title: Depletion region

text: In a semiconductor diode. There is a depletion region formed, it is formed when electrons from n type side migrate to p type side, now which electrons transfer? Valence band ones present at the boundary or the conduction band ones? If the conduction band ones only migrate, then are there enough present on the boundary such that they can create a field at the boundary that they can stop this diffusion process? By boundary i mean the border of n type and p type material.


title: Energy from the Feynman-Kikuchi Partition Function

text: The Feynman-Kikuchi Partition function is given as $$Z_{FK}=K_\beta \int dx \eta(x) \exp \left(-\frac{x}{\beta}\right) $$ where $K_\beta$ is a normalization constant and $$\eta(x)=\frac{1}{N!}\sum_P \delta\left(x-(R_0-PR_0)^2/4\lambda^* \right)$$ where $R_0$ is a set of particle coordinates for a perfect cubic lattice, the sum over P means over all perturbations, and $\lambda^*$ is defined as $$\lambda^*=\frac{\hbar}{2m^*}$$ where $m^*$ is the reduced mass. My questions is this: in Ceperley's notes "Path Integrals in the theory of condensed helium", he states that, if we differentiate the partition function above with respect to $\beta$, we obtain the energy, in the form $$\epsilon=\epsilon_0+\frac{3T}{2}-\frac{T^2 \langle x\rangle}{N}$$ I can't see how he would get that--I know that I have to calculate $$\epsilon=-\frac{1}{Z}\frac{\partial Z}{\partial \beta}$$ But wouldn't that just give me $$\epsilon=-\frac{1}{\beta^2}\frac{\sum_P (R-PR)^2\exp\left(-\frac{1}{\beta}(R-PR)^2 \right)}{\sum_P \exp\left(-\frac{1}{\beta}(R-PR)^2 \right)}$$ I just don't see how that would give me Ceperley's results. Thank you in advance.


title: Is the Planck length Lorentz invariant?

text: The planck length is defined as $l_P = \sqrt{\frac{\hbar G}{c^3}}$. So it is a combination of the constants $c, h, G$ which I believe are all Lorentz invariants. So I think the Planck length should also be a Lorentz invariant! But there seem to be some confusion about that, see e.g. the following paper Magueijo 2001: Lorentz invariance with an invariant energy scale: > The combination of gravity $G$, the quantum $h$ and relativity $c$ gives > rise to the Planck length, $l_p$ or its inverse, the Planck energy $E_p$ . > These scales mark thresholds beyond which the old description of spacetime > breaks down and qualitatively new phenomena are expected to appear. ... This > gives rise immediately to a simple question: in whose reference frame are > $l_P$ and $E_P$ the thresholds for new phenomena? But if $l_P$ is a Lorentz invariant their is no question about that. $l_P$ is the same in all reference frames! Another confusing issue is that the Planck mass (from which the Planck length is derived) is often derived by setting equal the Compton length $\lambda_C = \frac{h}{m_0 c}$ ( a Lorentz invariant 4-length) and the Schwarzschild length $r_{s} = \frac{2Gm}{c^2}$ (which I believe is not a Lorentz invariant, since in the derivation of the Schwarzschild metric it is assumed to be a 3-length, measuring a space distance). But since Compton wavelength and Schwarzschild radius are not lengths of the same kind I think such a derivation is not valid. So my question is: Is the Planck length a Lorentz invariant and if so, how to derive it then without using the Compton wave length and the Schwarzschild radius ?


title: Pauli principle for particles very far apart from each other

text: Can two electrons be in the _same state,_ when they belong to two different atoms, which are "far enough" (whatever that means) apart from each other? With "same state" I mean that (as far as specifiable) the states are really identical, except for the position of the electrons. More specifically:   I am still not clear of how the separation of particles is taken into account for the Pauli exclusion principle. E.g. in a crystal the electrons seem close enough for the exclusion principle to become meaningful, for particles "a universe apart" from each other, it seems that it is kind of redundant (is there specific maths to that?), but what about everything in between? (This is closely related to the talk Brian Cox gave, see this question: physics.stackexchange.com/q/18527/16689 )


title: In which direction does the electron move during an electronic transition?

text: Let's imagine a molecule and put it in a 3D space. Let's also imagine an electronic transition for this molecule. I know how an electronic transition works, how to value if it is possible (using the rules derived from the symmetry of the molecule and so on). My question is: how do I understand how the electron moves during the transition WITHOUT using the symmetry of the molecule? Can I "see" in which direction the electron moves in the 3D space without using the symmetry? (I hope I've been clear enough to understand the question! Sorry for my mistakes)


title: Why black body radiation is all over the frequency range

text: I was studying black body radiation and how quantization of energy solves the problem of ultraviolet catastrophe. But I have a very fundamental doubt. A black body can be assumed as a cavity with a small hole with radiation leaking out of it. As the temperature of the black body is increased we can assume the the charge particles, electrons, on the metal surface will behave as harmonic oscillators and the energy of the harmonic oscillation will be equal to the energy density of the radiation inside the cavity at thermal equilibrium. My doubt is that at thermal equilibrium, the cavity(assuming it to be uniform) will have a uniform temperature. Since the oscillations of the charged particles is due to the thermal agitation, how can the charge particles radiate all over the frequency range? The temperature is uniform over the cavity, wont they all be experiencing same thermal agitation and oscillate at same frequency


title: Is every ordinary coloured transparent paper an optical filter?

text: Can any red coloured transparent paper be used as an optical filter that only allows red colours through it?


title: Action-reaction principle in special relativity

text: Consider a relativistic train of weight $W_t$ in its rest frame. Force transformation equations say that in the planet's frame its weight is reduced to $W_p=\frac{W}{\gamma}$. My question is: what is the reaction force experienced by the ground? We cannot say that in every frame the reaction force is equal to the weight of the train, because these forces are applied to objects of different velocities, and so transform differently under Lorentz transformation. In this particular case - if in planet's frame $R_p=W_p=\frac{W}{\gamma}$, then Lorentz transformation of reaction force to train's frame gives $R_t=\frac{R}{\gamma}=\frac{W}{\gamma^2}$, which is different to $W_t$


title: Electron flow and electric potential

text: In the physics tutorial topic on current electricty it defines the positive terminal as the high potential terminal and the negative as the low potential. When talking about positive test charges this analogy makes sense, the charge moves from the +ve to the -ve where it loses potential till it has to be "topped" up by the cell. What confuses me when they begin talking electrons. Using this analogy would mean that the electrons move from the -ve to the +ve but this means it would work again the electric field so the potential energy increases till it reaches +ve terminal. My question is how does this analogy,if it does, change to accommodate electrons? Does the electric field reverse so the -ve terminal is the high potential and the +ve the low potential?


title: How to explain polarization in Zeeman effect

text: In the Zeeman effect, when we observe along the $B$ field, the polarization of light should be circular polarized. It can be understood by the conservation of angular momentum. $\Delta m=0$ can not be observed and $\Delta m=\pm1$ can be observed and thus circular polarization. However, I can't understand what we observe perpendicular to $B$ field. It should be linear polarized light. So the state of the photon should be something like $|\mathrm{linear~polarized}\rangle=\frac{1}{\sqrt{2}}|x_+\rangle+\frac{1}{\sqrt{2}}|x_-\rangle$($|x_\pm\rangle$ are circular polarized states in $x$ direction. We assume $B$ is parallel to $z$ axis) ,depends on $\Delta m$. How could it be like this? Why the light is linear polarized and can't be circular polarized? I am really confused.


title: Spreading UV light? Fresnel lenses?

text: I'm making a little UV exposure box and I'm looking for a way to evenly spread out the UV light. I've used Fresnel lenses to do just this in the past, but that was for visible light. But from what I've read, not all things work at all wavelengths and some materials will block UV. So my first question, would a Fresnel lens still work and evenly spread out UV light? What material should I look for? The lens I've seen are PVC and from what I've seen elsewhere, PVC has good UV resistance, but I don't know if that means it will just hold up to UV or if it will block UV all together. Any tips or suggestions would be great thank you


title: How do cathode ray tubes (CRT) synthesize colours?

text: I just read about how LCD displays were based only on black&white contrasts, and only colour filters on each subpixel generated the colour of a pixel. However, how does that happen for a CRT? Is it the same process? If not, during the screen sweep, is there one beam modulated differently during each third of a pixel period that react in some way with the screen? Or three identical beams targeted at each subpixel coated with a different reactor? Or even three different beams targeted at each subpixel made of the same reactor? Bonus question: how does the screen glow upon photons hitting its surface?


title: Physical meaning of isobaric line in T-v diagram

text: ![enter image description here](http://i.stack.imgur.com/fqcvC.png) Hi, I was just reviewing some slides from my thermodynamics class, and in one of the slides the professor asks what is the "physical meaning of the slopes" of the isobaric line in a T-v diagram. Can anyone help me understand this? I know what I am looking at is the constant pressure line from a T-v diagram, and that the slope of the line would be $m = \frac{P_2 - P_1}{v_2 - v_1} = \frac{\Delta P}{\Delta v}$ I haven't been able to find anything on the internet or in my book that can tell me what the physical meaning is. Can anyone help me with some insight?? Thx!


title: Angular momentum, what is it, is it conserved, and how do we know?

text: Firstly, most definitions of angular momentum assume a point about which you define angular momentum. I realize that you can consider the angular momentum about any point, and have many angular momentums, and that if they are all useful and all conserved then maybe it doesn't matter if there are lots of different ones. But it does make me wonder how fundamental it is, or even whether it always makes sense, or whether the term has the same meaning in every branch of physics. I'm used to momentum being frame dependent, but conserved. Its conservation just looks like three of the free components of $0 = T^{\mu \nu}_{,\nu}$ from Special Relativity. I'm also familiar with the fact that in Newtonian Mechanics, forces that act in the direction between two particles conserve angular momentum about any point. I'm also familiar with the fact that for any variable that the Lagrangian doesn't depend on, there is a corresponding momentum conjugate to that variable that is conserved (so if a variable deserves the name angular, then the corresponding conjugate momentum is both conserved and equally deserves to be called an angular momentum). I've also seen conservation of angular momentum from Special Relativity in the form $0 = (x^{\alpha} T^{\mu \nu} - x^{\mu} T^{\alpha \nu})_{,\nu},$ where this is the divergence of a rank 3 tensor. Though to be fair, this is usually derived from the symmetry of the stress-energy tensor $T^{\mu \nu}$ which itself is usually derived from considerations that angular momentum doesn't go crazy at fine scales. So it does make me wonder which ideas come first, conservation of angular momentum or the symmetry of the stress-energy tensor. Also, you can have angular momentum in electromagnetic fields (or in electromagnetic potentials, which depend on gauge this is even more than just depending on a point). An example is the angular momentum set up by having a linearly charged wire inside a solenoid inside a charged cylindrical shell. But if you turn the current off from the solenoid, it begins to rotate so it seems like mechanical angular momentum is only conserved when coupled to field (or potential) angular momentum. And in quantum theories you have intrinsic spin angular momentum as well as the regular angular momentum, and the regular one again depends on your origin or potentially also on the electromagnetic potentials (but here I'm less bothered by the apparent gauge dependence). With so many potentially (pardon the pun) different momentums, and so many derivations of special cases where some of them are conserved. The question is whether they are the same thing in these very different theories, and whether they are always conserved or just some of them or just in some situations, and in particular how this relates to General Relativity. For instance, a charged rotating black hole is often parameterized by a total angular momentum per unit mass. So it's common in General Relativity to talk about a total angular momentum. But does a total angular momentum make sense in General Relativity and what if anything from other theories does it correspond to (field angular momentum, potential angular momentum, mechanical angular momentum, intrinsic spin, combination of all or just some of those)? I'm not even aware of a general differential conservation law for angular momentum in General Relativity. And knowing whether it is stored in the fields or in mechanical angular momentums seems relevant to knowing how much is inside a region if we want to get a total for a region. As an example of how common the assumption of conservation of angular momentum is, a quick google search came up with http://csep10.phys.utk.edu/astr161/lect/solarsys/angmom.html as the second link and it says "all experimental evidence indicates that angular momentum is rigorously conserved in our Universe." The course is an Astronomy course, not a Cosmology or General Relativity course, but I'd still expect instructors and students in an Astronomy course to have some awareness of General Relativity and so wonder which things continue to hold in General Relativity. The first link from a quick google search is to the wikipedia article on angular momentum, which cites the Newtonian mechanical versions, the gauge dependent electromagnetic case, the quantum case, and no mention of General Relativity except to say that angular momentum is not, in general, conserved in General Relativity, but it is missing a citation for that. It also just assumes at the beginning that no net external torque means conservation of angular momentum, even though the proofs used requires that forces act along the line between the particles even for Newtonian Mechanics and the article itself admits that angular momentum (however it is defined) is not consider, in general, in General Relativity. I want to point out that one reason I express skepticism about the definition of angular momentum in General Relativity is because it does not appear to be conserved. If you had a quantity with a differential conservation law, then there are grounds to say we have something legitimate. If you define something and then see that it isn't conserved, then it begs the question about whether a different definition would have been conserved. And it seems strange to have a non-conserved thing exist in a theory that is supposed to be an experimentally verified limit of a quantum theory where thing you are supposed to correspond to is generally assumed to be conserved. In particular if every star is made up of ions and fields and you expect angular momentum to be conserved by them, then a definition that is supposed to correspond to angular momentum but isn't a conserved thing should be suspect. But the tensors I see for angular momentum in Special Relativity seem to generalize mechanical momentum, so don't include any intrinsic spin. So when a parameter refers to the total angular momentum of a system in General Relativity, it's not clear what is does or is supposed to correspond to. And I don't know if we are happy with our definitions as they are (say from a Lagrangian formulation of General Relativity), or whether this is an area of active research that goes by a name I'm not familiar with. In case this helps anyone answer, there are classical electromagnetic forces besides the Lorentz force law for objects with intrinsic magnetic moments, e.g. $\vec{F}=\vec{\nabla}(\vec{\mu}\cdot \vec{B})$. So the usual angular momentum tensors might simply be ignoring the stress-energy associated with this kind of work (which unfortunately is non conservative, so I'm not immediately aware of how to get a stress-energy tensor from those kinds of forces). I also am not looking just for a pointer to Einstein-Cartan theory (or any other alternative to General Relativity) without a discussion about whether current General Relativity is considered to be confused, wrong, or deficient or whether these are just two theories that appear to be equally valid and functional at this juncture in our studies of Physics. P.S. I wanted my fifth tag to be correspondence-principle but can't make a new tag.


title: How does bulk-boundary correspondence works for various cases of time-invariant system?

text: I was pondering this question after I read this review: > M. Zahid Hasan and Charles L. Kane. “Colloquium: topological insulators.” > _Reviews of Modern Physics_ **82** , no. 4 (2010): 3045. (arXiv) How do the edge modes on the interface between two domains with spin-Chern number n=1/-1 of a 2D system differ from the edge modes of the system with "n=2/0" interface? Specifically in the two cases, spin-Chern numbers differ by 2 across the interface; according to the bulk-boundary correspondence, there should be 2 edge modes of the same spin in each case. However, in Z2 classification of TI, case 1 is non-trivial in both domains (n=1 and n=-1), yet case 2 is trivial (n=2 and n=0). I'm guessing in case 1, we do have edge modes that are against impurity and do not backscatter, but in case 2, that is not the case. Am I thinking in the right path? Is there any materials that discuss about this question? Many thanks.


title: Classify equilibrium points and find bifurcation points of a non-linear dynamic system

text: **Context** : The question refers to computational physics of non linear systems with _Mathematica_. **Excercise** : Given the system $\\{f_1: \dot{x} = a x + y + x^3, f_2: \dot{y} = x - y \\}$:   1. Find the equilibrium points and classify them.   2. What kind of bifurcation is there and for what value of $a=a^*$ does it happen ?   3. Draw the phase portrait for one $a<a^*$ and for one $a>a^*$. **Solution** : I calculate the equilibrium points by solving the system of equations $\\{\dot{x} = 0, \dot{y} = 0\\}$. I get three solutions: $(0,0), (-\sqrt{-1-a},-\sqrt{-1-a}), (\sqrt{-1-a}, \sqrt{-1-a})$. My lecture notes mention that in order to classify an equilibrium point, I first need to find out the topology near the equilibrium points. This is done by calculating the eigenvalues of the following matrix: \begin{equation} A= \begin{pmatrix}\frac{\partial f_1}{\partial x} & \frac{\partial f_1}{\partial y}\\\\\frac{\partial f_2}{\partial x} & \frac{\partial f_2}{\partial y}\end{pmatrix}_{(x_0^*, y_0^*)} \end{equation} As can be seen from the following screenshot (L1 are the eigenvalues of matrix A calculated at the 1st equilibrium point, at (0,0)), we can see that:   1. for $a <-1$, it is $\lambda_1 < 0, \lambda_2 < 0$, therefore it is stable node.   2. for $a>-1$, it is $\lambda_1 < 0 < \lambda_2$, therefore it is saddle. **My 1st question is** : What does it happen for $a = -1$, where the eigenvalues become (-2, 0) ? My closest guess is to use the _Hartman-Grobman theorem_ and say that the equilibrium point is 'linearly stable'. Is there anything I am missing here ? **My 2nd questions is** : Regarding the bifurcation point, by looking at the equilibrium points of the system, we deduce that at value $a=-1$ they change (the stable node becomes saddle and vice versa). Does that suffice to say that there is a bifurcation point at $a=-1$ ? Also for $a=-1$ the three equilibrium points collide to just one. How does that this fact fit with everything else ? ![Eigenvalues of matrix for equilibrium point \(0,0\)](http://i.stack.imgur.com/HX4Xl.png) **Update:** These are the phase portraits for $a<a^*, a>a^*$ with $a^*=-1$ being the bifurcation point: ![enter image description here](http://i.stack.imgur.com/ijkzF.png)


title: What is the simplest way to prove that the Earth orbits the Sun?

text: Assume you're talking to someone ignorant of the basic facts of astronomy. How would you prove to them that the Earth orbits the Sun? Similarly, how would you prove to them that the Moon orbits the Earth?


title: Can entanglements themselves be entangled?

text: In other words, could there be higher dimensional entanglements between entanglements? For instance, this could allow us to entangle two entangled- far-away pairs to create a system of four entangled particles. I am sorry if this sounds like a silly question.


title: Average number of spin up particles

text: In a paramagnetic system, where $N = N_\uparrow + N_\downarrow$ is fixed, how does one calculate the average number of spin-up particles $\langle N_\uparrow \rangle$? You can assume we have the canonical partition function. The Hamiltonian for this system is: $$ H = \sum_{i=1}^N {\frac{(p_i - b s_i)^2}{2m} - h s_i} $$ where $s_i = +1$ for spin-up and $s_i = -1$ for spin-down. How do I actually go about calculating this? I know, in general for an operator $O$: $$\langle O \rangle = \sum_{\mu} P_\mu O_\mu = \frac{1}{Z} \sum_{\mu} O_\mu e^{-\beta E_\mu}$$ Can someone show me how to explicitly use the Hamiltonian to evaluate this? Also, can we use $\langle N_\uparrow \rangle$ to evaluate $\langle N_\uparrow - N_\downarrow \rangle$? Can you just say $\langle N_\uparrow - N_\downarrow \rangle = 2\langle N_\uparrow \rangle - \langle N \rangle = 2\langle N_\uparrow \rangle - N$ since $N$ is fixed?


title: Theory of Space Charge in a Planar Diode

text: A planar diode in a vacuum bulb can support current in only one direction. To even get current flowing, you have to heat up the cathode filament to excite the electrons. However, once current is flowing, there is a certain limit imposed on it, making the resulting graph of current versus voltage look like this: ![Graph of Current vs Vooltage](http://i.stack.imgur.com/y6yS0.png) You can observe an inflection point about half way through the graph, and you'll notice the curve slowly flattening out. What is the formal theory and terminology behind this limit? Is it because of the build up of space charge (slow moving electrons, in this case) in between the diodes, which would repel the electrons in the cathode and keep them from escaping across the diode, thus imposing a certain limit on current?


title: Boiling water reactor

text: What are the typical power densities in conventional boiling water nuclear reactor per cubic centimeter of fission material?


title: Inverse of gauge covariant derivative

text: Consider the gauge covariant derivative defined by $$ D_z = d_z + \Delta_z $$ or explicitly $$ (D_z)^a{}_c = \delta^a_c d_z + (\Delta_z)^a{}_c = \delta^a_c d_z + f_{bc}{}^a A_z^b $$ Here, $d_z$ is the covariant derivative w.r.t. to the metric $$ ds^2 = \gamma_{z{\bar z}} \left( dz \otimes d{\bar z} + d {\bar z} \otimes d z \right) $$ $f_{ab}{}^c$ are the structure constants of a Lie algebra satisfying $$ [T_a, T_b] = f_{ab}{}^c T_c $$ Let us assume that $d_z^{-1}$ is a well-defined operator on the background. Let us also assume that the gauge field satisfies $$ F^a_{z{\bar z}} = d_z A^a_{\bar z} - d_{\bar z} A^a_z + f_{bc}{}^a A_z^b A_{\bar z}^c = 0 $$ In matrix notation, we can write this equation as \begin{equation} \begin{split} D_z A_{\bar z} = d_{\bar z} A_z,~~ D_{\bar z} A_z = d_z A_{\bar z} \end{split} \end{equation} Under these conditions, I wish to write the inverse of the gauge covariant derivative $D_z^{-1}$ in terms of $d_z^{-1}$. Is this possible?


title: What does the complex electric field show?

text: We have a complex electric field. Is there any definition for absolute and imaginary part of a complex electric field? What do they stand for?


title: Units describing the behavior of neutrinos

text: http://proj-cngs.web.cern.ch/proj-cngs/Download/CNGSDGVE/cngsdgve.pdf on page 13 I read that "the present plan is to provide nu_mu neutrinos with an energy between 5 and 30 GeV." Wikipedia neutrino article https://en.wikipedia.org/wiki/Neutrino says that neutrinos are electrically neutral. What is the justification of giving a non-electrical particle the units of electricity?


title: The vector r points from $P'(x',y',z')$ to $P(x,y,z)$

text: For some reason this question is giving me a hard time :( The vector $r$ points from $P'(x',y',z')$ to $P(x,y,z)$. (a) Show that if $P$ is fixed and $P'$ is allowed to move, then $\nabla'(\frac{1}{r})=\frac{\hat{r}}{r^2}$ (b) (a) Show that if $P'$ is fixed and $P$ is allowed to move, then $\nabla(\frac{1}{r})=-\frac{\hat{r}}{r^2}$ I don't really understand the notation $\nabla'(\frac{1}{r})$ What is the prime? what exactly am I applying the gradient operator to? and how can I show this and answer the question proposed?


title: Coherent State in 2 dimensions

text: I am looking at a 2D harmonic oscillator $$H=\frac{1}{2m}(p_x^2+p_y^2)+\frac 12m(\omega_x^2x^2+\omega_y^2y^2)$$ Where $\omega_x=5\omega_y$. I am told that the oscillator is prepared in a coherent state with the following qualities: $$\langle x(0)\rangle=x_0$$ $$\langle p_x(0)\rangle=0$$ $$\langle y(0)\rangle=0$$ $$\langle p_y(0)\rangle=p_0$$ I am looking for the time dependent state of the system. My approach: Seeing as this is a coherent state, the expectation values act as classical variables, this means that the position and momentum will be given as $$q(t)=q(0)\cos(\omega t)+\frac{1}{m\omega}p(0)sin(\omega t)$$ $$p(t)=p(0)\cos(\omega t)-m\omega x(0)\sin(\omega t)$$ Additionally, the coherent state is given as: $$\alpha (t)=\frac{1}{\sqrt{2m\omega\hbar}}(ip(t)+m\omega x(t))$$ So if I want to find the state as a function of time, do I find the alpha for both x and y and add them together?


title: How would an electron bunch/beam look different in the rest and lab frames?

text: With respect to special relativity, I was wondering how the spatial dimensions would differ between the rest and LAB frame of an electron beam. System: Electron bunch/beam traveling in linear motion. How large/long would the bunch appear in the point of view of an electron (rest frame). How large/long would the bunch look in the lab frame? How would the answers change if the electron bunch/beam was accelerating?


title: Proving $RTln(a_i)$ equal to non PV work done by system,where $a_i$ is activity

text: First of all many here will might say that it is Chemistry question.If you think so then read this article and then answer question. Question is --- We know chemical potential is defined as $\mu_i =\mu_i^{standard}+RTln(a_i)$ .Here $a_i$ is activity of $i^{th}$ component of solution.In case of gases instead of $a_i$ it is $f_i$ that is fugacity of $i^{th}$ gas out of mixture of gases.Activity is here nothing but activity coefficient multiplied by concentration or in case of fugacity it is product of fugacity coefficient multiplied by partial pressure of gas. But now this is not the only definition of Chemical potential.Chemical potential also means **partial molar gibbs free energy** when pressure and temperature of system are constant.We know that Gibbs free energy of system at constant temperature and pressure also represents Non-PV work done by system.Now how can the term $RTln(a_i)$ can be correlated to Non-PV work done by system in order both two definitions will be equivalent.


title: Energy-Momentum Tensor under Lorentz Transformation

text: In relativity, the symmetric energy-momentum tensor is given by $$ T^{ij}, $$ where $T^{00}$ is the energy density and $\frac{1}{c}T^{10}$ is the momentum density. Thus: $$ \left(\frac{1}{c}T^{00}dV, \frac{1}{c}T^{10}dV\right)^{T}$$ is the 4-momentum. Under a Lorentz transformation, this should transform like 4-vectors where $$ \frac{1}{c}T^{00}dV= \left[\frac{1}{c}T'^{00}dV'+\frac{v}{c^2}T'^{10}dV'\right] \left( 1-\frac{v^2}{c^2}\right)^{-1/2}\\\dV=dV'\sqrt{1-\frac{v^2}{c^2}}.$$ After simplifications, we have: $$ T^{00}= \left[T'^{00}+\frac{v}{c}T'^{10} \right] \left( 1-\frac{v^2}{c^2}\right)^{-1}$$ But if we apply the Lorentz transformation to the tensor directly we get $$ T^{00}= \left[T'^{00}+\frac{v}{c}T'^{10}+\frac{v^2}{c^2}T\ ^{11} \right]\left( 1-\frac{v^2}{c^2}\right)^{-1}$$ What accounts for the difference? I think the first is wrong but have no idea why.


title: How to find solutions to the gravitational potential metric h

text: I'm working on a problem in which a star of mass M1, radius R1 is surrounded by a thin shell of mass M2, , radius R2. I want to find the solutions to the gravitational potential h in the region in between the star and the shell and region outside both. Can we assume the mass is stationary, and not too large, and thus use the analogous Poisson's equation (Laplacian of potential = 4*pi*G*rho)? Or is there some physical reason that we should use the potential equation in the Hilbert gauge (d'Alembertian of tensor potential = 4*pi*G*(energy momentum tensor)/(c^2)? My thinking is that Poisson's is totally appropriate for this simple situation of a stationary star with a shell mass around it. Now, I don't see how to use Poisson's equation. I'm getting tripped up with how to consider the two masses. What if, for example, it's a very dense shell which is in fact more massive than the star. That would give a potential radially outward. How do I find a solution to the Laplacian?


title: Mixing of quarks, neutrinos... and leptons?

text: This is a quite simple question: quarks do mix (through the CKM matrix), neutrinos do mix (through the PMNS matrix). Then **why do charged leptons not mix?**


title: Rigorous mathematical formalism of particle physics

text: Can anyone provide me with a rigorous mathematical definition of the fundamental particles (all fundamental bosons and fermions), reflecting the analogy of action of groups with interaction of particles, isospin and mediating forces? An answer would be preferable, but a link of an appropriate source would also suffice.


title: Precision and accuracy calculations

text: Precision is usually understood as the number or significant figures in some experiment. Accuracy is the difference between the best measurement and the real value. How are precision and accuracy determined from a given experiment? Or equivalently, how are systematical errors and statistical errors calculated. Statistical errors are usually treated approaching "the normal distribution". But I am not sure how systematics are determined...Perhaps making some "pattern" measurements?


title: Rank of the Poincare group

text: There are two Casimirs of the Poincare group: $$ C_1 = P^\mu P_\mu, \quad C_2 = W^\mu W_\mu $$ with the Pauli-Lubanski vector $W_\mu$. This implies the Poincare group has rank 2. Is there a way to show that there really are no other Casimir operators other than trying to build all possible combinations of generators and seeing them fail? To state it differently: Can one determine the rank of the Poincare group without explicit construction of the Casimir operators?


title: Steam Turbine Inlet Velocity

text: Consider an ideal steam turbine. The power generated by a steam turbine comes down to the torque and the angular velocity, which is ultimately dependent on the velocity at which the flow enters the turbine (and turbine factors, but will only consider the flow properties). Similarly, the power needed by a turbopump is dependent only upon the velocity at which we want the fluid to leave the pump. The velocity of the flow can be determined as such: $$ v = \dot{V}/A $$ $$ \dot{V} = \dot{m}/\rho $$ Now suppose I make a fluid loop, with a turbopump in series with a turbine of identical size and design. Continuity of the fluid loop gives us this equation: $$ \dot{m} = \rho_1v_1A_1 = \rho_2v_2A_2 $$ Assuming no loss to friction or anything else, you will get exactly as much power from the turbine as you put into the turbopump. Okay so far. However, then I simply make the turbine have a smaller cross section area. Since mass flow is constant through the fluid loop, $v$ increases, and you end up getting more power out of the loop than you put in. Since this is thermodynamically impossible, does this mean the flow density must decrease proportionally to assure that the power in and power out is the same? However, what if this was a liquid-only fluid loop, where the density remains roughly the same? Ok, now suppose we turn this into a Closed Brayton Cycle. In between the turbopump and turbine we add in a boiler which adds an amount of heat energy to the fluid. Is it correct to use the ideal gas law to determine the density of the flow? $$ P = \rho R'T $$ In this way, while $\dot{m}$ remains constant, $\rho$ would decrease based on the temperature, which would ultimately increase $v$. Is that correct, or does the addition of heat alter the flow velocity differently in order to convert heat energy into work?


title: 25 cm of vacuum correspond to what pressure?

text: I the following (american) test > Vacuum testing consists of placing samples from the packiging operation into > a jar filled with water. A lid is placed over the samples to fully immerse > them in the water. A container lid is applied to create a seal effective > enough to create approximately **25 cm of vacuum**. The vacuum pump is set, > and the samples are tested for approximately 1 minute... Is 25 cm of vacuum equal to 333 or -667 mbar (or are the +- opposite)? or does it just mean that there has to be 25 cm distance from the water front to the container lid **__**?


title: How can individual photons have different amounts of energy?

text: If photon is an elementary particle, how can different photons have different energy, if $E=mc^2$ and all photons have (or don't have) the same mass and the speed of photon is constant shouldn't it mean all photons have the same amount of energy?


title: Does supercavitation create vacuum?

text: While reading the amazing things that the mantis shrimp can do, such as moving two of their limbs so quickly that the water around them 'supercavitates', a friend of mine told me that those limbs were creating vacuum, in the water. Does this creation of vacuum happen? Or is it something else? If so, am I creating vacuum when boiling water?


title: Why is baryon or lepton violation in standard model is a non-perturbative effect?

text: The baryon number B or lepton number L violation in the standard model arise from triangle anomaly. Right? Triangle diagrams are perturbative diagrams. Then why the B or L violation in Standard model is said to be a non- perturbative effect? I'm confused.


title: How can we see that there is superconductivity/superfluidity in the boundary theory in the holographic principle?

text: For example in the models for holographic superconductors we can calculate the conductivity. Also there is an energy gap. I can understand that it describes a superconductor. However I have also heard people talk about superfluidity in holographic QCD. I believe they are talking about the condensates such as the pion or the chiral condensate. Where is the superfluid in holographic QCD (quark gluon plasma maybe?) and how do I know that it is a superfluid?


title: Energy of a string

text: What is the correct definition of the energy of a string ? I suddenly get confused with the definition of the energy of a string. Considering, for instance, a bosonic open string in the light-cone gauge, We have $H = p^-=p_+$, the hamiltonian, and we have $p^0$, which appears in the expansion of the string in light-cone gauge $X^o(\sigma, \tau) = x^0 + \frac{p^0}{p^+}\tau + ...$. a) In Zwiebach's First course in string theory (1), there is a calculus of the entropy of a string, which begins (Chapter $16.2$ p $354$) by a function partition for a non relativistic string with fixed points (the quantum violin string). For a given energy $E = N \hbar w_0$, where $N$ is the level, the entropy is computed (this is direcly linked to the number of partitions p(n)). More precisely, an expression for the entropy, $\sim T$, and the energy ,$\sim T^2$, are obtained, se we get a $S \sim \sqrt{E}$, and plugin the $E \sim N$ expression, we get a $S \sim \sqrt{N}$. "Suddenly", in chapter $16.3$ page $361$, it is said that "we now return to relativist strings that carry no spatial momentum, and this happens if the open string endpoints end on a D-0 brane, so the energy levels are given by the rest mass of its quantum states". Of course, we have $m^2 \sim N$ (for large $N$), so $m\sim \sqrt{N}$, so if the energy ($p^0 ?$) is identified with $m$, we have $S \sim E$ Now, if we take the hamiltonian $H$, for large $N$, we have $H \sim N$, so $S \sim \sqrt{H}$ So, this confused me, because I have $2$ law expressions for the entropy, depending on the definition of the energy. b) An other problem, is saying that the energy of the string is proportionnal to its length. If we look at $H$, we have $H=p^- \sim \frac{1}{l}$ (with zero trans verse momenta $p^i$), while $p^+ \sim \frac{l}{\alpha'}$. So, may I write that the correct energy is $p^o \sim (p^+ + p^-)?$


title: Scaling of non-gravitational energy in a black hole

text: When looking at a Schwarzschild black hole, for instance, we know that we may apply black hole thermodynamics. We may define a entropy of the black hole which scales like the area of the horizon : $$S \sim R_s^2$$. It is understood in the more general context of the holographic principle which states that " the description of a volume of space can be thought of as encoded on a boundary to the region—preferably a light-like boundary like a gravitational horizon" Now, the non-gravitationnal energy $E_{ng}$, so the mass $M$ for the Schwarzschild black hole, has a different scaling : $$E_{ng} \sim R_s$$ So, does that mean that the energy is encoded in a one-dimensional object (perimeter, loop, string, radius), and is it a different "holographic" principle ?


title: The Spinning Log "Perpetual Motion" problem, and my attempt at a solution

text: So I was introduced to this "perpetual motion" riddle a few weeks ago. The problem goes like this: we all know perpetual motion machines are not possible, but this riddle _seems_ like it should work as a perpetual motion machine - the problem is to explain _why it doesn't work_. **Here's the situation:** (1) You take a room, and put a wall down the middle of the room - splitting it in to two equal half-rooms. (2) Take a perfectly cylindrical log whose length is the same as the length of the wall we just put in to make two half-sized rooms, and cut out a place halfway up the wall so that the log slips right in, length wise (if you were looking at the middle wall from the inside of one of the rooms, you would not see the ends of the log at all, you would a half-cylinder running the length of the wall - if you looked from the other room, you would see the other side of the cylinders length). (3) Make sure the space between the log and the wall is air-tight (the log is treated to absorb no water, there is no friction between it and the wall, etc), and fill one of the half-sized rooms all the way to the top with water. Now you have one side of the log exposed to air, and one side totally surrounded by water. The idea is that, since everyone has experience that logs rise in water, the side exposed to the water should "rise" and the log should spin, creating a perpetual motion machine. Now, I have two explanations of why it _doesn't_ work. One I came up with myself, and one that is proposed in the solution on the page in which I found the problem. **My Solution:** Water pressure increases as you increase in depth, so the water molecules hitting the log deeper in the water would have more force that the water molecules hitting the log less deep, but at each infinitesimal depth increase, the water is hitting the log at all angles, and the force from the molecules coming on from all directions would cancel out to leave a force pointing toward the center, at every place the log is exposed to water. Since all of the net forces from the water at each point exposed to water is pointing toward the center, there can be no net torque and the log cannot spin. The **net** force would be a slight force directly upwards (due to the difference in size of the force vectors pointing inwards between the less dense water up top and the more dense water on the bottom), but since the log is held in place, it cant move upwards. **The Proposed Solution** "To understand why this would not work, we need to look at how buoyancy itself works. When a lighter than water object (lets say a hollow ball) is submerged it is pushed up by the water apparently in spite of gravity. In fact the opposite is true. It is gravity pulling down on the water that pushes the ball up. The kinetic energy to lift the ball comes from an equal volume of water falling to occupy the space where the ball just was. As the ball moves up, the evacuated area is filled with water from above it; therefore, this water is falling. When the water was above the ball it had potential energy that is exchanged for kinetic as it falls to fill the void left by the ball thus providing energy to lift the ball. The opposite is true for a heavier than water object (lets say a brick), but it's still the same principle. As the brick falls through the water, it is filling space that was once filled by water. As the brick falls, it is providing energy to lift the water to fill the space it just evacuated. Now back to our spinning log example. If the log were to spin, it would not be evacuating any space for the water to fill. Even though a different part of the log would be filling that space, it would still be the same space it was filling before. There would be not downward movement (falling) of water to convert potential energy into kinetic, so there is no energy to cause the log to spin." **Here's my problem** I've never understood this explanation of buoyancy. Why, just because an object is less dense than water, is the water "rushing below it" to fill the space evacuated by the object as it rises? Why doesn't the object just stay where it is? It seems like you're using buoyancy to explain buoyancy - the less dense object floats upward due to buoyancy and the water rushes in below to fill the evacuated space, which causes the buoyant force upwards... what? Why is the water rushing below less dense objects, and not rushing below more dense objects? Am I missing something in the explanation? I understand that when you submerge an object in water, it displaces an equal volume of water, and the weight of the water displaced equals the bouyant force upwards, but I just don't understand why the water is "rushing underneath and pushing the object up".


title: What is the closest general-relativistic equivalent of a "time slice"?

text: In a newtonian universe, one can talk of a "time slice", that is, the state of the universe at a given point in (global) time. In a "typical" classical universe, a time slice would contain enough information to fully compute the state of the universe at any other point in time, backwards or forwards. So, disregarding what we know of quantum effects and talking purely of a general-relativistic universe, which concept is closest to a time slice?


title: dynamic casimir effect

text: A few years ago, when i studied the casimir effect interpretation as the filtering out of vacuum modes with appropiate boundary conditions, i had the following dilemma; supposedly the derivation of the force between the walls was **entirely** equivalent to just calculating Van der Waals forces. This, as it was argued, compelled us to not take too seriously the idea of a negative energy density of the space between the walls. However, i was left wondering what would happen if one would take the interpretation based on vacuum modes a bit further, and we could somehow switch on and off the permitivity $\epsilon$ of the conductor; what sort of long range radiation we would expect to see? A possible experimental implementation would be if the layers where superconductive, and then a magnetic field would be switched on just to break the superconductive phase. We probably need to use the Dirac-Heisenberg-Wigner formalism for these sort of dynamic quantum systems, but what would be intuitively expected to be detected?


title: Is it possible for one side of the universe to "meet" the other?

text: I've variously heard the shape of the universe being described as multi- dimensional, like a helix or mobius strip, and super string theorem seems to say there are lots of universes all piled up next to each other in vibrating planes. My question is, can the edges of our universe ever join up? Are they already all joined up? as in is our universe really in the shape of a giant donut (yum). or is this shear lunacy? I guess I am assuming a model of the universe where the "edge" is not necessarily expanding constantly "out" into infinite amounts of empty vacuum, but rather one where the edges are the boundaries to that expansion (so that it seems from an observable position that the universe is expanding, when really its flexing inside these boundaries). As indicated in the comments no doubt an inaccurate description, but I was imagining it like a balloon.


title: How many joules of energy are required to convert 1 liter of water into hydrogen and oxygen, theoretically?

text: I've been reading http://en.wikipedia.org/wiki/Electrolysis_of_water and other related questions on physics.stackexchange, but I fear I don't have a decent enough understanding of math and science to understand and decipher into laymen terms. I basically just want to know how much energy in watts or joules it takes, taking efficiency out of the equation (assuming the perfect, yet realistically possible method / voltage / etc), it takes to convert 1 liter of water into hydrogen and oxygen (electrolysis).


title: What prevents the accumulation of charge in a black hole?

text: What prevents a static black hole from accumulating more charge than its maximum? Is it just simple Coulomb repulsion? Is the answer the same for rotating black holes? ### Edit What I understand from the answers given so far, is that **maximum charge** is a **moving target**. You can add charge to a black hole but Coulomb repulsion guarantees that you will do so in a manner than will increase "maximum charge value". Is this correct?


title: Superposition principle and polarization

text: I am reading an optics book ( _Physics of Light and Optics_ by Peatross and Ware) that asserts this: > A beam of light can always be considered as an intensity sum of completely > unpolarized light and perfectly polarized light: > > $$I=I_{Pol}+I_{Un}$$ How does this marry with the superposition principle? Why can we just add the intensities?


title: Neutrino mass and energy question

text: If a neutrino has mass then it travels less than the speed of light. Suppose I boost myself to the rest frame; i.e. bring it to rest in the laboratory. Now if it oscillates between different states and masses sitting there, where does the oscillating excess/loss energy and mass reside? In some internal state? I have a very limited knowledge of QM but in SR it would seem strange that "mass" would move into another place. A pointer to where I should start studying would (I hope) be sufficient.


title: Given the Nyquist plot, determine order and stability

text: ![enter image description here](http://i.stack.imgur.com/7GRZp.png) Is the nyquist plot of the open control loop. The starting point (not shown in the figure) is in the negative infinity of the imaginary axis. What is the order of the process ? What is the maximum gain of the controller, so that the control loop is still stable.


title: Why is the anti-symmetric tensor more important than symmetric tensors?

text: In differential geometry, the differential forms are anti-symmetric tensors. So, why is the anti-symmetric tensor like $ d x_1 \otimes dx_2 - d x_2 \otimes d x_1 $, more important than the symmetric tensor like $ d x_1 \otimes dx_2 + d x_2 \otimes d x_1 $ for the theory? Why do we need anti-symmetric tensors for the exterior algebra, the wedge? In quantum mechanics, we have fermionic wave functions and bosonic wave functions. Is the former more fundamental than the latter in a certain sense?


title: Density matrix formalism

text: The density matrix $\hat{\rho}$ is often introduced in textbooks as a mathematical convenience that allows us to describe quantum systems in which there is some level of missing information. $\hat{\rho} = \sum_{i=1}^N c_i \rvert\psi_i\rangle \langle \psi_i \rvert$ I have two questions regarding density matrices. First, it is clear that a generic pure state $\rvert \psi \rangle$ belongs to some Hilbert space $\mathcal{H}$. But what mathematical space do density matrices belong to? It is clear that the expression $\hat{\rho} \in \mathcal{H}$ is incorrect, as a mixed state cannot possibly be described in terms of a state in a Hilbert space. Can we think of the space of all possible density matrices (of a given dimension) as a metric space? Does this space have the topological properties of a manifold? Secondly, can the density matrix be considered 'physical'? For example, if we take a single photon described in the Fock basis (neglecting polarization), can the fundamental description of that photon ever be the completely mixed state $\frac{1}{2}(\rvert 0 \rangle \langle 0 \rvert+\rvert 1 \rangle \langle 1 \rvert)$? Or is this only a reflection of some ignorance on the part of the experimentalist, and in reality the photon must be described by a pure state?


title: Difference in decay for muon and anti muon

text: In a couple weeks, I will conduct a lab experiment where I measure the lifetime of the muons from the secondary cosmic radiation. For that, we have two detectors above each other, one will give a start signal, the other will give a stop signal, assuming the muon came to rest in the second detector and decayed. There is a preparation question that says: > Muons as well as anti muons arrive at the surface of earth. Which nuclear > process is possible for muons but not for anti muons? How does this > qualitatively affect the measured lifetime? As far as I know, the decay of the muon goes like this: $$ \mu^- \to \nu_\mu + \bar \nu_e + \mathrm e^-$$ And for the anti muon, it goes like this: $$ \mu^+ \to \bar \nu_\mu + \nu_e + \mathrm e^+$$ The Wikipedia page says: > The mean lifetime of the (positive) muon is 2.1969811±0.0000022 µs.[1] The > equality of the muon and antimuon lifetimes has been established to better > than one part in 104. What is that difference that they are asking about in the lab course manual?


title: A "global warming" problem

text: Suppose that a someone decided to calculate, for a given amount of heat added into a planet's atmosphere( **an ideal gas** ), how much is the corresponding temperature rising of the atmosphere, $\Delta T$. Assuming that the whole heat transforms into the internal energy of the gas and distributes uniformly (s)he got $\Delta T=1K$. A question rises: actually, some of the heat is transformed into gravitational potential energy of the atmosphere(we ignore kinetic energy of the atmosphere). If we take also this into account then how much is the temperature rising of the atmosphere, $\Delta T$ now? Maybe the effect due to the gravity is negligible? For simplicity let's assume that the pressure change is approximately exponential with height, $h$ $$P=P_0e^{-\frac{h}{H}}$$ where $P_0$ is pressure at $h=0$ and $H$ is called the scale height for pressure. For definiteness, let's take the Earth's atmosphere. In this case $P_0=101$ $kPa$ is average sea-level pressure. $H$ is about $8$ $km$.


title: Scale invariance Vs Conformal invariance

text: > **Possible Duplicate:**   >  Why does dilation invariance often imply proper conformal invariance? What exactly is the difference between the two? Can someone give an example of a theory which is scale invariant but not conformally invariant?


title: Physics II Video Courseware Recommendations

text: I'm looking for something to supplement my Physics II class. Last year I started using these video lectures to supplement my Calculus class and it helped tremendously. I also turned to this educational software when I was stuck in Physics I. I can't seem to turn up anything similar that deals with Physics II. I'm pretty much looking for something that I can watch repeatedly until it sinks in. Perhaps you know of something? Thanks


title: How could this person have discovered the resonant frequency from this string of magnets?

text: I stumbled onto this page http://mylifeisaverage.com/story/1364811/ and the post states that they were > all making strings and shapes with these sets of 216 really small spherical > earth magnets. What did I do? I strung them together and found the string's > resonant frequency. It was 15 Hz so I'm wondering how the poster could have figured this out?


title: Connection between particles and fields and spinor representation of the Poincare group

text: Let's have a definition of massive particle as an irreucible representation of the Poincare group. Then, let's have a spinor field $\psi_{\alpha \alpha_{1}...\alpha_{n - 1}\dot {\beta} \dot {\beta}_{1}...\dot {\beta}_{m - 1}}$, which is equal to $\left( \frac{m}{2}, \frac{n}{2}\right)$ representation of the Lorentz group. There is the hard provable theorem: $\psi_{\alpha \alpha_{1}...\alpha_{n - 1}\dot {\beta} \dot {\beta}_{1}...\dot {\beta}_{m - 1}}$ realizes irreducible representation of the Poincare group, if $$ (\partial^{2} - m^{2})\psi_{\alpha \alpha_{1}...\alpha_{n - 1}\dot {\beta} \dot {\beta}_{1}...\dot {\beta}_{m - 1}} = 0, $$ $$ \partial^{\alpha \dot {\beta}}\psi_{\alpha \alpha_{1}...\alpha_{n - 1}\dot {\beta} \dot {\beta}_{1}...\dot {\beta}_{m - 1}} = 0. $$ Can this theorem be interpreted as connection between fields and particles?


title: How can we create superposition in QM?

text: How can we force a particle (let's say that we know this particle has spin up) to be in a superposition of spin up and down? Wouldn't literally any interaction of it with anything cause it to be in superposition, since we can never know the exact position/speed of the thing it is interacting with, so surely the outcome is unknown?


title: Why can we use just one angular velocity vector to describe the rotation of a whole non-inertial reference frame?

text: The other day in class the professor was explaining non-inertial reference frames. We were working out how to find the acceleration of a point as measured from the non-inertial reference frame, and one problem was finding the time derivatives of the unit vectors as measured from the non-inertial reference frame. To explain how to do it, he drew a nice diagram that showed that, if $\mathbf{\Omega}$ is the vector representing angular velocity of the rotation of $\hat{\imath}$ (that is, its direction is the axis of rotation), then $\frac{\mathrm{d}\hat{\imath}}{\mathrm{d}t} = \mathbf{\Omega} \times \hat{\imath}$. This makes sense. Next, he said that we can use the same argument for the other unit vectors: $\frac{\mathrm{d}\hat{\jmath}}{\mathrm{d}t} = \mathbf{\Omega} \times \hat{\jmath}$ and $\frac{\mathrm{d}\hat{k}}{\mathrm{d}t} = \mathbf{\Omega} \times \hat{k}$. I don't understand why we use the same $\mathbf{\Omega}$ for all of them. Since their axes of rotation are different, shouldn't we use different angular velocity vectors?


title: How would an observer perceive movement on a train that's travelling near the speed of light?

text: Person A is on Earth and a train (or whatever you want to imagine) travels past him at near the speed of light. How would person A perceive movement on the ship? If time is slowed on the ship from the perspective of Person A, then if Person B were washing the windows or something (moving his hands up and down), would they look like they were going really fast, slow, or normal? If time is slowed from the perspective of Person A, then Person B's hands would be moving a certain distance (let's say the window is a meter) in a really short amount of time, no? And that would mean it looks like it is going fast. But then since Person A is perceiving the time as moving slower, it would look slower, no? Would they cancel and movement on the ship looks normal? I tried to see if the Wikipedia page could help, but it didn't seem to say anything about a stationary observer and a fast-moving vessel.


title: Partition function for classical particle and quantum particle are the same?

text: Permutation for classical particle $$\Omega=\frac{N!}{\Pi n_i!}$$ By using Lagrange method of undetermined multiplier, we get $$n_i=Ae^{\frac{-E}{kT}}$$ Probability, $$p=\frac{n_i} {Z}$$ where we define $Z$ as $$Z_{classical}≡∑e^{\frac{-E}{kT}}$$ But for fermion, $$n_i=(Ae^{\frac{E}{kT}}+1)^{-1}$$ My question is: why do we have the same partition formula for classical and quantum particle? If possible, please tell me how to define the partition function for fermion. Thanks


title: Why doesn't a neon sign seem that hot?

text: I heard that neon signs contain plasma, why aren't they hot? is it because the electrons and ions do not hit the lamp's wall? Is it because it is non thermal plasma and electrons and ions are not in thermal equilibrium? If that is the case do the electrons and ions and neutral atoms (all of them) hit the lamps wall?


title: Applying angular velocity to a rotation matrix

text: I have a very simple question. In our project we store an object's orientation as a 3x3 matrix which holds the orthonormal base of that object's local space. For instance if the object is aligned with the axis of the world, then its orientation matrix is :               1 0 0     0 1 0     0 0 1      Now we store the angular velocity as a vector that contains values in rad/s around each of the world's axis. For example if the object should rotate around the world's y axis at 20 rad/s, then the angular velocity is               0 20 0      My question is then, say we want to apply this velocity over 1 second to the orientation matrix, how would we do this?


I read that all bosons contribute to the casimir effect with an attractive force. http://math.ucr.edu/home/baez/physics/Quantum/casimir.html Does the Higgs boson contribute to the Casimir effect? Could it be the missing 5%?


A commercial airplane travels at a speed which is 85% of the speed of sound. The wings of the airplane are designed such that the bottoms of the wings are flat and the tops of the wings are curved so that air travelling above the wings follows a path which is 15% longer than the straight path of the air below the wings. The wings are approximately rectangular in shape with a length of 35 m and a width of 8 m. The thickness of the wings is negligible. What is the maximum mass that the airplane can have and still maintain enough lift to fly?


title: Why does $S = k_B \ln W$ not always apply?

text: I thought for a long time that the Boltzmann formula for entropy, $S = k_B \ln W$, was a universally true statement, or rather _the_ definition of entropy from the perspective of statistical mechanics. However, I have since come to understand that it is only applicable for an isolated system (i.e. the microcanonical ensemble) for which all microstates of the system are equally likely. The more general statement is the Gibbs entropy $$S = -k_B \sum_i P_i \ln P_i \,.$$ However, I have seen a derivation of the Boltzmann formula such that I can't quite see _why_ said formula doesn't always apply. I was hoping somebody could point out the error in the following reasoning. From classical thermodynamics we know that $$T = \left(\frac{\partial U}{\partial S}\right)_V \,.$$ Now let us consider two systems in thermal contact, allowed to exchange energy. If we suppose that all microstates of the _joint_ system are equally likely, we argue that equilibrium will be attained for a division of energy that maximises the number of possible corresponding microstates. So we have $$ \frac{d}{dE_1}\big(W_1(E_1)W_2(E_2)\big) = 0 \,. $$ Working through this, we get to the condition that $$ \frac{d \, \ln W_1(E_1)}{E_1} = \frac{d\, \ln W_2(E_2)}{E_2} \,,$$ but since equilibrium corresponds to equal temperatures, we make the natural definition that $$\frac{1}{k_B T} = \frac{d \, \ln W(E)}{d E}\,,$$ where we choose this particular form of the expression on the left since it results in energy flowing in the right direction (hot to cold) for two systems with very similar temperatures. Putting this together with the above result, identifying $U =E$, we must have that $$ S = k_B \ln W \,.$$ **Question:** where in this argument have I made any assumptions or mistakes, such that this formula applies only to a specific class of systems? Why can I not use this formula to determine the entropy of (say) one of the two systems I placed in thermal contact (in the discussion above of temperature)? Why is the Gibbs formula the correct one for systems allowed to exchange energy? We also use, I believe, this definition of temperature in the derivation of the canonical/Boltzmann distribution (see e.g. here), and yet in this case the reservoir is not an isolated system, and so I would have thought that this expression would not apply. Thank you.


title: Gravitational slingshot maximum

text: I have recently read an article about gravitation slingshot assist used by Voyagers 1-2, and was thinking on why this hasn't been used for travel between solar and other systems.   I mean sligshot can be done as many times as it is necessary to get speed of lets say half the speed of light that would allow to travel to Alpha Centauri in ~10-20 years can it not? There must be a flaw in my thinking that 3 or 4 planets can be re-used to get to necessary speed otherwise it would already have been done (drawing below). Even if planets would align differently I should always be able 'find' the planet that would allow me to jump to one that is closer to the sun, and repeat the acceleration again and again. ![enter image description here](http://i.stack.imgur.com/t52Vs.png) **What maximum (theoretical) speed could be achieved using planets of solar system as sligshot and how much would this speed wary from planetary alignment and what realistic speed could be achieved?** UPDATE: To be more specific on the second part of the question Lets say craft weight's 500kg at starting speed of 30,000 km/h initially it slings around Mercury (`radius 2440km`), Venus (`radius 6052 - 300 (atmosphere) = 5750 km`), and Earth (`radius 6378 - 300(atmosphere) = 6050km`) until diameter of planets is to wide to not to crash craft on surface. Then it flies to the moons of Saturn - Titan (`radius 5150km`), Rhea (`1527km`), Lapetus (`1470km`), Dione (`1123km`), Tethys (`1062km`), Enceladus(`504km`), Mimas (`396km`) and starts slinging there until diameter is to wide too. What approximate maximum speed could it get to leave the solar system?


title: Has the possibility of Higgs boson being a composite particle excluded already?

text: I heard some theory, such as technicolor, predicts the Higgs-like particle discovered at LHC should be a composite particle (correct me if I am wrong). Has this possible been completely excluded already?


title: Where does the steam pressure go?

text: Lets say you build a steam boiler and then you connect several turbines to the boiler in series. What limits the number of turbines that you can connect to the boiler in this way? In my mind, it seems that the steam still has to escape, so no matter how many turbines are in the way, the steam will still turn all of them in its attempt to escape the boiler.


title: How much electricity could be generated by cloths made of thermocouples?

text: Lets say that we made a glove, shirt, pants and a hat out of the most effective thermocouple material available today. How much electricity would be generated by each, by a healthy person, on a cool day (98 degrees F vrs 79 degrees F?)


title: How does a magnet work?

text: I'm having trouble understanding how a magnet (not the field that is generated as a result but the material itself) work. The particles are aligned in a specific direction to give rise to force but I don't see how this alignment gives rise to "attraction" or force.


title: what is the specific cause of permanent magnetism

text: Why can't we answer this simple question? Where does the magnetic field of a permanent magnet come from? What is different about a magnetizable atom that allows it? Why is it perpetual? Or is it recurrent? Charged particles in motion into wave-flux, then the wave-flux back to charged particles in motion? Primal energy still not explained. To the back side of Mars and physics still cannot explain the basis of a child's toy. tecvia


title: What crystal structure have electrons in Wigner crystal?

text: If electrons form a crystal in the Wigner crystal what is the structure of that crystal? cubic, bcc, fcc,...?


title: Black hole complementarity - absorption of Hawking radiation

text: I try to understand two principles formulated by Leonard Susskind in his book _The Black Hole War_ : 1, To any observer who remains outside a black hole, the stretched horizon appears to be a hot layer of horizon-atoms that absorb, scramble, and eventually emit (in the form of Hawking radiation) every bit of information that falls onto the black hole. 2, To a freely falling observer, the horizon appears to be absolutely empty space. [...] Now, in reaction to (an unanswered) question by Nathaniel, let us suppose that a distant observer A sees (according to the principle #1) Hawking radiation from a black hole (with Planck spectrum and say measurable temperature). Now suppose there is a freely falling gas somewhere between the event horizon and the observer. According to the principle #2, to the gas (considered an observer B) the horizon is empty space so there is no light coming from it to absorp. Would the observer A detect absorption lines in the black hole spectrum? How does it change with distance of the gas (or the observer) from the black hole? Will the distant observer see the lines if he is free falling?


title: How much of the universes mass is bound up in photons traveling between stars?

text: I was watching a Scott Manley video on youtube and he mentioned that the Sun was loosing 4 million tons of mass a second as it converts to energy. With a few trillion trillion stars also converting mass to energy, I wonder if that would be a component to the missing "dark matter" mass? I'm a complete layperson here, just very interested in astronomy and the physics involved.


title: what is "work"? (of a force)

text: I would like to know why two forces (of different magnitudes but same direction) that we'd use sequentially on an object to move it by the same amount twice, will produce two different values for the work quantity. Why is there a mismatch between the maths and the observed physical change? What exactly is the work of a force? (If possible, please avoid circular definitions like "work is the energy produced by a force" and "energy is the ability to do work".)


title: What is the reason for the $ i \tau_2 $ - factor in the higgs coupling with up-type quarks?

text: The quark mass term in the Standard Model Lagrangian looks like this: $$ L = - \lambda_d \bar{Q}\phi d_R - \lambda_u \bar{Q} i \tau_2 \phi^* u_R $$ What is the reason for the $ i \tau_2 $ - factor? What keeps me from writing undesirable things like $$\lambda_u \bar{Q} \phi u_R $$ ?


title: Spin in relativity

text: Mass and spin of the particle are used in classification of elementary particles. The mass is defined to be a Lorentz invariant quantity. On the other hand, the spin is a spacelike 4-vector and cannot be defined as an invariant quantity. My question is, what would be a convenient definition of a relativistic spin 3-vector? As is known, 3-vectors get contracted/dilated under Lorentz transformations. So, is the 3-spin defined to be equal to the rest frame 3-spin, or do we allow it to contract/dilate? It seems to me that the first option is generally used, especially when considering Thomas precession and similar effects. Although, I haven't seen any relevant discussion in textbooks on this topic.


title: Finding the current in a parallel circuit

text: Two resistors are joined as shown. The top resistor receives a current of 3 A. What is the current in the other resistor? What is the current that enters at junction A? I'm confused on how the two currents are 3A and 1A because I thought in a series circuit all the currents are the same. Why is the answer to the current entering junction A 4 A and not 1 A? ![](http://i.stack.imgur.com/XviUa.jpg)


title: Theoretically, is there anything in physics that prevents something like a transparent ipad or smartphone?

text: Also, is it possible to have electronic displays on the walls of a home? Is there anything in physics that prevents that? I guess I'm trying to ask if something like this is possible: http://www.techcrates.com/transparent- smartphones-our-new-reality/ http://www.whistleout.com.au/MobilePhones/News/CES-LG-steals-display-glory- with-futuristic-mirror-and-window-TVs


title: Do atoms produce synchrotron radiation?

text: Since synchrotron radiation is created when charged particles are radially accelerated and electrons are definitely orbiting a nucleus (assuming a Bohr model), electron should then logically emit synchrotron radiation. However, if it does, then it lose energy and would unfortunately spiral into the nucleus, which we know doesn't happens. So, is it that atoms doesn't produce synchrotron radiation or other mechanisms are compensating the synchrotron radiation effect?


title: Infinite heat capacity or susceptibility means fluctuation on all scales

text: I remember reading in an introductory text to phase transition (sorry I don't remember the name) that at a second order phase transition the specific heat and the magnetic susceptibility become infinite and that this implies fluctuations on all length scales. I understand how this implies strong fluctuations as they are respectively linked to the variance of the energy and of the magnetization, or that being infinite it's not possible to construct a preferred length scale with them; but I'm not sure to understand why it forces the fluctuation to happen at ALL the scales. Do you have an intuitive argument or example for that? Thank you in advance.


title: Why are the backs of airplanes curved?

text: I get the front part, but why is the back curved too? I do not see a problem with the back being flat.


title: Renormalizability of standard model

text: I'm wonder what precisely is meant by the renormalizability of the standard model. I can imagine two possibilities:   1. The renormalizability of all of the interaction described by the Lagrangian before spontaneous symmetry breaking (SSB) by the nonzero vacuum expectation value (VEV) of the Higgs field.   2. The renormalizability of the Lagrangian obtain from the initial one after SSB, expressed in terms of suitable new fields (which has direct physical interpretation contrary to the fields appearing in initial Lagrangian).  It seems that in case (2) we obtain an effective (nonrenormalizable) theory only and this precisely was the reason to introduce the mechanism of generating mass by nonzero VEV of Higgs field. The original Lagrangian (case (1)) contains only power counting renormalizable vertices so if there are no anomalies then SM befor SSB is renormalizable. However, in physical prediction (actual computations being performed), as far as I know, Lagrangian after SSB is used. Does is require infinite number of counterterms (is it effective theory)?


title: Does Heisenberg equation of motion imply the Schrodinger equation for evolution operator?

text: Let us choose to postulate (e.g. considering the analogy of the Hamiltonian being a generator of time evolution in classical mechanics) $$ i\hbar \frac{d\hat{U}}{dt}=\hat{H}\hat{U}\tag{1} $$ where $\hat{U}$ is the (unitary, linear) evolution operator and $\hat{H}$ the Hamiltonian (the most general version of which; i.e. time-dependent with instances at different times non- commuting). In S-picture, one can easily show that (1) is **equivalent** to $$ i\hbar \frac{d}{dt}|\psi_S(t)\rangle=\hat{H}_S|\psi_S(t)\rangle\tag{2} $$ where $\psi$ is a state, $|\psi_S(t)\rangle = \hat{U}(t)|\psi_S(0)\rangle \equiv \hat{U}(t)|\psi_H\rangle$ and $\hat{H}_S:=\hat{H}$. In H-picture, it is straightforward to show that (1) **implies** $$ \frac{d\hat{A}_H}{dt} = \frac{\partial\hat{A}_H}{\partial t}+\frac{1}{i\hbar}[\hat{A}_H,\hat{H}_H]\tag{3} $$ where $A$ is an observable and $\hat{A}_H(t)=\hat{U}(t)^\dagger \hat{A}_H(0)\hat{U}(t)\equiv\hat{U}(t)^\dagger \hat{A}_S\hat{U}(t)$ (and also $[\hat{H}_{H},\hat{H}_H]=0$ implying that the time dependence of $\hat{H}_H$ is purely explicit, i.e. $\hat{H}_H=\hat{H}_S\equiv \hat{H}$ with $[\hat{H},\hat{U}]=0$). **My question:** is it possible to obtain (1) from (3), i.e. to show that (1) is **equivalent** to (3)? Some thoughts on this: it is extensively mentioned in literature that both pictures yield same answers. Therefore, it should be possible to obtain (1) from (3) since (1) and (2) are equivalent. Assuming (3), the best I can get is that given an observable $A$, the operator $$ \hat{C}:= \hat{A}_S\left(\frac{d\hat{U}}{dt} \hat{U}^\dagger - \frac{\hat{H}}{i\hbar}\right) $$ must be skew-Hermitian.


title: Decuplet baryon resonances

text: Are all baryons from decuplet resonances?? Because all resonances decay by strong interaction; but omega minus baryon can't decay via strong, because any other bound state with strangeness -3 exists. Thanks


title: Why do string theory and Hopf algebra renormalization seem to have no intersection?

text: Hopf algebra appears in recent papers that systematize renormalization of quantum field theory (QFT). For example see Connes' work and citing papers or a paper referenced here on PSE: R. E. Borcherds, "Renormalization and quantum field theory"   http://arxiv.org/abs/1008.0129 This seems to be an effort that is separate from string theory, but I wonder if they are compatible. Since "compatible" is perhaps too loose of a term, I'm asking: > Why do string theory and Hopf algebra renormalization seem to have no > intersection? If the reader feels the question could be improved by rewriting, please so assist.


title: What is a good reference for the quantum mechanical description of lasers?

text: I am currently taking a graduate level class on lasers. The primary focus of the class is on the design and engineering aspects of lasers, e.g. resonator design. However the first portion of the class is an overview of the quantum mechanics of laser processes in semi-classical terms: transition probabilities for a 2-level dipole atom in an external (classical) field, perturbation theory, and matrix QM. I have an undergraduate degree in physics, so the basics of QM are familiar to me, but it has been years since I seriously applied any of that knowledge. In addition, the professor is moving through this material very rapidly, as it is not the primary focus of the class. I can follow well enough that I'm not worried about my grade, but I'm feeling a bit cheated out of a more thorough understanding of this topic, which I would really like to have. Not only is this subject matter an appropriate level of advancement over my existing knowledge, it is also the basis for something (lasers) which I will likely spend a career dealing with. Can anybody recommend a text which covers this topic, and would be appropriate for somebody with my background? I would value a clearly written description over absolute rigor, but I am by no means looking for a laypersons description of QM. For example, I am quite comfortable already with the notation and concepts in the textbook "Quantum Mechanics of atoms, molecules, solids, nuclei, and particles" By R. Eisberg and R. Resnick. The ideal reference would be at a similar level of sophistication, but provide a more in-depth look at laser processes in particular.


title: Helium Balloon Experiment

text: We tied a helium filled balloon to a counterweight and it floats aimlessly in the room. I asked if the balloon were to be outside the building (all things considered no wind same temperature) would the balloon float at the same level?


title: Does string theory provide a physical regulator for Standard Model divergencies?

text: In other question, Ron Maimon says that he thinks string theory is the _physical_ regulator. I did not know that string theory regularize divergencies. So, **Q1** : How does string theory regularize the ultra-violet divergencies of the "low-energy" (Standard Model) fields? And **Q2** : Why it doesn't regularize its own divergencies. By regularization I mean that the theory is ultra-violet finite before removing the regulator.


title: How to prove that proper orthochronous Lorentz transformations form a group?

text: ~~Proper orthochronous Loentz transform are Lorentz transforms that satisfy the conditions (sign convention of Minkowskian metric $+---$) $$\det \Lambda=+1, \qquad \Lambda^0{}_0 \geq +1.$$ How to prove they form a subgroup of Lorentz group? All books I read only give this result, but no derivation.~~ Why is this condition $\det \Lambda=+1, \ \Lambda^0{}_0 \geq +1$ enough for a Lorentz transform to be proper orthochronous? The temporal component of a transformed vector is $$x'^0=\Lambda^0{}_0 x^0+\Lambda^0{}_1 x^1+\Lambda^0{}_2 x^2+\Lambda^0{}_3 x^3,$$ the positivity of $\Lambda^0{}_0$ alone does not seem at first glance sufficient for the preservation of the sign of temporal component. And how to prove that all Lorentz transform satisfying such simple conditions can be generated from $J_i,\ K_i$? * * * For those who think that closure and invertibility are obvious, keep in mind that $$\left(\bar{\Lambda}\Lambda \right)^0{}_0\neq \bar{\Lambda}^0{}_0\Lambda^0{}_0,$$ but instead $$\left(\bar{\Lambda}\Lambda \right)^0{}_0= \bar{\Lambda}^0{}_0\Lambda^0{}_0+\bar{\Lambda}^0{}_1\Lambda^1{}_0+\bar{\Lambda}^0{}_2\Lambda^2{}_0+\bar{\Lambda}^0{}_3\Lambda^3{}_0.$$ And I'm looking for a rigorous proof, not physical "intuition".


title: How do people get the proton number for each element from experiment?

text: How did people determine the proton number for each element from experiment in each decade of 20th century?


title: experiment proposal to validate microcausality

text: I've been wondering about microcausality for some time now (a recent question of mine regarding the topic) and i'm wondering if its possible to devise an experiment to detect potential violations I thought something like this: the hypothesis to test is that if two atoms A and B are close enough, and by close enough i mean that the cross-section of atom B and photons emitted in all/random directions from A is roughly of the order $\frac{4 \pi}{100}$, so statistically, and assuming no dipolar couplings affecting the orientation, if A is excited and B is on the base state, 1% of photons emitted from A will be absorbed by B if microcausality holds, and any deviations from this average will suggest a causality violation (i.e: that the atom B at time $t + \tau_0$ absorbs an advanced photon that "causes" A at time $t$ to emit it, where $\tau_0 = \frac{d}{c}$) In order to detect when photons are emitted from each atom, actually we'll arrange so atom A favours a two-photon decay $\omega_0$ and $\omega_A$ from the excited state, and B, when in its base state, will accept $\omega_0$ but quickly decay to a metastable level sending a photon of $\omega_B$. We will use timings of $\omega_A$ and $\omega_B$ photons to roughly time the emission and absortion times of each atom But there is still a problem with the above setup: is entirely time-symmetric, so there is no good reason to expect that the atom B at the future, even if emitting advanced photons, will be more likely to interact with the past A atom (because both atoms will see each-other cross sections to be symmetrical). So in order to fix this, we actually arrange not one, but many atoms like A surrounding the atom B, so there is a probability $p_R$ of all the atoms A emitting a photon that is absorbed by B in the future (if only retarded causes are allowed), but there is a higher probability $p_A$, say, very close to 1, that the atom B will absorb an advanced photon from some of the A atoms. Any experimental outcome between $p_R \le p \le p_A $ would imply a mixture between retarded and advanced causation. Does this sounds like something that makes sense, and if it does, sounds like something that is doable? I've think about this from time to time, but never actually found a reason why it would not work. Obviously, sayings stuff like "this is nonsense because microcausality holds and i know it because etc." its useful but _not the point of the question_ , and with that i mean, if there are theory arguments against a positive result (beyond assuming causality holds ad hoc) it would be great to know them. But my question is regarding the experiment itself. For instance if you can sketch and argument about, say, $\Delta E \Delta t \ge \hbar $ implies that any correlations will be swamped by quantum noise, that is completely relevant to the question and you should write about that thanks!!!


title: Some questions about the free Fermionic partition function on a circle (Ginsparg's CFT lectures)

text: The following questions are based on these lectures, http://arxiv.org/abs/hep- th/9108028   * I would like to know what is the relationship between the last equation on page 82 ($(L_0)_{cyl} = L_0 - \frac{c}{24}$) and the last 2 sets of equations on page 84 (where one has derived the zero-point energies of $\pm 1/24$, $\pm 1/48$ of periodic and anti-periodic Fermions on the cylinder)  Can the equations below 7.6 (on page 84) be derived from the last equation on page 82?   * I can understand the derivation of the partition function in equation 7.5 at the top of page 84 where one gets that $Z = q^{-\frac{c}{24}}\bar{q}^{-\frac{\bar{c}}{24} } Tr q^{L_0}\bar{q}^{\bar{L}_0}$  But from there how does one derive the statements in the first paragraph of page 92 where he states that (1) when the Fermion is anti-periodic along both time and space the partition function is $q^{-1/48}Tr_A q^{L_0}$, (2) when its periodic in time and anti-periodic in space the answer is $q^{-1/48}Tr_A (-1)^Fq^{L_0}$ (3) when its anti-periodic in time and periodic in space the answer is $(1/\sqrt{2})q^{-1/48}Tr_P q^{L_0}$ and (4) when its periodic along both its $(1/\sqrt{2})q^{-1/48}Tr_P (-1)^Fq^{L_0}$ These 4 answers above look very mysterious to me at various levels - (1) I don't see how they can be read off from the general answer of $Z = q^{-\frac{c}{24}}\bar{q}^{-\frac{\bar{c}}{24} } Tr q^{L_0}\bar{q}^{\bar{L}_0}$, (2) how is the (-1)^F operator seeing the boundary conditions? (3) why do these answers have only the $q^{L_0}$ part and not the full $q^{L_0}\bar{q}^{\bar{L}_0} $


title: Reason for different type of energy transfer for two kinds of collisions

text: According to my physics book, if an electron were accelerated with 15 MeV of (kinetic?) energy and collided into a 100g thermally insulated copper block (not sure if the fact it is thermally insulated is relevant) the energy would transfer into thermal energy on the copper block and the copper block would heat up by a few degrees. However, if I were to collide into that same copper block with a bowling ball travelling with 15 MeV of kinetic energy then that energy would be transferred into kinetic energy on the copper block and the copper block would move. It seems to me, that in both cases, there is a thing smashing into another thing with 15 MeV of energy. When the first thing is small (e.g. electron) the energy is transferred into thermal energy. When the first thing is large the energy is transferred into kinetic energy. Why doesn't the electron's energy get transferred into kinetic energy as well?


title: Why do prisms work (why is refraction frequency dependent)?

text: It is well known that a prism can "split light" by separating different frequencies of light: ![prism diagram](http://i.stack.imgur.com/ZfYPs.gif) Many sources state that the reason this happens is that the index of refraction is different for different frequencies. This is known as dispersion. **My question is about why dispersion exists. Is frequency dependence for refraction a property fundamental to all waves?** Is the effect the result of some sort of non-linearity in response by the refracting material to electromagnetic fields? Are there (theoretically) any materials that have an essentially constant, non-unity index of refraction (at least for the visible spectrum)?


title: On what refractive index of medium depends ?

text: does the refractive index of a medium depends on wavelength of light passing through it ? If yes then why ?


title: Is it possible to reduce speed of light?

text: > **Possible Duplicate:**   >  What really cause light to appear slower in media? This came to me as a question from one of my friend while I was convinced that the answer is NO. But I have read in Special Theory of relativity that we CAN make light take more time to travel a distance.   And then there's also one more thing which is kind of misleading: > Light speed is slow in optically dense media So is it possible to slow it down?


title: Why does the light travel slower in denser medium?

text: Wikipedia says that "in general, the refractive index of a glass increases with its density." And the refraction index of water vapor is less than ice, and even less than liquid water. Is there any simple explanation to that?


title: What is the mechanism behind the slowdown of light/photons in a transparent medium?

text: So light travels slower in glass (for example) than in a vacuum. What causes light to slow down? Or: How does it slow down? If light passes through the medium, is it not essentially traveling in the "vacuum" between the atoms?


title: Difference in velocity of light in change in medium

text: It is often seen that according to physics the light changes it's velocity according to the medium through which it is traveling. So can it be explained that why so happen?


title: What causes refraction of visible light?

text: Refraction is the bending of a wave when it enters a medium where its speed is different. The refraction of light when it passes from a fast medium to a slow medium bends the light ray toward the normal to the boundary between the two media. My question is what is the fundamental reason for this refraction?is it something related to the atomic force?


title: Does the light speed change in media?

text: Does the light speed change? The Theory of Relativity says that the speed of light in vacuum is the same and unchangeable, while I read that the speed of light in glass is lower than in air, and that it's the maximum in vacuum. If so that the speed of light in media changes, then we can conduct the following suppose a media as air, and a body is moving through it in a uniform velocity, we get that time relative to this body dilates because of it's motion, $t = \dfrac{t'}{ \sqrt{1- \dfrac{v^2}{ c^2}}}$ Now suppose another media as $CO_2$ gas, and the same body is moving through it with the same regular velocity, time also dilates relative to it. So, since the velocity is the same, Time must dilate with the same value, but the speed of light is different, So time dilation has to be different. So which one is correct? Does this dilation of time change or no? I'm really confused.


title: Why does the speed of the Electromagnetic wave in the material depend on the frequency of the wave where as they are constant in vacuum (freespace)?

text: I am confused on why would the propagation speed of any EM waves at ANY frequency is constant in the free space (vacuum) but they seem to disperse in any other materials as the propagation speed of EM waves in other materials depends on the frequency? Can someone give me a good proof of this phenomena with appropriate formulas?


title: What really cause light/photons to appear slower in media?

text: I know that if we solve the maxwell equation, we will end up with the phase velocity of light is related to the permeability and the permittivity of the material. But this is not what I'm interested in, I want to go deeper than that. We know that the real speed of light is actually not changing, the decrease in speed is just apparent. Material is mostly empty, the light will still travel with c in the spacing. The rare atoms will disturb the light in some way. So I am interested in how the atoms affect the light. Some textbooks that I read explain it in a way kind of like this: In a material the photons is absorbed by atom and then re-emitted a short time later, it then travels at a short distance to the next atom and get absorbed&emitted again and so on. How quickly the atoms in a material can absorb and re-emit the photon and how dense the atoms decides the speed of light in that material. So the light appears slower because it has a smaller “drift speed”. But recently I realize an alternative explanation: Atoms respond to the light by radiating electromagnetic wave. This “new light” interferes with the “old light” in some way that result in delayed light(advanced in phase),this can easily shown by using simple phasor diagram. Consequently effectively the light covers a smaller phase each second, . Which gives the impression of a lower phase velocity . However the group velocity is changing in a complicated way. I think that the first explanation does not explain the change in phase velocity of light. if we consider light traveling into a slab of negative refractive index non-dispersive material, let’s say the light is directed perpendicular to the slab. The phase velocity’s direction will be flipped, but group velocity’s direction in the material will not change. Only the second explanation can explain the flipped phase velocity direction. I guess that the velocity that we get in the first explanation is actually belongs to the group velocity. It makes sense to me that the front most of the photon stream determines the first information that the light delivers. So the question is What really cause the phase velocity of light to be decreased? 1."drift velocity" of photons ( they aren't the same photons, they are re- emitted all the time) 2\. phase difference between absorbed and emitted light 3\. something else And also, I still don't really understand about the detail of the absorption- emission process for small light's wavelength ( for large lambda compare to the atoms spacing, the photons will be absorbed by the phonons). The dispersion relation that we know is continuous and also some material is non- dispersive, therefore the absorption process must occurs in all frequency for a certain range. So definitely it doesn't involve the atomic transition, otherwise it will be quantized. My guess is that the relevant absorption process gets smooth out by the dipole moment. What makes the spectrum continuous? EDIT: link for dispersion relation: http://refractiveindex.info/?group=CRYSTALS&material=Si


title: Speed of Light in a Medium

text: For light travelling in a medium with refractive index greater than one: The "average" speed of light is slower than the speed of light in a vacuum. As far as I know, the instantaneous speed of light in a medium is equal to the speed of light in a vacuum. Is this correct? If so, is this because it takes some non-zero time for photons to be absorbed/remitted? Is this because of some backscattering? Is there some other phenomenon at work? PS: We're Bringing back Joel Spolsky's subreddit -- joel.reddit.com.


title: Refractive index variation with colors

text: To explain the spectrum formation in the prism, my teacher said that different colors have different speeds in a medium, so, refractive index is different for all. So, angle is different. But I don't think that the speed can be different, because, if this is the case, we shouldn't be able to see white light under water...what do you think is right? Also, what is the reason for different deviation angles?


title: Virtual Work: How is the applied force related to the coordinates chosen?

text: I have a question after reading a section from Goldstein's Classical Mechanics. The question deals with equation 1.43 in the text (given below): $$ \tag{1.43} \sum\limits_{i} {\bf F}_i^{(a)}\cdot \delta{\bf r}_i ~=~ 0.$$ Just below the equation in the text, Goldstein says > [...] _in general ${\bf F}_i^{(a)} \neq 0$, since the $\delta {\bf r}_i$ are > not completely independent but are connected by the constraints. In order to > equate the coefficients to zero, we must transform the principle into a form > involving the virtual displacements of the $q_i$, which are independent._ I do not understand what the fact that the $\delta {\bf r}_i$ are not completely independent has to do with the applied force ${\bf F}_i^{(a)}$. Moreover, I'd like to see how transforming into generalized coordinates $q_i$ can send the applied forces to zero.


title: Dna Fingerprint

text: I recently came across an article indicating that the half life of DNA in the most ideal situations is 521 years (http://www.nature.com/news/dna- has-a-521-year-half-life-1.11555). However, since human's DNA is 99.9% identical (and 98% identical to chimps), I can't quite understand how DNA evidence is often used in old samples. There has been DNA extracted from 100,000 year old Neanderthals, and DNA evidence has been used in cases over 50 years old. However, with a 521 year half life, after only 0.75 years, there would only be 99.9% of the DNA left. Surely if 99.9% of a human's DNA is identical after only this short amount of time wouldn't it be impossible to distinguish any sample from any other? After 50 years only 93% would remain the same, meaning it would be much less similar to a human than just any chimp I would think. And the 100,000 year old Neanderthal would only have 1.6*10^-57 % of the original sample left. How is there possibly any sort of useful information left? All in all, how can DNA be distinguished from other samples when they are all so similar and the half life is so low?


title: How to derive Eq.(1) in Prof. A Leggett's lecture note?

text: I came across a lecture note on weak localisation by Prof. A. J. Leggett. In this note, he wrote down his first equation as follows: $$\sigma = \lim_{\omega\rightarrow 0} \lim_{q\rightarrow 0} \frac{1}{i\omega}\frac{\delta j(q\omega)}{\delta A(q\omega)} = \lim_{\omega\rightarrow 0} \lim_{q\rightarrow 0}\frac{1}{i\omega}\langle[j_q(t),j_q(0)]\rangle_{\omega}\\\ = \int^{\infty}_0 dt\int\int\frac{dr dr'}{V}\langle[j(rt),j(r'0)]\rangle.$$ I understand the first two equalities. But how do I prove the third one? Your help is greatly appreciated.


title: What is the expected distance of the electron from the nucleus in the hydrogen atom?

text: Specifically, I would like to know the general formula, in terms of $n$ and $l$, assuming the electron is in an orbital (i.e. simultaneous eigenstate of $H$, $L^2$, and $L_z$). I understand that it involves integrating an associated Laguerre polynomial, but I wasn't able to find the formula for the integral. I looked on Wikipedia and in Abramowitz & Stegun, but no luck.


title: Newton's Third Law Clarification

text: Assuming you place an object so heavy on a table that it breaks it, then according to newtons third law the forces must cancel out (equal magnitude and opposite direction), but if this is true, then how can the object break the table in the first place?


title: Two suns, one moon, and one planet?

text: I have a question about how would seasons and the moon cycle be affected in a system where one planet orbits Sun #1, and Sun #1 orbits a second sun. Online I found this description: > "Type II: "Close Stars" - The suns may occupy different parts of the sky, > indicating that one sun orbits the other farther out than the planet." (I > had a sketch, but didn't know how to upload it) To make it easier to explain I'm going to assume this system's Sun #1, planet, and moon are equal mass and size of Earth, it's moon and it's sun. The planet is in a habitable zone of Sun #1 but outside the gravitational pull of the second sun. So I guess my questions are:   * How would the moon cycle be affected when the planet orbits between both suns? Would there be times when it seemed like the sun never set or would days be twice as short, because there is a sun on both sides of the planet for 3-4 months out of the year?   * Would the same situation described above cause a significant shift in seasons(summer, winter,spring) when the planet is between two suns?   * Does the size of a sun, or it's mass more dictate it's gravitational pull? Is it even possible for a system such as this to survive or would it eventually get stuck in between both suns gravitational pull, or switch to a figure 8 orbit, and eventually be thrown out of both suns orbit.


title: What is the Landé g factor?

text: What is the Landé g factor? I know that it gives the relation between magnetic moment and angular moment, but i wanted to know why are those magnitudes related to each other and why is the magnetic moment operator defined proportional to he angular moment and how is the gyromagnetic ratio calculated theoretically.


title: How does Infrared Relate to heat?

text: I never understood the relationship between Infrared and Heat. Is IR emitted when heat is generated, is heat generated when IR is made, and how do the two relate?


title: Minimal coupling of an atom to the EM field

text: The Hamiltonian of an atom coupled to an EM field, both described quantum mechanically is: $$H = \frac{1}{2m}(\hat{p}-q\hat{A})^2 = \frac{\hat{p}^2}{2m} -\frac{q}{m}\hat{p}\hat{A}+\frac{q^2}{2m}\hat{A}^2$$ Under the condition of transversality ($\hat{p}$ and $\hat{A}$ are vectors). I have only seen it treated in the dipole approximation where the last term becomes a constant and so unphysical; but what is its general interpretation? Is it shifting the energy of the free photons, it doesn't look diagonal in number of occupation basis.