title: superposition versus statistical mixture interpretation

text: I have an interpretation problem here. It is about the coherent state $$\left|\left.\alpha,\frac{\pi}{2\Omega}\right.\right\rangle = \frac{1}{\sqrt{2}}\left(e^{-\frac{i\pi}{4}}\mid\alpha\rangle+e^{\frac{i\pi}{4}}\mid-\alpha\rangle\right)$$ The properties of this state are now studied in a "macroscopic" situation, $|\alpha|\gg1$ and $\alpha=i\rho$ . The wavefunctions are $$\Psi_{\alpha}\left(x\right)=Ce^{-\frac{1}{2}\left(x-\alpha\sqrt{2}\right)^{2}}$$ and $$\Psi_{\alpha}\left(p\right)=C\text{´}e^{-\frac{1}{2}\left(p-i\alpha\sqrt{2}\right)^{2}}.$$ Now one calculates the propìbability distribution and finds out: $$P\left(x\right)\varpropto e^{-x^{2}}\cos^{2}\left(x\rho\sqrt{2}-\frac{\pi}{4}\right)$$ and $$P\left(p\right)\simeq e^{-\left(p-\rho\sqrt{2}\right)^{2}}+e^{-\left(p+\rho\sqrt{2}\right)^{2}}$$ If one types this into wolfram alpha for $\rho = 5$ one finds interference fringes for $P(x)$ and 2 Gaussian peaks for $P(p)$. From these facts it should be visible that whereas in $k$ space the 2 microscopic separated peaks are visible (as it is supposed to be) if $\alpha$ is pure imaginary and $\rho$ is macroscopic, the interference fringes in $x$ space are practically not visible. My question is now why these fringes are not visible? I see them for $\rho=5$?


title: Any easily available ferromagentic material with $T_c$ in room temperature?

text: I want to experiment it in my house or office. I think it would be of great fun to see the transition in real life.


title: Data/signal from a black hole to observe a singularity

text: I wonder if a situation is possible where, we measure some signal/property concerning a black hole. Supposing the measurement we make with some telescope, gets us the Fourier transform coefficients of the signal, and then by using the Fourier coefficients...we make some calculations which suggest a possibility of a singularity in the signal/parameter. I know its not possible to confirm a singularity by finite computations experimentally but I guess its innteresting to see the data suggesting something like a singularity. I hope to use this result on the measured Fourier coefficients. All I need is a data measurement of this type, if possible and appreciate some suggestions of possibility.


title: All objects radiate energy, but we cannot see all objects in the dark. Why?

text: We claim that all objects radiate energy by virtue of their temperature and yet we cannot see all objects in the dark. Why not?


title: How to solve Schrödinger's equation for this potential algebraically?

text: I want to solve Schrödinger's equation with the potential $$V(x)=\frac{1}{2}mx^2+\lambda x$$ algebraically? Is there any way to construct ladder operators that are similar to the one for the harmonic oscillator in order to get the full Hamiltonian written as $H=k_1(b^*b+k_2)$? A good reference may be suitable as well.


title: Lightest flying solid material

text: I need to use a number of lanterns, to write something with it! I would like to use a light material which will bind the lanterns together (separated by space) so it looks like dotted letter. I know that lanterns will lift up like helium balloon, but I don't know if I can think of a material which will be strong enough to not ruin/mis-shape the letter (one letter at a time), but not heavy enough to stop the lanterns from raising up or causing harm to life on earth (people mainly) when they fall back down to earth (gently). I hope I have given all the variables for the above.


title: About parametrizing quadratic fluctuations in the metric about $AdS_2 \times S^2$

text: I am referring to the contents of page 20-23 of the paper, http://arxiv.org/abs/1108.3842.pdf   * Equation 4.5 seems to suggest that one wants to restrict the metric fluctuations $h$ to a subset such that, $D^\mu h_{\mu \rho} = (1/2)D_\rho h^\mu _ \mu$. (where the $D$ is presumably calculated on the background metric and the indices of $h$ are raised and lowered w.r.t the background metric) What is the motivation for this and how does one see that this fixes all the gauge freedom?    * Can someone explain (or give reference) to how equation 4.6 and 4.7 were derived? (I tried a lot but couldn't get this simplification)    * In equation 4.6 and 4.7 is it possible to impose the condition that the background metric is whatever is standard on $AdS_2 \times S^2$? (..is there a covariant way of putting in this background?..)  [...the following questions can be asked about equation 4.22, 4.23 and 4.30 too .. ]   * In equation 4.14 it seems that one parametrized the 10 components of the 4 dimensional metric $h$ via 10 scalars $B_0,..,B_9$. How has this been done?  There $u$ seems to be a chosen scalar harmonic on $AdS_2 \times S^2$ and it seems to suggest that somehow the metric components can be expressed in terms of derivatives of such a harmonic scalar. How? Can someone help derive this $4.14$? In 4.14 the variables used $\kappa_1$ and $\kappa_2$ are defined in 4.11 and its not well motivated to me. Also these have such a singular dependence on the variable "a" - which as can be seen in 4.12 parametrizes the Abelian gauge field strength. Then is it obvious from here as to what will the parametrization be if it were just pure gravity? It looks mysterious to me that when 4.14 is substituted into 4.6 all dependency on $u$ seems to be have vanished to give a far simpler looking 4.15, How has this "magic" happened?   * Also if the base manifold had been just $AdS_2$ then what would have been the parametrization? Is that easily readable from 4.14?    * Is there some general principle at play here by which one can automatically generate such 4.14 like parametrizations for say any $AdS_n \times S^m$?


title: What exactly is implied by Einstein's insight in this scene from the NOVA series "$E=mc^2$ Einstein's Big Idea?"

text: In the documentary movie "$E=mc^2$ Einstein's Big Idea?", the narrator says _"Einstein had a monumental insight",_ when Einstein and his close friend were looking at the clock towers in the city. Einstein seems to be so excited here and thanks to his friend to go home and work as I guess but I am very curious what is so exciting here but I can't get what exactly is triggered in Einstein's mind here which is implied by the narrator. Einstein says "I've completely solved the problem." What is the implication here? What is its relation to clocks at different distances? Is it possible for you to describe it to a person who is not a physicist but knows basic undergraduate physics?


title: Origin of Overpotential in Electrochemical Cell

text: I'm well aware of the significance of overpotential and how it is defined to be the difference between the thermodynamically determined equilibrium potential and the experimental potential required to get a cell reaction going. What I don't fully understand is the origins of said overpotential. Since the exact value of overpotential will vary between cell setup even for the same reaction I assume it is mainly dependent on the specific resistance of the circuit ? Any insights would be welcome.


title: Excitations of a quantum spin-liquid

text: I have a simple (yet perhaps profound ?) question related to the physics of quantum spin liquids (QSL). These states of matter are "defined" as state that do not break any symmetry of the Hamiltonian of the system and consequently are not described by any local order parameter. I was reading through the excellent book of XG Wen on QFT of many-body systems and he I stumbled over a statement he makes in Chapter 9: > The excitations in spin liquids always carry fractional quantum number [...] Now my question is simple : **why is this statement true ?** My reasoning to understand this statement is the following : if excitations out of a QSL had only non-fractional quantum numbers, then the excitations would be described by the fluctuations of an order parameter, which is exactly what is absent in a QSL. **Is this reasoning valid ?**


title: Can a hybrid vehicle ever be more efficient than a hydrocarbon-only vehicle built with the same parts?

text: Based on the laws of thermodynamics, shouldn't it be theoretically impossible for a non plug-in hybrid vehicle to ever be more fuel-efficient than a vehicle that connects the same engine directly to the wheels without converting it into electricity first? The only energy input on a non plug-in hybrid is the gas tank, and therefore any energy leaving the vehicle in the form of torque must have originated in that tank. When the vehicle is driven, electrical energy leaves the battery and is converted into kinetic energy by the motors. Once the battery is depleted, the gasoline engine begins turning, converting chemical energy into kinetic energy, which turns the wheels directly as well as turning an alternator which converts the kinetic energy into electrical energy, charging the battery. But since all this energy is, no matter what, coming from that gas tank, it really doesn't matter what form it's in when it's turned towards moving the vehicle; it still started out as the same amount of chemical energy! Furthermore, no conversion of energy can ever be one hundred percent efficient: some is inevitably lost to heat and other forms of unusable energy. So chemical -> kinetic -> electrical -> kinetic will always be less efficient than chemical -> kinetic. The only ways I can think of a hybrid being more efficient is through energy recovery; the regenerative breaking system turns kinetic energy back into electrical instead of turning it into heat and wasting it like in normal brakes. But will the energy recovered by this method ever be greater than the energy lost by the extra conversions, as well as the increase inertia from the heavy components?


title: Distance from redshift

text: I am looking for a exact derivation of a relation between redshift $z$ and distance $d$. What I know is the definition $$z=\frac{\lambda_{\text{observed}}}{\lambda_{\text{unshifted}}}-1=\sqrt{\frac{1+\frac{v}{c}}{1-\frac{v}{c}}}-1$$ and that the Hubble constant $H$ as a function of $z$ is: $$H^2=H_0^2\left(\Omega_m\left(1+z\right)^3+\Omega_{\Lambda}\right)$$ How can I use this to derive the distance?


title: Does "dark matter" explain how I can have -1 apples?

text: If I have 3 apples and you take 4 of them, that means I have -1 apples... is that apple made of dark matter?


title: Resources for theory of distributions (Generalized functions) for physicists

text: I am looking for tutorials, articles or books containing theory of distributions in context of mathematical physics. Please suggest.


title: Does a permanent magnet loose its magnetism?

text: The question is slightly deceiving however still simple for those academics therefore, here I present my question: If I take a para-magnetic material and put it very close to a homogeneous magnetic field, the para magnetic material will start to accelerate towards the magnetic body thus it has kinetic energy (but I ensure it does not hit the magnet and stop it with my fingers) so does the magnetic field or body loose it magnetism or will the same magnetism stay? Because in Wikipedia, it says: > Energy is needed to generate a magnetic field both to work against the > electric field that a changing magnetic field creates and to change the > magnetization of any material within the magnetic field However I'm sure that para-magnetic materials are not magnetized, so where is this energy coming from? Or is my understanding wrong if so can someone link me to a good website which uses mathematics and theory to show how para magnetic material works? Thanks!


title: What is the basis of gauge theory?

text: I’m learning about gauge concepts. I’ve always had the idea that by looking at a phenomenon from different viewpoints, that symmetries could be derived – in fact, that was what an equal sign signified. In other words, the underlying phenomenon remained constant, only the viewpoints changed, and therefore could be equated. But it seems that gauge theory is the opposite – identical observable quantities are seen, although the configurations of the underlying fields change. Is this right, or am I completely off-track?


title: Strength of Magnetic Field Around a Superconductor

text: I recently learned that the strength of a Magnetic field around a conductor is proportional to the current flowing in it. So if we have a Mercury wire at absolute zero and pass a current through it (Resistance = 0) and then toss some iron filings at it, will the filings cylindrically float around the conductor (along the Magnetic field lines) due to the extreme strength of the field or just behave the way they do for an ordinary conductor (like Copper), in which case we must rest the filings on cardboard and then give it a jerk to align it along the lines.


title: Is there a difference between "average acceleration" and centripetal acceleration?

text: Question adapted from Examkrackers MCAT prep book: > A particle moves along a half circle (diameter=$10\text{ m}$) at a constant > speed of $1\text{ m/s}$. What is the average acceleration of the particle as > it moves from one side of the half circle to the other side? > > A. $0$   >  B. $0.2/\pi$   >  C. $0.4/\pi$   >  D. $1$ The book says C is correct. Acceleration is change in velocity divided by time. Initial velocity is $1\text{ m/s}$ up; final velocity is $1\text{ m/s}$ down. The change in velocity is therefore $2\text{ m/s}$. The time is found from speed equals distance divided by time. Distance is $2\pi r/2$. Thus $$a= \frac{2}{(2\pi(5)/2)/1} = \frac{2}{5\pi} = 0.4/\pi$$ I thought all of the answers were wrong because I thought they should've used the centripetal acceleration equation: $a= v^2/r$ SO my question like the title: Is there a difference between " ** _average acceleration_** " and centripetal acceleration? I searched for a couple hours and couldn't find this issue directly addressed.


title: What are the topics of string theory that are comprehensible with only a mathematical background on Manifolds and Algebraic Topology?

text: What are the topics of string theory that are comprehensible with only a mathematical background on manifolds and algebraic topology? Also, I have read only the first four chapters in Peskin & Schroder. And I have studied QM.


title: Why will the increase in the sun's luminosity accelerate with respect to time?

text: From http://en.wikipedia.org/wiki/Future_of_the_Earth#Solar_evolution > At present, nearly half the hydrogen at the core has been consumed, with the > remainder of the atoms consisting primarily of helium. As the number of > hydrogen atoms per unit mass decrease, so too does their energy output > provided through nuclear fusion. This results in a decrease in pressure > support, which causes the core to contract until the increased density and > temperature bring the core pressure in to equilibrium with the layers above. > The higher temperature causes the remaining hydrogen to undergo fusion at a > more rapid rate, thereby generating the energy needed to maintain the > equilibrium. ![enter image description here](http://i.stack.imgur.com/DSWv4.png) So mathematically speaking, what in the fusion-burning equations suggests that the Sun's luminosity will accelerate with respect to time?


title: How to properly construct the electromagnetic tensor in curved space-time? (Part II)

text: In this question, I am testing what was previously discussed. I can't seem to get my results to match D'Inverno's electromagnetic tensor for a charged point (page 239 of his book - _Introducing Einstein's Relativity_ ). **Here are D'Inverno's steps:**   * The line element in spherical coordinates is ($\eta$ and $\lambda$ are functions of $r$ only) $$ \mathrm{d}s^2 = \mathrm{e}^\eta \mathrm{d}t^2 - \mathrm{e}^\lambda \mathrm{d}r^2 - r^2 (\mathrm{d}\theta^2 + \sin^2\\!\theta\ \mathrm{d}\phi^2). $$   * He defines this covariant electromagnetic field tensor: $$ F_{\mu\nu} = E(r) \begin{pmatrix} 0 & -1 & 0 & 0 \\\ 1 & 0 & 0 & 0 \\\ 0 & 0 & 0 & 0 \\\ 0 & 0 & 0 & 0 \end{pmatrix}. $$   * He then proceeds to find the electric field, and consequently the electromagnetic field tensor by using the source-free Maxwell equations: \begin{align} \nabla_\nu F^{\mu\nu} & = 0 \\\ \partial_{[\lambda} F_{\mu\nu]} & = 0. \end{align}   * Solving the differential equation that appears from the equations above, he finds the electric field: $$ E(r) = \mathrm{e}^{(\eta+\lambda)/2} \varepsilon/r^2. $$   * He then notes that this field is that of a point charge at infinity ($\eta$ and $\lambda$ go to zero at infinity) where $\varepsilon$ is the electric charge. I managed to reproduce these steps. **Now, here are my steps, using the four-potential procedure (the line element is the same):**   * I define my contravariant four-potential (there is just the first element which is the electric potential of a point charge, just as D'Inverno found): $$A^\mu = (\varepsilon/r, 0, 0, 0). $$   * Then I lower the index of this four-potential to find the covariant one: $$A_\mu = (\mathrm{e}^\eta \varepsilon/r, 0, 0, 0). $$   * Finally I apply this equation to build the covariant electromagnetic tensor: $$ F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu. $$   * The result is $$ F_{\mu\nu} = \frac{\mathrm{e}^\eta \varepsilon}{r^2}\\! \left(r\frac{\partial\eta}{\partial r}-1\right) \begin{pmatrix} 0 & -1 & 0 & 0 \\\ 1 & 0 & 0 & 0 \\\ 0 & 0 & 0 & 0 \\\ 0 & 0 & 0 & 0 \end{pmatrix}. $$   * Where: $$\frac{\mathrm{e}^\eta \varepsilon}{r^2}\\! \left(r\frac{\partial\eta}{\partial r}-1\right) = E(r)$$ And this is different from D'Inverno's electric field. I don't know what I am doing wrong. The calculations are not difficult for this simple case. The question is, due to these calculations: Do my contravariant four-potential needs to contain my metric funcions in some way? I was assuming it is just the four-potential for a electric charge in the flat space: $$A^\mu = (\varepsilon/r, 0, 0, 0). $$ If everything is right, the wrong assumption must be here.


title: Do objects gain and lose heat at the same rate?

text: For example, if I take an item out of the refrigerator, set it on the counter for a period of time, allow it to warm up a bit (but not so long that it reaches room temperature and stabilizes), and then put it back in the fridge, will it reach its original temperature in the same amount of time that it was on the counter? For the sake of simplicity, assume that both the fridge and the air temperature in the kitchen are perfectly normalized, i.e., no fluctuations.


title: Measuring acceleration of earth due to its fall around the sun

text: Every orbiting of a satellite around a mass is nothing else but a constant fall - and therefore acceleration - towards this mass. In a way it is a "falling around" that mass. **My question**   Is it possible to measure this acceleration on earth due to its "falling around" the sun?


title: what about there is no time, only space

text: what about a theory where no time exists, only discrete space, steady states and occasion chains. what would speak against this? ;-)


title: Lattice Gas Cellular Automata - HPP model square lattice

text: In the HPP model of LGCA, a square lattice is used and there is only one collision configuration as mentioned in figure (taken from the book Lattice Gas Cellular Automata and Lattice Boltzmann models by Dieter A. Wolf-Gladrow pp. 50) My questions are: _1._ Why is this the **only** collision configuration? There can be others like when all four cells are filled ![enter image description here](http://i.stack.imgur.com/xQojR.png) _2._ After the collision, the system looks like its just the **same as previous configuration** (rotation by 90 degrees); so isn't this again a collision configuration? (if it is, the collisions will continue forever and it will just oscillate between these two states) _3._ The figure also shows the momentum vectors. In the collision configuration the momentum vectors show that the particles are moving **away** from each other instead of towards each other. So why will they collide?


title: How Did Paul Dirac Predict The Existence of Antiproton?

text: > The existence of the antiproton with -1 electric charge, opposite to the +1 > electric charge of the proton, was predicted by Paul Dirac in his 1933 Nobel > Prize lecture.   * Quotation by Wikipedia. **Question.** How did Paul Dirac predict existence of antiproton?


title: Dangerous dose of I-131?

text: I was searching a lot and could only find dosages for curing cancer and allowed emission, but no Iodine-131 dose that could be connected with increased thyroid cancer risk (like, 10mSv is the radiation dose resulting with measurable increase of cancer). The Central Laboratory of Radiological Protection publishes Iodine-131 content in air measured in $\mu$Bq/m$^3$. The number has grown by 3 orders of magnitude since the Fukushima accident, and while I strongly believe it's still good 6 orders of magnitude below dangerous levels, there's no convincing of some people without solid numbers - and finding these has proven quite hard. So how can I then get either the safe levels of I-131 in air in ($\mu$Bq/m$^3$) , or a way to (indirectly - this gonna be convoluted) convert the content in air ($\mu$Bq/m$^3$) to dose equivalent radiation ($\mu$Sv) from absorbed Iodine?


title: Total internal reflection and waveguides

text: I'm looking at a model of a planar dielectric waveguide along the lines of this picture: ![Sketch of a planar dielectric waveguide](http://i.stack.imgur.com/sTLxJ.gif) For the wave inside the $n_1$ dielectric slab to totally internally reflect $\theta_M$ needs to be smaller than $cos^{-1} (n_2 / n_1)$. In my textbook it says that the treatment of the planar dielectric waveguide is completely analogous to the case of two planar mirrors with perfect reflectivity SHORT OF ONE difference: Due to the evanescent wave that travels a very short length into the $n_2$ medium the reflected wave experiences an angle-dependent phase shift given by $tan(\phi /2) = \sqrt{\frac{sin^2(\theta_C)}{sin^2{\theta}} -1} $, where c denotes the critical incident angle and $\phi$ is the phase shift, that the reflected wave experiences. There is no explanation of this in the book and I couldn't figure out how to get to that result. Anybody care to explain? Cheers


title: Essential background for QFT study

text: The preface to Mark Srednicki's "Quantum Field Theory" says that to be prepared for the book, one must recognize and understand the following equations: $$\frac{d\sigma}{d\Omega} = |f(\theta,\phi)|^2, \qquad (1)$$ $$a^{\dagger}|n\rangle = \sqrt{n+1} \space |n+1\rangle, \qquad (2)$$ $$J_{\pm} |j,m \rangle = \sqrt{j(j+1)-m(m\pm 1)} \mid j,m \pm 1 \rangle, \qquad (3)$$ $$A(t) = e^{+iHt/\hbar}Ae^{-iHt/\hbar}, \qquad (4)$$ $$H = p\dot{q}-L, \qquad (5)$$ $$ct'=\gamma (ct-\beta x), \qquad (6)$$ $$E=(\mathbf{p}^2c^2+m^2c^4)^{1/2}, \qquad (7)$$ $$\mathbf{E} =-\mathbf{\dot{A}}/c-\mathbf{\nabla} \varphi. \qquad (8)$$ I am certainly not ready to dive into this book, so I would like some help identifying these equations and learning more about their fundamental usefulness. I don't recognize (1), but (2) looks like a quantum mechanical creation operator? I thought those were only really useful in the context of the harmonic oscillator problem, but maybe everything is just a complicated HO problem? (3) has to do with angular momentum? (4) is a plane wave solution to the Schrodinger Eqn? (5) is the classical mechanics Hamiltonian, with cannonical coordinates? (6) is the relativistic Lorentz transformation. (7) is the general form of mass energy equivalence from SR. (8) is the electric field expressed as vector and scalar potentials? Is that really the only E&M machinery required? Any insight as to why these particular expressions are relevant / important / useful to QFT is also appreciated. Also, where are the statistical mechanics ideas hiding? In the QM?


title: Unit conversion from km to mm

text: I'm a bit rusty on my unit conversion skills and I need help figuring out what I did incorrectly converting 9.06 x 10^8 km into mm. Here's what I did: 9.06 x 10^8 km x (1 meter/10^3 km) x (10^-3 km/1 m). I'm pretty sure my last step is the incorrect one, but I'm not certain.


title: Physical interpretation of the relation $\dot{x}(t) \rightarrow i \omega \tilde{x}(\omega)$

text: If $x(t)$ is some time-dependent real quantity i can interpret its Fourier Transform $\tilde{x}(t)$ as representing, in a generic sense, the frequency components of $x(t)$. What about the FT of $\dot{x}(t)$? Can this quantity be correlated in some intuitive way to the "motion" $x(t)$ itself? In other words, is there some intuitive picture explaining the fact that $\dot{x}(t) \longrightarrow i \omega \tilde{x}(\omega)$? This discussion should, of course, not be limited to functions that depend on time (only).


title: Apply Gauss' Law to find Electric Field around nonconducting plastic sheets

text: > _The question: Two very large, nonconducting plastic sheets, each 10.0 cm > thick, carry uniform charge densities $\sigma_1$,$\sigma_2$,$\sigma_3$ and > $\sigma_4$ on their surfaces, as shown in the following figure. These > surface charge densities have the values $\sigma_1= -6.30$ $\mu > \text{C}/m^2$ , $\sigma_2= 5.00$ $\mu \text{C}/m^2$, $\sigma_3= 2.10$ $\mu > \text{C}/m^2$, and $\sigma_4= 4.00$ $\mu \text{C}/m^2$. Use Gauss's law to > find the magnitude and direction of the electric field at the following > points, far from the edges of these sheets._ ![enter image description here](http://i.stack.imgur.com/cPXlE.jpg) Attempt: So, using Guass' law, I attempt to place the charge densities into a surface and use $E = \frac{\sigma}{2\epsilon_0}$, representing the sheets as infinite thin sheets. For A, I have a surface encompassing every $\sigma$. $\sigma_1$ is negative, so it goes into the surface, and the other three are inside the surface and positive, so they are going from inside to outside the surface. Hence, for A, it should be $\frac{\sigma_2 + \sigma_3 + \sigma_4 - \sigma_1}{2\epsilon_0}$. For B, I have a surface encompassing $\sigma_1, \sigma_2$ and one encompassing $\sigma_3, \sigma_4$. Well, again $\sigma_1$ goes into the surface, and the other three go out of their surfaces, so it would be the same answer as for A, but apparently this isn't right. I would use the same methodology for C, but it would be wrong. I don't understand how to approach this. Edit: I have read solutions to this elsewhere, but they aren't in-depth enough and aren't satisfactory.


title: Transformations of a left-handed gauge field

text: In a set of lectures I'm watching on Effective Field Theory the professor introduces a spurion vector field, $\ell_\mu$. He then says that we take it to transform as a "left handed gauge field" and writes, \begin{equation} \ell ^\mu \rightarrow L (x) \ell ^\mu L ^\dagger (x) + i \left(\partial _\mu L (x) \right) L ^\dagger (x) \end{equation} where $L (x)$ is the transformation matrix. I've never encountered a left-handed gauge field and I would have naively only included the first part, $L \ell L^\dagger$. How did he get this transformation law? Though the question is self contained, for more context feel free to take a look at my lecture notes under Effective Field Theory (pg 53).


title: How can I find the amplitude?

text: Prove that the motion of a mass $m$ on a linear spring with constant $k$, has the form $$y (t) = A \sin(wt+f),$$ where $t$ is the time and $A, w, f$ are constants. We know that for $t = 0, y(0)=y_{0}$ and $y'(0)=v_{0}$. If, in addition, the mass is subject to external force $F (t) = F_{0} \sin (w_{0}t)$, where $F_{0}$ the amplitude and $w_{0}$ the cyclic frequency, then calculate the amplitude of the motion. When the mass is subject to external force $F (t) = F_{0} \sin (w_{0}t)$,we get this differential equation: $$y''+w^{2}y=\frac{F_{0}}{m} \sin(w_{0}t),$$ which has the solution: $$y(t)=c_{1} \cos(wt)+c_{2} \sin(wt)+\frac{F_{0}}{m(w-w_{0}^{2})} \sin(w_{0}t),$$ where $c_{1}=y_{0} $ and $ c_{2}=\frac{v_{0}}{w}-\frac{F_{0}w_{0}}{mw(w-w_{0}^{2})}$. Right? But how can I find the amplitude of the motion?


title: Finding the height of a power-line

text: **Question:** To find the height of an overhead power line, you throw a ball straight upward. The ball passes the line on the way up after 0.80sec , and passes it again on the way down 1.5sec after it was tossed. What is the height of the power line? What is the initial speed of the ball? **What I know/What I don't get:** I am having trouble finding the height of this power line with the given information and the information I conceptually know: acceleration =-9.8 m/s^2 time of the first pass =0.8 sec time of the second pass =1.5 sec height of the power line=? Initial Velocity = ? Initial Speed = ? I suspect I must use the formula $$x= x_0 + v_0 t + \frac{1}{2}at^2$$ but I don't quite know for sure what the two different times should be defined as or what values should I plug in. I asked myself these questions in advance. Should I set the formula equal to each other with the two different times plugged into either the left or right equation? Another thought I have is, should I subtract the two different times and solve from there? Please me know if I am asking the right questions and the content of this post is grammatically correct. If not please edit this post for the sake of the quality of the content. **Thanks to all that help in advance.** **Disclaimer:** I'm not looking for the answer, I am only looking for help to set me in the right direction to solve this problem on my own to further understand how to deal with a problem like this.


title: Coin on an turntable | Exact description of forces

text: Does more static friction between coin and turntable means that more it will slip off Or Just Exactly opposite of it.When I make picture of situation in my brain I am getting first statement but I myself doesn't seem that conclusion logically correct.Please help me in solving problem of coin to be getting slipped on rotating turntable regarding static friction between surfaces and centripetal and centrifugal forces. EDIT:Sorry For asking such a poor question I got myself answer.I am writing down it below.


title: Properties of liquid and air bubbles

text: What is/are the properties of a liquid (e.g. viscosity, surface tension) which is conducive to formation of stable bubbles floating in air (not the bubble inside the liquid)? E.g., if soap dissolved water is bubbled through a straw, big-small air bubbles are formed which comes out of the liquid and keep floating in the air. Edit:   This Wikipedia page (http://en.wikipedia.org/wiki/Surface_tension#Surfactants) attributes this to low surface tension of the liquid, but I can enumerate several liquids like spirit, petrol which have quite low surface tension, still if you blow through a straw into it, you won't get bubbles.


title: The Origins of Instantons from Path Integrals

text: I know that you can come across non-perturbative effects in QFT, particular when the coupling constant lies outside the radius of convergence of the asympototic perturbation series. From the perspective of canonical quantization I understand how and why such instantons crop up. Specifically, they give (non-trivial) exact solutions of the equations of motion for the full interacting theory. These may be canonically quantized to produce a quantum instanton. My question relates to the origins of instantons from the path-integral approach. Weinberg (in Volume II of his QFT text) claims that in perturbation theory > we expand the action $S$ around spacetime independent vacuum values of the > fields, keeping the leading quadratic term in the exponential $\exp(iS)$ What does he mean by this? I've definitely never done any expansions around vacuum values to derive perturbation theory for the path integral. Doesn't one usually just split e.g. $\exp(iS_{free})\exp(iS_{int})$ and that immediately gives you the series? He goes on to say that non-perturbative effects > arise because there are stationary points of the action which are spacetime > dependent I understand this in the context of canonical quantization, but it doesn't make much sense to me with the path-integral. Unless of course one introduces the stationary phase approximation, but I don't know its relation to usual pertubation theory. Hopefully this will be clearer once the previous paragraph makes sense to me!


title: Angular Momentum of Two Non-interacting Particles

text: I'm reading a book (An Introduction to Mechanics by Kleppner) where they calculate the angular momentum $l$ of a system of two non-interacting particles, but I don't understant what are they doing. Consider two non-interacting particles with $m_1$ and $m_2$ moving toward each other with constant velocities $\mathbf{v}_{1}$ and $\mathbf{v}_{2}$. Their paths are offset by distance $b$, as shown in the sketch. ![enter image description here](http://i.stack.imgur.com/15iNW.jpg) In general the energy of a system of two particles, relative to the center of mass, can be written $$E=\frac{1}{2}\mu v^{2}+U(r), \qquad (1)$$ being $\mu$ the reduced mass. Or, using $v^2=\dot r^2+r^2 \dot \theta ^2$ and $l=\mu r^{2}\dot{\theta}$, $$E=\frac{1}{2}\mu\dot{r}^{2}+\frac{l^{2}}{2\mu r^{2}}+U(r), \qquad(2)$$ So the book calculates the angular momentum $l$ of the system using (1) and (2) and the fact that for a system of two non-interacting particles $U(r)=0$. The book says: The relative velocity is $$\mathbf{v}_{0}=\dot{\mathbf{r}}:=\dot{\mathbf{r}}_{1}-\dot{\mathbf{r}}_{2}=\mathbf{v}_{2}-\mathbf{v}_{1} $$ with $\mathbf{v}_{0}$ is constant since $\mathbf{v}_{1}$ and $\mathbf{v}_{2}$ are constant (which I think it's right). The energy of the system relative to the center of mass is (see equation 1) $$E=\frac{1}{2}\mu v_{0}^{2}$$ or (from equation 2) $$E=\frac{1}{2}\mu\dot{r}^{2}+\frac{l^{2}}{2\mu r^{2}}$$ Now here comes the argument I really don't understand: **Argument a):** When $m_1$ and $m_2$ pass each other, $r=b$ and $\dot r = 0$ (But the book said earlier that $\mathbf{v}_{0}=\dot{\mathbf{r}}$, a constant vector in time!). Hence $$\frac{l^{2}}{2\mu b^{2}}=\frac{1}{2}\mu v_{0}^{2} \qquad (3)$$ I'm confused because I could apply the same condition in an arbitrary point, say $r=2b$, then $\dot r=0$, and (3) wouldn't be valid. So, my **question** is: Is the argument a) wrong? If not, please help me to clarify the ideas behind this.


title: Explosively Pumped Flux Compression Generator = EMP strong enough to destroy my phone from a mile away?

text: Explosively Pumped Flux Compression Generator detonated a mile away or so, what damage would be done to my phone, and other electronics? I would imagine it would be similar to a CD in the microwave where every trace would act as an antenna and then a fuse? Or am I thinking about a much stronger nuclear EMP? I have read about Starfish Prime, but that was in 1962 and we didn't have the microelectronics that we do now, basically anything with transistors would be fried or what?


title: Electric & Magnetic fields - clarification

text: If (changing electric field)=(magnetic field) and (changing magnet field)=(electric field), then why is there a constant magnetic field around a current carrying wire when the electric field does not change?


title: Maximum theoretical bandwidth of fibre-optics

text: Ignoring hardware at either end and their technological limitations, what is the maximum theoretical bandwidth of fibre optic cables currently in use / being deployed in a FTTH type situations? I understand there's a limit to the number of freqencies or channels we can have in fibres, and each channel would have a theoretic max bandwidth too, I'd imagine? I'm asking particularly to find out more about the current plan for a National Broadband Network in Australia, which is supposed to roll out fibre optics to almost every premises in the country. I'm interested in finding out how much data we can fit down the fibre before we have to dig it all up and replace it with newer fibres with higher bandwidth or some new medium we haven't started talking about yet. More general answers are interesting too.


title: Ferromagnetism and electrical conductivity

text: Are ferromagnetic materials good conductors of electricity? If yes, please explain with proper examples.


title: Does a constant of motion always imply a Hamiltonian formulation?

text: If a continuous dynamical system has a constant of motion that is a function of all its variables, and is not already evidently Hamiltonian, is it always possible to use a change of variables and obtain a Hamiltonian system?


title: How can the Hall effect ever show positive charge carriers?

text: The Hall effect can be used to determine the sign of the charge carriers, as a positive particle drifting along the wire and a negative particle drifting the other direction get deflected the same (as $F = q \vec{v}\times\vec{B} = (-q) (-\vec{v})\times\vec{B}$). But I don't understand how positive charge carriers are ever possible. As I've understood, a positive hole is nothing more than the absence of an electron. As all the electrons in the valence band are still negatively charged, why would this hole behave in a magnetic field as if it were positive? Also, a hole is created if an electron is excited into the conductance band. If there is always the same numbers of holes as electrons, how can any Hall effect ever occur? Thank you very much


title: What is a virtual photon pair?

text: When describing a black hole evaporation in the hawking black body radiation it is usually said that is due to a virtual photon pair, is it this what happens? And what is virtual photon pair, does the photon has anti particle or it is its own ? I am not looking for a deep theory, just the general picture.


title: Why do birds, sitting on electric wires, not get shocked?

text: If we would touch electric wires, we would get a shock, even if we are not touching ground (so that no connection is complete form wire to ground through us). I always see that birds sit on electric wires. Why don't they get shocked?


title: Electric shock from a power line

text: A bird perches on a bare high power line and nothing happens to it. A man standing on ground touches the same line and gets a fatal shock. why?


title: How does a crow sitting on one of the electric lines attain the same potential as that of the line so as to prevent a charge flow through its body?

text: The crow doesn't get an electric shock while sitting on only one electric line because it has the same potential as that of the line. Since there is no potential difference the charge does not flow through the crow's body unless the crow touches the neutral line and creates a potential difference between one end of its body to the other. How could the crow attain this much electrical potential? The crow was at a zero potential when it came and sat on the line. Was there any charge flow to the crow's body from the line? If there was, how could the crow survive such a potential hike?


title: Why do sound waves travel at the same speed moleculewise? (Same medium)

text: I don't understand what happens in reality (outside of wave theories). If I clap my hands I invest energy in the nearby air molecules, which move and transfer their energy to nearby molecules which move...and so on. Why then, when I clap louder instead of investing more energy in the same particles making them move faster I simply involve more of them? Or am I wrong? If I'm not, then what if I clap louder in a vacuum with limited air in it?


title: Doubts concerning Wigner's classification

text: Wigner classified particles in function of the eigenvalues of $P_\mu P^\mu$ and $W_\mu W^\mu$. Then, it can be proved that for massless particles spin values can be only $\pm s_{max}$. But for a particle with mass could have intermediate spin values.   1. If we think that a massless particle is the limit where $m\rightarrow0$ (very small mass), how can we have this sudden change of the spin values (I think this is just an intuition error)?   2. What is the difference between polarization and spin (or helicity)?   3. Is it reasonable to say that massless particles have no spin but just helicity (I've read that this is because that don't have a center of mass and also because spin could point in any direction)?   4. Should we consider photons of different helicity different particles?   5. Is there any nice demonstration of why $W^2$ eigenvalues are $-m^2s(s+1)$, most books just refer to Wigner (1939). Comment: I'm not sure whether I should divide this into several questions.


title: quantization of Dirac field

text: The general solution to the Dirac equation is a sum of plane wave solutions $$ \psi(x) \sim \int d^3k \sum_r b_r(k) u_r(k)e^{-ikx} + d^\dagger_r(k) v_r(k)e^{+ikx} $$ The basis spinors $u_r$ and $v_r$ are not eigenstates of chirality (though they are eigenstates of helicity). I can construct left- and right-handed chirality eigenstates of the $\gamma_5$ operator from linear sums of left- and right-handed helicity spinors. From this, I can expand left- and right-handed chirality spinors $\psi_L$ and $\psi_R$ in plane wave solutions with creation and annihilation operators. $\psi_L$ contrains operators that annihilate a left-handed helicity particle, and create a right-handed helicity anti-particle.   1. Why doesn't $\psi_L$ create/destroy only left-handed (anti)-particles? (or does the RH anti-particle have LH helicity, i.e. anti-particle here is CP)   2. Why doesn't $\psi_L$ create/destroy particles with definite chirality (only definite helicity)?


title: Experiment to find Ammeter Resistance

text: I've been studying experimental electrodynamics and I needed to describe an experiment to find the resistance of an ammeter if I just have the ammeter together with one voltmeter and a protection resistance. My idea was the following: since the ammeter has much less resistance than the voltmeter we connect them in paralel and connect this in series with a protection resistance connected to the battery. The idea is that the current will be reduced by the protection resistance avoiding damages to the ammeter. Then the current will divide and a much greater part will flow through the ammeter. Since the ammeter and the voltmeter are in parallel the voltage seen in the voltmeter is the same on both devices, and the ammeter will show the current that flows through it so that we can use Ohm's Law to find it's resistance: $R=V/I$. If we knew then the errors of the ammeter and of the voltmeter we could propagate to find the error in the resistance. Is this correct? Is this a valid way to measure the resistance of the ammeter? Thanks very much in advance for the help.


title: Can we make a space elevator/lift?

text: Is it possible to connect a long rope (excluding the danger of it's breaking) to a geostationary space station and use it to transport food and other necessary item's to the station and then delivering it to international space station or any other place where it is required?


title: Simple uncertaintly calculation of the center coordinates of a Landau Level

text: I am reading the following review paper on the Quantum Hall Effect. I am sorry for the extremely stupid question, but I have been stuck on this very easy equation for long. In equation 2.39, the author derives the following commutation relation between the coordinates of the center of a Landau Level. $$[X,Y]=il_B^2$$ $l_B$ called as the magnetic length is $\sqrt{\frac{\hbar}{eB}}$. From this equation, the author says using the uncertainty relation(eqn 2.40) $$\Delta X \Delta Y = 2 \pi l_B^2$$ How do you get this equation. I know that the general uncertainty relation equality is $\Delta A^2 \Delta B^2=<\frac{1}{2i}[A,B]>^2$, but this is obviously not giving the required answer. $\frac{1}{2i}il_B^2=l_B^2/2=\frac{h}{4 \pi eB}$. Why is this not the correct answer? Have they used some stronger form of the uncertainty principle?


title: Movement of the particle in a magnetic field

text: This is the numerical question. > _A particle having a mass of 0.05 mg carries a charge of $5\times10^{-6}C$. > The particle is given an initial horizontal velocity of 5000 m/s, what is > the magnitude and direction of the magnetic field that will keep the > particle moving in a horizontal direction?_ Well I didn't understand this question. what I think is once the charged particle enters the magnetic field whatever magnitude it is it will travel in a circular path. I don't understand why this question is asking for the magnitude of magnetic field so that it will keep moving in horizontal direction.whatever the magnitude is,it should move in a circular path is what i have read but why doesn't it do so?


title: Does one second exist?

text: Let us assume .5 second has passed. Now similarly let .9 seconds also passed.. Then let's say that .99 second has passed...we're still not done because 1 second hasn't passed. Then follows .999 seconds, after that .9999 seconds.....goes on and on to .9999999999999999999999999999999999.....the numbers aren't gonna give up, so 1 second is never reached because no matter how many 9s you add, it's never gonna be equal to 1. How can we solve this problem?


title: Finding speed of light by $c=f\lambda$?

text: When considering EM radiation as waves it is said that it is electric and magnetic fields that oscillate with time. Therefore $f$ is not frequency of distance but of electromagnetic fields. I have also been taught to derive wavelength from the equation $c=f\lambda$. However this raises a question: if $f$ is not frequency of oscillation of distance and $\lambda$ is measure of distance isn't the equation $c=f\lambda$ bogus in the first place?


title: Faraday's law for a current loop being deformed

text: I'm working through Jackson's Classical Electrodynamics text (3rd version, chapter 5.15) about Faraday's law: Faraday's law is pretty familiar: $\int_c E \cdot dl = -\frac{d}{dt}(\int_s B \cdot n da)$ which states that the contour integral of the electric field around a bounded surface is equal to the negative time rate of change of the flux bounded by that surface. No problem. Here we have a total derivative and Jackson addresses 2 cases. "The flux though the circuit may change because (a) the flux changes with time at a point, or (b) the translation of the circuit changes the location of the boundary" In either of these cases, it is easy to show that: $\frac{d}{dt}(\int_s B \cdot n da) = \int_s(\frac{\partial B}{\partial t} \cdot n da) + \int_c(B \times v)\cdot dl$ by using $\frac{d}{dt} = \frac{\partial}{\partial t} + v \cdot \nabla$ and then applying a "product rule", $\nabla \cdot B = 0$ and stokes theorem. i.e.: $\frac{d}{dt}\int_s B \cdot n da = \frac{\partial}{\partial t}\int_s B \cdot n da + (v \cdot \nabla) \int_s B \cdot n da$ **In the case where the surface is constant** , we can exchange the order of differentiation and integration: $\frac{d}{dt}\int_s B \cdot n da = \int_s \frac{\partial}{\partial t} B \cdot n da + \int_s (v \cdot \nabla) B \cdot n da$ I still have no problem with this. Now we can work on the second term on the right (apply a product rule, drop terms with $\nabla \cdot B$ and then apply stokes theorem) to get the $\int_c(B \times v)\cdot dl$ term. which, when applied to faraday's law gives the nice result: $\int_c [ E' - (v \times B)] \cdot dl = - \int_s \frac{\partial B}{\partial t} \cdot n da$ The problem arises when you allow the shape of the bounded surface to change. (Consider a circular wire loop in a magnetic field that you then hit with a hammer causing a dent in one side). How does changing the surface influence the result that we derived above? It seems to me that this would prevent us from being able to exchange the order of differentiation and integration which seems like it leaves us in a pretty tight spot ...


title: What determine whether the dynamical equations are tensor equations or vector equations?

text: Newton's 2nd law which is central to Newtonian dynamics, is a vector equation $\sum\textbf{F}_{external}=m\textbf{a}$ Same with Maxwell's equations in the covariant form. On the other hand, general relativity is governed by the tensor equation $\displaystyle R_{\mu\nu}-\frac{1}{2}R~g_{\mu\nu} =\frac{8\pi G}{c^4} T_{\mu\nu}$ My questions are:   1. Is there a deep reason why some dynamical equations are tensor (of the 2nd rank) equations and others are vector equations?    2. Is Schrodinger equation a scalar equation?    3. Are there dynamical equations in physics which are tensor equations of rank higher than 2?    4. Is there an upper bound on the highest rank a physics tensor equation can have? from some physical argument may be?


title: How does cryptochrome produce quantum entanglement?

text: I found this video on the net, which explains how a bird called "European robin" uses a trick based on quantum entanglement to see the earth's magnetic field. I am just asking if some of you know any research paper or reference on this subject. What I want to know is how exactly the molecule Cryptochrome in the bird's eye produces quantum entanglement after being hit by a photon.


title: Find out the dimension of $\frac{a}{b}$

text: > $E=b-\frac{x^2}{at}$ [x=distance,t=time, E=energy] I have tried following but don't know whether I am correct or not $$\frac{x^2}{at}=E$$ $$\frac{L^2}{aT}=ML^2T^{-2}$$ $$a=M^{-1}T^{1}$$ Please help to solve these problem.


title: Flattening rate of a parabola

text: If you have a piece of paper that is furled and unfurled so that it's in the shape of part of a parabola, and knowing that if you leave it, it'll flatten itself after time, would it flatten faster if you leave it as a right-way-up parabola, or an upside-down parabola? More particularly, I was wondering what factors would influence the flattening rate. Note: It would be $y = kx^2$ and $y = -kx^2 + l$ on a 3-D graph, but extended perpendicularly into the $z$-axis


title: Expectation value - Zetilli vs Griffith

text: I know that an inner product between two vectors is defined like: $$\langle a | b\rangle = {a_1}^\dagger b_1+{a_2}^\dagger b_2+\dots$$ but because a transpose of a component for example $a_1$ is again only $a_1$ the above simplifies to: $$\langle a | b\rangle = \overline{a_1} b_1+\overline{a_2} b_2+\dots$$ Where $\overline{a_1}$ is a complex conjugate of $a_1$. Furthermore we can similarly define an inner product for two complex functions like this: $$\langle f | g \rangle = \int\limits_{-\infty}^\infty \overline{f} g\, dx$$ * * * In the Griffith's book _(page 96)_ there is an equation which describes expectation value and we can write this as an _inner product of a function $\Psi$ with a $\widehat{x} \Psi$_ : \begin{align*} \langle x \rangle = \int\limits_{-\infty}^{\infty}\Psi\,\,\widehat{x}\Psi\,\,dx = \int\limits_{-\infty}^{\infty} \Psi\,\,(\widehat{x}\Psi)\,\, dx \equiv \underbrace{\langle\Psi |\widehat{x} \Psi \rangle}_{\rlap{\text{expressed as an inner product}}} \end{align*} In Zettili's book _(page 173)_ the expectation value is defined like a fraction: \begin{align*} \langle \widehat{x} \rangle = \frac{\langle\Psi | \widehat{x} | \Psi \rangle}{\langle \Psi | \Psi \rangle} \end{align*} **Main question:** I know the meanning of the definition in Griffith's book but i simply have no clue what Zetilli is talking about. What does this fraction mean and how is it connected to the definition in the Griffith's book. **Sub question:** I noticed that in Zetilli's book they write expectation value like $\langle \widehat{x}\rangle$ while Griffith does it like this $\langle x \rangle$. Who is right and who is wrong? Does it matter? I think Griffith is right, but please express your oppinion.


title: Magnetic flux and transformers

text: I have two linked quesitons, 1. what is magnetic flux linkage's formal definitaiton and 2. when people say the flux flows better thru a soft iron core then the air what do they mean by flux flowing (in a transformer)???


title: Exciton energy splitting in indirect semiconductor

text: Let's take an indirect bulk semiconductor and imagine the exciton that build up at the indirect gap. Since we have a bulk semiconductor we should find three p-like orbitals: $p_x,p_y,p_z$. Would their energy levels be degenerate due to the mass anisotropy which we have at the indirect gap? My intuition screams yes. But if I try to bring it on the paper I actually find the opposite. Any suggestions, ideas or helpful tips?


title: Problem of Landau, Lifshitz - Mechanics - Integration of the Equations of motion

text: I'm studying Landau, Lifshitz - Mechanics. Could someone help me with this problem ? =) **Problem 2** _(Page 27 3rd Edition)_ Determine the period of oscillation, as a function of the energy, when a particle of mass $m$ moves in fields for which the potential energy is $(b)$ $U=-U_{0}/\cosh^{2}\alpha x$ **Solution.** The period is given by $ T=4 \sqrt{\frac{m}{2}}\int\frac{dx}{\sqrt{E+\frac{U_{0}}{\cosh^{2}\alpha x}}}$ How can I evaluate this integral? I know that the answer is $T=(\pi/\alpha)\sqrt{2m/|E|}$


title: Lamb-Chaplygin dipole

text: There is an interesting issue of hydrodynamics: whirlpools. I want to learn about the Lamb-Chaplygin dipole. Lamb-Chaplygin dipole corresponds to a steady solution of the two dimensional Euler equations. Can someone know where I can find a full development of Lamb-Chaplygin dipole?


title: Could there be massive gravitons and in our spacetime?

text: My question is: Could there in principle be sub-groups of gravitons (if they existed) that has mass (and in our spacetime dimensions) such as those affected by electric or by strong force or other forces which could be above the energy threshold of our current particle accelerators. (Perhaps they could be created, say, near pulsars due to high-energy collisions)?


title: One Loop Higgs Mass Correction

text: I am attempting to compute the one loop correction to the Higgs mass, which requires the evaluation of a scattering amplitude, namely $$\require{cancel} \mathcal{M} = (-)N_f \int \frac{\mathrm{d}^4 k}{(2\pi)^4} \mathrm{Tr} \, \left[ \left( \frac{i\lambda_f}{\sqrt{2}}\right) \frac{i}{\cancel{k}-m_f} \left( \frac{i\lambda_f}{\sqrt{2}} \right) \frac{i}{\cancel{k} + \cancel{p}-m_f}\right]$$ which corresponds to the Feynman diagram: ![enter image description here](http://i.stack.imgur.com/pX1RH.png) After combining constants, and rationalizing the denominators, I obtain, $$-\frac{N_f \lambda_f^2}{2} \int \frac{\mathrm{d}^4 k}{(2\pi)^4} \frac{\mathrm{Tr}\left[ \cancel{k}\cancel{k} + \cancel{k}\cancel{p} +2m_f \cancel{k} + m_f \cancel{p} + m_f^2\right]}{\left(k^2-m_f^2\right)\left((k+p)^2 -m_f^2 \right)}$$ Computing traces, via the relation $\mathrm{Tr}[\cancel{a}\cancel{b}] = 4(a\cdot b)$ yields, $$-2N_f \lambda_f^2 \int \frac{\mathrm{d}^4 k}{(2\pi)^4} \frac{k^2 +k\cdot p + m_f^2}{\left(k^2-m_f^2\right)\left((k+p)^2 -m_f^2 \right)}$$ At this point, I employed dimensional regularization, followed by Feynman reparametrization to combine the denominators, and then completed the square, yielding $$-\frac{2^{2-d}\pi^{-d/2}}{\Gamma (d/2)}N_f \lambda_f^2 \int_{0}^1 \mathrm{d}x \int_0^\infty \mathrm{d}k \frac{k^{d-1}(k^2 +kp + m_f^2)}{\left[ \left(k-(x-1)p\right)^2 +p^2(x-x^2 -1)\right]^2}$$ **Additional Calculations (Edit)** I attempted to further simplify the integrand using a substitution in only the first integral, namely $\ell = k-(1-x)p$ which implies $\mathrm{d}\ell = \mathrm{d}k$, yielding (after several manipulations), $$-\frac{2^{2-d}\pi^{-d/2}}{\Gamma(d/2)}N_f \lambda_f^2 \int_0^1 \mathrm{d}x \, \int_{(x-1)p}^{\infty} \mathrm{d}\ell \frac{(\ell + (1-x)p)^{d-1}[(\ell + \frac{1}{2}p(3-2x))^2 - \frac{1}{4}p^2 + m_f^2]}{[\ell^2 + p^2(x-x^2-1)]^2}$$ **N.B.** Mathematica evaluated the original integral over $k$, and outputted a combination of the first Appell hypergeometric series, which possess the integral representation, $$F_1(a,b_1,b_2,c;x,y) = \frac{\Gamma(c)}{\Gamma(a)\Gamma(c-a)} \int_0^1 \mathrm{d}t \, t^{a-1}(1-t)^{c-a-1}(1-xt)^{-b_1}(1-yt)^{-b_2}$$ with $\Re c >\Re a >0$, which has a structure similar to the beta function. If I can express the loop integral in a similar form, I may be able to express it in terms of these functions. At the end of the calculation, I will take $d \to 4-\epsilon$ to obtain poles in $\epsilon$, using the usual expansion $$\Gamma(x) = \frac{1}{x} -\gamma + \mathcal{O}(x)$$ and a similar expansion should the final answer indeed contain the Appell hypergeometric series. **Passarino-Veltmann Reduction (Edit):** Based on my understanding of Veltmann-Passarino reduction, it is not applicable as the numerator contains an arbitrary power of loop momentum. I could plug in $d=4$, and impose a high momentum cut off, but this has already been done in many texts. As aforementioned, I would like a dimensionally regularized amplitude. I am stuck at this point, can anyone give some details as to how to proceed? In addition, I have a query regarding the hierarchy problem. If using a simple cut-off regularization, the one loop correction can be shown to be quadratically divergent. But why is this an issue that needs to be remedied, by for example, the minimally supersymmetric standard model? Can't the divergence be eliminated by a regular renormalization procedure?


title: What is the term for heat generation by a flowing fluid?

text: I would like to know more about the heat distribution over time in a flowing liquid. To this end, I consider the Navier-Stokes equation (where the coefficients may be temperature dependent) and the heat equation $$ \rho c_p \left(\frac{dT}{dt} + u \cdot \nabla T\right) = \sigma + S(u) $$ where $T$ is the temperature and $u$ the flow velocity. The term $\sigma$ may be some artificial heat source. The term $S(u)$ is now of interest to me: It corresponds to the heat generated by the friction of particles (I think), and I believe it must be proportional to the dynamic viscosity of the medium. I don't recall the specifics though. Can anyone help out on what $S(u)$ should be?


title: Finding a discontinuity and jump (time independent Schrodinger eqn)

text: A particle of mass $m$ is confined to move in a one-dimensional and Dirac delta-function attractive potential $$V(x)=-\frac{\hbar}{m}\alpha\delta(x)\text{ $,\alpha>0$ }.$$ Show that the function $\psi(x)$ is discontinuous at $x=0$, and determine its jump at the point. Showing its discontinuous at zero is easy. It follow from the following: $\frac{-\hbar^2}{2m}\int_{-\epsilon}^{\epsilon}\frac{d^2\psi}{dx^2}+\frac{-\hbar^2\alpha}{m}\int_{-\epsilon}^{\epsilon}\delta(x)\psi(x)=0\implies \lim_{\epsilon\to 0} \frac{d\psi}{dx}\bigg|_-+\frac{d\psi}{dx}\bigg|_+=2\alpha\psi(0)$ As for the jump, that's where I'm stuck. I assumed it was at $y=\alpha$. But I believe I'm wrong.


title: Heat preserving performance of container relative to content

text: This question has been addressed in the case of a thermos bottle: Performance of a thermos bottle relative to contents I am asking the question again without the hypothesis that it is a thermos bottle. Given a container with a warm liquid inside ("warm" meaning warmer than the medium surrounding the container), will it cool faster, slower or equally, when it is half-full than when it is full. To simplify the analysis, it is assumed that the opening and cap of the container have the same properties with respect to heat as the rest of the container, so that they may be ignored in the analysis.


title: Direction of tension?

text: ![enter image description here](http://i.stack.imgur.com/Iy0YY.jpg) If you draw the free body diagram of the frame above, what direction would the tension force acting on the frame be - to the right or down? Because the rope it horizontal at some points but vertical at others. THanks


title: Dimensions in lagrangian potential

text: According to Mankowski flat space dimensions We can write, $$L= \int \text{dt} \text d^d{x} \left[ \frac{1}{2} \dot\phi^2 - \frac{1}{2} \left(\frac{\partial \phi}{\partial r} \right)^2 -V(\phi)\right] \tag{1}$$ Where $V$ can be written as $$V = \frac{1}{8} \phi^2 (\phi-2)^2$$ But the author wrote in his article in the equation (1) including dimensions. $$V= \frac{1}{2} m^2 \phi^2+ \frac{\lambda_3}{3!}m^\frac{5-d}{2} \phi^3+ \frac{\lambda_4}{4!}m^{3-d} \phi^4$$ My question is how the dimensions incorporated with the potential?


title: Are there functions of the metric that are scalars under spatial diffs up to total derivatives?

text: Let $g_{\mu\nu}$ be a metric on a manifold with a time direction $x^0$ singled out. I'm wondering if there exists a function $F(g_{\mu\nu},\partial_\rho g_{\mu\nu},\ldots)$ that transforms under spatial diffeomorphisms as \begin{align*} F(g_{\mu\nu}'(x'),\partial_\rho g_{\mu\nu}'(x'),\ldots)=F(g_{\mu\nu}(x),\partial_\rho g_{\mu\nu}(x),\ldots)+ \nabla_\mu \Lambda^{\mu}(g_{\mu\nu},\partial_\rho g_{\mu\nu},\ldots,x'), \end{align*} where $\Lambda$ is some functional of the metric and $x'$. This would imply that the integral \begin{align*} \int d^dx\, \sqrt{-g}F \end{align*} is invariant under spatial diffs. Any ideas?


title: How to test whether galaxies are moving away from each other in a static background or whether space is being created beween them?

text: Observations show that galaxies are moving away from one another on the macroscopic scale. Now, scientists interpret this by saying this happens not because galaxies are really moving away from each other in a static background but because more and more space is created between galaxies. Now how scientists can distinguish between the 2 scenarios? what made them believe in the 2nd and discard the 1st?


title: Why can't we dope the intrinsic semiconductors with sixth or second group elements?

text: Suppose we want to make an n-type silicon semiconductor why do we use arsenic or phosporous as an impurity? Why cant we use selinium or terellium? Infact my doubt is when we use sixth group element it can donate two free electrons so that we can have higher carrier concentration even with lesser doping concentration . Why it is not done so?


title: How does Bloch's theorem generalize to a finite sized crystal?

text: I would be fine with a one dimensional lattice for the purpose of answering this question. I am trying to figure out what more general theorem (if any) gives Bloch's theorem as the number of unit cells-->infinity.


title: When is the principle of virtual work valid?

text: The principle of virtual work says that forces of constraint don't do net work under virtual displacements that are consistent with constraints. Goldstein says something I don't understand. He says that if sliding friction forces are present then the principle of virtual work fails. But then he proceeds to say that this doesn't really matter because friction is a macroscopic phenomena. The only way I can interpret this is for the friction forces to be a constraint force. But I thought constraint forces were pretty much always forces whose net effect is known but their exact force exerted is difficult to know. For friction we know its force exerted, so why would you treat it as a constraining force? I also don't understand why friction being a macroscopic phenomena means it doesn't matter for this. Is it because we considering a system of particles?


title: Does alternative reality theoretically prove?

text: Does Hugh Evert theory means Alternative-Reality (or in general, Parallel- Worlds) and Many-Worlds-Interpretation theoretically prove? Or are scientists have any progress to prove it?


title: Noether Theorem and Energy conservation in classical mechanics

text: I have a problem deriving the conversation of energy from time translation invariance. The invariance of the Lagrangian under infinitesimal time displacements $t \rightarrow t' = t + \epsilon$ can be written as \begin{equation} \delta L = L\left( q(t),\frac{dq(t)}{dt},t\right) - L\left( q(t+ \epsilon),\frac{dq(t+ \epsilon)}{dt},t+\epsilon \right) = 0. \end{equation} Using Taylor series, keeping only first order terms this gives \begin{equation}\rightarrow \delta L =- \frac{\partial L }{\partial q} \frac{\partial q}{\partial t} \epsilon- \frac{\partial L }{\partial \dot{q}} \frac{\partial \dot{q}}{\partial t} \epsilon - \frac{\partial L }{\partial t} \epsilon = 0. \end{equation} Using the Euler Lagrange equation and assuming that the Lagrangian does not depend explicitly on time we get \begin{equation}\rightarrow \delta L =- \frac{d}{d t} \left(\frac{\partial L(q,\dot{q},t)}{\partial \dot{q}} \right) \frac{\partial q}{\partial t} \epsilon- \frac{\partial L }{\partial \dot{q}} \frac{\partial \dot{q}}{\partial t} \epsilon =0. \end{equation} Which we can write as \begin{equation}\rightarrow \delta L = - \frac{d}{d t} \left(\frac{\partial L(q,\dot{q},t)}{\partial \dot{q}} \frac{\partial q}{\partial t} \right) \epsilon = - \frac{d}{d t} \left(p \frac{\partial q}{\partial t} \right) \epsilon = 0. \end{equation} But unfortunatly this is not the Hamiltonian. This computation should yield \begin{equation} \rightarrow \frac{d}{dt} \left( p \dot{q} - L \right) = 0. \end{equation} But i can't find no reason why and how the the extra $-L$ should emerge. I can see that this term can be written at the place where it is written because we have $\delta L = - \frac{d L}{dt } \epsilon$ and therefore \begin{equation} \rightarrow \delta L = - \frac{d}{d t} \left(\frac{\partial L(q,\dot{q},t)}{\partial \dot{q}} \frac{\partial q}{\partial t} \right) \epsilon = - \frac{d L}{dt } \epsilon. \end{equation} And then the desired equation would only say $0-0=0$. Any idea where i did a mistake would be much appreciated.


title: Hawking radiation and quark confinement

text: The simple picture of Hawking radiation is that a pair-antiparticle pair is produced near the event horizon, then one falls into the black hole while the other escapes. Suppose the particles are quarks-antiquarks, which experience quark confinement thanks to QCD. If one of them is swallowed by the black hole, its partner is left alone. Eventually the quark gains enough energy and turns into a hadronic jet. Is my line of thinking correct? If yes, is it (or QCD in general) taken into account when calculating Hawking radiation?


title: Relationship between measures of light (candela and lux)

text: I am trying to understand the relationship between two measures of light: lux and candela. It is probably easy for those with longer experience, but here it goes: I have learnt (here for instance) that lux equals candela per square meter. But this feels too "simplified" - doesn't the distance from the light source to the area have any impact? A thought experiment: ![enter image description here](http://i.stack.imgur.com/5EbtL.png) Imagine that we have a hollow cylinder (yellow in the figure above) with a length of **L** (say 1 meter) and a inner diameter of **D** , say 0.1 meter. In the end of the cylinder there is a lux-meter, measuring the amount of light reaching the end of the cylinder (as measured in lux). If we would place the tube in front of a light source with a light intensity of 1 candela, what would the lux-meter show? If we set the inner diameter **D** to 0.1 meter, the end of the cylinder will have an area of ~0.0078 square meters. But is the lux then simply 1 candela divided with this area? Doesn't the distance from the light source to the area have any impact?


title: Soliton mechanics

text: How much of classical mechanics can be modelled with solitons? What I am aware of is that single solitons behave in a way like free particles: they move along as stable entities with constant velocity. But:   * Can solitons interact with an external field?   * Can solitons collide?   * Can solitons interact with each other by general (central) forces? That means: Are there PDEs with solutions that can be interpreted as   * a particle in an external field    * two colliding particles   * two generically interacting particles  obeying appropriate laws of classical mechanics? How would the mass/momentum of a soliton come into play? Pointers to references would be very welcome!


title: Home experiments using wireless LAN or mobile phones about electromagnetism?

text: Are there any nice experiments using wireless LAN access points or routers or mobile phones to demonstrate physical features of electromagnetic fields, especially em-waves? More precisely I am looking for experiments which use one of the devices above as the main tool and some materials that can be purchased at a reasonable cost from, say, a hardware store or electronic supply store only. I do not impose any restrictions on the cognitive level the experiments aim at; undergraduate level is fine as well as graduate level in em-theory. Please use separate answers for different experiments.


title: Why rendezvous attempt failed on Gemini 4?

text: It is said in Wikipedia, that > On the first orbit, McDivitt attempted to rendezvous with the spent Titan > second stage. This was unsuccessful for a number of reasons: > > NASA engineers had not yet worked out the idiosyncrasies of orbital > mechanics involved in rendezvous,[citation needed] which are counter- > intuitive. Simply thrusting the spacecraft toward the target changed its > orbital altitude and velocity relative to the target. When McDivitt tried > this, he found himself moving away and downward, as the retrograde thrust > lowered his orbit, increasing his speed. I don't understand this. Is there any explanation, given in local reference frame? Referring "orbital altitude" referrers global reference frame and is OK. But any set can be possibly regarded in any reference frame. Local reference frame should be inertial with tidal, Coriolis and other forces. How to describe the situation with this? **UPDATE** Suppose we are inside giant closed spacecraft like Rama or O'Neill cylinder. This spacecraft is on Earth orbit, but we are inside and don't know this. We feel weightlessness. Now, If Rama is rotating, we can feel some non-inertial effects like centrifugal or Coriolis forces. But suppose Rama is not rotating. Then, the only strange thing we will feel is Earth tidal force. The tidal force mean that all objects will periodically distracted along axis, directed to (invisible) Earth. So, you want to say, that McDivitt failed due to tidal forces? Hard to believe.


title: Calculate pressure of air trapped underwater

text: Seal top end of straw, partially submerge straw in water, how do I calculate the air pressure inside the straw? Bonus points for compensating for any secondary factors like temp, humidity or flex of the straw. Application: measuring water level in 5 gallon aquarium to accuracy of about 5mm.


title: What's the first physics textbook for undergraduate self-learner？

text: I am kind of studying physics on my own now. I choose University Physics (13th Edition) for myself,is it fine? I am also studying Calculus using Thomas' textbook. http://www.amazon.com/University-Physics-13th-Edition- Young/dp/0321696891/ref=sr_1_1?ie=UTF8&qid=1352260382&sr=8-1&keywords=university+physics+13th+edition


title: What process could produce two neutral pions only?

text: Any examples? $$? \rightarrow \pi^0 \pi^0$$ If such a process exist, could there be nonzero total orbital angular momentum in the final states of the two neutral pions? But then how to understand the interaction between the two neutral pions that make them rotate with each other to contribute the non-zero L?


title: Question about conservation of energy and force

text: Feynman mentions in his book, _The Feynman Lectures on physics_ > Let us now illustrate the energy principle with a more complicated problem, > the screw jack shown in Fig. 4-5. A handle 20 inches long is used to turn > the screw, which has 10 threads to the inch. We would like to know how much > force would be needed at the handle to lift one ton (2000 pounds). If we > want to lift the ton one inch, say, then we must turn the handle around ten > times. When it goes around once it goes approximately 126 inches. The handle > must thus travel 1260 inches, and if we used various pulleys, etc., we would > be lifting our one ton with an unknown smaller weight W applied to the end > of the handle. So we find out that W is about 1.6 pounds. This is a result > of the conservation of energy. I don't know where does this 126 inches would come from? And what does '10 threads to the inch' means?


title: Online physics diagram tool

text: What is a suitable tool for drawing physics diagrams online? I would love to know if there is a program like the excellent schematic drawing tool that is integrated into the electrical engineering stack exchange, only for physics. This would really help in answering questions more graphically and in an intuitive fashion, without having to rely on formulas. I am aware of the offline possibilities, but for most problems, I just need to draw simple shapes such as lines, circles, boxes, and squiggly lines. I work from several machines, so an online tool would be a lot more flexible.


title: Is there a derivation of all possible Bravais lattices in 2D and 3D?

text: Do any of you know a (possibly elegant) derivation of all possible Bravais Lattice and the Crystallographic Group of Lattices in 2 and 3 dimensions? Group theory /abstract nonsense approaches are welcome.


title: Two Photon Absorption dependence on Semiconductor Band Gap

text: It's well known that Two Photon Absorption coefficient is scaling with $E_g^{-3}$. Does anyone know what is the physical reason for this scaling? What is the physical theory behind this mathematics?


title: Why don't we normally see the Higgs boson?

text: I am a physics student and my dad just asked me about the Higgs Boson. I've told him the little I know, that the Higgs field is a field that is supposed to give mass to elementary particles, and that finding the boson was crucial to see if this mechanism actually did exist. After telling that, a question has come into my mind. I have heard that it is very difficult to create the circumstances where we would have Higgs Bosons. I have been told also that this boson is the carrier of the field responsible of giving mass to other particles. So the question is quite natural to me, how come something which "is not there" in "normal" conditions happens to do its work? I mean, the carrier of the electromagnetic force are the photons and we have photons in "normal conditions"(by this I have meant not the conditions we have inside the LCH, for example) so it is natural to see their effects, but how is this possible with the Higgs?


title: On the Axial Anomaly

text: I know that if we start with a massive theory, the chiral states $L$ and $R$ remain coupled to each other in the massless limit. Because a charged Dirac particle of a given helicity can make a transition into a virtual state of the opposite helicity by emitting a real photon (this is the physical origin of the anomaly). Also, the masslessness of a Dirac field theory is expressed by the invariance under a chiral transformation $$\psi(x)\rightarrow\mathrm{e}^{-i\omega\gamma_5}\psi(x).$$ From Noether theorem, the chiral invarince gives a conserved axial-vector current $$j_{5}^{\mu}=\bar{\psi}\gamma^{\mu}\gamma_{5}\psi,$$ and from the E.o.M. for Heisenberg fields we find $$\partial_{\mu}j_{5}^{\mu}=2mj_{5},$$ where $j_5$ is the chiral density and $m$ is the mass. Here is what I don't understand: I expect that in the limit $m\rightarrow 0$ this should be true $\partial_{\mu}j_{5}^{\mu}\rightarrow 0$. But its not. All the text books give $$\partial_{\mu}j_{5}^{\mu}=2mj_{5}+\frac{\alpha_0}{2\pi}\bar{F}^{\mu\nu}F_{\mu\nu}.$$ I can't figure out this result. I don't know how to derive it or what its physical interpretation is.


title: How did the scientific community receive this measurement of speed of gravity

text: This link and this one concern a recent measurement, by Chinese scientists, of the speed of gravity using Earth tides. They find it is consistent with a speed equal with the speed of light, with an error of about 5%. Is it real? Was it done before another way, with better precision? Is it something particularly important for the validation of gravitation theories?


title: Direction of Current, according to Lenz's law, from induced EMF

text: My textbook (NCERT Physics Class 12th) has the following figure (Pg 212). ![enter image description here](http://i.stack.imgur.com/PDv9V.jpg) The Arm PQ is moved to the left side thus decreasing the area of the rectangular loop. This results in a decrease in magnetic flux in the Downward (inside the page) direction. Hence, according to lenz's law, the induced current should be in Clockwise direction but the book says the opposite. Am I missing something? or is the book wrong?


title: Kahn-Penrose metric

text: the following line element defines the Kahn-Penrose metric with coordinates $(u,v,x,y)$ and constraints $u \geq 0$, $v < 0$ $$ds^2=-2dudv+(1-u)^2dx^2+(1+u)^2dy^2$$ If we restrict ourselves to the following convention, in which the Minkowski metric $\eta$ is mostly plus $(-,+++)$ and $ds^2>0$ for space like intervals. The reasoning goes as follows: in order to determine the nature of a coordinate, we produce some small but otherwise arbitrary variation of that coordinate while keeping the rest constant (which can be done if we assign algebraically the valor $0$ to the $x^μ$ not being varied. Then the sign of $ds^2$ will tell the space-time-null character of the coordinate in question. My question is if the following statements are correct:   1. $x$ and $y$ are space like coordinates.   2. $v$ is a space like coordinate   3. $u$ is a time like coordinate


title: Concerning Mie scattering and phase function dependance

text: In general, scattering intensity is written as $I = I_{0} \frac{\pi a^2 Q_{sca} P(\theta)}{r^2 4 \pi}$ It is also written as $I = I_{0} \frac{i_1 + i_2}{2 k^2 r^2}$ I am sort of confused, given the expressions of $i_1$ and $i_2$, is the phase function $P(\theta)$ ONLY dependant on the scattering angle $\theta$? It is my understanding that the scattering efficiency $Q_{sca}$ incorporates depends on index of refraction, sphere size and wavelength, but the phase function only depends on the scattering angle. Is my understanding correct?


title: How do Aharony et. al conclude that all scalar fields in the supergravity multiplet are periodic?

text: This question is for anyone who has read/gone through the paper above or knows anything about AdS/CFT. The paper can be found here. On page 46, eq. (2.33), the author finds solutions to the scalar field equation $(\Delta- m^2 ) \phi = 0$ in $AdS_{p+2}$ background as $\phi = e^{i \omega \tau} G(\theta) Y_l(\Omega_p)$, with the functions $G$ and $Y$ defined just below this equation. On page 51, in equation (2.54) the author shows that when $p=3$, $\omega$ is quantized in multiples of $\frac{1}{R}$, i.e. $\omega R \in {\mathbb Z}$. He then goes on to state the following > This means that all the scalar fields in the supergravity multiplet are > periodic in $\tau$ with the period $2\pi$, ... I do not see how he comes to that conclusion. Am I missing something?


title: How Does force relate to velocity

text: I had originally asked this question on math overflow and it was suggested that I ask it here. So I know that a force will change the magnitude of velocity if it is at an angle other that 90 degrees. If the force is perpendicular to the velocity it will cause the path of the object to curve in a circular motion and the magnitude will remain constant. My problem is that that is all I know. I'm working on a program to model a charged particle in a magnetic field. I'm using the equation for the Lorentz force to calculate the force on the particle but I don't know where to go from there. I have searched online with no success. All I can find is a general description of how a force will change velocity. My question is how does a force vector affect a velocity vector mathematically and how can I use a known force and velocity to determine a new velocity?


title: Theoretical power limit of nuclear bomb

text: Is there any limits of power or power to mass (or any other) limits for nuclear bombs? I found this wiki article: Nuclear weapon yield: Yield limits. Is the information provided is correct? If yes, from where this limits come from?


title: Applications of Quantum Walks

text: Can someone explain to me what real world applications could potentially benefit from the study of quantum random walks? I have researched a fair amount on how quantum walks operate and their properties but I would like to know the end goal of studying them at a deep level? Is it simply for the sake of it in terms of mathematics or physics, or are there actual real-world applications that can benefit from this research?


title: Frequency Response RLC circuit - Current against Frequency graph - Symmetry?

text: I understand that in a Frequency Response experiment dealing with an RLC circuit, the graph of Current against Frequency is supposed to be symmetrical about the resonant frequency theoretically. However, experimentally it is not the case. Could anyone explain why this happens?


title: Spontaneous conversion of heat into work at negative temperatures

text: Consider a heavy macroscopic object moving in a gas. Friction causes its kinetic energy to be converted into heat. Thermodynamically, there is (effectively) no entropy associated with the kinetic energy because all the energy is concentrated in a single degree of freedom. Therfore, if an amount $J$ of energy is converted from kinetic energy into heat, the total entropy change is $J/T$, so we can see that this is a spontaneous process. But now consider an object moving relative to a gas with negative temperature. Such a thing has been created in the laboratory, so this is not just idle theoretical speculation. If an amount $J$ of kinetic energy gets converted into heat, the total entropy change is still $J/T$, but now this is negative. This seems to mean that the opposite process - conversion of heat into kinetic energy, accelerating the object - would be spontaneous. This generalises to all other processes that convert work into heat. For example, performing the Joule heating experiment with a negative-temperature gas should cause the paddle to turn, and negative-temperature gas flowing through a pipe should experience an accelerating force rather than a decelerating one. Just as superfluids have zero viscosity, it seems that negative-temperature fluids must have negative viscosity. I realise that this does not lead to perpetual motion. As heat is converted into work the inverse temperature ($1/T$) will increase until it reaches zero. But what does look odd is that in some ways the arrow of time appears to be reversed. I realise that experimentally we're very far from being able to produce the macroscopic quantities of negative-temperature fluids that would be required in order to observe these things. But is it possible in principle? And if it is, would we actually see the phenomena I described, or is there some fundamental reason why they wouldn't happen after all? And has such a connection between negative temperatures and the arrow of time been discussed or debated in the literature?


title: Work of an adiabatic/ compresion: sign?

text: As far as I am concerned the adiabatic work it is given by: $$ W_{adi}=\frac{P_{f}V_{f}-P_{i}V_{i}}{\gamma -1}$$ While I have seen in other examples that it is used with $1-\gamma$ in the denominator changing the sign of the result. Is there something that I should have in mind when using it or am I missing something? Thank you!


title: Invariants of Connection Form

text: I am somewhat going out "on a limb" here, since I am much more grounded in the physics side of things than I am in mathematics. Nonetheless, I am wondering if someone is able to comment on the following : given a connection form defined on a differential manifold, one can derive a tensorial invariant of this, the curvature form. If there is a soldering form present, one can also define torsion as an invariant of the connection form. My question now is - are there any invariants other than curvature and torsion that can be derived from a connection on a differentiable manifold? My interest here is again motivated by physics, specifically gravitation theory; here using the torsion-free Levi-Civita connection will give us GR, and allowing torsion into the picture will give us Einstein-Cartan gravity. To me, this raises the question if - mathematically speaking - there are other options based on other connection invariants that I am not myself aware of. I apologise if the question turns out to be meaningless - I am not a mathematician, and was not immediately able to find conclusive answers to this through a Google search.


title: Proofs on Quantum Inequalities?

text: Are there any experimental proofs witch support The Quantum inequalities and The Quantum interest conjecture?


title: Basic introductory quantum mechanics question

text: > Given that $\psi = \frac{1}{\sqrt{32 \pi > a_{0}^{3}}}(2-\frac{r}{a_0})exp(\frac{-r}{2a_0})$ is a wavefunction of the > hydrogen atom, write down the probability density for r and calculate the > ratio for $a_0$ and $3a_0$. I'm just wondering if someone could point out where i went wrong, the book says that the answer is $\frac{e^2}{9}$ but I get $\frac{e^2}{3}$, here's what I did: The probability density for $r$ is $\langle r\rangle = \psi^{*}r\psi$ and since $\psi$ is real this becomes $$\langle r\rangle = \frac{1}{32\pi a_{0}^{3}}r(2-\frac{r}{a_0})^2exp(\frac{-r}{a_0})$$ Therefore: $$\langle a_0\rangle = \frac{1}{32\pi a_{0}^{2}}e^{-1}$$ $$\langle3a_0\rangle = \frac{3}{32\pi a_{0}^{2}}e^{-3}$$ So the ratio is $$\frac{\langle a_0\rangle}{\langle3a_0\rangle} = \frac{e^2}{3}$$ Can anyone see where I went wrong, or is the answer in the book wrong? I feel like my answer makes more sense, the particle is more likely to be found at $\langle a_0\rangle$ than it is at $\langle 3a_0\rangle$ so the ratio should be greater than one.


title: Chiral Spin Liquid(CSL), Chern number, and the ground state degeneracy(GSD)

text: Consider a 2D _gapped_ CSL with a _nonzero_ Chern number $m$, then is the GSD of the system on a torus directly related to the Chern number $m$? For example, see this article, in the last paragraph on page 7, the authors give the **4-fold GSD** from the Chern number $m=\pm2$ for a CSL. I can not understand the explanation, can anybody present an intuitive illustration or a simple mathematical proof ? I will be very appreciated, thank you very much.


title: Real World measurements related to phonon dispersion

text: I hope no one objects to the style of this question. Let me explain my motives in asking the question first.   Condensed matter physics is one of the most beautiful subjects around. But those who are trained as Theoretical Condensed Matter physicists tend to be abstracted away from the realities of labs and neutron diffraction measurements.   Which is why I wanted to do a section in my thesis where the theoretical output, in my case the Phononic Dispersion of a particular Material, is numerically obtained.   I would like to ask what other properties / measurables depend on the phonon dispersion relation (doesn't have to be exclusively dependent).   The ones I found out out myself are -   1. Thermal Conductivity/Specific Heat   2. Neutron Diffraction  You don't have to expand on the answers if you don't want to .. I will EDIT the list and put in descriptions myself.   The problem is that there cannot be ONE right answer to this. So I wouldn't be able to mark a certain answer as correct. All the same it is a valuable question to ask .. and I will curate the answers myself. _bows_


title: Is There a List of Physics Formulas?

text: I am looking for a list of _simple_ physics formulas \- like $F=ma$. Is there any website/book that is a list of such equations?


title: Connecting global conservation laws to local features of a function

text: I have a heuristic question about using global constraints of a problem to make local estimates of geometric features of a curve, such as its local slope. Consider a suitably well behaved function on $\mathbb{R}$, denoted $f(x,t)$, where $t\in \mathbb{R}^+$ denotes time. Now, $f(x,t)$ is oscillatory, and due to symmetries of the problem, I know that the number of waves (in this context I believe it's the same as saying number of zero crossings) of the function remains constant with respect to time. Quantitatively this is the statement that $$\frac{d N(t)}{dt} \equiv \frac{d}{dt} \int_{-\infty}^{\infty} |f(x,t)|^2 \ dx =0$$ Next, by exploiting other conservation laws, I know that the variance of the signal goes to zero, i.e. $$V(t) = \int_{-\infty}^{\infty} (x-\bar{x})^2 |f(x,t)|^2 \ dx \to 0 $$ in $t$ finite, where $$\bar{x} = \int_{-\infty}^{\infty} x |f(x,t)|^2\ dx$$ Now, I want to make a quantitative statement about the local magnitude of the slope of the function $f'(x,t)$, where primes $'$ denote differentiation with respect to $x$. My question is, how can I use the information I have about $V(t)$, as well as $N(t)$, to make a statement about the local slope of $f(x,t)$? Naively, it seems easy to see that as $V(t)$ goes to 0, the slope must get large, as the number of oscillations stays fixed, but exactly how to quantify this beyond a simple dimensional argument is beyond me. Any insights will be appreciated.


title: Does evaporation decrease entropy?

text: I have a container of water in thermal equilibrium (there are no temperature fluctuations inside the container). Some molecules will evaporate out of the container of water thus decreasing its temperature. This is clearly a spontaneous process. But doesn't this mean that the entropy of the system is decreasing? The molecules that evaporated were clearly in thermal equilibrium with the other water molecules before. But now they gained some energy and the other water molecules lost energy. How can this be?


title: Basic concepts in quantum theory to start learning semiconductor physics

text: What basic concepts in quantum theory one must known to start learning semiconductor physics?


title: How many years of expansion occurred during inflation?

text: During inflation, expansion happens at a very rapid rate. How many years of expansion did it fast forward through? Meaning, if it weren't for inflation, how many years would it take for the universe to expand at its basic rate from the big bang to the size of the universe after inflation?


title: Covariant derivative

text: I would very much appreciate some help in The following: What is 2nd order covariant derivative $$\nabla_i\nabla_jf(r)$$ in terms of $r,\theta, g(r)$ and partial derivative, given that the metric takes the form $$ds^2=dr^2+g(r)d\theta^2$$ and $f$ is a scalar function of $r$? For Cartesians, I know that the covariant derivatives reduce to partial derivatives. However, since this is in polar coordinates...


title: The angle to shoot moving object

text: I am obliged to count very simple problem (at least it seemed that it is simple, I hope it isn't to simple for this site). So i got observer who is standing $H$ below the object. The object is fired horizontaly with the speed $V_{0}$. Assuming that the observer has the reaction time that equals $T$, on what angle he should fire his bullet to destroy the object. We also know that bullet is fired with the speed $V_{1}$. So I drew picture (I changed the orientation because the computation seems simplier): ![enter image description here](http://i.stack.imgur.com/O8m7I.jpg) And I write equations for bullet: $x(t)=V_{1}\cos(\alpha)\cdot t$ $y(t)=V_1\sin(\alpha)\cdot t + H - \frac{gt^2}{2}$ And for object: $x_{1}(t)=V_{0}(t+T)$ $y_{2}(t)=\frac{g(t+T)^2}{2}$ Now I do $x(t)=x_{1}(t)$ : and i get: $\cos (\alpha)=\frac{V_0\cdot(t+T)}{V_{1}t}$ From the second equation $y_{1}(x)=y(x)$ I count $t$ (the time they will meet) and move it to the equation $\cos (\alpha)=\frac{V_0\cdot(t+T)}{V_{1}t}$. I got very complicated and hard equations that I am not able to solve, so I am asking is it simplier way to this kind of task? Or did I miss something that makes me equations very complicated?


title: How does one determine ladder operators systematically?

text: In textbooks, the ladder operators are always defined," and shown to 'raise' the state of a system, but they are never actually derived. Does one find them simply by trial and error? Or is there a more systematic method/approach to obtain them?


title: I don't understand the relationship between electron indistinguishability and the Pauli exclusion principle

text: I know I'm wrong but this is my line of thought: If electrons are indistinguishable, then why do we have an exclusion principle? If we have two electrons in an s orbital, the Pauli exclusion principle says that they can't have the same set of quantum numbers, but then what does that say about electrons being indistinguishable? So we have these two electrons that are supposed to be indistinguishable, but then we say, no they can't have the same set of quantum numbers, isn't this making them distinguishable then?


title: Adiabatic expansion of steam through a valve

text: I'm working on a homework problem, and I have a suspicion the textbook is trying to trick me. The question is: "Steam at 20 bar and 300 C is to be continuously expanded to 1 bar. Compute the entropy generated and the work obtained per kilogram of steam if this expansion is done by passing the steam through an adiabatic expansion valve." My energy and entropy balances give: $$\dot{W} = \dot{M}(\hat{H}_2 - \hat{H}_1)$$ $$\dot{S}_{gen} = \dot{M}(\hat{S}_2 - \hat{S}_1)$$ where $$\dot{W} = \dot{W}_s - P\frac{dV}{dt}$$ ... but, there is no $\dot{W}_s$ since it's an expansion valve, right? If it was a turbine (which the next question asks), then there would be shaft work, but unless I'm mistaken, there is no shaft work for an expansion valve, right?


title: First and second fundamental forms

text: I'm writing notes about the 3+1 formalism in general relativity, for myself. Inevitably I came across the notions of first and second fundamental forms. Mathematically, it is clear how these objects are defined: ($M$ is a 4-dim manifold with metric $g$, $\Sigma$ a hypersurface of $M$) The _first fundamental form_ is the induced metric on $\Sigma$, also given by the pullback of the spacetime metric $g$. The _second fundamental form_ $K: T_{p}(\Sigma)\times T_{p}(\Sigma)\rightarrow \mathbb{R}$ is given through the Weingarten map $\chi$, i.e. $(u,v)\mapsto -u\cdot \chi(v)$. Now, I'm having difficulties with underlying physcial intuition for these two objects (especially the second fundamental form). Is there a way for a physicist to "visualise" them in a way? What kind of objects are these forms exactly?


title: Volume of gas at which relative fluctuation of gas density occurs

text: I have the following question: > In what volume of gas occurs 10 % relative fluctuation of gas density under > pressure of $10^5\text{ Pa}$ and temperature of $293.15\text{ K}$? I don't understand the topic but I assume this is about ideal gas. Can you please explain this to someone who has just high school physics knowledge?


title: Potential of a spherical conductor next to a point charge

text: Griffiths, when he's talking about the method of images, shows how to calculate the potential distribution when given a point charge that is outside of a spherical conductor which is held at a fixed potential. If you are given a similar scenario, where a point is outside a spherical conductor but the conductor is not held at a fixed potential, how would you calculate the potential of the conductor? It's basically solving laplace's equation when the boundary conditions aren't known. But intuitively, it seems like a solution would exist, so I'm guessing the boundary condition can be extracted somehow.


title: Are $A$ and $J$ in Black's equation (for electromigration reliability of wires/interconnections) independent?

text: Black's equation is an empirical model to estimate the MTTF (mean time to failure) of a wire, taking electromigration into consideration. Since then, the formula has been getting popularity in semiconductor industry. \begin{eqnarray} MTTF = AJ^{-n}e^{\frac{E_{a}}{kT}} \end{eqnarray} where $A$ is a constant, $J$ is the current density ($A/cm^2$), $n$ is a model parameter, $E_{a}$ is the activation energy ($eV$), $k$ is the Boltzmann constant, and $T$ is the temperature ($K$). Since the current density $J$ is obtained as the ratio of current and cross- sectional area of the wire, and since $A$ is a constant which is obtained based on metal line properties and is cross section area dependent, are these two parameters ($A$ and $J$) correlated factors or not really? I am confused because, based on their definitions, they sound to be dependent, however, this dependence is never considered in relevant researches. Essentially, people try to widen the wires to decrease the current density but at the same time they use a same constant $A$ for different wire widths. Especially, when $n$ is considered as $2$, their correlation will be very significant. P.S. I also did a quick research and noticed that the values for $n$ and essentially $A$ are found by fitting the model to experimental data.


title: CMB anisotropies and tightly coupled limit

text: Sorry if this is a technical question. I am studying the origin of CMB anisotropies and the **tightly coupled limit** of the Boltzmann equations. We have a fluid composed of ionized electrons and photons. In this limit, I read that, in the Legendre expansion of $\Theta = \delta T / T$, in the Fourier space : $$\Theta_{\ell}(k,t) = \int_{-1}^{1} \dfrac{d\mu}{2}P_{\ell} (\mu)\ \Theta(k,\mu,t) $$ only the monopole and dipole are non-zero. Why ? I understand why very small scales are damped (essentially because the wavelength is much smaller than the damping scale that relates to mean free path, density and so on). But large scales are beyond horizon, so should have very little anisotropy ? And scales between sub-horizon scales and small damped scales should exist. As I see the things, $\Theta_{\ell}$ should be non zero only at medium scales, but here it's only $\ell =0,1$ that are non zero. I think that I am confusing the wavelenght of perturbation with the order of multipole in Legendre expansion. For me, large scale = $\Theta_1 \to \Theta_{\ell\sim 10}$, etc, a bit like in the power spectrum.


title: Nonablianity of Wilczek Zee Potential

text: Consider a hamiltonian with degeneracies of energy in parameter space $(R)$.Now the Geometric phase(Wilczek Zee Potential) will acquire a non abelian nature. To prove the non abelian nature of Wilczek Zee potential one generally take an unitary transformation of wave function $\psi(t)$ to $\Omega(t)\psi(t)$ and show the potential has a non abelian gauge transformation,Is there any other way to show the non abelian nature of the Wilczek Zee potential? ($\psi$ is the wave function of the hamiltonian $\psi_{a}=U_{ab}(t)\eta_{b}(t)$ where $\eta(t)$ is the basis vector at each point of degeneracy. )