title: Can spacetime exist in the absence of matter and energy?

text: I'm pretty sure Ernst Mach would have said that spacetime cannot exist without matter in it. But I'm also pretty sure that a black hole can be described as a self- sustaining gravitational field, which is (I guess?) a case of spacetime sustaining itself without the matter meaning much anymore. So, what is the current theoretical view on this?


title: Are electronic wavefunctions in band gap insulators localized? is a single-particle picture sufficient in this case?

text: I am having trouble understanding the physics of band gap insulators. Usually in undergrad solid state physics one looks at non-interacting electrons in a periodic potential, with no disorder. Then, if the chemical potential lies in the gap between two bands, the material is insulating. At least in this derivation, the individual electronic wavefunctions composing the bands are not localized. However, when talking about insulators, people often think about localized electrons. Do the electronic wavefunctions become localized in band gap insulators? If they are, is it because of interactions? I was thinking that perhaps, since screening is not effective in insulators, the role of interactions is increased, and therefore perhaps the entire non-interacting, single-particle picture used to construct the band structure breaks down. Similarly, an impurity potential will not be screened and could localize the states. So which is it?


title: How many ways are there to distribute M excitations of N identical particles among K=3 quantum harmonic oscillators?

text: I'm trying to numerically calculate a partition function of N non-interacting but identical particles in a 3D SHO. To do this, I'd like to know the degeneracy of $M$ excitations, $N$ _indistinguishable_ particles, in $K=3$ _distinguishable_ oscillators. In other words, what's the degeneracy of $E = \hbar \omega (M + 1/2)$? I want to capture the Bose statistics and calculate something like a few-particle Bose-Einstein condensate. For example, if I have two excitations, two particles, and three oscillators, the degeneracy is nine (please correct me if I'm wrong!):   * Both excitations in one oscillator: 3 ways   * One excitation in each of two oscillators: 6 ways My current approach is to write an algorithm that counts all of these combinations, but I feel like there must be a simpler way. Alternatively, say we only have one oscillator. How many ways can I distribute M excitations among N identical particles? Edit: I originally miscounted the degeneracy of my example. I think it's 9, not 6.


title: Why is the nucleus of an Iron atom so stable?

text: Lighter nuclei liberate energy when undergoing fusion, heavier nuclei when undergoing fission. What is it about the nucleus of an Iron atom that makes it so stable? Alternatively: Iron has the greatest nuclear binding energy - but why?


title: Why do we define such a thing as singularity?

text: According to general relativity, an observer not in the close proximity of a black hole, observing a mass fall into that black hole, will never see that mass cross event horizon(it will gradually fade but never actually pass through the event horizon) as it requires an infinite amount of time to do so. So, effectively, for the observer, the mass of the object falling in will never reach the middle of the event horizon, or the so called "singularity", as it will require more-than-infinite amount of time to do so. Then why do we even define such a thing as a singularity? I am alright with the singularities in space-time curvature, but not with a finite amount of mass being compressed into zero volume, not because it is silly, because it would never happen in a finite time for a casual observer outside of black hole.


title: Calculating M squared

text: I have a laser beam which has a wavelength of 1064nm. The beam quality is 12 mm mrad (SPP). The diameter of the beam is 10 mm. I was told to calculate: $$M^2$$ I used this formula: $$M^2=SPP*\pi*\frac{1}{\lambda}$$ My result is: $$M^2=35,4$$ But usually, the type of laser that was used has a value between 1 and 20. Did I calculate something wrong, or is this just a very poor beam quality?


title: Creating a state of definite momentum and position(within uncertainty limit)

text: **claim:** $a^{\dagger}$= $\int d^{3}kf_{1}(k)a^{\dagger}(k)$ Creates a state with Localized momentum $k_{1}$and localized position near origin; where $f_{1}(k)$ $\propto exp[\frac{-(k-k_{1})}{4\sigma^{2}}]$ If $|0\rangle $ is a vacuum state in free(klein-gordon) field theory then 1) $a(k)|0\rangle$$=0$ 2) $a^{\dagger}(k)|0\rangle = |k\rangle$ 3) $\langle 0|0\rangle = 1$ 4) $\langle k_1|k\rangle = (2\pi)^{3} 2\omega\delta^3(k - k_1)$ How should i prove it? Should i try to expand $a(k)$ etc in terms of $\Phi(\bf{x})$ fields and try to find position and momentum eigenvalues? That way i have to know the meaning of $\Phi(\bf{x})|0\rangle$ etc.


title: Can false origin be anywhere in a graph?

text: Can I start labeling my $x$ and $y$ axes from non-zero values when drawing a graph? Or is there any convention to only label $x$-axis from non-zero value when using a false origin and not $y$-axis? Please clarify this.


What is the basic difference between absolute pressure and gauge pressure? I have a great confusion regarding these two pressures. Please someone explain it to me


title: Why Won't a Tight Cable Ever Be Fully Straight?

text: I posted this picture of someone on a zipline on Facebook. ![picture of a sagging zipline](http://i.stack.imgur.com/uazFd.jpg) One of my friends saw it and asked this question, so he could try to calculate the speed at which someone on the zipline would be going when they hit the water. The only answer, which was accepted, includes the disclaimer, " _Assuming the pulley being used to slide to be friction less.Though not possible.Also the rope is assumed to be in-extensible and straight._ " I used to have a zipline of about the same length in my back yard as a kid and even when I was young, I noticed that we could never straighten the line completely, even when it was slack, we could not make it completely straight. And, naturally, once weight was added, there was a curve where the weight pulled the line down. One of the comments from the member providing the answer is " _Well i can show you why the string cannot be ever straight._ " I know that from experience. We could never make it completely straight with no sagging. I asked the reason for this and was directed to a book on Amazon. Having just spend $50 on a number of books for summer reading, my book budget is gone for a while. So can someone answer that? Why will the line never be straight when it's set up (and when there is no load on it)?


title: Why isn't there heterotic holographic QCD?

text: I think the question speaks for itself... Top-down holographic QCD, like Sakai-Sugimoto, always involves the Type II string. There are one or two papers on hQCD using the Type 0 string. But I can't see anyone making hQCD models using the heterotic string. Why not? I may as well explain how the question arose: I was thinking about two forms of conventional string phenomenology - M-theory on G2 manifolds, and E8xE8 heterotic models - and was looking at possible beyond-standard-model symmetries, like flavor symmetries, family symmetries, etc. Somehow I stumbled into a time warp and ended up in 1965, looking at "The Covariant Theory of Strong Interaction Symmetries" by Salam, Delbourgo, and Strathdee, in which "A classification of particles is suggested based on a U(12) symmetry scheme". They want to make hadrons out of a "12-component (Dirac) quark" (i.e. _u_ , _d_ , _s_ as a single "quark"), and propose representations for vector and pseudoscalar mesons, and spin 1/2 and spin 3/2 baryons. It occurred to me that it would be rather eccentric to think about ways to embed a variation of their scheme into E8xE8, but that someone working on holographic QCD might well be interested in forgotten approaches to the strong interactions; and yet, top- down hQCD employs Type II string theory, which is dual to the heterotic string. Why haven't insights from heterotic string theory played a role in holographic QCD so far?


title: Testing conservation laws experimentally

text: How conservation laws are tested experimentally independently from each other? what do I mean by that question? It seems that to test one conservation law experimentally, such as conservation of energy, we will have to assume other conservation laws are correct such as conservation of charge and conservation of momentum and angular momentum...etc. But to really test any conservation law one has to check if one of them is valid independent from any other one. Is this possible experimentally? if not then how scientists check conservation laws?


title: How long should I leave the light on?

text: I remember my old physics teacher at school saying that if you switch a light on in a room and you're planning to return relatively quickly, it uses less electricity if you leave the light on until you return as switching the light on uses more eletricity than leaving it on for a relatively short period, but how long before leaving it on becomes more energy consuming than switching it off until you return and switching it on again?


title: How does current work

text: In all diagrams i have ever seen of circuits it's implied that electrons travel through load/bulb and bulb makes light/heat. Now there is a problem here...because of this implied logic i assumed more electrons that follow more power and therefore after power is used in light their speed reduces. But that doesn't make sense what's happening with clogged up electrons. -coming electron and leaving electron both are the same. I assumed, that unused electron somehow was bigger and puffed and when it gets used its small/slow/ or less charged..But it's none of the above. It's actually the same...So i learned energy is carried via waves that travels through electrons. Now i don't know what is current or electricity. If used and unused electron both are the same what provided the energy. please explain how number of electrons, speed of electrons, wave(?) of Energy, and all other stuff fits into how electricity works.


title: Conservation of mass-energy and nuclear transmutation

text: This may just be beyond the grasp of the everyman, but I'm trying, and failing, to grasp how conservation of mass-energy works in cases of beta decay and electron capture. A neutron has a mass of one dalton. The combination of a proton and an electron also has a mass of one dalton. This allows us to assign an overall "mass number" to elements and their specific isotopes. We can say, for instance, that because Carbon-14 and Nitrogen-14 have the same mass number, they have equivalent masses, even though nitrogen has one more proton; it also, in its charge-balanced state, has one more electron than the carbon, and so is equivalent in mass to the carbon atom, just distributed differently. However, Carbon-14 directly decays into Nitrogen-14 by $\beta^-$ decay of one of its neutrons. The "reactant" is one neutron from the carbon nucleus, and the products are one proton, one electron, and one electron antineutrino. This extra particle, a lepton similar to the electron, is not massless (and mass is an absolute quantity so you can't have "negative mass"), and therefore it would seem that the combined mass of a proton and electron are _not_ in fact equal to that of a neutron. It gets even weirder when you see it work in reverse. Atoms that have too few neutrons can lower their atomic number by electron capture; the electron is "absorbed" by a proton in the nucleus, forming a neutron, and this interaction results in the emission of an electron neutrino (also not massless). So, if you were to see a neutron decay to a proton, then capture an electron to become a neutron again, you would have seen it emit a lepton and anti- lepton, both of them having a small but nonzero mass, and yet because a neutron is a neutron, the mass of the particle before these two transformations should be equivalent to the one after. Obviously I'm missing something. Is there an energy input required (possibly from the neutrino-antineutrino annihilation but it could come from anywhere) that is omitted from the basic nuclear chemistry equations, that because of mass-energy equivalence is re-adding the mass lost as leptons? Is there indeed such a thing as "negative mass", and the mass of the anti-lepton is somehow modified by its spin sign? Or is this a subatomic case of the Second Law of Thermodynamics, and in fact not all neutrons are created equal?


title: Is there 100% reflective mirror...i just want one

text: I have heard there are none right now...but i saw something that said something about 100% reflection. Forgotten completely. sorry. I want it, i think it'd be very amazing to save sunlight in it from day and let little out in night. But i have heard it's theoretically not possible to have surface that's 100% reflective. I don't believe that. Would it be possible to reflect light without absorbing or normal reflection like using some technique to bend it 180degrees. explain please...


title: The physics equation for a perpetual seesaw

text: I am trying to create a game that envovles two or more people jumping on a seesaw and propelling each other in the air. I need help with the physics equations Assume no friction no air resistance for now. Ideally when you hit the sweet spot of the seesaw at the very end person B should go a little bit higher than person A who just came down. And if you hit close to the piviot, then the reverse should happen. (I don't know if that is really physics but that is what I am going for) I know the classic example of two forces on each side and t =r x F so if you had a mass heavier on one side that heavier mass would have to be closer to the piviot ![enter image description here](http://i.stack.imgur.com/wX3kw.gif) What I am not grasping is if F=ma. And assuming that the masses are all equal then F=a. Now gravity is 9.8/m/s/s on both objects. Is this true even if one of them is at rest on the ground so the seesaw is tilted up? ![enter image description here](http://i.stack.imgur.com/YLQ1c.jpg) So because Force is not related to velocity the following doesn't make sense I would have thought dropping an object from 10 meters would have created more F then from 5 meters if they hit the same spot and have the same mass. So while the velocity increases the acceleration is constant isn't the force constant? What am I not grasping? Thank you


title: How do you keep a fridge full of food cooler?

text: I just graduated and a mate moved into a flat (none of us are physicists by the way). So, were graduates, we got a new flat, and were broke. So, were now having a debate on how to keep a fridge cool... well, cooler. I suggest that a fridge full of water will keep the food colder, as objects get cooler when water evaporates, (like when we sweat). He says that the fridge just has to do extra work to keep the water cool, but I said it doesn't matter as it's already colder when it enters the fridge, and it's properties should mean that the overall effect, is a colder fridge. Another conundrum, if I AM right, is it more effective to store the water at the bottom, middle, or top shelves? Also, what size/shape should the water be in, should it be stored in glasses, bowls, or in the form of cloths, where the surface area can be distributed, a lot more widely.


title: Difference between Non-Polar and Dipole moment $\vec\mu$=0

text: Is there any difference between a molecule having $\vec\mu=0$ and being Non- Polar?


title: Why do Fermi liquids have T^2 resistivity?

text: I have often read that metals that are Fermi liquids should have a resistivity that varies with temperature like $\rho(T) = \rho(0) + a T^2 $. I guess the $T^2$ part is the resistance due to electron-electron interactions and the constant term is due to impurity scattering. Is there a simple argument to show this? Or maybe you could point me to a nice reference? Also, it seems that for electron-electron interactions to introduce a finite resistivity, some umklapp scattering is necessary (to break Galilean and translational invariance). Is this correct? Which of these symmetries (Galilean or translational) has to be broken?


title: The rule breaker, emissivity + reflectivity = 1

text: If emissivity and reflectivity are inversely proportionate, why does glass have a high emissivity of around 0.95-0.97 as well as being very reflective for IR Radiation? normally it works but not with glass! Can anyone explain this?


title: Bohr-van Leeuwen theorem and quantum mechanics

text: **Preamble** : If one considers an ideal gas of non interacting charged particles of charge $q$ in a uniform magnetic field $\mathbf{B} = \mathbf{\nabla} \wedge \mathbf{A}$, then the classical partition function in the canonical ensemble reads (in SI units): $Q(\beta,V,N,\mathbf{B}) = \frac{1}{N!}q(\beta,V,\mathbf{B})^N$ where $q(\beta,V,\mathbf{B}) = \int \frac{d\mathbf{p} d \mathbf{r}}{h^3}\:e^{-\frac{\beta}{2m}(\mathbf{p}-q\mathbf{A}(\mathbf{r}))^2}$ If we integrate first with respect to momenta over all possible values from $-\infty$ to $+\infty$ for each component, a simple change of variable leads to $q(\beta,V,\mathbf{B})=\frac{V}{\Lambda^3}$ which is the ideal gas result and where $\Lambda$ is the thermal de Brooglie wavelength. If one then wants to get the magnetization per particle $\mathbf{\mu}$ induced by the field $\mathbf{B}$ it is simply: $\mathbf{\mu} = -\frac{\partial \langle \epsilon \rangle}{\partial \mathbf{B}} = \frac{\partial }{\partial \mathbf{B}}\left( \frac{\partial \ln(q(\beta,V,\mathbf{B}))}{\partial \beta} \right) = \frac{\partial }{\partial \beta}\left( \frac{\partial \ln(q(\beta,V,\mathbf{B}))}{\partial \mathbf{B}} \right) = \mathbf{0}$ This is one way to state the Bohr-van Leeuwen theorem. Now, I physically understand this result as coming from some symmetry associated with the momenta (it is as likely to go to the right as it is to go to the left) and the fact that the boundaries of the integral over the momenta are infinite. If the problem is treated quantum mechanically, the eigenstates of one charge particle are discretized Landau levels with a typical spacing between two neighbouring levels that is $\hbar \omega_c$ where $\omega_c = qB/m$ is the cyclotron frequency and one finds that the sum over these states depends on the magnetic field $\mathbf{B}$. **Question(s):** I am lost in my interpretation of the quantum to classical limit for this system...so far I thought that the quantum -> classical limit for the statistical properties of an individual particle was related to the way of counting the number of states for this particle i.e. whether we consider the set of states as a continuum or as a discrete set. This analogy seems to work in this case as well since the classical limit arises if $k_B T \gg \hbar \omega_c$. However two major points differ from what I am used to:   * The quantum treatment of this system yields a non zero magnetic moment (although it vanishes at infinite temperatures) in the limit where $k_B T \gg \hbar \omega_c$ while the classical treatment gives strictly zero.   * I do not understand how does the left-right symmetry argument used in the classical partition function disappear in the quantum treatment to yield a partition function that depends on $\mathbf{B}$.   * Is there any classical way to assess that quantum corrections will be of order $\mathcal{O}(\Lambda/R_c)$ where $R_c \sim \sqrt{m k_B T}/(qB)$ is the typical size of the radius of the helical paths taken by a charged particle? Sorry if my questions seem confused, I will try to improve them if they are not clear enough. **EDIT** : I realize that one of my points is not very clear and shall explain it with the example of a true harmonic oscillator. If I consider classical statistical mechanics, I know that $\langle \frac{1}{2}m\omega^2 x^2 \rangle = \frac{1}{2}k_B T$. This tells me that the typical uncertainty on the position of my particle is $\sigma_x = \sqrt{k_B T/(m\omega^2)}$. Incindently this length is also the typical confinement length scale owing to the harmonic potential. One way to semi-classicaly probe the validity of the classical limit is to imagine the particle as a non dispersive wave packet of width $\Lambda = h/\sqrt{2\pi m k_B T}$ and to realize that interferences (ultimately leading to quantization) are not important if $\Lambda \ll \sigma_x$. This is very appealing because one can then probe the validity of a classical approximation using a $\sigma_x$ that comes from a classical treatment. My biggest problem with a charged particle in a magnetic field is that the Bohr-van Leewen theorem apparently prevents this typical length scale (that I know for sure is $R_c$) to be found with a classical statistical treatment.


title: Hamiltonian constraint in spherical Friedmann cosmology

text: I'm taking a GR course, in which the instructor discussed the 'Hamiltonian constraint' of spherical Friedmann cosmology action. I'm not quite clear about the definition of 'Hamiltonian constraint' here. I searched internet but cannot find a satisfying answer. Could anybody help me a bit?


title: Why does the road look like it's wet on hot days?

text: Often, I'll be driving down the road on a summer day, and as I look ahead toward the horizon, I notice that the road looks like there's a puddle of water on it, or that it was somehow wet. Of course, as I get closer, the effect disappears. I know that it is some kind of atmospheric effect. What is it called, and how does it work?


title: Animating the Bosonic String

text: I am interested in studying the classical solutions to the Bosonic string in flat 3+1 dim. spacetime by having them rendered a moving picture on a computer. This is partly for fun, and partly to develop an intuition for their kinematical behavior. But, I'm not sure where/how to start. Should I start from the **Nambu-Goto Action** or the **Polyakov Action**? And, should I use the **light-cone** gauge or the **covariant** gauge? Of course, whatever my choice, I should get the same results, but I want to know the choice that is _most convenient to obtain solutions that can be visualized_. Finally, what interesting **initial-values** should I use? I would like to investigate the motion of the string doing all sorts things. Any suggestions here? Eventually I hope to animate the classical scattering of two strings, if possible (I'm not too sure if classical strings interact...).


title: Deriving equations of motion using integration

text: Please refer to my school textbook pg48 (of the book, and not the pdf counter) here: http://ncertbooks.prashanthellina.com/class_11.Physics.PhysicsPartI/ch-3.pdf My doubt is in this context: (right side column) $a = dv/dt = v dv/dx$ then integrating $v$ with respect to $dv$, and $a$ with respect to $dx$. Now, when we integrate $a$, either we say it is a constant, and give $a\times(x-x_o)$ as a result of the integration, or we say it is not constant, and is a function of time, in which case, it can not be integrated like this. But the book integrates it like this, and then it is written: > The advantage of this method is that it can be used for motion with non- > uniform acceleration also. Can you please explain how it can be used with non uniform acceleration while it has been assumed while integrating that $a$ is constant? And moreover, does this line seem to apply to only this derivation, or does it apply to other two before it as well?


title: Understanding the cause of sidebands in Amplitude Modulation

text: I've read it many places that Amplitude Modulation produces sidebands in the frequency domain. But as best as I can imagine it, modulating the amplitude of a fixed-frequency carrier wave just makes that "louder" or "quieter", not higher-frequency or lower-frequency. That is, I believe I could sketch, on graph paper, a path of a wave function that touches a peak or a trough exactly every 1/f increments, regardless of the "volume". Why do the sidebands appear?


title: Lepton Number Conservation

text: What is the global symmetry of the electroweak Lagrangian that gives rise to lepton number conservation? As I understand it, electric charge is some linear combination of the conserved quantities corresponding to the $SU(2)$ symmetry and the $U(1)$ symmetry of the electroweak theory. If something similar is true for lepton number, then perhaps it would not be easiest to give the global symmetry directly. This is perfectly fine of course.


title: Solution of motion in hamiltonian formalism

text: I have these canonical equations: $$\dot p = - \alpha pq$$ $$ \dot q =\frac{1}{2} \alpha q^2$$ I have to find $q(t)$ and p$(t)$, considering initial conditions $p_0$ and $q_0$. I thought to simply integrate with respect time both members of the equations, but something must be wrong because the solutions are:$$q(t)=\frac{q_0}{1- \frac{1}{2} \alpha q_0 (t-t_0)}$$ $$p(t)=p_0[1-\frac{1}{2} \alpha q_0 (t-t_0)]$$ and I don't know how obtain them...


title: "as measured in a local Lorentz frame"?

text: I've seen the phrase "as measured in a local Lorentz frame" tagged on the end of so many sentences. What does it mean precisely? To give an explanation with an example, consider the context of measuring the speed of an object, if this applies.


title: Electron configurations and $M_{s}$ quantum numbers

text: I have a little question that are confusing me at bit. I have to argue, from the Pauli Exclusion Principle, that $M_{s} = 2$ is the maximum $M_{s}$ quantum number for the $nd^{6}$ configuration. Now, from what I can find, $M_{s}$ is given by: $M_{s} = \sum\nolimits_{i} m_{s_{i}}$ And with 6 valence electrons, that can have $m_{s} = \pm \frac{1}{2}$, I can't seem to see how I end up with a maximum of 2 ? So anyone who could give me a little hint ? Thanks in advance.


title: Does lightning have mass?

text: My chemistry teacher/book states that lightning is just light, and therefore has no mass and takes up no space (we're not very far through the book yet, it's defining matter). However, I take issue with this statement - it feels _wrong_. My reasoning is this: Lightning, as far as I know, is simply electrical energy. Also, I believe that electrical energy needs a medium to travel through, right? If this is true, wouldn't the air the lightning travels through be the mass and space the lightning takes up?


title: Temperature dependance of silicon oxide band gap

text: What is the temperature dependance of silicon oxide band gap?


title: Violation of Newton's Second Law (?)

text: Here the big circle denotes the circular path of a stone (small circle on path) tied to a string from the centre of the circular path. This is COMPLETELY HORIZONTAL ![This is COMPLETELY HORIZONTAL](http://i.stack.imgur.com/A1tHB.png) ![enter image description here](http://i.stack.imgur.com/5yNrc.png) At an instant the velocity in horizontal direction is $0$, and tension acts horizontally. Then it is said tension will do no work. Now, at the very next instant of time, ![enter image description here](http://i.stack.imgur.com/c2yse.png) Supposedly, all this is horizontally happening, this is a horizontal circle. Then, tension increased velocity in horizontal direction, no one decreased velocity in vertical direction. Then as a result, net velocity should increase. No other force, other than tension is acting. Then it means as net velocity didn't increase, this is a violation of newton's second law (?) Now in the vertical direction velocity didn't change and in the horizontal direction velocity increased since $a=\frac{T}{m}$ where $T$ is Tension in string. So as it acted for a very small amount of time, it did increase velocity in horizontal direction. and velocity in vertical direction remained same, thus, $|v| = \sqrt{(v_y)^2+(v_x)^2} > v_{earlier}$, thus speed must have increased in the next instant a/c to newton's laws. But it didn't according to the fact that tension did no work--why? _When I say x $/$ horizontal direction, it means parallel to the tension shown in the image. And vertical $/$ y means perpendicular to tension shown in image ._


title: Interpretation of a quantity from the reduced three body problem

text: Consider two bodies $S$ and $J$ that gravitate about their center of mass $O$, under the influence of mutual attraction. A third massless body $P$ is moving in the same plane. What is the interpretation of the quantity $$F={m_1\over SP}+{m_2\over JP}$$ where $m_1$ and $m_2$ are the masses of $S$ and $J$ respectively? $EDIT:$ And to answer my post, $F$ is the sum of the quotients of the masses of the two massive bodies by their respective distances to the massless body. This is its physical interpretation. It is some sort of funny averaging that seems non-sensical at first sight. But forgive my poorly crafted question. What I really want to know is whether this quantity has a name or some notable significance.


title: Coulomb interaction and conservation laws

text: In many-body solid-state physics, the Coulomb interaction term in the Hamiltonian usually implies the momentum conservation law in indicies: $$H_c=\frac{1}{2} \sum_{\mathbf{k},\mathbf{k}',\mathbf{q} \neq 0} V_{\mathbf{q}} a^{\dagger}_{\mathbf{k}'-\mathbf{q}} a^{\dagger}_{\mathbf{k}+\mathbf{q}}a_{\mathbf{k}'} a_{\mathbf{k}},$$ where $\mathbf{k},\mathbf{k}',\mathbf{q}$ are quasi-momenta and quantum numbers for the continuum spectrum of electron gas simultaneously. In quantum chemistry textbooks, the Coulomb term usually looks like: $$H_c=\sum_{i,j,k,l} V_{i,j,k,l} a^{\dagger}_{i} a^{\dagger}_{j}a_{k} a_{l}$$ Numbers $i,j,k,$ and $l$ are running over some discrete energy spectrum. Is it possible to state any conservation laws for quantum numbers $i,j,k,$ and $l$ in the expression above? Should not they obey any conservation law, selection rules or any additional restrictions? I will appreciate any references to textbooks or papers.


title: The approximate uncertainty in $r$

text: The volume of a cylinder is given by the expression $V=\pi r^2 h$ The uncertainties for $V$ and $h$ are as shown below $\begin{align} V&\pm 7\%\\\ h&\pm 3\% \end{align}$ > What is the uncertainty in $r$? Now, the obvious answer would be $2\%$, from $$\frac{dV}{V}=\frac{dh}{h}+2\frac{dr}{r}$$ However, rearranging to $r^2=\frac{V}{\pi h}$ gives $$2\frac{dr}{r}=\frac{dV}{V}+\frac{dh}{h}$$ which gives a different answer of $5\%$. Thus, by simply rearranging the formula, we get different values for uncertainty in $r$.       How do you explain this? _(The mark scheme lists $5\%$ as the correct answer)_


title: What is the maximum non-fatal force withstandable by a human being for a short period?

text: Assuming any (optimal) body orientation/position relative to the direction of force and mechanism of death. I'd be interested in a rough estimate of threshold force and a suggested mechanism of death. A person experiencing a force just below this threshold should be able to survive for a few minutes. To give more context the question derives from: 'How much centripetal acceleration is fatal?', or 'How fast and tight must a rollercoaster go to kill you?'. I realise there may be some biological considerations for this question, but giving bounds on these seems within the remit of back-of-the-envelope physics.


title: A rod filled with water and charged particles is rotated in magnetic field

text: Assume we have a rod of diameter $d$ and length $l$. It is filled with a mixture of water and, say, sodium chloride. Thus there are positively charged sodium ions and negatively charged chloride ions in the water. Now, assume that the rod is rotated perpendicular to the ground. I should add that the rotation happens at the mid point, not at one of the ends. Earth's magnetic field $B$ thus asserts force to the moving particles according to $q (v \times B)$ (Lorentz force). This means (I think) that the particles should start to move to the ends of the rod. If the rotation is in counterclockwise direction, the right hand rule states that positive particles should move to the right side and opposite for negative. (I think) Question is, does this really happen or is the centrifugal force going to push the particles to the sides? If so, how strong magnetic field $B$ or the velocity $v$ should be in order for this to happen?


title: Gauge covariant derivative in different books

text: It puzzles me that Zee uses throughout the book this definition of covariant derivative: $$D_{\mu} \phi=\partial_{\mu}\phi-ieA_{\mu}\phi$$ with a minus sign, despite of the use of the $(+---)$ convention. But then I see that Srednicki, at least in the free preprint, uses too the same definition, with the same minus sign. The weird thing is that Srednicki uses $(-+++)$ I looked too into Peskin & Schröder, who stick to $(+---)$ (the same as Zee) and the covariant derivative there is: $$D_{\mu} \phi=\partial_{\mu}\phi+ieA_{\mu}\phi$$ Now, can any of you tell Pocoyo what is happening here? Why can they consistently use different signs in that definition?


title: Measuring acceleration of a bus using water between two sheets of glass

text: I was riding a bus one day and noticed that the double windows had some water between them. As the bus accelerated, the water collected to the sides, first forming a trapezoid and then a right triangle. I begun wondering how it would be possible to measure the acceleration using only the geometry of the form of the amount of water. Like, assume that at $a=0$ the water has height $h$ and width $w$. As the bus accelerates, at some time $t_1$ the water forms a trapezoid with the shorter side $h_1$ ans longer side $h_2$. The bottom has same width $w$. At time $t_2$ water has formed a triangle with height $H$ and width $w$ (the last moment it touches the other side of the glass). And finally at time $t_3$ the height is $y$ and the width is $W$. In each situation, what is the acceleration of the bus?


title: Why is electric potential scalar?

text: I can't conceptually visualize why it would be so. Say you have two point charges of equal charge and a point right in the middle of them. The potential of that charge, mathematically, is proportional to the sum of their charges over distance from the point ($q/r$). But intuitively, my thought process keeps going back to the concept of direction and how the electric field at that point would be zero. So why would the electric fields cancel while the electric potentials just add up algebraically?


title: If a pendulum is on a rotating table, will a torque be generated?

text: Here is the set up. Very simple. A flat (i.e. horizontal table, there is no gravity) and rounded table that spins on its axis (through the center of the table). A spring mass system is now put on the table inside a slot along the radius of the table. One end of the spring is attached to the origin of the table, and the other end is the small bob mass. Lets assume the relaxed length of the spring is half way along the radius. So the above is the physical set up. Here is a picture, showing the spring with a little bit of tension in it. ![enter image description here](http://i.stack.imgur.com/u5ijO.png) So, the pendulum can only move along the radius, as it is inside a slot in the table. Now I spin the table, say clockwise with some initial angular speed, and at the same time release the bob so it also starts to vibrate back and forth on the radius as the table rotates. What I am not sure about is that whether a torque be generated due to the reaction of the bob on the edge of the slot as the disk spins or not? This reaction force has an arm length from the center, hence a torque will be present. In otherwords, will this system's angular momentum be constant or not? If there is no torque generated, then the angular momentum is ofcourse constant. Thanks **Edit** May be I should explain why I thought at first there might be a torque generated. When the bob is rotating along with the table, it will have 2 components of accelerations. One component is along the radius direction, which is `(x'' - x (theta')^2)` where `x` is the bob coordinate along the radius, and `theta(t)` is the angle of rotation. In the above, the term `x(*theta')^2` is the centripetal acceleration and `x''` is the acceleration of the bob along the radius. There is also an acceleration perpendicular to that, which is `(x theta'' + 2 x' theta')`. In the above, the second term is the coriolis acceleration and the first term is the standard Euler acceleration. I made this diagram ![enter image description here](http://i.stack.imgur.com/GVb2S.png) Now, whenever I see something accelerating, then there must be a force along that direction. Right? This is the force that I took as generating a torque on the disk (actually it is the reaction to this force, ie. back to the disk). This is why I thought there is a torque. But when I did the same derivation using Lagrangian method `L=T-V`, I got different equation of motion. (no torque). Hence I thought to ask the experts here. **update 4** I have finally I think sorted this whole thing out!. I have updated the derivation. Solved this problem using Newton and Lagrangian methods, and both give the same equation of motions, and verified that the angular momentum remain constant. I have a PDF and HTML page and the applet here http://12000.org/my_notes/mma_demos/slot_on_disk/index.htm This is not as easy as it first seemed. There is actually a torque involved. Due to Coriolis force. This torque however changes in value such that the angular momentum remain constant. The critical thing for me to see was that the spring in infinitly rigid against rotation. This was mentioned below by Shaktyai. This is critical, as it means a slot is not even needed to keep the spring stright, and hence there is no Euler acceleration term perpendicular to the radial direction. The only side acceleration is the Coriolis one. This does generate a torque. I am not sure what this torque is called ( Coriolis torque?). It all now fits nice. I removed the slot from the applet since it is not needed, and updated it. If someone can verify it for me it will be appericated. The applet runs in the browser. Written in Mathematica. Wiki has a nice illustration of the Coriolis effect here http://en.wikipedia.org/wiki/Coriolis_effect


title: Symmetries, Generators, Commutators and Observables

text: I'm learning about generators and conservation laws and have derived the equation (1) $$[Q,A]=-i\hbar f(A)$$ which is satisfied by the observable generator $Q$ for a transformation group with elements of form $$g_a(A)=af(A)+\textrm{O}(a^2)$$ The lecture notes I'm reading say that this equation (1) defines $Q$ provided we know $f(A)$ for all observables $A$. Why is this true mathematically? And what does all observables mean? The example in the notes applies it to transformations along the $k-$axis for a system of $r$ particles, obtaining $$[Q,\hat{x}_i^r]=-i\hbar\delta_{ik}$$ $$[Q,\hat{p}_i^r]=0$$ It then states that $Q = \hat{P}_k=\sum_r \hat{p}_k^r$. This is obviously a solution, but do I know that it's the only one? Many thanks in advance.


title: Why does a wobbly metal sheet make the sound of thunder?

text: In other words, what is the similarity between a lightning bolt and a wobbling sheet that make them sound alike? It seems to me that the two systems have a much different way of moving the air, and probably if you could hear it up close, a lightning bolt wouldn't make the sound of a wobbly metal sheet. I think maybe the many reflections and how they interact with the cylindrical shape of the thunder-wavefront may be what gives the similarity.


title: Does a fluid's temperature affect the way light passes through it?

text: For example, if I were to supercool water would it's refractive index still be 1.33 or would it be 1.31, the same as water-based ice even though it's still in liquid form?


title: Yukawa Coupling of a Scalar $SU(2)$ Triplet to a Left-Handed Fermionic $SU(2)$ Doublet

text: Suppose we have a field theory with a single complex scalar field $\phi$ and a single Dirac Fermion $\psi$, both massless. Let us write $\psi _L=\frac{1}{2}(1-\gamma ^5)\psi$. Then, the Yukawa coupling of the scalar field to the left-handed Fermion field should be of the form $$ g\overline{\psi}\psi \phi, $$ where $g$ is the coupling constant. So far, so good (at least I think; please correct me if this is incorrect). Now, we introduce gauge invariance into the theory and demand that $\phi$ transform as a triplet and $\psi _L$ transform as a doublet under the gauge group $SU(2)$. What form does the Lagrangian take now? My confusion is arising because now, in particular, although $\phi$ transforms as a scalar under the Lorentz group, it must be described by $3$-components so that it can transform as a triplet under $SU(2)$. But then, the Yukawa coupling term listed above, as written, is not a number! I know this has something to do with the fact (I think) that $\overline{2}\otimes 2$ decomposes into something involving the triplet representation of $SU(2)$. Unfortunately, I don't know enough about the representation theory of $SU(2)$ to turn this into a Yukawa coupling term that makes sense. Once again, because of my lack of knowledge of the representation theory of $SU(2)$, I don't know how to write the gauge covariant derivative corresponding to the triplet representation of $SU(2)$. If we use the Pauli matrices as a basis for $\mathfrak{su}(2)$, how is the triplet representation of $SU(2)$ described in terms of the Pauli matrices acting on a three- dimensional complex vector space? I am also unsure as to exactly what should happen to the kinetic term for the Fermion field. Before insisting upon gauge invariance, this term should be of the form $$ i\overline{\psi}\gamma ^\mu \partial _\mu \psi . $$ However, because (I believe) gauge invariance is only being demanded for $\psi _L$, presumably something more complex happens to this term than just $i\overline{\psi}\gamma ^\mu D_\mu \psi$, where $D_\mu$ is the appropriate gauge covariant derivative. Instead, should this kinetic term be written in the form $$ i\overline{\psi_L}\gamma ^\mu D_\mu \psi _L+i\overline{\psi _R}\gamma ^\mu \partial _\mu \psi _R? $$


Would a Helium powered balloon vehicle work on Mars? Things that might need to be answered:   1. How much liquidHelium would you need to pack in order to fill a balloon with enough helium gas to travel the planet.   2. How much energy does it take to re-compress Helium per/litre at low temperatures on the surface of mars and would a solar cell be enough?   3. Would a "Balloot" (balloon/parachute) be enough to complete spacecraft re-entry on arrival to mars without really needing to immediately "touch down"?   4. Any other things to think about by anyone else?


title: How exactly does mass affect speed?

text: I konw that mass affects weight (force), so how does that relate to speed? F=ma. so how does all this affect speed?


title: Why is the atmosphere transparent in the visible spectrum?

text: One of the great _coincidences_ in physics is that the Sun happens to shine most brightly at exactly the wavelengths our eyes can see; it's an easy explanation that our eyes evolved to make the most out of what was available. However, this spectral window is also one of the few at which our atmosphere is transparent, with Rayleigh scattering and the ozone layer protecting us from the UV and strong water vapour absorption in the IR until way into the radio range. My question is, is _this_ a coincidence? Were there geological factors that made it so? Or is it just lucky chance? (I also notice other planets in our neighbourhood weren't quite so lucky.) Or had atmospheric absorption been different our eyes would have tuned to whatever spectral window were brighter as a whole?


title: DFT breakdown for effects that depend on phase?

text: Given the Hohenberg-Kohn theorem, a given ground-state density uniquely defines _all_ ground state properties, because in principle the external potential $V_{ext}$ can be understood as a functional of the ground-state density $n_0(r)$. Now the Schrödinger equation gives me - again, _in principle_ , the ground- state wave function $\Psi_0(r)$, so if I knew the functional $V_{ext}[n(r)]$, I could go from a ground-state density to the ground-state wavefunction. This might seem counter-intuitive at first, because the density itself seems to not contain any phase information, but Hohenberg-Kohn says that, in fact, all the information is there (just very well hidden, and not readily extracted). The question is now: Is the missing phase information contained more in the density or more in the mysterious functional? If it's the latter, I would expect that DFT using approximations to the exchange-correlation functional should break down for phenomena that rely somewhat strongly on (local) phase, such as Anderson localization or the Aharonov-Bohm effect?


title: Open Ended/ Close Ended instruments?

text: Close ended instruments have twice the wavelength, because the wave must travel twice the distance to repeat itself. Why must a wave reach a lower density medium (air in this case) to repeat? When it collides with the closed end (same density), what happens instead of repetition?


title: Commutators with a density matrix

text: The equation describing the evolution of our system is as follows: $ \dot{\rho} = u_1(t)(a^\dagger a \rho - 2a\rho a^\dagger +\rho a^\dagger a) + u_2(t)(a a^\dagger \rho - 2a^\dagger\rho a +\rho a a^\dagger) $ where $\rho$ is the density matrix and $a$ is the photon annihilation operator. I need to rearrange this equation to look something akin to: $\dot{\rho} = u_1(t) \hat{A} \rho + u_2(t) \hat{B}\rho $ How do I handle the commutators and the like in order to rearrange it?


title: What is the "foamy space" hypothesis that has been debunked recently?

text: In "Space-Time Is Smooth, Not Foamy", a Space.com article, it is stated: > In his general theory of relativity, Einstein described space-time as > fundamentally smooth, warping only under the strain of energy and matter. > Some quantum-theory interpretations disagree, however, viewing space-time as > being composed of a froth of minute particles that constantly pop into and > out of existence. What are these "quantum-theory interpretations" the article is talking about? Can you quickly summarize the physics behind this experiment?


title: What is the exact definition of center of gravity?

text: I've come across many definitions. Is it 1) The point from which the weight of the body acts, i.e., the point at which if the entire mass of the body is assumed to be concentrated, the **gravitational force** acting on the body remains the same 2)The point at which if the entire mass of the body is assumed to be concentrated, the **gravitational potential energy** of the body remains the same 3)The point at which if the entire mass of the body is assumed to be concentrated, the **gravitational torque** acting on the body about the origin remains the same 4) All/some of the above? I came across this while trying to answer this question.


title: Positivity for the level of Chern-Simons theory

text: In many classical papers about Chern-Simons theory (see, e.g. [1]), it is claimed that the Chern-Simons theories with gauge group $G$ are classified by an element of $k\in H^4(BG,\mathbb Z)$, the so-called "level". (I'm not assuming here that $G$ is simply connected, or even connected) Now, I have reasons to believe that this description is not quite accurate. Rather, I suspect that only the levels subject to a certain **positivity condition** yield actual quantum field theories. The exact positivity condition I have in mind can be formulated by saying that the bilinear form on $\mathfrak g$ which is the image of $k$ under the Chern-Weil homomorphism should be positivie definite . (Here, the Chern-Weil homomorphism goes from $H^*(BG)$ to $Sym(\mathfrak g^*)$) **Question:** Is there any reference on Chern-Simons theory in which the above positivity requirement on the level is noted? * * * **Reference:** [1] R. Dijkgraaf and E. Witten, Topological Gauge Theories And Group Cohomology, Commun. Math. Phys. 129, 393 (1990).


title: How is the charge concentrated at the centre of shell for calculating potential from it?

text: While calculating electric potential due to a spherical shell at a point outside the shell, why do we say that the entire charge is present at the centre of shell? Is this true even if the charge on the shell is distributed non-uniformly?


title: Splitting of degenerate energy levels with a perturbed particle in a box

text: Suppose you have a particle in a square box $[0,L]\times[0,L]$. As the box is a square, the (2,1) and (1,2) eigenfunctions will have the same energy. If you were to apply an oscillating electric field in the $x$-direction so that the Hamiltonian would become $$ H \psi=-\frac{\hbar^2}{2m}\Delta \psi -e\,\vec{x}\cdot \vec{E}(t)\psi $$ (where, again, the field is in the $x$-direction--say $\vec{E}(t) = (A cos(\omega t),0)$ ), the second energy level would split in two. I'd like to know what the two lowest energy levels of the perturbed Hamiltonian are, and, more importantly, whether the second energy level corresponds to (2,1) or (1,2).


title: Is time nothing but the speed of light (or the light itself)?

text: With regard to relativistic effects on time, all the examples and explanations revolve around light and its speed. Especially in explanatory situations that explain this using photon clock, it seems that when the clock moves near the speed of light, the photon has to take a long path to reach the other mirror. So all of these explanation are driving me to conclude or believe that the time is nothing but the speed of light (or the light itself) and all this special relativistic effects are built around the assumptions that only light (or its speed) are superior in this universe.   * What if there is another thing (probably invisible so we never found it?) which travels faster than light? Would we need to change all our laws of relativity?   * Coming back to the photon clock example, who decided that only photons traveling between two mirrors can be used to define a tick (I mean one second)? Would we replace the photon with some other thing if it is found in the future that the thing actually moves faster that light? **More questions:** Is it possible to understand this with out using any Math? It appears to me that constancy of **_C_** is a result of our Math. So universal properties(like the speed of light never changes) are decided using Math? What if an Alien Math equation differs from ours theory? I think I'm not able to understand a simple key between time and its relation between speed of light(and probably accelerating bodies with respect to other bodies). And in the problem of twin paradox, the person who travels near speed of light and comes back to Earth would actually see his friend aged more than him. so how does the speed/acceleration would slows down his aging process? I would be very thankful if you can explain this with out using any Math. Even Einstein must started with a thought process before using Math to solve/prove this problem.


title: Capacitor-like-thing for controlling temperature of fluid?

text: I want to minimise the Gibbs' phenomenon like thing i.e. sudden peaks (temperature peaks here) in a container. Assume you have a cone where you want to block the transmittance of the temperature into the cone like current by capacitors. The ideal situation would be that the material by which the cone is made contains naturally things like capacitors. However, I have not seen such a material and so small capacitors. The other design is to have the cone surface insulator while at the ends of the cone to have things like capacitors to store heat (instead of electric charge) for fluid. **Are there any things like capacitors for fluid in such a situation?**


title: What happens when a slow wave reaches lower hybrid resonance?

text: Lower hybrid resonance occurs when $n_{\perp}^2$ goes to infinity, and it occurs only for the slow wave solution, not the fast wave. Since $n_{\perp}$ is proportional to $k_{\perp}$, and $k = \frac{2 \pi}{\lambda}$, it means that the wavelength of the wave goes to zero. But what physically happens when the slow wave reaches the lower hybrid resonance? I should mention that I'm talking about in the cold plasma model, where the fast and slow wave modes have meaning.


title: Understanding and deriving ellipsoidal coordinates geometrically

text: If one were to read old texts on mathematical physics, like Maxwell, Morse & Feshbach, Hilbert and Courant, Jacobi, etc... they'd find ellipsoidal coordinates popping up, but the authors derive the coordinates using un- illuminating algebra. **Is there a quick, geometric, way of writing down the Cartesian coordinates** $x^2 = (a^2+\xi)(a^2+\eta)(a^2+\zeta)/(b^2-a^2)(c^2-a^2)$ $y^2 = (b^2+\xi)(b^2+\eta)(b^2+\zeta)/(a^2-b^2)(c^2-b^2)$ $z^2 = (c^2+\xi)(c^2+\eta)(c^2+\zeta)/(a^2-c^2)(b^2-c^2).$ **of theellipsoidal coordinate system by drawing a picture, analogous to the way one derives the Cartesian components of the spherical system by drawing a sphere and projecting:** ![enter image description here](http://i.stack.imgur.com/73S9A.jpg) **(It can be done quickly algebraically, but there's no geometry there)?** There seems to be an unintelligible, un-intuitive trigonometric method for the special case of oblate-spheroidal coordinates ![enter image description here](http://i.stack.imgur.com/joBrK.png) ![enter image description here](http://i.stack.imgur.com/29Aso.png) (Pages 139 - 145 here give another un-intuitive description with a bad picture) but this isn't the general case, thus I'm hoping for a picture that that literally copies the spherical picture above, including the details of what the $x$, $y$ & $z$ are by projections are, which these oblate pictures don't do. If you have similar quick intuition as to why the scale factors and Laplacian take the forms they do, that would be a fantastically helpful bonus, thanks very much. References:   1. Morse, Feshbach - "Methods of Mathematical Physics", P. 512   2. Featherstone, Claessens - "Closed-Form Transformation Between Geodetic and Ellipsoidal Coordinates".   3. Mathworld Confocal Ellipsoidal Coordinates Article   4. Landau, - "Electrodynamics of Continuous Media", Section 4   5. Tang, - "Mathematical Methods for Engineers and Scientists 2", P. 139 - 145


title: Boosts are non-unitary!

text: The boost transformations are not unitary unlike rotations, the boost generators are not Hermitian. When this induces transformations in the Hilbert space, will those transformation be unitary? I think no. If that is the case, what is the physical significance of such non-unitary transformation corresponding to boosts in the Hilbert space?


title: Interpretation of the Einstein-Hilbert action

text: Everyone knows the famous Einstein-Hilbert action $S_{EH} = \int d^4x \sqrt{-g} R$. I'd like to know if, after we first explicit the Ricci scalar in terms of the metric, it could be possible to interpret each one of the terms as a kinetic or potential term, and even more read what could be the free propagator of the graviton, and the graviton-graviton vertexes. If yes, what are the form of those vertexes (how many? are some of them derivatives couplings? how many gravitons per vertex?) Thanks.


title: Velocity of an object undergoing homogenous acceleration

text: So I was considering the following problem within the context of Special Relativity: Given an object O, with initial velocity v, undergoing constant acceleration at a rate of a, I want to express the velocity as a function of time. So from newtonian mechanics: velocity (S) = initial-velocity (v) + acceleration*time (a*t) However this makes no sense in context of special relativity since it suggests that given a particular acceleration and enough time it is possible to exceed the speed of light. What I realized I needed was a mapping from newtonian velocity to its special relativistic equivalent. Which I derived as follows: Kinetic Erel = $m_0*c^2/(1 - v_{\text{rel}}^2/c^2)^{1/2} - m_0*c^2$ Kinetic Enewt = $1/2 m_0 v_{\text{newt}}^2$ Where $v_\text{rel}$ = relativistic velocity, $v_{\text{newt}}$ = newtonian velocity, $m_0$ = rest mass, $c$ = speed of light. Setting both equal to each other and dividing by $m_0$ I find that: $$\frac{c^2}{\left(1 - \frac{v_{\text{rel}}^2}{c^2}\right)^{\frac{1}{2}}} - c^2 = \frac{1}{2} v_{\text{newt}}^2$$ Adding $c^2$ to both sides and raising to the power -1 I find: $$\frac{\left(1 - \frac{vrel^2}{c^2}\right)^{\frac{1}{2}}}{c^2} = \frac{1}{c^2 + 1/2 v_{\text{newt}}^2}$$ multiplying both sides by c^2, squaring both sides, subtracting 1, multiplying by -1, and taking the square root I now have: $$v_{\text{rel}} = c\cdot \left(1 - \frac{c^2}{c^2 + 1/2 v_{\text{newt}}^2}\right)^{\frac{1}{2}}$$ So given a velocity from a newtonian problem ex: 5 m/s I can convert to its energy equivalent in special relativity via this formula. Note that as newtonian velocity goes to infintiy relativistic velocity approaches c and at 0 both quantities are 0. Given this framework I know for fact from earlier that newtonian velocity is given as: v(initial) + at or using our previously defined units: v + at. Therefore the relativistic velocity can be expressed as: $$v_{\text{rel}} = c \cdot \left(1 - \frac{c^2}{c^2 + 1/2(v + at)^2}\right)^{\frac{1}{2}}$$ Is this correct?


title: Formulas for kinetic energy

text: I was reading _ABC of relativity_ from Bertrand Russell and some formulas about kinetic energy caused me some problems. Here is the extract : > The kinetic energy is, in the usual form $\frac{m}{\sqrt{-1-v^2}}$. As we've > seen before, energy can be gained or lost, so if we want to, we can add an > arbitrary quantity. Therefore, we can take the energy using > $\frac{1}{2}mv^2$. (...) But it is better to use the formula as > $\frac{m}{\sqrt{1-v^2}}$, because it is valid for the conservation law. First of all I have to admit that I am just a high school student and I've only seen the 2nd formula to calculate the kinetic energy. Supposing that the other forumulas are equal to the second (even if only in special cases) seems to be impossible, just because of the square-root wich value will always be negative in the first one and negative for $v>1$ for the third one, wich of course will give a complex number as an energy. Don't know about high-level physics but this has no meaning to me. Can soneone explain me what are these formulas and where they come from?


title: Loop-Quantum Gravity versus String Theory

text: Basically asking what were the motives behind each theory. What was it that lead physicists toward these ideas?


title: Show that charge conservation $\partial_\mu J^\mu = 0$ implies global U(1) invariance?

text: The $U(1)$ global gauge symmetry of electromagnetism implies - via Noethers theorem - that electric charge is conserved. Actually, it implies a continuity equation: $$ \psi \rightarrow e^{i\theta}\psi \quad\Rightarrow\quad \partial_\mu J^\mu = \frac{\partial}{\partial t} \rho + \vec\nabla\cdot\vec j = 0 $$ Is it possible to turn this around, and starting from $\partial_\mu J^\mu = 0$ (maybe $J^\mu$ interpreted as classical field), show that the responsible symmetry is $U(1)$? I tried to write down Noethers theorem for an appropriate Lagrangian, and compare terms, but it led to nothing. **I'd greatly appreciate a simple derivation. Literature tips and solution hints would also be helpful.** Note this is not homework, but rather a case of yak shaving. The question is a corollary to my other question "Understanding the argument that local U(1) leads to coupling of EM and matter" (renamed to make the difference clear). I'm writing a piece and trying to justify local gauge theories from simple physics (like classical electromechanics), rather than letting them fall from the sky. The part that's missing is where the $e^{i\theta}$ comes from.


**This question is Edited after recieving comments.** What is the definition of momentum when a mass distribution $\rho(r,t)$ is given?   Assuming a particle as a point mass we know the definition of momentum as $p = mv$.   I need a definition where it is assumed that point masses are not present.


title: minimizing the Higgs potential equivalent to finding the minimum?

text: When my advisor tells me to "minimize the Higgs potential", is she asking me to find the minimum (take the derivative of the potential and set it equal to zero)?


title: Properties of electric field

text: I have the following vector field: $E=E_0(-sin(\phi),cos(\phi),0)^T$ and $E_0$ is some constant. Does anybody here have an idea what this electric field does to a metallic sphere that is in it? Thereby I mean, does it create a dipole moment or does it create a current and so on?


title: Why is Physics so hard?

text: I'm in a classical mechanics class now. On our exams, most questions are quantitative. And in general, besides the theory part, all physics problems just require you to gather formulas, manipulate them a little bit by using other formulas, and come up with an appropriate value. In the question, you are given all the information you need. For our exams, they even give us a formula sheet, with every formula and equation we can possibly need. But even after all this, physics remains to be one of the most challenging subject for many students, including myself. I just dont get it. It seems like it should be so easy. Grab a formula or two, and manipulate it until all the given quantities can fit into these formulas. But yet, even after all this, its not easy to do, but it _should_ be. Sorry if I'm not making any sense. I'm a student, and most of you already know this stuff, so try thinking as a newbie to physics. Edit: Although this seems like a general study question, its actually a very specific question. I'm a really good student, but I think I've met my match with physics. It seems you just cant 'memorize' things in this class, like you can with any other class. You really just have to understand it. And lately, I've just been so discouraged to study since no matter how hard I study, I do really bad on the tricky exams. My study routine goes a little something like this:   1. Watch pre-lecture video that explains the current chapter, and take notes on that.   2. Do the weekly homework, which itself is pretty challenging.   3. To study for the exams, I do a BUNCH of practice problems from previous semester exams that they supply us with. I literally go over and do 5 old exams, one question at a time, and I understand everything perfectly-atleast I think I do, until the REAL exam comes around, and I do absolutely terrible.  I dont know what else to do at this point :(


title: A question about Virasoro algebra

text: (skip disclaimer) I have a question in Polchinski's string theory book volume 1 p54, related to the Virasoro algebra. Introducing complex coordinates $$w=\sigma^1 + i \sigma^2 $$ $$z=\exp (-i \omega) =\exp( -i \sigma^1 + \sigma^2) $$ and $$T_m=L_m - \delta_{m0} \frac{c}{24}$$ $$ \tilde{T}_m=\tilde{L}_m - \delta_{m0} \frac{\tilde{c}}{24}$$ It is said > The Hamiltonian H of time translation in the $w=\sigma^1+i\sigma^2$ frame is > $$H=\int_0^{2 \pi} \frac{ d \sigma^1}{2 \pi} T_{22} = L_0 + \tilde{L}_0 - > \frac{ c +\tilde{c}}{24} (2.6.10) $$ My questions are (sorry ask two questions in a thread, since they are closely related) (i) Does $T_{22}$ mean $T_{zz}$ or $T_{\sigma^2 \sigma^2}$? (ii) How the anti holomorphic operator $\tilde{L}_0$ are obtained in Eq. (2.6.10)?


title: Do all impacts create a wave-like disturbance in the medium through which they travel?

text: There is a scene in the first Matrix movie, where a helicopter strikes a skyscraper. The most interesting part is the 'slow-motion' bit where, as the helicopter strikes the building, a wave first propogates from the impact zone, outward along the facade, before the glass finally shatters. I was wondering whether this same thing happens across the board. Is a wave produced _every time_ two surfaces come into contact (AKA collide)? This is the clip I'm referring to: https://www.youtube.com/watch?v=2E3JDtqHaVM&t=3m56s


title: General relativity (gravitation) in time and one spatial dimension

text: I don't have any idea of general relativity but intend to learn. Is it a good idea to study general relativity in two dimensions (time and single spatial dimension) in the begining to get good idea on the subject? If it is, then please give some references for such a treatment.


title: Free neutrons pass through most materials but how do they damage materials?

text: Neutrons has no charge so they are hard to interact with materials. That's what I'm thinking. But they collide with atoms and it causes nuclear fission. And they can also be a dangerous radioactive which bring out biological damage. Why? You know that the most of the atom is empty space.


title: What is the percentage of useful energy do we get from matter-antimatter annihilation?

text: This is a theoretical question since we haven't made enough antimatter to try it in reality of course. But I am asking about the physics part in this. Also, by "useful energy" I mean the energy we are able to use either as a heating source for something like a nuclear reactor, or as energy for an explosion like nuclear explosions. If I am not mistaken, a large part of the energy we get from the annihilation is in the form of neutrinos, which we for some reason can't consider them useful energy. So now, if we subtract the energy of the neutrinos, is it safe to consider the rest as useful energy as I explained ? Please try to be as simple as possible because I don't speak English very well.


title: Path integral & Gaussian integration

text: The following is from Ref. 1. Given the (Euclidean) action for a particle ($q$) coupled to a bath of harmonic oscillators $q_\alpha$. Goal is to find an effective action for the particle, e.g integrate out the bath degrees of freedom. $$ S_{particle} = \int_0^\beta d\tau'\left(\frac{m}{2}(\partial_{\tau'}q)^2+V(q) \right)$$ $$ S_{bath} = \sum_\alpha\frac{m_\alpha}{2}\int_0^\beta d\tau'\left((\partial_{\tau'}q_{\alpha})^2+\omega_\alpha^2q_{\alpha}^2 \right)$$ and coupling $$ S_c = \sum_\alpha\int_0^\beta d\tau'\left( f_\alpha[q]q_\alpha+\frac{f_\alpha[q]^2}{2m_\alpha\omega_\alpha^2}\right) $$ $f_\alpha$ are some generic functions and the $q$ and $q_\alpha$ obey periodic boundary conditions. My first instinct is switching to frequency space: $$ S_{bath} = \sum_{\alpha,n} \frac{m_\alpha}{2}q_{\alpha,n}\underbrace{(\omega_n^2+\omega_\alpha^2)}_{A_{n,-n}}q_{\alpha,-n} $$ My problem is a) The matrix $A$ is not diagonal and b) the coupling term still reads $$ S_c = \sum_{\alpha,n} \int_0^\beta d\tau' f_\alpha[q]q_{\alpha,n}\exp(i\omega_n\tau') $$ and I don't know how to get rid of the integration. Does somebody have a hint on how to perform the gaussian integration? The effective action should read $$ S_{eff} = S_{particle}[q] + \frac{\beta}{2}\sum_{\alpha,n}\frac{\omega_n^2f_\alpha[q(\omega_n)]f_\alpha[q(-\omega_n)]}{m_\alpha\omega_\alpha^2(\omega_\alpha^2+\omega_n^2)} $$ References:   1. A. Altland and B. Simons, _Condensed Matter Field Theory,_ Second edition (2010), p.130.


title: Does the Lennard-Jones force equation give its answer in Newtons?

text: I'm trying to do the dimensional analysis of the Lennard-Jones force to work out what units are being used in my MD simulation. The lennard Jones force is given as the negative derivative of the potential: $F = 24e/r*[2*(s/r)^{12}-(s/r)^6]$ This article: http://chemwiki.ucdavis.edu/Physical_Chemistry/Quantum_Mechanics/Atomic_Theory/Intermolecular_Forces/Lennard- Jones_Potential suggests that the units of r and sigma are in angstroms and epsilon is in kj/mol. Upon performing dimensional analysis of the bracket I get 0 implying that the force is dimension-less? or 0? This cannot be correct; could someone tell me where I am going wrong? Thanks


title: Visualizing irreps of SU(N)

text: What physical system can one use as an example while considering irreps of SU(N)? What is the correspondence between the system and the irreps?


title: Qustion about the appearance of $\Delta_{FP}[A_{\mu}]$ in the path integral of gauge field

text: Why is the Faddeev-Popov quantization of a $U(1)$ gauge field not the naive solution $$\int {\cal D}A \, \, \delta\left[F(A_\mu) \right]\exp \left\\{ -\frac{i}{4}\int \mathrm{d}^4 x \, F_{\mu\nu}F^{\mu\nu}\right\\} $$ rather than $$\int {\cal D}A \, \, \delta\left[F(A_\mu) \right]\, \Delta_{FP} \exp \left\\{ -\frac{i}{4}\int \mathrm{d}^4 x \, F_{\mu\nu}F^{\mu\nu}\right\\} $$ I know how to derive the correct one, but I want to know the significance/reason of the appearance of the term $\Delta_{FP}[A_{\mu}]$.


title: How can black hole increase its mass?

text: From observer point of view an object, which falls into black hole never crosses its horizon. Then how does black hole appears and grows its mass? Or does any black hole looks (and feels by all other information sources) for us like a void sphere with all mass around its horizon?


title: What portion of impulse is received by the cylinder?

text: Condition: A cylinder with a piston are situated along Y axis. Piston receives an impulse (P1) along Y axis and transmits it to the cylinder through a gas medium with which cylinder is filled. The impulse received by the cylinder along Y axis is P2. Question: Is P1 = P2? Note: we are only considering the Y component of kinetic impulse.


title: Can a flying bullet be caught with your hands?

text: Many films show heroes catching fired bullet with their fingers. Is it practically possible? Is it possible to do like that or just in movies? Can we catch a bullet that is fired towards catcher?


title: How much computer power would we need to model every physical aspect of the universe

text: So I know there are equations that give you the total energy in the Universe(Friedman). So about if we we to simulate every single interaction from neurons to physical forces, strong and weak, gravity, electromagnetic, every single interaction between bosons, quarks, electrons, neutrons, atoms, molecules, etc. Basically everything with regard to the knowledge that we currently have. For example in the wiki page for mind uploading, there are estimates that to computationally model the neurons for one person using a 'Stochastic behavior of single molecules' algorithm will take approx, 10E43(CPU FLOPS) and 10E14(Memory) and it wont be till about the year 2111 till his can be done, now that is for the neurons of one person, multiply that by all the billions of people etc.. Then take into account everything else, absolutely everything that can be computationally modeled by an equation. Yes it would be quite some amount of processing power, is there anyway to approximate how much we would need?


title: Geometric quantization AND nuclear physics

text: Classical mechanics has a natural mathematical setting in symplectic geometry and it may be asked if the same is true for quantum mechanics. Geometric quantization is one formalization of the notion of quantization of a classical mechanical system/classical field theory to a quantum mechanical system/quantum field theory. Geometric quantization play an important role in Nuclear Collective Model. see here or restricted symplectic model of collective nuclear motion. I am looking for recent applications of geometric quantization theory in nuclear physics.


title: Radiation pressure thermodynamic paradox

text: Could the radiation pressure of a black body (theoretically) perform work on the perfectly reflecting apparatus in the figure below? Assume that the block does not hinder the passage of light through the aperture. ![enter image description here](http://i.stack.imgur.com/yLDvT.jpg) (The source of the radiation is to the left of the block.)


title: 'There shall be reheating' and other inflation-related questions

text: I've some basic questions about inflation:   1. What is reheating exactly? What triggered the reheating? Is it something that one pulls out of a hat? Through which force did the decay of the inflation occur ?    2. How can I picture the inflaton? It is a scalar field, like the Higgs, could we ever hope to observe the inflaton? Can we assign to the field excitation a mass?    3. On the blog Quantum Frontier I read the following: > "A possible solution the the super-Planckian value of the change of the > inflaton field during the inflation is that the field didn't go in a > straight line but somehow turned in circles." (Preskill's post on Quantum > Frontiers) What is meant by "going into circles" ? That it oscillated around a minimum of the potential (but isn't that the phenomenon associated to reheating ?) Or is it an identification, e.g. $\phi(x)= \phi (x+2\pi)$? (This seems even stranger :S.) Thanks in advance.


title: Question about Cars: Momentum

text: > **Car B rests at the bottom of a frictionless inclined plane. In order to > travel a height of 0.6m and maintain a speed of 2 m/s at the end of the > track it needs to start with 4 m/s.** > > a) If car B perceives the speed through a fully elastic impact from car A, > with what speed does car A have to hit car B if the mass of car A is twice > the mass of car B? > > b) What's the velocity of car A after the collision? How I tried to solve it: a) $$m_A \cdot v_{A,End} = m_B \cdot v_{B, Beginning}$$ $$2m_B \cdot v_{A,End} = m_B \cdot 4 \frac{m}{s}$$ $$v_{A,End} = 2 \frac{m}{s}$$ Therefore $v_{A}$ has to be 2 $\frac{m}{s}$ when hitting car B. b) $$v_{A, End} = \frac{(m_A-m_B) \cdot v_{A, Beginning} + 2 m_B v_{B, Beginning}}{(m_A + m_B)}$$ $$v_{A, End} = \frac{(2m_B-m_B) \cdot 2 \frac{m}{s} + 2 \cdot m_B \cdot 4 \frac{m}{s}}{3 m_B}$$ $$v_{A, End} = \frac{m_B \cdot 2 \frac{m}{s} + 2 \cdot m_B \cdot 4 \frac{m}{s}}{3 m_B}$$ Which gives me 2 $\frac{m}{s}$ to the right. **According to the solutions, however, I should get $3 \frac{m}{s}$ for a) and $1 \frac{m}{s}$ for b).**


title: Transforming a lagrangian to hamiltonian and vice versa

text: I am not refering to Legendre transform, but to something more simple. In analytical mechanics, the Lagrangian can be described as $L=T-V$, and the Hamiltonian is if the Lagrangian doesn't explicitly depend on time, then $H=T+V$. There a simple change of functions which I am contemplating here, basically if I write: $U=i \sqrt{V}$, the the Lagrangian becomes: $L=T+U^2$, and the Hamiltonian becomes $H=T-U^2$. I know it looks like meaningless, but also going from Minkowskian metric from Euclidean metric and vice versa doesn't seem like such a big deal to me, but physicist use it. So is this change of variables between Lagrangian and Hamiltonian being used in theoretical physics? Does it have any meaningful applications?


title: is lightning attracted to pointy objects?

text: Advice I was given when growing up: in a thunderstorm, stay away from tall, pointy things. It's pretty obvious why lightning would prefer to strike tall objects, but _pointy_ ones? Is there any truth to this, or is it simply that tall objects tend to be pointy? I imagine a 100ft copper spike would be _less_ receptive to strikes than a 100ft copper sphere.


title: Is it proven that all "solar" neutrinos are coming from the Sun direction?

text: In "Observation of 8B solar neutrinos in the Kamiokande-II detector" (Phys.Rev.Lett., 63, 16(1989), http://prl.aps.org/pdf/PRL/v63/i1/p16_1) the Figure 2 shows that only small percentage of registered neutrinos came from the Sun direction. The rest of registered events have random direction. Can it potentially mean that majority of the registered neutrinos came not from the Sun? If not how this background noise can be explained?


title: The process $\mu^+\mu^-\rightarrow hh$

text: I am doing some calculations in the Standard Model. I have a question that seems rather simple but makes me think a lot. I want to compute the cross section of the following process at the leading order $$ \mu^+\mu^-\rightarrow hh. $$ This can be interesting for a mu-collider. The Feynman diagram I can consider has an intermediate higgs particle decaying into a couple of higgs particles (this vertex exists in the Standard Model) while a $Z^0$ is excluded due to conservation of angular momentum. Is this analysis correct? Otherwise, what is the right Feynman diagram? Also references could be part of a good answer.


title: Degrees of freedom of the photon in $d=n$

text: It is well known that in ordinary $4$ dimension, the photon has on shell only two physical degrees of freedom. Physically this means its elicity is either $\lambda=+1$ or $\lambda=-1$ but cannot assume the value $0$, or that out of the four components of $A^{\mu}$ only two are really independent. Now, is it possible to work how many degrees of freedom does the photon have (on shell) in an arbitrary number $n$ of dimensions?


title: What does QM observable operator actually DO?

text: I know, that each observable is represented by some linear operator, while it's eigenvalues represent probability amplitudes of possible observable values. But what if I ACT with this operator on some state. Will this action have some physical sense?


title: Compute distance between planes in a crystal

text: I want to compute the distance between two (111) planes in a cubic crystalline structure, in order to do some computations involving Bragg reflection. I have a sketch of which the (111) planes are, however I don't know what these Miller-indices really tell me. Could you explain this to me? Furthermore, is there an easy way to compute the distance between such planes in general, given the distance of two adjacent atoms?


title: How does tempered glass work?

text: I've read that tempered glass work by heating and then rapidly cooling the glass so that the outer layer 'freezes' (stops expanding/contracting) but the core contracts further as it cools. I'm told that this causes compression in the outer layer and tension in the core but I can't see how this could be. Surely, if the still-contracting core were 'pulling' on it, the outer layer would be under tension. For the outer layer to be compressed, surely the core must continue to expand after the outer layer 'freezes', pushing on the outer layer. Any help would be greatly appreciated.


title: How can gravity affect light?

text: I understand that a black hole bends the fabric of space time to a point that no object can escape. I understand that light travels in a straight line along spacetime unless distorted by gravity. If spacetime is being curved by gravity then light should follow that bend in spacetime. In Newton's Law of Universal Gravitation, the mass of both objects must be entered, but photon has no mass, why should a massless photon be affected by gravity in by Newton's equations? What am I missing?


title: Two-Dimensional Tight-Binding Dispersion Relation

text: As in my last post, I am doing out a calculation in Giamarchi's Many-Body text: http://dpmc.unige.ch/gr_giamarchi/Solides/Files/many-body.pdf. This time, I am going through the derivation of the dispersion relation for the 2D tight-binding model. I know that, for the 1D model, that $$ \xi(k)=-2t\cos(ka)$$ This might be derived from the tight-binding Hamiltonian: \begin{align} H&=E_0\sum_j c_j^\dagger c_j -t\sum_{\langle j,\,\ell\rangle}c_j^\dagger c_\ell\notag \\\ &=E_0 \frac{1}{N_s}\sum_{j}\sum_{k_1,\,k_2}e^{-ik_1r_j}e^{ik_2r_j}d_{k_1}^\dagger d_{k_2}-t \frac{1}{N_s}\sum_{\langle j,\,\ell\rangle}\sum_{k_1,\,k_2}e^{-ik_1 r_j}e^{ik_2 r_\ell}d_{k_1}^\dagger d_{k_2} \notag \\\ &=E_0 \frac{1}{N_s}\sum_{j}\sum_{k_1,\,k_2}e^{ir_j(k_1-k_2)}d_{k_1}^\dagger d_{k_2}-t \frac{1}{N_s}\sum_{\langle j,\,\ell\rangle}\sum_{k_1,\,k_2}e^{-ik_1 aj}e^{ik_2 a\ell}d_{k_1}^\dagger d_{k_2} \notag \\\ &=E_0 \sum_{j}\delta_{k_1,\,k_2} d_{k_1}^\dagger d_{k_2}-t \frac{1}{N_s}\sum_{\langle j,\,\ell\rangle}\sum_{k_1,\,k_2}e^{ia(k_2\ell- k_1j)}d_{k_1}^\dagger d_{k_2} \notag \\\ &=E_0 \sum_{j}\delta_{k_1,\,k_2} d_{k_1}^\dagger d_{k_2}-t \frac{1}{N_s}\sum_{j}\sum_{k_1,\,k_2}\left(e^{ia(k_2(j+1)-k_1j)}+e^{ia(k_2j-k_1(j+1))}\right)d_{k_1}^\dagger d_{k_2} \notag \\\ &=E_0 \sum_{j}\delta_{k_1,\,k_2} d_{k_1}^\dagger d_{k_2}-t \sum_{k_1,\,k_2}\frac{1}{N_s}\sum_{j}\left(e^{ia(k_2(j+1)-k_1j)}+e^{ia(k_2j-k_1(j+1))}\right)d_{k_1}^\dagger d_{k_2} \notag \\\ &=E_0 \sum_{j}\delta_{k_1,\,k_2} d_{k_1}^\dagger d_{k_2}-t \sum_{k_1,\,k_2}\delta_{k_1,\,k_2}\left( e^{ia(k_2(j+1)-k_1j)}+e^{ia(k_2j-k_1(j+1))} \right)d_{k_1}^\dagger d_{k_2} \notag \\\ &=E_0 \sum_{j} d_{k}^\dagger d_{k}- \sum_{k}t\left( e^{ia(k(j+1)-kj)}+e^{ia(kj-k(j+1))} \right)d_{k}^\dagger d_{k} \notag \\\ &=E_0 \sum_{j} d_{k}^\dagger d_{k}- \sum_{k}t\left( e^{iak}+e^{-iak} \right)d_{k}^\dagger d_{k} \notag \\\ &=E_0 \sum_{j} d_{k}^\dagger d_{k}- \sum_{k}2t\cos(ak)d_{k}^\dagger d_{k} \notag \\\ &=E_0\sum_k d_k^\dagger d_k+\sum_k \xi(k)d_k^\dagger d_k\notag \\\ \end{align} Notice that I have used the relation $$\sum_{\langle j,\,\ell\rangle}c^\dagger_ic_j=\sum_jc^\dagger_{j+1}c_j+c^\dagger_jc_{j+1}$$ For the nearest-neighbor sum. For the 2D tight-binding model, I do the same thing, except now I generalize the summation relation above to separately sum over x and y components. This gives me, for the latter term in the 2D Hamiltonian, \begin{align} &-t\frac{1}{N_s}\sum_{\langle j,\,\ell\rangle}\sum_{k_1,\,k_2}\left(e^{ia k_2(j+1)-k_1j}+e^{ia(k_{2}j-k_1(j+1))}\right)d_{k_1}^\dagger d_{k_2}\notag \\\ &=-t_x \sum_{k_{1x},\,k_{2x}}\delta_{k_{1x},\,k_{2x}}\left( e^{ia(k_{2x}(j_x+1)-k_{1x}j_x)}+e^{ia(k_{2x}j_x-k_{1x}(j_x+1))} \right)d_{k_{1x,\,1y}}^\dagger d_{k_{2x,\,2y}}\notag \\\ &-t_y \sum_{k_{1y},\,k_{2y}}\delta_{k_{1y},\,k_{2y}}\left( e^{ia(k_{2y}(j_y+1)-k_{1y}j_y)}+e^{ia(k_{2y}j_y-k_{1y}(j_y+1))} \right)d_{k_{1x,\,1y}}^\dagger d_{k_{2x,\,2y}} \notag \\\ &=- \sum_{k_x,\,k_y}t_x\left( e^{ia(k_x(j_x+1)-k_xj)}+e^{ia(k_xj_x-k_x(j_x+1))} \right)d_{k_x,\,k_y}^\dagger d_{k_x,\,k_y} \notag \\\ &\- \sum_{k_y}t_y\left( e^{ia(k_y(j_y+1)-k_yj_y)}+e^{ia(k_yj_y-k_y(j_y+1))} \right)d_{k_x,\,k_y}^\dagger d_{k_x,\,k_y} \notag \\\ &=- \sum_{k_x}t_x\left( e^{ia(k_x)}+e^{-ia(k_x)} \right)d_{k_x,\,k_y}^\dagger d_{k_x,\,k_y}- \sum_{k_y}t_y\left( e^{ia(k_y)}+e^{-ia(k_y)} \right)d_{k_x,\,k_y}^\dagger d_{k_x,\,k_y} \notag \\\ &=- \sum_{k_x}2t_x\cos(a k_x)d_{k_x,\,k_y}^\dagger d_{k_x,\,k_y}- \sum_{k_y}2t_y\cos(a k_y)d_{k_x,\,k_y}^\dagger d_{k_x,\,k_y} \notag \\\ &=-\sum_{k_x,\,k_y}\left(2t_x\cos(ak_x)+2t_y\cos(ak_y)\right)d_{k_x,\,k_y}^\dagger d_{k_x,\,k_y}\notag \\\ &=\sum_{k_x,\,k_y}\xi(k_x,\,k_y)d_{k_x,\,k_y}^\dagger d_{k_x,\,k_y}\notag \\\ &\rightarrow \xi(k_x,\,k_y)=-2t_x\cos(k_xa)-2t_y\cos(k_ya) \end{align} I know that this is the right answer, but is my method correct? More precisely, am I correct in splitting the nearest-neighbor sum to calcualte the x and y components separately?


title: What are phase conventions in angular momentum and rotation calculations?

text: I work with complicated angular momentum calculations related to atomic physics; nevertheless, I never need to use anything related to a phase convention (apparently because it's taken care of in a consistent way everywhere with me not noticing it), but I keep reading this term within subjects related to Clebsch-Gordan coefficients, Wigner D-matrices, 3j and 6j symbols, and Racah W-Coefficients, and more. Can someone please explain what a phase convention is and how to use/utilize it? Or guide me to some good book that would explain that?


title: Does a local symmetry transformation cause a change in kinetic energy?

text: Consider a local transformation $$ \varphi_i^{\prime}= \varphi_i + \alpha(x) \delta\varphi $$ If this is a symmetry of the action, the Lagrangian is given by $$ \mathcal{L}^{\prime}=\mathcal{L}+ j^{\mu} \partial_{\mu} \alpha(x) $$ where $j^{\mu}$ is the Noether current corresponding to the field transformation with $\alpha(x)$ constant. We now have a different Lagrangian (density) than before the transformation. Since the Lagrangian is the kinetic energy minus potential energy, and $\delta \mathcal{L}$ features a derivative, it seems that such a transformation gives rise to a change in kinetic energy. Is this indeed the case? Are local symmetry transformations generally associated with an increase in kinetic energy?


title: Experiment dropping electrons into glass of protons

text: So, when you drop dye into a glass of water the dye spreads out. Now I realize you cant simply replace the water in the glass with protons (or a pure concoction of electrons) but I am wondering... What is the closest real world experiment to doing just that: mixing a drop of electrons with a glass of protons, or vice versa.


title: Acceleration of a car required to reach point B with velocity D from point A with velocity C in time T

text: How do I get the acceleration of a car required to reach point B with velocity D from point A with velocity C in time T? I understand the acceleration is not constant but how can I calculate the required acceleration as a function with time as its parameter.


title: Polarization true or false: Repeating circular filters is different from repeating linear filters?

text: Begin with standard abbreviations for polarized filters:   H = Horizontal, V= Vertical, D = Diagonal, R = Right circular.   HHH means three horizontal filters in a row.   Start with totally unpolarized light. H transmits 50% as does HH, HHH, HHHH, etc.   Simiarly, V, VV, VVV, etc. And D,DD, DDD, etc.   HD transmits 25%. As do DV, HR, and RD.   But HV transmits 0%, and HDV counterintuitively transmits 12.5%.   ( I call HDV the famous three filter trick.)   Now, if R consists of H followed by a quarter wave plate, then R transmits 50%, but RR transmits 25%, RRR transmits 12.5% RRRR transmits 6.25%, etc., unlike V,VV,VVV,VVVV etc.   Is this true? Or do RR, RRR, RRRR, etc. all transmit 50%?   Is there another form of circular polarizer that does work like linear polarizers when repeated?


title: How large would the electric field need to be to split the mj -/+1 626.6 nm line of neon?

text: I am an undergraduate considering an extension on a quantum beats experiment with neon. The premise is to excite neon atoms coherently from a metastable 'ground' state into a superposition of excited states (transistion wavelength 626.6 nm) which have been zeeman split by a magnetic field. The decay of the coherently excited states shows beats in a time domain analogue of Young's double slit experiment. I would like to do essentially the same experiment but with the Stark effect instead of the Zeeman effect but I have been unable to find any literature which I can understand that would help me get a feeling for the electric fields involved. If i can get away with something below the kV range then I can probably do it.


title: What is the nature of the correspondence between unitary operators and reversible change?

text: Why does the formalism of QM represent reversible changes (eg the time evolution operator, quantum gates, etc) with unitary operators? To put it another way, can it be shown that unitary transformations preserve entropy?


title: What is the result of a classical collision between THREE point particles at the same precise instant?

text: Classical Mechanics is said to be deterministic, a statement that nearly always is followed by that quote from Laplace, something like > If at one time, one knew the positions and velocities of all the particles > in the universe, the laws of science should enable us to calculate their > positions and velocities at any other time, past or future. I always scratch my head after hearing/reading that. If 3 or more rigid point particles happen to collide elastically _at the same precise instant_ , is it not impossible to predict the resulting trajectories? If it is possible, how?


title: Energy spectrum of a tight-binding model

text: Consider the one-dimensional tight-binding Hamiltonian $$\mathcal{H}=t\sum_m\left(a^\dagger_m a_{m+1}+a^\dagger_{m+1} a_{m}\right).$$ With the lattice constant set to 1, the energy spectrum is given by $$\epsilon(k)=2t\cos{k},$$ for $k\in[-\pi,\pi]$ in the first Brillouin zone. However, if we repeat the calculation with each unit cell containing two sites, we would obtain an energy spectrum of $$\epsilon(k)=\pm2t\cos{k}$$ instead, with the size of the first Brillouin zone reduced by half. Since the energy spectrum of a system can be determined unambiguously by experiments such as neutron scattering, it seems that we have a contradiction. What am I missing here?


title: What is the change in flux through a loop that has been rotated?

text: We have a number of field lines perpendicular to one loop of wire with an area $A = 10\textrm{ cm}^2$. The magnetic field is$B= 7.2\times10^{-5}\textrm{ T}$. You turn the loop and the flux decreases with 30%. Calculate the angle at which the turn is turned. My idea was: $\sin( \alpha ) = \dfrac{B_n}{B}$. If I continue this way I get $\alpha = 44^\circ$ My book however goes with: $B_n = B\cos \alpha$, $\cos \alpha = 0.7$. If you continue this way you get $ \alpha = 46^\circ$. Why is the latter correct? I have no idea at all how this is done. I will draw a picture to show my confusion: ![enter image description here](http://i.stack.imgur.com/lcWm2.jpg) Why is this incorrect?


title: Klein-Gordon inner product

text: Studying the scalar field and Klein-Gordon equation in quantum field theory I came across this definition for the inner product in the space of the solutions of the K.G. equation: $\langle \Phi_1 | \Phi_2 \rangle = \int \mathrm{d}\vec{x} \Phi_1 ^* i \overleftrightarrow{\partial_0}\Phi_2 = \int (\mathrm{d}\vec{x} \Phi_1 ^* i \partial_0\Phi_2 - \Phi_2 i \partial_0\Phi_1^* $ I see that this definition should be invariant under Poincaré transformations, but I couldn't prove it. Do you know some references about this? Moreover I couldn't find the reason why such a scalar product is introduced? Aren't there other possible scalar products? Why choose this one?


title: Friction + Bouncing of an Object against an Elastic Wall

text: I am trying to create the formula for applying a bouncing effect to an element which is already slowing down by friction. At the moment I have an element which moves in one dimension at speed "S" and every 20ms. The friction is applied multiplying "S" by 0.9 every frame (20ms). What happens when this element collides with an elastic rubber band? To be honest, after bouncing I would like the object to stay connected to the wall.


title: Can anyone solve this simple static equilibrium problem?

text: A bridge is 50m long, has a mass of 20,000kg, and rests on two pivots, A and B. The distance between A and the left side of the bridge is 10m. The distance between B and the right side of the bridge is 10m. A lorry drives across the bridge. i) What is the maximum possible mass of the lorry? I worked this answer out to be 30,000kg. (10 × M = 20,000 × 15) But I'm struggling with the second part of the question: ii) Derive an expression for the force acting upon A and the force acting upon B as a function of x (x = the distance of the lorry from one side of the bridge). An explanation of how you reached the answer, or some big (:P) hints on how to reach it would be much appreciated.


title: Simple: What do these ballistic impact numbers represent?

text: I bought physics for game programmers, found here, today to study ballistic impact, but I'm confused as to where a few numbers are coming from. **I'm basically looking to understand what the values represent** since the book does a bad job at telling me. I get that $E_k = \frac{1}{2}mv^2$. In talking about the F-Formula, developed by Dr. L. Thompson (neither of which I can find anything about) "The equation was presented as a ratio between the plate thickness and the diameter of the projectile." That equation: $$ \frac{t}{d} = 0.0623 \frac{(mv)^2}{d^3F^2}cos^2(\alpha)$$ I realize that the above was just an example of the equation in the book, but .0623 is pissing me off. I have no idea where they get this from. Probably some number relating to steel armor since that's what he's talking about... so I'll continue... "The coefficient F is a measure of the penetration resistance of the armor" That equation: $$ F = 1.8288(\frac{t}{d} - 0.45)(\alpha^2 + 2000) + 12192.$$ Can someone tell me what those numbers are? I don't care where they got them from, but what are they suppose to be? I need to know so I can plug my own in... Apparently it's also possibly to "rearrange the Thompson formula so that it is in the terms of the minimum projectile energy, $E_k$, necessary to penetrate the armor. $$Ek = \frac{1}{2}mv^2 = 8.025\frac{(td^2)(F^2)}{cos^2 \alpha} $$ Again... another imaginary number. What is this 8.025??? What should go there? It says the right hand side of the last equation there tells which factors influence steel armor pen, so I'm guessing the 8.025 is relating to that, but what factor? Thickness? To further confuse me, it seems like he then tries to take everything he's taught me so far and then simplify it. Assuming the impact is head on and uses the second to last equation I provided to compute the F coefficient: $$F = 1.8288(\frac{.01}{.009} - .45)(2000) + 12192 = 14610$$ So then taking that number $$\frac{1}{2}mv^2 = 8.025 * .01 * .009^2 * 14610^2 = 1387.5 J.$$ It footnotes and says a 9mm bullet has a mass of .0082 kg and a muzzle velocity of 440 m/s. The bullet only does 794 J, so the armor stops the bullet. **I just need someone to tell me what those numbers represent. Much thanks!**


title: How much light is there in space and how heavy is it?

text: Our night sky is filled with stars. On a dark night a significant fraction of the sky is light. This light, we are told, has been in transit for many millions of years. There must therefore be quite a lot of light in transit at any one time. There are many questions here where we are reminded that light has relativistic mass and therefore has a gravitational effect. How much light is there in transit at any one time? Perhaps a measure per cubic something would be interesting. And, of course, what gravitational effect would this light exert? Has it's effect been taken into account when the calculations of the amount of dark matter/energy is made?


title: Bose-Einstein condensate for general interacting systems

text: There is Bose-Einstein condensate (BEC) for non-interacting boson systems. Can we prove the existence of BEC for interacting systems?