even diffused throughout the volume of the atom) I think that the deflection of incident alpha particles would be caused by the electrons and not the positive charge at all. Is this correct? Quantitatively, for a detector at a specific angle with respect to the incident beam, the number of particles per unit area striking the detector is given by the Rutherford formula (source linked here): $N(\theta)=\frac{N_inLZ^2k^2e^2}{4r^2KE^2\sin^4{(\theta/2)}}$ where $N_i$ is the number of incident alpha particles, $n$ is the number of atoms per unit volume in target, $L$ is the thickness of the target, $Z$ is the atomic number of the target, $k$ is Coulomb's constant, $e$ is the electron charge, $r$ is the target-detector distance, $KE$ is the kinetic energy of the alpha particles, and $\theta$ is the scattering angle. For a plum pudding atom, would this formula still hold as is or would it need to be modified?


title: Aside from experimental evidence, is there any reason to model space as Euclidean?

text: Obviously experiment is the end-all-be-all of any science, but I'm curious if there's any a priori reason to model space as Euclidean three-space (from a pre-relativity viewpoint, of course; I'm ignoring the fact that space is only locally Euclidean). I'm comfortable with differential geometry if it's necessary.


title: Integrating with respect to time a double derivative $\ddot\phi + \frac bm\dot\phi = \frac{F}{mr}$

text: $$\ddot\phi + \frac bm\dot\phi = \frac{F}{mr}$$ I want to integrate this to get an equation for $\dot\phi(t)$ but I don't know how to integrate double derivatives. The answers say that the homogeneous equation associated with it is: $$\ddot\phi + \frac bm\dot\phi = 0$$ and by integrating that you get $$\dot\phi(t) = A \exp(−bt/m)$$ How do they integrate this homogeneous equation to get $A \exp(−bt/m)$? I am only familiar with integrating first derivatives. Also if it gives $A \exp(−bt/m)$, then the non homogeneous equation is according to the solutions that $$\ddot\phi + \frac bm\dot\phi = \frac F{mr}$$ integrates to give $$\dot\phi(t) = A\exp(−bt/m) + \frac F{br}.$$ I also don't get this, because wouldn't the non homogeneous equation give $A\exp(−bt/m) + F/mr$ instead of $F/br$??


Can two particles remain entangled even if one is past the event horizon of a black hole? If both particles are in the black hole? What changes occur when the particle(s) crosses(cross) the event horizon? I have basic Physics knowledge, so I request that answers not assume an in- depth understanding of the field. Thank you all in advance!


title: Does corrosion of iron increase its mass?

text: Does corrosion of a sample of iron increases or decreases its mass? I think that the mass will increase because of additional oxygen atoms.


title: Does a mirror help a near-sighted persion see at a distance clearer?

text: A near-sighted person without eye-glasses can not clearly see things at distance. If he takes a photo of the things at distance, he can see the things from the photo much clearer, because he can place the photo much closer to his eyes. If he turns his back at the things at distance, and holds a mirror close to his eyes in a position so that the mirror reflects the things at distance behind him, will he see the things much clearer than if he looked at the things at distance directly?


title: Is it possible to restart formal higher education in physics at a later age?

text: I would begin by apologizing for a soft question. Reading the FAQ, I have tried to formulate my question precisely in the last paragraph and you may skip the following paragraph where I make subjective observations, which I think may be relevant and any comments would be more than helpful. By formal education I mean registering in an academic department in a university as a grad/Phd Student. Subjective belief: I feel I am personally not upto the mark of high-quality physics students. I need to learn a lot more, and do a lot more problems to be any good. I've noticed that Physics particularly is a very selective field dominated by hard-working and primarily young people who accomplish a lot at an early age (olympiads/contests, early phd's from very good places, etc). Even among this bunch a lot get filtered out and wouldn't get faculty positions. Theoretical Physicists tend to be outliers in a field almost entirely composed of outliers. And to learn from such a community, I feel an academic environment is essential, and not just reading books in isolation. So I beleive academia is the only legitimate environment that functions to produce sane physicists who know their field. However, if one has been set- back for some reason but carries the self-beleif that he can be smart enough to do it, would ageing be a parameter. I read a statement in Kleppner's undergrad mechancis book that "by finishing all this at the age of 26, Newton set the trend which follows today, that great advances are made by young physicists". I haven't found a counterexample so I would trust academic departments have the same criterion. Q: Do people who missed out all the action in their 13-20 ages miss out on having a good career? What are the possible reasons to take a hiatus of "self- study" for a 22-year old so that he may be a good candiate for a respectable physics career (Phd/postdoc etc) later.


title: Is wave–particle duality considered a valid interpretation of the behavior of photons?

text: There are a number of questions on this site that explain the many wave-like behaviors of photons by making reference to wave-particle duality. However, I have just finished reading Feynman's book _QED_ , where he seems to go to great effort to explain things without resorting to this wave-particle duality. He explains the double-slit interference pattern entirely as a matter of the interaction of probability amplitudes. If anything, he seems to consider wave-particle duality to be an obsolete concept that physicists used to rely on to explain photon's behavior, before we "knew better". I know that Feynman's book is > 20 years old by now, but even the recent new introduction doesn't try to explain how things have evolved since then. Am I interpreting Feynman's explanation correctly, and if so, is this still considered the "best" way of explaining the wave-like behavior of particles?


title: Aharonov-Bohm Effect and Integer Quantum Hall Effect

text: What is the relationship between Aharonov-Bohm effect and Integer Quantum Hall effect?


title: How cold should it be outside for a hot coffee mug to break?

text: So I like to go outside for a morning coffee with a cigarette. In winter here it's usually between -5C to -25C, and sometimes it gets down to -30C and colder. Assuming that my coffee is about 75-80C, what are the chances that the coffee mug will shatter in the cold? My mug is made from some kind of organic glass with a seam in the middle like it's two halves glued together. I don't know if this question is serious enough for this site, but, basically, I wanted to know which precautions should I take to avoid breaking the mug?


title: NASA's "Impossible" Space Engine

text: Recently, there was some news that said that the researchers at NASA have come across some impossible kind of space engine which does not require any fuel. I have read at a few places like here, here and here. They all say that this engine seems to violate the laws of physics including conservation of momentum. The original paper is here. Is the statement that it violates conservation of momentum true?


title: Double dot product in Cylindrical polar coordinates - Strain Energy

text: I'm working with a problem in linear elasticity, and I have to calculate the strain energy function as follows: $$2W=σ_{ij}ε_{ij}$$ Where σ and ε are symmetric rank 2 tensors. For cartesian coordinates it is really easy because the metric is just the identity matrix, hence: $$ 2W=σ_{xx}ε_{xx}+σ_{yy}ε_{yy}+σ_{zz}ε_{zz}+2σ_{xy}ε_{xy}+2σ_{xz}ε_{xz}+2σ_{yz}ε_{yz} $$ My question is how the expression should be for cylindrical polar coordinates $(r,θ,z)$


title: A few questions on passive vs active Lorentz transformations

text: 1.) How do we physically interpret an active Lorentz transformation? The passive transformation seems simple enough: you view a fixed object from the perspective of a new observer. When we actively Lorentz transform a vector are we interpreting this as moving the vector to a new point in spacetime considered from the perspective of a single observer? 2.) I am reading David Tong's QFT notes, and am having a hard time interpreting what he means by active transformations. The notes in question can be found here: http://www.damtp.cam.ac.uk/user/tong/qft/qft.pdf The notes in question are on pg 11-12 as labeled by the book or pages 17-18 as labeled by the PDF. In his notes, Tong states that we can transform a scalar field as follows: $$\phi (x) \rightarrow \phi'(x) = \phi \left(\Lambda^{-1} x \right).$$ When he does this, I'm interpreting this as $$\phi (x) \rightarrow \phi'(x') = \phi \left(\Lambda^{-1} x' \right),$$ where $x'=\Lambda x$. From what I understand, the advantage of using the inverse Lorentz transformation on the primed system is that we can use the same functional form of $\phi$. However, when moving to the primed system we have still used $\Lambda$, not its inverse. Can anyone tell me if I'm correct in my understanding up to this point? If my understanding is correct up to this point then I really don't understand the next section in his notes. He states that under this transformation derivatives transform as $$(\partial_\mu \phi) \rightarrow \left( \Lambda^{-1} \right)^\nu_{\phantom{\nu} \mu} (\partial_\nu \phi)(y),$$ where $y=\Lambda^{-1}x$ (where this $x$ is primed, right?). But we've still gone from $x \rightarrow x'$ where $x'=\Lambda x$ (again based on my understanding which may be terribly wrong). Using $\Lambda^{-1}x'$ was simply a mathematical trick to allow us to use the same functional form of $\phi$. If that's the case, why are the derivatives transforming as $\Lambda^{-1}$ instead of just $\Lambda$? I'm sorry -- I know this is a little convoluted, but I'm having a really hard time getting my head around this, especially with his notation. I really wish he would have used primes or something... Am I completely lost? Someone please rescue me.


title: Is in this case distance same as displacement?

text: I have this graph, and I need to know the **_displacement_** in the interval **from 0 to 5 seconds** , so I found the area of the rectangle that you can see there. A = w*h, which means X = (5 seconds)*(20 m/s), then **displacement = 100 meters**.![enter image description here](http://i.stack.imgur.com/XeqZW.jpg) But now I have this other problem, in the following graph I need to found the distance in the interval of **0 to 5 seconds** , in the previous graph I had to found the **displacement** , but now I need the **_distance_** , but should it be the same? because actually it is just a magnitude, so maybe I need to found the area again, or should I use pythagoras?, I hope you can answer this easy question, I would appreciate if you explain why, thank you all.![enter image description here](http://i.stack.imgur.com/ccmvJ.jpg)


title: Is airplane moving slower when air is less dense?

text: Is airplane moving slower at higher altitude ? If air is less compressed shouldn't propeller move less air ?


title: Deriving the rocket equation

text: Recently I decided to brush on the those parts of classical mechanics that always gave me trouble. One of those parts was the rocket equation so I figured that I would try to derive the appropriate equations from scratch. Evidently I ran into trouble somewhere, so I am hoping that someone can point out my errors and steer me in the right direction. Anyway here goes ... Suppose a rocket expels exhaust behind it with constant velocity (relative to the rocket) $\mathbf{u}$. Fix an inertial reference frame and consider the rocket + exhaust system. If $m_R$ denotes the instantaneous mass of the rocket, then the momentum of the rocket can be written as $m_R\mathbf{v}$, where $\mathbf{v}$ is the instantaneous velocity of the rocket. The momentum of the exhaust ought to be given by the integral: $$\int (\mathbf{v}+\mathbf{u})\frac{\mathrm{d}m_E}{\mathrm{d}t} \; \mathrm{d}t$$ where $m_E$ denotes the instantaneous mass of the exhaust. Combining these expressions for the momentum, we arrive at the total momentum for the system: $$\mathbf{p} = m_R\mathbf{v} + \int (\mathbf{v}+\mathbf{u})\frac{\mathrm{d}m_E}{\mathrm{d}t} \; \mathrm{d}t$$ Now according to the 2nd Law of Motion, the force acting on the rocket + exhaust system is obtained from $\mathbf{p}$ by taking time derivatives. In other words: $$\mathbf{F} = \dot{\mathbf{p}} = \frac{\mathrm{d}m_R}{\mathrm{d}t}\mathbf{v}+m_R\frac{\mathrm{d}\mathbf{v}}{\mathrm{d}t}+(\mathbf{v}+\mathbf{u})\frac{\mathrm{d}m_E}{\mathrm{d}t} = m_R\frac{\mathrm{d}\mathbf{v}}{\mathrm{d}t} - \frac{\mathrm{d}m_R}{\mathrm{d}t}\mathbf{u}$$ This expression superficially matches the ones in my college mechanics books, but there is one big difference. In this expression $\mathbf{F}$ is the total force acting on the rocket + exhaust system, while in my texts this $\mathbf{F}$ is just the force acting on the rocket alone. So based on this I have made some fundamental error somewhere in my argument. As I asked before, if anyone could help me identify this error and correct it, I would be much obliged.


title: Two-block system connected to a spring

text: Say you have two blocks with masses $m_1$ and $m_2$, where $m_1>m_2$. The smaller block sits atop the larger block. The larger block is connected to a spring, which is then connected to a wall a distance $x$ away. What must the coefficient of friction be in order for the smaller block to stay on top of the larger block as the spring pushes the masses back and forth. So I get $$f\geq m_1\ddot{x},$$ which says that the friction experiences by the smaller block, must be greater than or equal to the force experienced by the larger block. We know that $$f=\mu_sN,$$ so it becomes $$\mu_sN \geq m_1\ddot{x}\Rightarrow \mu_s \geq \frac{m_1\ddot{x}}{N}.$$ Is this wrong?


title: Reference for the polar parameterization of quaternions

text: I would like to find the original reference in which the polar parameterization of quaternions was given (i.e. the relationship between the components of a unit quaternion and the polar angles of an axis and rotation angle of a three-dimensional rotation). I assume it would have been written by Hamilton, but the physics literature from the 60's doesn't really point back any further (See, e.g. Biedenharn LC. J Math Phys 1961;2:433, equation 10). Any suggestions?


title: Is it possible to shield a camera so as to record from the inside of a running microwave oven?

text: Would it be possible to create shielding for a camera, allowing it record food being cooked from the vantage point of the inside of a consumer microwave oven without the camera being damaged? Wikipedia says a consumer oven usually uses 2.45 gigahertz (GHz)—a wavelength of 12.2 centimetres. So surely you could create some kind of cage around it which allows light to pass through to the lens but not allow microwaves?


title: How does Hawking radiation grow as a black hole evaporates?

text: The temperature of Hawking radiation is inversely proportional to the mass of a black hole, $T_{\rm H}\propto M_{\rm BH}^{-1}$, and so as the black hole shrinks the temperature of the radiation should grow. What happens as the mass shrinks to zero? Naive application of the above formula leads to arbitrarily energetic Hawking radiation. Do assumptions in the Hawking Radiation derivation just break down in this limit? If so, is it known how to handle this limit?


title: The Emperor's New Mind or Shadows of the mind?

text: OK - I have never posted on this site before, and I hope this question isn't closed as "request for reference" but here goes: I'm a software guy who likes to dabble in math, physics and cosmology. Have read "The History of Time" and some books about information theory. Last night, I started reading Penrose's The Emperor's New Mind. In the preface to the current edition, written ten years after the first edition was published, Penrose says that in Shadows of the Mind he came up with some new ideas as the result of working with Stuart Hameroff,  an anesthesiologist by trade. Penrose says these new ideas improve on and surpass the ideas he presents in Emperor. **So I'm wondering: Should I just go to Shadows of the Mind and skip Emperor's, or do I need the material he presents in Emperor's, so I can to move on to Shadows?**


title: Tachyonic complex structure directions in flux vacua

text: In flux compactifications to 4D, e.g. Type IIB on a CY orientifold $X$, one uses fluxes to stabilize the axio-dilaton $\tau$ and the complex structure moduli $z_a$ - the periods of the holomorphic three-form $\Omega$ over the basis three-cycles: $z_a=\int_{\alpha_a}\Omega$, by solving a system of equations for a supersymmetric extremum: $\partial_{\tau}W+W\partial_{\tau}K=0$, $\partial_{z_a}W+W\partial_{z_a}K=0$, where $W$ is the flux superpotential $W=\int_X G_3\wedge\Omega$ and $K$ is the Kahler potential. Since in Type IIB the supergravity scalar potential has a no-scale structure, i.e. $V=e^KD^iW{\bar D_i \bar W}$ where $i$ runs over the complex structure and the axio-dilaton, the potential is positive-definite and therefore the supersymmetric extremum appears as an actual minimum, although the Kahler moduli are still unfixed. However, once we include the non- perturbative corrections to $W$ to stabilize the Kahler moduli and break supersymmetry by e.g. an anti-D3 brane, the scalar potential is no longer positive-definite. So, my question is: is there a proof that the eigenvalues of the Hessian of the scalar potential with respect to all the moduli are positive-definite once all the moduli are fixed by the combination of fluxes and non-perturbative terms? In particular, there seems to be no good reason to believe that there are no tachyons in any of the complex structure directions.


title: Specific heat and temperature

text: This is a two-part question... Firstly, models of the specific heat capacity $C$ (i.e. Debye, Einstein) in relation to the temperature $T$ give $C$ as steadily increasing with $T$. I assume that the change in $C$ is due to the heat the system gains being stored in its degrees of freedom- so why doesn't it increase in steps? Secondly, what value does the temperature refer to? If the heat supplied raises the temperature from $T_1$ to $T_2$, does $C$ refer to $T_1$, $T_2$ or somewhere in between? If, in experiment, a large temperature difference was used, can it be assumed that $C$ changes over the range and so the answer is an average/useless?