title: Any examples of commensurable subgroups appearing in physics?

text: I am a mathematician. I am studying and working on Hecke pairs which I am going to give the related definitions in the following. But first let me explain what I am looking for to learn by asking this question. I have observed that the following algebraic notions naturally appears in many different areas of mathematics such as number theory, representation theory, geometric group theory, hyperbolic 3-manifolds, etc. Even Hecke pairs appear in the Bost-Connes quantum statistical approach to class field theory of $\mathbb{Q}$. The basic definitions involved in Hecke pairs are rather elementary, so I suspect these notion may have appeared in various areas of theoretical physics as well. The things is people (even in mathematics) use different terminology to describe these notions, so one cannot Google to find all instances of Hecke pairs in other sciences. > Therefore I give several elementary definitions about Hecke pairs, to see if > they ring any bells and hopefully to get some clues how they appear in > physics. Two subgroups $H$ and $K$ of a (discrete) group $G$ are called **commensurable** if $H\cap K$ is a finite index subgroup of both $H$ and $K$. One easily checks that commensurability is an equivalence relation among subgroups. A subgroup $H$ of a group $G$ is called a **Hecke subgroup** of $G$ if $H$ is commensurable with $gHg^{-1}$ for all $g\in G$. One observes that this condition is milder than normality condition for subgroups, so some author refer to $H$ as an almost normal subgroup of $G$. There are also at least two more name for the same notion! Let me give one more definition. Given a subgroup $H$ of $G$, we define the **commensurator of $H$ in $G$** by the following formula: $$ \operatorname{Comm}_G(H) := \\{ g\in G\ |\ H\ \mathrm{and\ }gHg^{-1}\ \mathrm{are\ comensurable} \\} $$ Obviously, $H$ is always a Hecke subgroup of $Comm_G(H)$, and so $H$ is a Hecke subgroup of $G$ if $G=\operatorname{Comm}_G(H)$.


title: Linearized mass conservation equation

text: I'm working on global seismology and I'm currently facing troubles understanding how an equation is obtained. The equation concerned is the following one : $$ \rho^{E1} = -\nabla \cdot (\rho^0\mathbf{s}) $$ (From the book Theoretical Global Seismology, Princeton Press : Google Book Link equation (3.46) p.65) Where $\rho^{E1}$ is the first order perturbation of the density and $\rho^0$ the density in the reference state. $\mathbf{s}$ is the displacement field. It comes from the mass conservation equation **which has been integrated in time** : $$ \partial_t\rho^E + \nabla \cdot (\rho^E \mathbf{u}^E) = 0 $$ and the following decomposition : $$ \rho^E(\mathbf{x}, t) = \rho^0(\mathbf{x}) + \rho^{E1}(\mathbf{x}, t) $$ Where $\rho^0$ is independent of time and $\mathbf{u}$ is the velocity field. How do you obtain the first equation from the second one ? If someone could help me out on this one it's be really nice ! Cheers


title: Can gamma rays be used to investigate heavenly bodies?

text: This is a very layman level question. I watch a lot of documentaries, that's my only qualification. I have seen in documentaries that gamma rays have an extraordinary penetrating power. So if we could have a gamma ray gun on one end of planet Jupiter and something analogous to an X-Ray film of the other end, would it be possible to take an X-Ray ? Can or Is this penetrating property of gamma rays being used in science?


title: Quantum teleportation - the alternative to destroying atoms

text: Can't you just disassemble, not destroy our atoms and transport them to another teleporter via networking or telecoms? This is my thinking of how to keep the same person, not an exact copy of him/her, alive when teleporting. Please answer to tell me if it's possible or not.


title: Accelerated charged particles produce electromagnetic radiation, but holes (the charge carriers) do not. Is this correct?

text: Holes are treated as particles in solid-state physics, so I've had some trouble with reasoning through this properly.


title: Why is the candela dimension J, not W?

text: According to the table at the bottom of the Wikipedia page for the candela, the dimension for candelas is J (joules). Why is this not W (watts)? > The luminous intensity for light of a particular wavelength λ is given by > > $I_v(\lambda) = 683.002·\bar{y}(\lambda)·I_e(\lambda)$ > > where $I_v(λ)$ is the luminous intensity in candelas, $I_e(λ)$ is the > radiant intensity in W/sr and is the standard luminosity function. Since $\bar{y}(\lambda)$ seems to be unitless and between 0 and 1, and $I_e(\lambda)$ is expressed in W/sr, why is $I_v$ not also W/sr (or J/S since sr is dimensionless)?


title: Laser Coherence Length/Time

text: Scenario: Imagine a collimated beam of white light falling on one refracting face of a prism. Let the light emerging from the second face be focused by a lens onto a screen. Suppose there is linear dispersion at the screen. Now by introducing a narrow "exit slit" (on the order of 0.01cm) in the screen, one has a type of monochromator that provides a nearly monochromatic beam of light. The white light has a mean wavelength of 500nm. Questions: What is meant by linear dispersion at the screen. Furthermore how does the slit effect the coherence length and coherence time of the light.


title: Why can't we obtain a Hamiltonian by substituting?

text: This question may sound a bit dumb. Why can't we obtain the Hamiltonian of a system simply by finding $\dot{q}$ in terms of $p$ and then evaluating the Lagrangian with $\dot{q} = \dot{q}(p)$? Wouldn't we obtain then a Lagrangian expressed in terms of $t$, $q$ and $p$? Why do we need to use $$H(t, q, p) = p\dot{q} - L(t, q, \dot{q})?$$ Or is it that whatever the Lagrangian is the method of finding $\dot{q}=\dot{q}(t,q,p)$. Will give us that equation for $H$?


title: Sign in the photon propagator

text: The Klein Gordon propagator is given (I use Peskin and Schroeder's conventions, if it matters...), \begin{equation} \frac{ i }{ p ^2 - m ^2 + i \epsilon } \end{equation} The photon propagator (using Feynman gauge) is \begin{equation} \frac{ - i \eta^{\mu\nu}}{ k ^2 + i \epsilon } \end{equation} The time-like component of the photon field propagates with a different sign then the scalar field and the spatial components propagate with the same sign. Is there a physical significance to the difference in sign between the two or it is just a consequence of our conventions?


title: Gaussian Integrals : Functional determinant expressed as a trace

text: Be $A_{ij}$ a symmetric matrix. Then I can easily write $$ \int \exp\left(-\frac{1}{2}\sum_{i,j}x_i A_{ij} x_j+\sum_{i} B_i x_i\right)\; d^nx= \sqrt{(2\pi)^n}\exp\left\\{-\frac{1}{2}\mathrm{Tr}\log A\right\\}\exp\left\\{\frac{1}{2}\vec{B}^{T}A^{-1}\vec{B}\right\\}. $$ and this formula holds both in the finite and inifinite dimensional case. If I consider a functional integration, and a differential operator $\hat A$, the result is usually expressed as $$ \int \exp\left( - \frac 1 2 \varphi \hat A \varphi +J \varphi \right) D\varphi \; \propto \; \exp \left( {1\over 2} \int d^4x \; d^4y J\left ( x \right ) A^{-1}\left ( x - y \right ) J\left( y \right ) \right) $$ or in some cases as $$ \int \exp\left( - \frac 1 2 \varphi \hat A \varphi +J \varphi \right) D\varphi \; =\\\\\mathcal{N}\,\exp\left\\{-\frac{1}{2}\mathrm{Tr}\log \hat A \right\\}\, \exp \left( {1\over 2} \int dx \; dx^\prime J\left ( x \right ) A^{-1}\left ( x - x^\prime \right ) J\left( x^\prime \right ) \right) $$ where $\varphi \hat A \varphi = \int dx dx^\prime \varphi(x) A(x,x^\prime) \varphi(x^\prime) $ and $J\varphi = \int dx \varphi(x) J(x) $. A more complicated situation I have to face now would be $$ \int D\psi D\varphi \exp\left\\{ - \frac 1 2 \left[ \varphi \hat A \varphi + \psi \hat B \psi \+ \varphi \psi \hat C \psi \varphi \right] \right\\} $$ where $\varphi \psi \hat C \psi \varphi=\int dx dx^\prime \varphi(x) \psi(x) C(x,x^\prime) \psi(x^\prime) \varphi(x^\prime)$ where $\psi$ is a Grassman-odd field. Anyway, the problem should be the same in the case the $\psi$ was a simple Grassman-even field. My idea in this case would be rewriting it as $$ \int D\psi \exp\left\\{ - \frac 1 2 \psi \hat B \psi \right\\} \int D\varphi \exp\left\\{ - \frac 1 2 \varphi \left[ \hat A \+ \psi \hat C \psi \right]\varphi \right\\} $$ in such a way to obtain something similar to $$ \int D\psi \exp\left\\{ - \frac 1 2 \psi \hat B \psi \right\\} \exp \left\\{ - \frac{1}{2} \mathrm{Tr} \log \left[ \hat A \+ \psi \hat C \psi \right]\right\\} $$ but I don't know how to treat the Trace in this case. Would this procedure be acceptable, and how should I deal with the trace operator?


title: Homotopy $\pi_4(SU(2))=\mathbb{Z}_2$

text: Recently I read a paper using $$\pi_4(SU(2))=\mathbb{Z}_2.$$ Do you have any visualization or explanation of this result? More generally, how do physicists understand or calculate high dimension homotopy group?


title: "Inverted" quantum oscillator

text: I'm trying to understand the problem of the "inverted" oscillator, which has the following Hamiltonian: $$ \hat{H}=\frac{\hat{p}^{2}}{2m}-\frac{k\hat{x}^{2}}{2} $$ Suppose that a particle at the initial time is in a state $$ \Psi(x)=\frac{1}{(\pi b^{2})^{\frac{1}{4}}}\exp^{-x^{2}/2b^{2}} $$ We want to find $\langle\Psi\mid\hat{x}(t)\mid\Psi\rangle$ and $\langle \Psi \mid \hat{x}^{2}(t) \mid \Psi \rangle$ and their asymptotic when $t \rightarrow +\infty$. Heisenberg equation for operator $\hat{x}$ (we use that $\hat{x}$ does not depend explicitly on time, $[\hat{x},\hat{x}]=0$, $[\hat{x},\hat{p}]=i\hbar$ and Leibniz rule:$[A,BC]=[A,B]C+B[A,C]$): $$ i\hbar \dot{\hat{x}}=\left[\hat{x},\hat{H}\right]=\left[\hat{x},\frac{\hat{p}^{2}}{2m}-\frac{k\hat{x}^{2}}{2}\right]=\left[\hat{x},\frac{\hat{p}^{2}}{2m}\right]=\frac{\hat{p}}{m}\left[\hat{x},\hat{p}\right]=i\hbar\frac{\hat{p}}{m} $$ $$ \dot{\hat{x}}=\hat{p}/m $$ Similarly, Heisenberg equation for operator $\hat{p}$: $$ i\hbar\dot{\hat{p}}=\left[\hat{p},\hat{H}\right]=\left[\hat{p},\frac{\hat{p}^{2}}{2m}-\frac{k\hat{x}^{2}}{2}\right]=-\left[\hat{p},\frac{k\hat{x}^{2}}{2}\right]=-k\hat{x}\left[\hat{p},\hat{x}\right]=i\hbar k\hat{x} $$ $$ \dot{\hat{p}}=k\hat{x} $$ So, we have following set of equations: $\dot{\hat{x}}=\hat{p}/m$; $\dot{\hat{p}}=k\hat{x}$. Using them, we can write: $$ m\ddot{\hat{x}}=k\hat{x} $$ Solution of this differential equation is $\hat{x}=A\cosh(\omega t)+B\sinh(\omega t)$, where $\omega=\sqrt{\frac{k}{m}}$. Initial conditions: $A=\hat{x}(0)$; $B=\frac{\dot{\hat{x}}(0)}{\omega}$. Remember that $\dot{\hat{x}}=\hat{p}/m$. Then $B=\frac{\hat{p}(0)}{m\omega}$. So we have $$ \hat{x}=\hat{x}(0)\cosh(\omega t)+\frac{\hat{p}(0)}{m\omega}\sinh(\omega t) $$ Then $$ \hat{x}^{2}=\hat{x}^{2}(0)\cosh^{2}(0)+\frac{\hat{p}^{2}(0)}{(m\omega)^{2}}\sinh^{2}(0)+\frac{1}{m\omega}(\hat{x}(0)\hat{p}(0)+\hat{p}(0)\hat{x}(0))\sinh(\omega t)\cosh(\omega t) $$ Now we want to calculate $\langle\Psi\mid\hat{x}\mid\Psi\rangle$ and $\langle \Psi \mid \hat{x}^{2} \mid \Psi \rangle$: $$ \langle\Psi\mid\hat{x}(t)\mid\Psi\rangle=\langle\Psi\mid\hat{x}(0)\cosh(\omega t)+\frac{\hat{p}(0)}{m\omega}\sinh(\omega t)\mid\Psi\rangle=\langle\Psi\mid\hat{x}(0)\mid\Psi\rangle\cosh(\omega t)+\langle\Psi\mid\hat{p}(0)\mid\Psi\rangle\frac{\sinh(\omega t)}{m\omega} $$ Similarly $$ \langle \Psi \mid \hat{x}^{2}(t) \mid \Psi \rangle=\langle\Psi\mid\hat{x}^{2}(0)\mid\Psi\rangle\cosh^{2}(\omega t)+\langle\Psi\mid\hat{p}^2(0)\mid\Psi\rangle\frac{\sinh^{2}(\omega t)}{(m\omega)^{2}}+\frac{1}{m\omega}\langle\Psi\mid(\hat{x}(0)\hat{p}(0)+\hat{p}(0)\hat{x}(0))\mid\Psi\rangle\sinh(\omega t)\cosh(\omega t) $$ Lets introduce the notation:$x_{0}=\langle\Psi\mid\hat{x}(0)\mid\Psi\rangle$, $p_{0}=\langle\Psi\mid\hat{p}(0)\mid\Psi\rangle$,$x_{0}^{2}=\langle\Psi\mid\hat{x}^{2}(0)\mid\Psi\rangle$, $p_{0}^{2}=\langle\Psi\mid\hat{p}^{2}(0)\mid\Psi\rangle$. Then: $$ \langle\Psi\mid\hat{x}(t)\mid\Psi\rangle=x_{0}\cosh(\omega t)+\frac{p_{0}}{m\omega}\sinh(\omega t) $$ $$ \langle\Psi\mid\hat{x^{2}}(t)\mid\Psi\rangle=x_{0}^{2}\cosh^{2}(\omega t)+p_{0}^{2}\frac{\sinh^{2}(\omega t)}{(m\omega)^{2}}+\frac{1}{m\omega}\langle\Psi\mid(\hat{x}(0)\hat{p}(0)+\hat{p}(0)\hat{x}(0))\mid\Psi\rangle\sinh(\omega t)\cosh(\omega t) $$ **Here is the first big trouble** : i've found some lecture notes in the internet, where they say that $\langle\Psi\mid(\hat{x}(0)\hat{p}(0)+\hat{p}(0)\hat{x}(0))\mid\Psi\rangle=0$ for real $\Psi(x)$. I tried to prove this statement, but i can't. **Can you explain why it is equal to zero?** Well, suppose that this statement is true. Then: $$ \langle\Psi\mid\hat{x}(t)\mid\Psi\rangle=x_{0}\cosh(\omega t)+\frac{p_{0}}{m\omega}\sinh(\omega t) $$ $$ \langle\Psi\mid\hat{x^{2}}(t)\mid\Psi\rangle=x_{0}^{2}\cosh^{2}(\omega t)+p_{0}^{2}\frac{\sinh^{2}(\omega t)}{(m\omega)^{2}} $$ Thus, we expressed values through their meanings at the initial time. So, lets find $x_{0}$, $p_{0}$, $x_{0}^{2}$, $p_{0}^{2}$. From this point we will work in the coordinate representation. $$ x_{0}=\langle\Psi\mid\hat{x}\mid\Psi\rangle=\int\Psi^{\ast}x\Psi dx=\int\Psi x\Psi dx=\int\Psi^{2}xdx $$ Remember that $\Psi(x)=\frac{1}{(\pi b^{2})^{\frac{1}{4}}}\exp^{-x^{2}/2b^{2}}$ : $$ x_{0}=\frac{1}{(\pi b^{2})^{\frac{1}{2}}}\int x\exp^{-x^{2}/b^{2}}dx $$ We integrate in domains from minus infinity to plus infinity. But this integral is equal to zero! So $$ x_{0}=0 $$ Similarly: $$ p_{0}=\langle\Psi\mid\hat{p}\mid\Psi\rangle=-i\hbar\int\Psi^{\ast}\frac{\partial}{\partial x}\Psi dx=\frac{2}{b^{2}}i\hbar\frac{1}{(\pi b^{2})^{\frac{1}{2}}}\int x\exp^{-x^{2}/b^{2}}dx $$ The domains of integration are the same:from minus infinity to plus infinity. But this is also is equal to zero! So, we have : $$ \langle\Psi\mid\hat{x}(t)\mid\Psi\rangle=0\cosh(\omega t)+\frac{0}{m\omega}\sinh(\omega t)=0 $$ **I'm confused by this result! Is there a mistake?** If not, can you explain me this result? Now we are going to calculate $x_{0}^{2}$ and $p_{0}^{2}$: $$ x_{0}^{2}=\langle\Psi\mid\hat{x}^{2}\mid\Psi\rangle=\int\Psi^{\ast}x\Psi dx=\frac{1}{(\pi b^{2})^{\frac{1}{2}}}\int x^{2}\exp^{-x^{2}/b^{2}}dx=\frac{1}{(\pi b^{2})^{\frac{1}{2}}}\frac{\sqrt{\pi} (b^{2})^{\frac{3}{2}}}{2}=\frac{b^2}{2} $$ $$ p_{0}^{2}=\langle\Psi\mid\hat{p}^{2}\mid\Psi\rangle=-\hbar^{2}\int\Psi^{\ast}\frac{\partial^{2}}{\partial x^{2}}\Psi dx=\frac{\hbar^{2}}{2b^{2}} $$ Finally: $$ \langle\Psi\mid\hat{x^{2}}(t)\mid\Psi\rangle=\frac{b^2}{2}\cosh^{2}(\omega t)+\frac{\hbar^2}{2b^{2}}\frac{\sinh^{2}(\omega t)}{(m\omega)^{2}} $$ Using the equation $\cosh^{2}(x)-\sinh^{2}(x)=1$ we get: $$ \langle\Psi\mid\hat{x^{2}}(t)\mid\Psi\rangle=\frac{1}{2}\left(b^{2}+\frac{\hbar^{2}}{b^{2}(m\omega)^{2}}\right)\cosh^{2}(\omega t)-\frac{\hbar^{2}}{2b^{2}(m\omega)^{2}} $$ Thus we see that when $t \rightarrow +\infty$ dispersion of coordinate $\langle\Psi\mid\hat{x^{2}}(t)\mid\Psi\rangle$ tends to infinity. To sum up, I have these several difficulties:   1. why $\langle\Psi\mid(\hat{x}(0)\hat{p}(0)+\hat{p}(0)\hat{x}(0))\mid\Psi\rangle=0$ for real $\Psi(x)$?   2. why $\langle\Psi\mid\hat{x}(t)\mid\Psi\rangle=0$?   3. why when $t \rightarrow +\infty$ $\langle\Psi\mid\hat{x^{2}}(t)\mid\Psi\rangle$ tends to infinity? Also i would be happy if you provide some articles(or books) where i can read about it.


title: Exact diagonalization of graphene's tight binding Hamiltonian

text: While directly diagonalize graphene's tight binding Hamiltonian, which is numerical. We have to use a finite-sized graphene. So how to deal with boundary conditions? The usual solutions are zigzag or armchair condition, but to make our model more realistic to real infinite graphene plane, how about using periodic condition at boundaries while exactly diagonalizing the tight binding Hamiltonian?


title: Constructing the exponential form of a unitary operator

text: I think I've got this figured out but wanted to make sure I'm doing this right. Working with operators that satisfy bosonic commutation relations $[b,b^\dagger] = 1$, I define a very general unitary transformation on them: $$\gamma^\dagger = ub^\dagger + vb + w$$ where the coefficients $u$, $v$, and $w$ are real. My goal is to find the unitary operator $U$ written in the form $e^S$ where $S$ is anti-Hermitian, so that $\gamma^\dagger = e^S b^\dagger e^{-S}$. Here's my attempt at this: The constant $w$ is straightforward using the expansion of $e^{S} A e^{-S}$ in terms of the commutator, i.e. $$e^{S} A e^{-S} = A + [S,A] + \frac{1}{2}[S,[S,A]] + \dots $$ and we see immediately that setting $S = w(b-b^\dagger)$ satisfies this. The more interesting part comes next: The requirement that the transformation is unitary is equivalent to demanding that $[\gamma, \gamma^\dagger] = 1$ as well. This leads to the requirement $u^2 - v^2 = 1$, which in turn means that we could write these coefficients as $$u = \cosh(x), \quad v = \sinh(x)$$ for some parameter $x$. The _infinitesimal_ transformation, $x = \epsilon \ll 1$, would then read, up to first order in $\epsilon$: $$\gamma^\dagger = b^\dagger + \epsilon b$$ Comparing this with the commutator-expansion above, we now just need to find an anti-Hermitian operator whose commutator with $b^\dagger$ is $b$. We don't even worry about the higher orders of the expansion since those are second order in $\epsilon$, but here they even vanish exactly: We simply set $$S = \frac{\epsilon}{2}(b^2 - (b^\dagger)^2)$$ since the commutator $[b^2, b^\dagger] = 2b$, and $[b,b]=0$. I think then that I'm done: For the full transformation, I just set $$S = w(b-b^\dagger) + \frac{x}{2}\left(b^2 - b^{\dagger, 2}\right), \quad \cosh(x) = u$$ This should be true if it's true that in order to reach "angle" $x$ with our "rotation", we just have to rotate $N$ times with "angle" $x/N$, and hence in the limit $N \rightarrow \infty$ we can use the infinitesimal generator. I know this argument works for rotations in space, where instead of $\cosh$ and $\sinh$ we'd be working with $\cos$ and $\sin$, but I'm sure this also works for the hyperbolical functions. EDIT: The final question, now, is: Is the above reasoning sound?


title: Bloch oscillations - Scattering to other bands

text: In the free electron approximation, a Bloch state $|k\rangle$ is the linear superposition of free plane wave states $\sum_G C_G(k) |k+G\rangle$, where $G$ are the conjugate lattice. Since the coefficients $C_G$ varies with $k$, under an electric field the electron will not merely shift inside the same band, but will scatter to other bands. Is this effect significant in practice? Also, does the total scattering loss during one complete Bloch oscillation tend to zero as the applied electric field tends to zero?


title: Why the pion does not get mass under Spontaneus breaking of chiral symmetry, but the quarks do?

text: Some sources state that when the mass of a quark goes to zero, it allows for Spontaneous Breaking of Chiral Symmetry and gets a constituent mass of about $200\, \mathrm{MeV}$. Other sources state that when the masses of the light quarks go to zero, so does the pion mass. In this case, the explicit breaking of chiral symmetry disappears, but the spontaneous chiral symmetry breaking is triggered too. So, in this limit, is the pion massless but their quarks carry a $200\, \mathrm{MeV}$ mass? What is wrong, or misunderstood, here?


title: Equivocal boundary conditions for Laplace equation of 2D "V"-shape conductor

text: A two dimensional infinite "V"-shape wedge conductor is earthed, wherein $\beta$ is the intersection angle. We can solve Laplace equation so as to get the electric potential inside the "V" zone, as is shown in the figure. $a_n$'s are actually determined by some other boundary condition far away from origin and we just preserve $a_1$ here. It is noteworthy that we make use of the equipotential condition (the 1st yellow formula), consequently, we have the exponents of $r^\nu$ to be $\nu=\frac{n\pi}{\beta}$. However, how about adding an additional $2\pi$ periodic boundary condition (which seemes to be physically reasonable here)? If we did it, then the exponents of $r^\nu$ ought to be $\nu=\text{integer}$. The form of solution appears to be rather different from the original one in the figure. Are they equivalent, or at least equivalent inside the "V" zone? Or is this wrong? ![enter image description here](http://i.stack.imgur.com/72z5L.jpg)


title: Can space be torn?

text: If space is warped by objects in space, and black holes are made of infinitely heavy objects, can space be torn by black holes?


title: Do photons and cosmic rays radiate energy through gravitational waves? If not, why not?

text: Due to the mass-energy equivalence, both matter and EM radiation bend spacetime, and both are capable of forming singularities (black hole, white hole/kugelblitz). In light of this, why do photons traveling from the most distant reaches of the observable universe not lose energy due to the gravitational radiation they must emit? Furthermore, mustn't cosmic rays (e.g. protons) slow down or stop as they lose energy through the same mechanism?


title: Why does a light wave invert at a boundary with greater index of refraction?

text: Is there a reason why a EM wave reflects invertedly when it meets a boundary point with a greater index of refraction. In the case of ropes, if remember correctly, the reason why it inverts is to agree with Newtons third law. But, its unclear why for EM waves, upon reflection, a phase change of $\pi$ occurs.


title: Can spin be infinite?

text: Can spin of a particle or a group of particles become infinity? Explain plz. Is there any representation for spins like dot(for $S=0$) and arrow (for $S=1$)? If so what for $S= \infty$?


title: Is an electron attracted to one of the magnetic poles in this scenario?

text: Do magnets attract electrons? I don't think so, but maybe in certain cases they can be? I guess it would depend on what direction the velocity and magnetic field is in so that the force acting on the charge is towards one of the poles but I haven't tried that yet. If an electron is shot between a north pole and a south pole, is it attracted to any of them? Well the resulting force would either be into or out of the page (right?) so it wouldn't be attracted to any. Can anyone confirm/explain this if I'm wrong?


title: Radiation Resistance

text: When a charge is accelerated, it radiates and loses kinetic energy. This can be modeled by having another force act on the charge, which is proportional to the derivative of the acceleration. So if this force is included the equations of motion, we end up with 3rd order equations. Is this allowed? Why don't we need to fix another constant to determine the motion? Is it because we usually neglect this force because of its small magnitude? Also, the electromagnetic Lagrangian only contains up to first derivatives (of the 4-potential in this case). How can this result in 3rd order equations of motion?


title: Quantum Entanglement: how to generate 2 entangled particles?

text: I get quantum entanglement but I don't quite get how one would go about generating two complementary particles that are entangled (a photon and its entangled sibling, an electron and its entangled sibling, etc.). Can anyone explain this referencing how this could be achieved in a lab? Thanks!


title: What controls the heating ability of steamers?

text: Pictured below is a standalone steamer used for foaming and heating milk for milk-based coffee drinks like cappuccinos and lattes. They can be filled with water up to the pressure relief incorporated in the handle which is about 32 US fluid oz (1L) worth. Once the pressure relief valve starts kicking in, a cup of cold milk can be steamed to about 150°F/65°C in about 60-75 seconds. Assuming the steamer is already on the most power gas burner, can: a) The 60-75 seconds it takes the milk to be steamed to 150°F/65°C be lowed?, and b) The time it takes the whole process, including initially boiling the water, be lowered? For example: a) Would the water level make a difference? b) How about the amount of steam let through the wand (controllable by the round knob)? c) How about the size of the hole at the end of the wand that the steam comes out of? Would more holes or different diameters make a difference? ![enter image description here](http://i.stack.imgur.com/2cAQy.jpg)


title: In Klein-Gordon, why should infinite downwards photon cascades be possible?

text: Here is a simple point about the standard interpretation of the Klein-Gordon equation that for the life of me I've never been able to understand: > Why would the existence of true negative energy states result in an infinite > downward cascade of photon emissions? I believe it was Dirac who originated this concept, but no matter how I turn it around in my head, I don't understand the logic of it. A suite of true negative energy particle solutions to Klein-Gordon (not antimatter, which experimentally shows positive energy) would by exact symmetry behave identically to their positive energy equivalents. Thus they would radiate negative energy photons until they reach their own ground states. So, if viewed from that true negative energy perspective, a downward cascade of positive energy photons would look like an ordinary electron raising its energy indefinitely by emitting negative energy photons. But we don't see that from within out positive particle perspective, so why should we expect the downward version to exist, either? Is there some critical conceptual and/or mathematical rationale that I'm missing for why infinitely-downward photon cascades should exist? I guess by implication that I may also be asking this: Why is Klein-Gordon not simply interpreted as having four solutions, rather than just two? They would include the known positive energy "pro" and "anti" matter solutions that share positive energy photons, but also a set of true negative energy versions of matter, antimatter, and photons. Surely this idea has at least been explored and then dismissed for some specific reason? I've looked some, but have not had any luck finding references. To balance out the total energy of the universe, most current theories seem to assign the negative energy to space itself, again for reasons I truly do not understand. Such an approach leads unavoidably to complex balancing acts of trying to get the very differently structured positive and negative energies always to match in magnitude. Wouldn't it be both easier and simpler to have a true, exactly symmetric negative-energy universe scurrying away from us in the $t^-$ direction, with space simply as the zero-energy balancing point between the two? Again, there must be some specific analysis that led to the current reliance on the more complicated "no true negative energy particles allowed" approach to balancing the energy of the universe (multiverse really), but I've had no luck uncovering it. What am I missing? Who made the decision to go to negative-energy-space balancing, and why? * * * **2013-02-26 - Some relevant references** To my surprise, Paul Dirac's 1933 Nobel Lecture does a clear and remarkably succinct job of explaining his logic not just about negative energy states, but how he wound up at antimatter by contemplating them. At one point I immediately thought of tachyons. After checking a bit, sure enough, those ideas later became at least an aspect of tachyon theory. Odd how that feature of his negative energy sea is almost never mentioned. As you can likely guess from the tachyon connection, Dirac interpreted the kinetics of negative energy states in an intriguing way that I was not expecting at all: > If one looks at Eq. (1), $[\frac{W^2}{c^2} - p_r^2 - m^2c^2]\psi = 0$, one > sees that it allows the kinetic energy $W$ to be either a positive quantity > greater than $mc^2$ or a negative quantity less than $-mc^2$. This result is > preserved when one passes over to the quantum equation (2), $[\frac{W}{c} - > {\alpha_r}p_r - {\alpha_0}mc]\psi = 0$, or (3) {$\alpha_\mu=I$; > $\alpha_{\mu}\alpha_{\nu} + \alpha_{\nu}\alpha_{\mu} = 0$; > $\alpha_{\mu}\neq\alpha_{\nu}$; $\mu,\nu = 0, 1, 2, 3$}. These quantum > equations are such that, when interpreted according to the general scheme of > quantum dynamics, they allow as the possible results of a measurement of $W$ > either something greater than $mc^2$ or something less than $-mc^2$. > > Now in practice the kinetic energy of a particle is always positive. We thus > see that our equations allow of two kinds of motion for an electron, only > one of which corresponds to what we are familiar with. The other corresponds > to **electrons with a very peculiar motion such that the faster they move, > the less energy they have, and one must put energy into them to bring them > to rest.** (I must remark that it is delightful to observe the little mutations in notation, with $W$ in place of $E$ and $\alpha$ in place of $\gamma$. Einstein's famous $E=mc^2$ is another example; it looked quite different in the original paper.) Seeing the original version also reminds me of the remarkable number of choices that Dirac had to make to get to his matrices. The reasons for some of his choices were obvious, but for others they were not. In any case, his choice of an inverted-velocities interpretation of the kinematics provides some opportunity for understanding his original logic more meaningfully.


title: D'Alembert's Principle: Where does $-Q_j$ come from?

text: This is a follow-up question to D'Alembert's Principle and the term containing the reversed effective force. From the second term of Eq. (1.45) $$\begin{align*} \sum_i{\dot{\mathbf{p}}_i \cdot \delta{\mathbf{r}_i}} &= \sum_i{m_i\ddot{\mathbf{r}}_i \cdot \frac{\partial\mathbf{r}_i}{\partial q_j} \delta q_j} \end{align*}$$ Starting from Eq. (1.50), I was able to follow that $$\begin{align*} \sum_i{m_i\ddot{\mathbf{r}}_i \cdot \frac{\partial\mathbf{r}_i}{\partial q_j}} &= \sum_i{\left[ \frac{d}{dt} \left( m_i\mathbf{v}_i \cdot \frac{\partial\mathbf{v}_i}{\partial\dot{q}_j} \right) - m_i\mathbf{v}_i \cdot \frac{\partial\mathbf{v}_i}{\partial q_j} \right]} \end{align*}$$ Goldstein substituted the above equation to (1.45) by saying: > ... and the second term on the left-hand side of Eq. (1.45) can be expanded > to By "second term", I understood it to be the very first equation I mentioned above. Therefore this is how I understood it: $$\begin{align*} \sum_i{\dot{\mathbf{p}}_i \cdot \delta{\mathbf{r}_i}} &= \sum_i{m_i\ddot{\mathbf{r}}_i \cdot \frac{\partial\mathbf{r}_i}{\partial q_j} \delta q_j}\\\ &= \sum_i\left[ \frac{d}{dt}\left[ \frac{\partial}{\partial\dot{q}_j} \left( \frac{1}{2}\sum_i{m_iv_i^2} \right) \right] - \frac{\partial}{\partial q_j} \left( \frac{1}{2}\sum_i{m_iv_i^2} \right) - Q_j \right]\delta q_j\\\ &= \sum_i{\left[ \frac{d}{dt} \left( \frac{\partial T}{\partial\dot{q}_j} \right) - \frac{\partial T}{\partial q_j} -Q_j \right] \delta q_j} \end{align*}$$ I was able to follow $$\begin{align*} T &= \sum_i{m_i\mathbf{v}_i} \cdot \partial\mathbf{v}_i\\\ T &=\frac{1}{2} \sum_i{m_iv_i^2} \end{align*}$$ But I am at a loss: **Where does $-Q_j$ come from?**


title: What's the principle used in infant breath monitor devices?

text: Such a device consists of a sensor pad that is placed under a mattress: ![sesnor pad placing under a mattress](http://i.stack.imgur.com/OMVui.gif) What's the principle it uses to detect breathing?


title: Optical trapping problem

text: Can we make light slower by applying optical trapping (I mean applying laser beam to lower the speed of light)?


title: Oscillon and soliton

text: I want to know the major difference between oscillon and soliton in terms of radiating energy with respect to time and position. And what about their localization?


title: Why atomic bomb makes a mushroom?

text: Atomic bombs make mushroom. It's well-known and we've seen many images of atomic experiments around the world. My question is, why mushroom? Does TNT and Grenade also make mini-mushrooms? Is it a physical attribute of explosion, or something specific to atomic energy? Thanks


title: How to get an integral formula for the flux time derivative

text: $$\frac{d}{dt}\int \limits_{A} \mathbf B d \mathbf A = \int \limits_{A} \left( \frac{\partial \mathbf B}{\partial t} + \mathbf v (\nabla \cdot \mathbf B ) + [\nabla \times [\mathbf v \times \mathbf B ]\right)d\mathbf A, $$ where $d \mathbf A$ - infinitesimal element of the vector of a surface $\Sigma$ bounded by circuit $d\Sigma$ (look at the picture). ![1](http://i.stack.imgur.com/uscCQ.png) Who can help me to explain the derivation of this expression? It's not homework. $d\mathbf A $, as I understand, is equivalent to the "allocation" surface of segment $d \mathbf l$ of the curve $d \Sigma$ fo the time $dt$ for a motion with the speed of $\mathbf v$: $$ d \mathbf A = [d \mathbf l \times \mathbf v dt]. $$


title: Oberth Effect in deep space

text: Does the Oberth effect only apply when in orbit of a planet or would a rocket generate more and more thrust (if kept on) even in deep space? Wikipedia explains that the faster the rocket goes, the more useful energy it generates. But since speed depends from what point of reference you are measuring it, how can that be possible since the fuel doesn't know the rocket's speed?


title: Stopping distance of two objects with equal Kinetic Energy

text: I'm working on a problem regarding two objects with the same kinetic energy. > Two objects with masses of $m_1$ and $m_2$ have the same kinetic energy are > both moving to the right. The same constant force F is applied to the left > to both masses. If $m_1=4m_2$, the ratio of the stopping distance of $m_1$ > to that of $m_2$ is: I believe the key information here is that both objects have the same kinetic energy. This must mean that an equal amount of work (force $\times$ distance) must be applied to each object in order to bring it to a complete stop ($KE + q = 0$)? Is the correct answer $1:1$?


title: Books describing motion of objects in fluid

text: I'm looking for any resources that would help me model the behavior of objects moving in fluid. My end goal is to be able to describe the motion of irregularly shaped objects in a river environment. Any recommendations are welcome.


title: I can't figure out crystal planes with negative intercepts

text: ![enter image description here](http://i.stack.imgur.com/OZeCm.jpg) As seen above, I don't follow how you figure out those planes. It seems they're not using the origin labeled. I'm not really sure I understand spatially what's going on in the left figure so let's focus only on the right figure. Why is one of the intercepts not really touching the xyz axes? Where is the origin in this thing?


title: Lambda transition data points of $\require{mhchem}\ce{^4He}$

text: I'm looking to get some data on the lambda transition of $\require{mhchem}\ce{^4He}$. I need the data points of the specific heat vs. temperature graph, if that makes sense.


title: Can we say there are 2 EM radiation types?

text: EM radiation seems to come from two different sources:   1. According to the Maxwell, by acceleration of electrons   2. According to the Bohr, by jumping of electrons between energy levels? Are these two, completely different things?


title: Why is $I_0\sin\left(\dfrac{\pi}{2}-\omega t\right)$ not used as a solution of the purely capacitive AC circuit?

text: Why is $I_0\sin\left(\dfrac{\pi}{2}-\omega t\right)$ not used as a solution of the purely capacitive circuit? **My Research** I've searched on Google, Phys.SE and many school level Physics books but couldn't find the answer. If the reader thinks that the question can be answered with a quick google search then please give me that link.


title: Singularity and evaporation of black hole

text: A question I have found that, in a question it is saying that what will be happened to the singularity when black hole evaporates. But recently I have checked that Stephen Hawking denied the singularity. Is there anythng wrong in my understanding?


title: What is the rotational velocity of Andromeda?

text: Can someone give me a reference which gives the rotation curve of the Andromeda galaxy (as well as its radius)?


title: Can two entanglement particles satisfy at same time two different wave functions?

text: The wave function determines the probabilities of an outcome, and the wave function is parameterized "shaped" depending on the measurement set. i.e. two different measurement set would lead to two different wave function description. Let's call A measurement set=>wave function A, and B measurement set=>wave function B. Having two entangled particles X and Y, there is something I can't understand, Supose we make successive measurement with A experimental set to successive generated entangled particles X : X, X',X'', X''' ... So measurements should match with the expected probabilities outcomes predicted by the wave function for A. Then the entangled counterpart measured with B experimental set to successive generated entangled particles Y : Y, Y',Y'', Y''' ... So measurements should match with the expected probabilities outcomes predicted by the wave function for B. How can the entangled particles satisfy both wave equation and at the same time, satisfy the entanglement condition? If it can't satisfy both, so decoherence of the entangled counterpart could be detected by a deviation of the outcomes and the expected probabilities. If it can satisfy both, then there is no effect in being entangled. Where is the problem? Thanks in advance for any help about understanding this


title: Help for density of charges in electric field

text: This is for a homework assignment so it would be great to have a hint or an explanation of how to do it. In a certain region in space, the electric field is given by $\vec{E}(r) = \dfrac{A}{r} \hat{r} + \dfrac{B \sin{\theta} \cos{\phi}}{r} \hat{\phi}$ What is the charge density in the region?


title: What happens if you connect a hot resistor to a cold resistor?

text: Kind of an extension to this question: If you heat up an object, and put it in contact with a colder object, in an ideal insulated box, the heat from one will transfer to the other through thermal conduction and they will eventually reach an equilibrium temperature at the midpoint, correct? Now if you have a hot resistor (electrical component) and a cold resistor, and connect them by their leads, so that they make a circuit: ![hot and cold resistors](http://i.stack.imgur.com/E1MCl.png) there will be the same conduction and radiation heat transfers. But also, the hotter resistor will have a larger noise current, right? So will there additionally be a transfer of electrical energy from one resistor to the other? Would completing the circuit allow them to reach equilibrium temperature faster than if they were just touching through an insulator with the same thermal conductivity?


title: Mathematically: What is SUSY?

text: Wikipedia says: > _In particle physics, supersymmetry (often abbreviated SUSY) is a symmetry > that relates elementary particles of one spin to other particles that differ > by half a unit of spin and are known as superpartners. In a theory with > unbroken supersymmetry, for every type of boson there exists a corresponding > type of fermion with the same mass and internal quantum numbers (other than > spin), and vice-versa. There is only indirect evidence for the existence of > supersymmetry [...]_ I want a mathematical explanation of SUSY.


title: Optical explanation of images of stars?

text: Very often when viewing pictures of the cosmos taken by telescopes, one can observe that larger/brighter stars do not appear precisely as points/circles on the image. Indeed, the brighter the light from star, the more we see this effect of four perpendicular rays "shooting" out from the star. ![Stars1](http://i.stack.imgur.com/LAA8M.jpg) (Taken from this page.) My question is: what are the optics responsible for this effect? I would suppose that both the hazy glow around the stars and the rays shooting outwards are optical effects created by the camera/imaging device, but really can't surmise any more than that. (The fact that the rays are all aligned supports this, for one.) A proper justification, in terms of (geometrical?) optics for both the glow and the rays would be much appreciated. Here are a few other examples of such images:   * http://grizzlyhugs.files.wordpress.com/2009/12/stars-sky-lg.jpg   * http://spacespin.org/images/articles/hubble-large-small-stars-harmonious_3.jpg


title: Could the Heisenberg Uncertainty Principle turn out to be false?

text: While investigating the EPR Paradox, it seems like only two options are given, when there could be a third that is not mentioned - Heisenberg's Uncertainty Principle being given up. The setup is this (in the wikipedia article): given two entangled particles, separated by a large distance, if one is measured then some additional information is known about the other; the example is that Alice measures the z-axis and Bob measures the x-axis position, but to preserve the uncertainty principle it's thought that either information is transmitted instantaneously (faster than light, violating the special theory of relativity) or information is pre-determined in hidden variables, which looks to be not the case. What I'm wondering is why the HUP is not questioned? Why don't we investigate whether a situation like this does indeed violate it, instead of no mention of its possibility? Has the HUP been verified experimentally to the point where it is foolish to question it (like gravity, perhaps)? # Edit It seems that all the answers are not addressing my question, but addressing waveforms/commutative relations/fourier transforms. I am not arguing against commutative relations or fourier transforms. Is not QM the theory that particles can be represented as these fourier transforms/commutative relations? What I'm asking this: is it conceivable that QM is wrong about this in certain instances, for example a zero state energy, or at absolute zero, or in some area of the universe or under certain conditions we haven't explored? As in: Is the claim then that if momentum and position of a particle were ever to be known somehow under any circumstance, Quantum Mechanics would have to be completely tossed out? Or could we say QM doesn't represent particles at {absolute zero or some other bizarre condition} the same way we say Newtonian Physics is pretty close but doesn't represent objects moving at a decent fraction of the speed of light? # Example **EPR Paradox:** "It considered two entangled particles, referred to as A and B, and pointed out that measuring a quantity of a particle A will cause the conjugated quantity of particle B to become undetermined, even if there was no contact, no classical disturbance." "According to EPR there were two possible explanations. Either there was some interaction between the particles, even though they were separated, or the information about the outcome of all possible measurements was already present in both particles." These are from the wikipedia article on the EPR Paradox. This seems to me to be a false dichotomy; the third option being: we could measure the momentum of one entangled particle, the position of the other simultaneously, and **just know both momentum and position** and beat the HUP. However, this is just 'not an option,' apparently. # Clarification I'm not disputing that two quantities that are fourier transforms of each other are commutative / both can be known simultaneously, as a mathematical construct. Nor am I arguing that the HUP is indeed false. I'm looking for justification not just that subatomic particles can be models at waveforms under certain conditions (Earth like ones, notably), but that a waveform is the only thing that can possibly represent them, and any other representation is wrong. You van verify the positive all day long, that still doesn't disprove the negative. It is **POSSIBLE** that waveforms do not correctly model particles in all cases at all times. This wouldn't automatically mean all of QM is false, either - just that QM isn't the best model **under certain conditions**. Why is this not discussed?


title: Is the normal force a conservative force?

text: Most of the time the normal force doesn't do any work because it's perpendicular to the direction of motion but if it does do work, would it be conservative or non-conservative? For example, consider the following block-incline system where the incline is on a friction-less surface. Now as the block moves down, the incline itself starts moving in the opposite direction. Here, the normal force acting on the incline by the block does do work. ![Block-incline](http://i.stack.imgur.com/35dSa.png) This question was on some website (it asked something about the final velocities of the block and the incline) and they solved this using mechanical energy and momentum conservation which confused me as I couldn't understand why the normal force here is conservative. So,   1. Is the normal force always conservative or is this only in some cases?    2. If yes, how do we deduce that like in the above example?   3. Also, if it is conservative, what would be its corresponding potential energy function?


title: Swords, impacts and elasticity for a noob

text: So I'm a game developer and I'm trying to understand some (extremely) basic facts of impact mechanics. I had read something entitled _Dynamics of Hand- Held Impact Weapons,_ but it was a bit too complicated for my feeble mind, and really more than I needed to understand for a game. I'm assuming as average a hit as possible, envisioning the problem as little more than a flat block hitting another flat block; bludgeoning at first, and cutting later after I understand that. From what I understand, elasticity is a range from 0.0 to 1.0, yes? Assuming the sword (basic flat block for now) has zero elasticity for simple gameplay purposes, how does elasticity affect the Time variable for acceleration? And what force is it that causes the target (a variably squishy flat block) to resist further deformation and retake its shape? If it sounds like I'm just misguided, guidance in what I ought to know would be fine too. Thanks!


title: Bra space and adjoint vectors

text: If I'm not wrong, a bra, $ \langle \phi_n | $, can be thought as a linear functional that when applied to a ket vector, $| \phi_m \rangle$, returns a complex number; that is, the inner product it's a linear mapping that $\xi \rightarrow \mathbb{C} $; yet, exist a bra to each ket, and in discrete basis, the reverse it's valid too. Well, thinking in the scope of discrete basis, my question, then, is: when we take a adjoint of a vector, we go from a vector space to a "linear functional" space? That is, when we want to calculate the inner product of $|\phi_n \rangle$ with itself, we are, as a matter of fact, applying the bra associated with the vector in the same vector?


title: What is the difference between UT0, UT1 and GMT time?

text: Every reference I find says that they are **_"essentially"_** the same, which we all know really means that they are not the same, but different only by a some small amount that someone else other than me decides is negligible. They also like to say that UT1 and UTC are the same, but if you look into it you find that they differ by as much as 0.9 seconds. This is, admittedly, insignificant to me. But the practicality of it is not my concern here. The technical reality of it is. I want specifics and exactness where applicable here.


title: Calculating spacecraft water leak flow rate

text: A large container of heated water is floating in space. How can I calculate leak rate if a small puncture is opened? I'm assuming the water is kept at 298K and in the liquid state inside the chamber. I guess that upon depressurization the water will immediately vaporize and then take some time to freeze and cool. I'm not sure if the phase change is important for estimating the leak flow rate. I also expect that the flow rate will be proportional to the pressure difference, viscosity of water, and the size of the hole. Just not sure how to put this all together.


title: Resistors in Parallel

text: From my book: "A length of wire is cut into five equal pieces. The five pieces are then connected in parallel, with the resulting resistance being 2.00 Ω. What was the resistance of the original length of wire before it was cut up?" I don't understand what exactly I'm looking for. What resisters? This isn't like any other problem or example in my book. I don't even know what a parallel with five pieces looks like. I tried using the parallel equation, assuming there are five resistors: $\ \frac{1}{2}=\frac{1}{x}\cdot 5$ Then each resister would have would be 0.40Ω, but that can't be right. I've only been given the equations for $\ R_{eq}$ in both series and parallels.


title: Definition of Casimir operator and its properties

text: I'm not sure which is the exact definition of a Casimir operator. In some texts it is defined as the product of generators of the form: $$X^2=\sum X_iX^i$$ But in other parts it is defined as an operator that conmutes with every generator of the Lie group. Are these definitions equivalent? If the answer is yes, how could I prove it (I'm thinking in using Jacobi's identity)?


title: How can I solve for time without knowing the vertical velocity?

text: A guy posted this problem on a forum: > _There is a bird sitting on a pole of height h. you throw a rock at it and > the moment the rock leaves your hand the bird starts flying horizontally > away from you at 10m/s, the rock passes the point where the bird was when > you threw it and then reaches twice the height of the pole. on its descent > it touches the bird. what is the horizontal velocity of your rock._ Promise it's not homework. How would you solve it? This is what I've done so far: Let's say were a distance $d$ away from the pole, and that the bird travels distance $x$ before we hit it. Then if $v_h$ is the horizontal velocity, we know that $d + x = v_ht_h$, and that $x = 10t_h$. The problem is a bit vague but we can assume that $2h$ is the vertical peak of the projectile. Then, we have that $2h = v_vt_{up} - (1/2)gt_{up}^2$. The rock will travel down $h$ before it hits the bird, so another equation is $h = (1/2)gt_{down}^2$. Finally we have that $t_{up} + t_{down} = t_h$. So $t_{down} = \sqrt{h/(4.9)}$. I'm not sure how to solve for $t_{up}$, especially because we don't know that the vertical component of velocity is. Any help would be appreciated!


title: Explanation for Negative $\rho$ (radial distance) in Cylindrical Coordinates

text: **My question :** What does it mean when we arrive at **negative** values for _distance_ variables like $\rho$ in cylindrical coordinates? (after some discussion here,I revised the question, at the end of the post) **My specific example:** For the motion of an ion in a quadrupole Paul trap, we arrive at the same equation (mathieu equation) (below) for both $\rho$ and $z$ components (the potential has only $\rho$ and $z$ dependance ): $$\frac{d^2u}{d\tau^2} + [a_u-2q_u\cos (2 \tau) ]u=0 \qquad\qquad $$ here $u$ represents a coordinate variable and may be $r$ , $\rho$ in cylindrical coordinates or $x , y ,z$ in Cartesian coordinates. **The problems is that , this equation has an oscillatory behavior and it's solution has negative values too. What is the meaning of negative $\rho$ in cylindrical coordinates?** Here is a sample solution for $a_z=0$ and $q_z=0.4$ ![A sample solution for $a_z=0$ and $q_z=0.4$ ](http://i.stack.imgur.com/eJSCb.jpg) **EDIT 1:** after some answers posted , I found out that the problem has something to do with our derivation of equations.Here is how _correct_ equations are. The potential is only a function of $r$ (or $\rho$) and $z$ in cylindrical coordinates,so we don't have any force in $\hat \theta$ direction: $$\mathbf F=-\nabla V(r,z)=-(\hat r{\partial V \over \partial r} +\hat \theta \frac{1}{r} {\partial V \over \partial \theta} +\hat z {\partial V \over \partial z})$$ so the equations of motion will be (I assumed the mass to be 1) $$\ddot r - r\dot{\theta}^2=-{\partial V \over \partial r} ***$$ $$r \ddot{\theta} +2 \dot r \dot \theta=0 \to \dot{(r^2\dot \theta)} = 0$$ $$\ddot z=-{\partial V \over \partial z}$$ so the equations for $z$ and $r$ **will not** be the same. Now , **the problem is : how can we arrive at those equations (mathieu equation, above)(in the literature on Paul traps they use this equations) from these equations of motion ; eliminate $\theta$ dependance and instead of that let $r$ get negative values and interpret it as a positive $r$ with $\theta_0+\pi$?**


title: What {R,G,B} values would represent a 445nm monochrome lightsource color on a computer monitor?

text: Is it possible to answer my question definitely (assuming the monitor is perfect)? What would be the formula for calculating RGB values for a visible monochrome light with given wavelength?


title: Diffeomorphism invariant Quantum field theories

text: In QFT , The action should be invariant under poincare symmetry $g_{\mu\rho}(x)=g_{\mu\rho}^{\prime}(x^\prime)$ . We can generalize this by considering theories invariant under conformal symmetry $g_{\mu\rho}^{\prime}(x^\prime)=A(x)g_{\mu\rho}(x)$ . If we consider more general variations ( Diffeomorphisms on spacetime manifolds ) What will happen ? If we gauge fix the redundant variables , Can we remove the metric completely from the action? The theory .Will the theory be background independent ? Can we classify topological invariant properities using correlation functions from that theory ?


title: Confusion about proper time

text: A spacecraft flies away from earth with a speed of 4.8 million meter per second relative to the earth, and then returns at the same speed. The spacecraft carries a clock that has been synchronised with a clock on earth, When the spacecraft returns to its starting point 1 year later as measured by the clock on earth, what is the difference in elapsed times in the two clocks ? Ok, so the problem for me here is determining the proper time, I figured that since the spacecraft is accelerating the correct proper time $t$ should be the time on earth ( 1 year ) and $ t prime $ should be the time elapsed on the rocket. But when I use the formula for time dilation this way I get the wrong answer however if I assume that the proper time is the one on the spacecraft I get the right answer. My question is why is the proper time the one on the spaceship


title: Theoretically speaking what are the limits for super-capacitors in terms of their storage capability?

text: According to wikipedia, commercially available super-capacitors in 2010 can store 30Wh/kg, with 85Wh/kg achieved in a lab in 2011. Have any theoretical calculations been done to show what can be achieved as an upper limit to super-capacitors?


title: Ejected Electrons with 0 KE?

text: So I was taught that: Kinetic Energy (of electron) = Energy (of photon) - Ionization Energy If E(photon) = IE, then KE=0 of the electron. What does this physically/theoretically mean? My current thoughts/interpretation is that enough energy/force is applied to ionize the electron so it is 'sufficiently far' from the atom, and then I guess it just moves with whatever speed it is moving at with natural kinetic laws, since no more energy/force is being applied....? Any clarifications would be greatly appreciated, thanks in advance.


title: transition between extremal and nonextremal black hole states

text: Extremal black holes are at zero temperature, hence they do not radiate. my question is twofold: 1) **is extremality of micro black holes a stable property?** electric charge is quickly emitted from sub-atomic black holes due to Schwinger pair production, what about angular momentum? does a black hole loses its extremality quickly or slowly (compared to its evaporation time)? 2) **can a black hole extremality state be changed by external sources?** can we add matter and angular momentum appropiately to a schwarszchild black hole in order to become extremal or nearly-extremal? can we do the inverse process?


title: Kinetic Energy And Rotational Motion

text: > _The problem is, "A metal can containing condensed mushroom soup has mass > 220 g, height 11.0 cm and diameter 6.38 cm. It is placed at rest on its side > at the top of a 3.00-m-long incline that is at 30.0° to the horizontal and > is then released to roll straight down. It reaches the bottom of the incline > after 1.50 s._ > > _(a) Assuming mechanical energy conservation, calculate the moment of > inertia of the can._ > > _(b) Which pieces of data, if any, are unnecessary for calculating the > solution?_ My attempt at solving (a): I figured that I could use the equation $\Sigma W=\Delta K=1/2I\omega_f^2-1/2I\omega_i^2$ Since the force of gravity that is acting along the incline is applied constantly over a distance, $W_g=mg\cos(60^{\circ})(3.00~m)$; and since the can rolls down the incline in 1.50 s, $v=3.00/1.5 \implies 2~m/s$, which means that $\omega_f=2/0.0319 \implies 62.695925~rad/s$ With this, and knowing that $\omega_i=0$, $mg\cos(60^{\circ})(3.00)=1/2I(62.695925)^2 \implies I=\frac{(6.00)mg\cos(60^{\circ})}{(62.695925)^2}$ When I calculated this, I got $I=0.00165~kg\cdot m^2$; however, the true answer is $I=0.000187~kg\cdot m^2$ I've re-worked my solution several times, what am I doing incorrectly? As for (b), the answer is that the height of the can is an irrelevant piece of information, why is that?


title: A problem from Pathria, canonical ensemble, how to calculate $\left\langle \left(\Delta E\right)^{3}\right\rangle $

text: This is problem 3.18, from the book Statistical Mechanics by Pathria. Show that for a system in the canonical ensemble $$\left\langle \left(\Delta E\right)^{3}\right\rangle =k^{2}\left\\{ T^{4}\left(\frac{\partial C_{V}}{\partial T}\right)_{V}+2T^{3}C_{V}\right\\} $$ Verify that for an ideal gas $$\left\langle \left(\frac{\Delta E}{U}\right)^{2}\right\rangle =\frac{2}{3N},\qquad\left\langle \left(\frac{\Delta E}{U}\right)^{3}\right\rangle =\frac{8}{9N^{2}}$$ I have no Idea how to solve it. I found that $$\left\langle \left(\Delta E\right)^{3}\right\rangle \equiv\left\langle \left(E-\left\langle E\right\rangle \right)^{3}\right\rangle =\left\langle E^{3}\right\rangle -3\left\langle E\right\rangle \left\langle (\Delta E)^{2}\right\rangle -\left\langle E\right\rangle ^{3}$$ And using the fact that $$\int\left[U-H(q,p)\right]e^{-\beta H(q,p)}d\omega=0$$ I obtain $$\frac{\partial^{2}U}{\partial\beta^{2}}=\left\langle \left(\Delta E\right)^{3}\right\rangle $$ But, now I'm stuck. How can I link that who results to show the first statement?


title: Accelerating expansion of universe - entire universe a non-inertial frame of reference?

text: If the expansion of the universe is accelerating, doesn't that mean that the entire universe is a non-inertial frame of reference? And if so, what implications does this have (if any)?


title: Directionality of angular momentum

text: I was told that the sum of linear and angular momentum is conserved. Given that angular momentum's direction as a vector is completely arbitrary (I believe there is no physical reason for choosing the cross product going the way it does (that is, perpendicular _up_ or _down_ , it just matters that it's perpendicular), correct me if I'm wrong), and linear momentum is often interchanged into angular momentum, surely the vector sum cannot be conserved? Or am I wrong, and the meaning is that the _magnitude_ is conserved? As a sidenote, I suppose this would be a good place to ask _why_ such an odd system was devised for calculating the direction (not magnitude) of angular momentum. Why not have the arrow pointing in the direction of motion, with its base at the point that we are considering angular momentum about? Is it useful simply to compute the conservation of it about a point, as it just involves addition?


title: How do physicists and astronomers handle leap seconds?

text: I'm confused by the many contradictory descriptions I see about how UTC leap seconds are accounted for. I understand that there are various ways to handle them in common practice, and I've seen a variety of formal definitions. But it seems that in scientific practice, they are simply _omitted_ : there is no UTC Julian Day (that is no JD(UTC) value) corresponding to any times during a leap second, and things like ephemeris are generally not reported for leap seconds. There are, of course, events that take place during leap seconds, but if one wants to refer to the time at which they occur, one uses a different timekeeping system (e.g. UT1 or TT). Is that right? It makes perfect sense as a way to accommodate the ambiguities that leap seconds introduce, and it in fact corresponds to the way some systems (e.g. POSIX) implement them; but it doesn't quite match the definitions I've seen.


title: Strange Behavior in Spring Computer Model

text: To learn more about oscillatory motion which I am learning about in my high school physics class, I have created a computer model of a damped spring where the damping force is proportional to velocity. The graph of position vs time appears as expected, a sine wave with amplitude decaying over the course of time. I was interested in seeing the curve encompassing the peaks of the graph (the green line in this image). To do so, I created a new column in Excel which calculated the total energy of the system using the values of position and velocity the computer program outputted, then expressed that energy as a displacement from equilibrium using x = sqrt(2E/k). When I graphed the results, I expected to see a smooth curve. However, I instead saw strange curves in between each peak: ![Graph produced](http://i.stack.imgur.com/iOBQ4.png). I am somewhat unsure about the origin of these. Should the "total energy" curve (the red one; although this isn't truly the energy) flatten out at each point at which the velocity equals zero in the real world? Or was my assumption that damping force is proportional to velocity a faulty one? If the latter is true, what would be a more accurate method of representing the damping force?


title: Flow of water in a lake

text: You are 10 miles upstream from a dam. A 50 sq ft gate 30 feet above the lower water level is opened and water starts to flow from the dam. How long is it until your boat starts moving towards the dam? The lake is 300 feet across, 30 feet deep. You will be pulled towards the dam. Thinking I will need the volume of water to the boat, the cross section area of the lake, area of the gate, height of the gate above the lower level, etc....I can calculate velocity out of the gate, v=sqrt (2gh) and using A1v1=A2v2 find v2 and setting t=10/v2 solve for t. But I don't think it works this way. Perhaps an integral calculus problem.   This is what happens when fish are not biting!!


title: Software to calculate forces between magnets

text: I am working on a complex configuration of magnets and every time I make an experiment something unforseen happens. Now I believe I could speed up the development by sitting down and calculating the configuration in all scenarios. My question is however: Is there any software that can help me to calculate the forces between magnets. I am already working with AutoDesk as my CAD software.


title: Why is gas(oline) in gas stations sold by volume (as opposed to mass)?

text: Fluids (including natural gasoline/petroleum) have variable volume based on the ambient temperature for the same mass of fluid. So, really, the amount of gas that you're filling your car with depends on the temperature because it's not the _volume_ of fuel that makes your car run, but the _mass_ which is combusted. In aircraft, aviation fuel is always measured by the kilogram. So, why is it measured by volume for cars?


title: Superposition principle

text: If $S=(v_{1},v_{2}......v_{n})$ is a basis for vector Space V, then every vector v in V can be expressed in the form of $v=c_{1}v_{1}+.......c_{n}v_{n}$ in an unique way. Explain the significance of this result in Quantum Mechanics. The answer to the first one is easy as its proof is based on Linear algebra and I have already done it. However, I am struggling for the explanation of the second one, though I think that it is based on the superposition principle.


title: Octave equivalence: biological or more?

text: I'm a graduate student in mathematics doing a bit of research in signal processing and Fourier analysis and I've come across a question that could probably be better answered by a physicist: Is the phenomenon of octave equivalence (the psycho-acoustical sensation that pitch is subject to an equivalence relation whereby two frequencies are considered "the same" if they differ by a factor of a power of 2) a product of our biology alone or is it an artifact of something more basic in acoustics? That is, could someone point me to a mathematical statement from which the sensation of octave equivalence could be "read off" in the matter that, say, the phenomenon of beats can be found in (I believe) the double angle formula? Or, does the sensation merely arise from the structure of our cochlea? Vaguely speaking, is the phenomenon somehow external or internal to our biology? This is all very ambiguous, but I hope that something meaningful can be extracted from it. If it helps, I know that thus far octave equivalence has been found in rats and human infants. Edit: Also, if anyone can point me to any pertinent literature, preferably in physics or applied mathematics, I'd be very grateful. Thank you.


title: In a column of rising hot air, is the velocity higher at the top?

text: Since the movement of the air is induced by buoyancy, i. e. there's a constant force acting on the air, so I would expect the velocity to increase during ascent, much like an object falling down due to gravity accelerates. But wouldn't a velocity gradient tear the column apart along the axis? Or would it just lead to a corresponding pressure gradient and a deformation of the column (thinner at the top)? If there is acceleration, there's probably a terminal velocity, just like with objects falling down through the atmosphere. What would that be limited by? There's two effects that I would like to neglect for the purpose of this question, if possible, the dissipation of the warmer air into surrounding air and the cooling of the air as it rises. I suspect with a high enough $\Delta T$ and a column that is wide enough and not too tall, we can ignore those (for a bit, anyway)? And just for context, what brought me to this was the idea of having hot air rising through a chimney to act as a pump via a constriction in the chimney that creates a Venturi effect. I was wondering whether it would be better to put that constriction further up in the chimney, so I get a higher pressure difference because of a higher velocity.


title: Pluto's gravitational pull on a person on the Earth's surface?

text: My Physics teacher stated that Pluto has a gravitational pull on objects on Earth, namely humans. Is this true? What is the free-fall acceleration of Pluto with respect to being on the Earth's surface? (i.e. the Earth's free- fall acceleration is 9.8 m/s*s).


title: A classically trivial quantum field theory of electromagnetism

text: Presumably there is a field theory of electromagnetism that classically gives trivial equations of motion, but when quantized shows interesting topological phenomena. I am talking about the Lagrangian density $$\mathcal{L}=\epsilon^{\mu\nu\lambda\sigma} F_{\mu\nu}F_{\lambda\sigma}\propto \vec{E}\cdot\vec{B}$$ where $F_{\mu\nu}$ is the field strength tensor. Unfortunately, I don't know the name of this theory or what keywords to search for, so I am unable to dig up any references that treat this theory. It may well be that this question is a duplicate. In that case, a link to an old question would suffice. So my "question" is: Please give me a reference where I can read about this. Thank you for your time.


title: Effective operator in four-fermion interaction

text: In one book, I have got the following lines which I found myself unable to understand what is effective operator? The paragraph is given below: > _The weak interaction describes nuclear beta decay, and at low energy it is > given by an effective four fermion interaction. Since the effective operator > has dimension 6, the coupling constant has inverse mass-squared dimension._


title: Why does the bubbling of a soda bottle pulsate?

text: If I take a bottle of carbonated water and open the cap slightly to allow the gas to escape, there will be a sudden rush of bubbles foaming to the top of the liquid for a few seconds, then a partial lull for a few seconds, then another smaller rush of bubbles, then a smaller lull, etc, like a damped oscillation. What causes this pulsation?


title: When moving something by entrainment in a fluid stream, where does the energy come from?

text: The particular Something I had in mind here would be air bubbles that are pulled downwards against their buoyancy by a stream of water falling down a shaft. The work required to push those bubbles down must decrease energy somewhere else, so where does it come from? Is the stream's velocity reduced?


title: Normal force of loop-the-loop at the side of the circle

text: In the loop-the-loop ride a car goes around a vertical, circular loop at a constant speed. The car has the mass of 230 kg and moves with the speed of 300 m/s. The loop-the-loop has a radius R=20 m. What would then be the magnitude of the normal force on the car when it is at the side of the circle moving upward? I tried to solve this problem, by:   1. gravitational force = $mg$ = 230*9.8 (downward)   2. centripetal force = $mv^2/r$ = 300^2/20 (toward circle, which is horizontal) and by vector addition/subtraction, magnitude of normal force would be $\sqrt {(mg)^2+(mv^2/r)^2}$ . But the answer I got is wrong, so this approach must be wrong... What did I do wrong here?


title: Cyclist's electrical tingling under power lines

text: It's been happening to me for years. I finally decided to ask users who are better with "practical physics" when I was told that my experience – that I am going to describe momentarily – prove that I am a diviner, a psychic, a "sensibil" as we call it. The right explanation clearly needs some electrodynamics although it's "everyday electrodynamics" and theoretical physicists are not trained to quickly answer such questions although each of us has probably solved many exercises that depend on the same principles. ![enter image description here](http://i.stack.imgur.com/azt0a.jpg) When I am biking under the power lines – which probably have a high voltage in them – I feel a clear tingling shocks near my buttocks and related parts of the body for a second or so when I am under a critical point of the power lines. It is a strong feeling, not a marginal one: it feels like a dozen of ants that are stinging me at the same moment. It seems almost clear that some currents are running through my skins at 50 Hz. I would like to know the estimate (and calculation or justification) of the voltage, currents etc. that are going through my skin and some comparison with the shock one gets when he touches the power outlet. Now,   * my bike that makes this effect particularly strong is a mountain bike, Merida;   * the speed is about 20 km/h and the velocity is perpendicular to the direction of the current in the power line;   * the seat has a hole in it and there is some metal – probably a conducting one – just a few centimeters away from the center of my buttocks. It's plausible that I am in touch with the metal – or near touch;   * my skin is kind of sweating during these events and the liquid isn't pure water so it's probably much more conductive than pure water;   * the temperature was 22 °C today, the humidity around 35%, clear skies, 10 km/h wind;   * the power lines may be between 22 kV and 1 MV and at 50 Hz, the altitude is tens of meters but I don't really know exactly. What kind of approximation for the electromagnetic waves are relevant? What is the strength? How high currents one needs? Does one need some amplification from interference etc. (special places) to make the effect detectable? (I only remember experiencing this effect at two places around Pilsen; the most frequent place where I feel it is near Druztová, Greater Pilsen, Czechia.) Is the motion of the wheels or even its frequency important? Is there some resonance? Does the hole in the seat and the metal play any role? Just if you think that I am crazy, other people are experience the effect (although with different body parts), see e.g. here and here. This PDF file seems to suggest that the metals and electromagnetic induction is essential for the effect but the presentation looks neither particularly comprehensive nor impartial enough. An extra blog discussion on this topic is here: > http://motls.blogspot.com/2012/05/electric-shocks-under-high-voltage.html


title: How does a roof top ventilator work?

text: ![enter image description here](http://i.stack.imgur.com/8QAKW.jpg) I have spotted this on the rooftops of industries, workshops, etc. I guess this is used for providing ventilation. This is not electrically powered. But then how does this work??


title: Why does gravitational lensing bend light rather than absorbing it?

text: I've read that gravitational lensing bends the light; I am a little confused about black holes and why lensing produces different results. Black holes absorb light whereas lensing should have also done the same but why does gravitational force help bend the light?


title: Higher-Dimensional Metrics in (Hyper)-Spherical Coordinates

text: I want to compute the components of the Riemann curvature tensor (for a case similar to the Schwarzschild solution) in 4 + 1 dimensions, but I want to use a higher-dimensional analogue of spherical coordinates. I first want to investigate a metric for the Euclidean case, i.e. "flat" space-time with no matter or energy present. How would I write this Euclidean metric using a higher-dimensional analogue of spherical coordinates?


title: Tracking photon color in Bell experiments

text: In parametric down-conversion, it is said that a driving photon is converted into two entangled photons whose frequencies add up to the driving frequency. Yet in discussions about entanglement experiments, I have not seen anything about the frequency at the point of detection. What is the story here? Do you have pairs of red and green entangled photons? If Alice detects a red photon does Bob detect a green one and vice versa? Is the color of the photon even known or measured at the point of detection? EDIT: Thanks to Slavic for the awesome picture posted below. I am struggling now to understand what I see, so let's start with an easy question: obviously, the blue is the driving frequency; what is the mechanism for the cone separation (just ordinary prism effect??) and if so, why are the color rings reversed in the complementary light cones?


title: Physical significane and context in which Dirac introduced the Dirac Delta function

text: I'd like to know the exact context in which Paul Dirac introduced the Dirac delta function. What was the physical significance of the Dirac delta function when he first used it in Physics ?


title: How to get statics out of a dynamic force concept?

text: If one defines force as the time derivative of momentum, i.e. by $$ \vec{F}= \frac{d}{dt} \vec{p} $$ how can this include static forces? Is there a generally accepted way to argue in detail how to get from this to the static case? If not, what different solutions are discussed? Edit: I shoud add, that by static forces I mean forces involved in problems where bodies don't move.


title: Stephen Hawking says universe can create itself from nothing, but how exactly?

text: Stephen Hawking says in his latest book _The Grand Design_ that, > Because there is a law such as gravity, the universe can and will create > itself from nothing. Is it not circular logic? I mean, how can gravity exist if there is no universe? And if there is no gravity, how can it be the reason for the creation of universe? Also, if the universe doesn't exist, how can _it_ create _itself_? The very sentence doesn't make sense to me. It seems so absurd and illogical that I've never heard such sentences even in philosophy. On what ground, Stephen Hawkings _claims_ this?


title: Is the electron magnetic moment responsible for the Lorentz force?

text: My question about the sum of the electrons' magnetic moments in a wire(What is the sum of the electrons' magnetic moments in a wire?) had an answer which disappeared later. The answer was - if I understood right - that were is a moment in sum. If there is a moment, could this explain the Lorentz force?


title: Is there a strong evidence of antineutron existance?

text: Wikipedia explains: > _The antineutron was discovered in proton–proton collisions at the Bevatron > (Lawrence Berkeley National Laboratory) by Bruce Cork in 1956, one year > after the antiproton was discovered._ Where is (preferably open accessible) literature about the discovery of the antineutron? Or more quick I ask:   * Was the result with strong evidence of neutron-antineutron annihilation to pure energy?   * Was the result in the way, that they get a particle shower (particle decay)? Which particles they discovered?


title: Electromagnetic Wave Penetration into Conductors

text: Right now, I'm looking at two straightforward derivations of the dispersion relations of electromagnetic waves in conductors, and they seem to disagree on the low frequency attenuation. Could someone supply some insight into the regimes of validity of these very basic methods? ## **In my electromagnetism class** , we derived the dispersion relation of waves in homogeneous highly conductive media as follows: > We have Ampere's Law and Faraday's Law > > $\nabla\times H=\partial_t D+ J$ > > $\nabla\times E=\partial_t B$ > > Use the constitutive relations $D=\varepsilon E ; B=\mu H$, and Ohm's law > $J=\sigma E$ > > $\nabla \times H = (\varepsilon\partial_t + \sigma)E$ > > $\nabla\times E=\mu\partial_t H$ > > For purposes, let's just assume exponential solutions > $E=\hat{x}E_0e^{i(\omega t-kz)}$ > > $\nabla \times H = (i\varepsilon \omega + \sigma)E$ > > $\nabla\times E=i\mu \omega H$ > > Curl of Ampere's Law, plug in Faraday's, use vector identity, as usual, and > arrive at > > $k^2 = (\epsilon \mu \omega^2 -i\mu\omega\sigma)= \varepsilon\mu\omega^2 > (1-i\frac{\sigma}{\epsilon\omega}) $ > > Suppose we have a really good conductor ($\sigma\gg\epsilon\omega$), then > the right-hand side is almost pure imaginary, and we can write $k\approx > \sqrt{\mu\sigma \omega/2} (1-i)$. **One can easily see that increasing > $\omega$ increases (the imaginary part of) $k$, which increases the > attenuation of the wave. Higher frequencies attenuate more sharply in good > conductors.** ## **In my solids class** , we derived the dispersion relation of waves in highly conductive media in the free electron gas model. > See _Introduction to Solid-State Physics_ by Kittel (pg 396-398) > > Starting from the free electron in an E-field > > $m\partial_t^2x=-eE$ > > Solve for the magnitude of the oscillations: > > $x=eE/m\omega^2$ > > Relate to the polarization > > $P=-nex=-\frac{ne^2}{m\omega^2}E$ > > Then get the relative permittivity from that relation > > $\varepsilon_r = 1 -\frac{ne^2}{m\omega^2} = 1 -\frac{\omega_p^2}{\omega^2}$ > > Which leads to the dispersion > > $k^2 = \mu\epsilon (\omega^2-\omega_P^2) $ > > For, say copper, $\omega_P$ should be something like 6 PHz. For frequencies > above that (like UV), $k$ is real and the light propagates. For frequencies > below that, $k$ is imaginary and the waves must be attenuated/reflected. > > I want to compare with the result from above. I assume for my E&M class, > we're dealing with frequencies much lower than PHz, since these are > electrical engineers. So, looking at this result in the $\omega \ll > \omega_P$ limit, we see that increasing $\omega$ decreases the magnitude of > $k$, which means that **waves will be attenuated less sharply at higher > frequencies.** These two models seem to disagree on the $\omega \ll\omega_P$ behaviour. Why is that/ how does the latter model break down these frequencies? Thanks!


title: Technical detail in the solution of the hydrogen atom

text: I'm trying to do an exercise in which you solve the Schrödinger equation for the hydrogen atom. Through the exercise, I've already shown that the wavefunction is: $$ \psi_{n\ell m}(r,\theta,\varphi) = R_{n\ell}(r)Y^m_\ell (\theta,\varphi)$$ and that $Y^m_\ell (\theta,\varphi)$ are the spherical harmonics. Then, when solving for the radial part, the exercise tells me we need to study the asymptotic behaviour of $R(r)$ for large and small $r$. I had no problem showing that $R(r) \overset{\underset{+\infty}{}}{\sim} e^{-kr}$ but then the exercise gets completely bananas for the asymptotic part for small $r$. It tells me to introduce $u(r) \doteqdot rR(r)$, and then I managed to show the Schrödinger equation becomes: $$-\frac{\hbar^2}{2m}{\mathrm{d}^2 u(r) \over \mathrm{d}r^2} - \frac{e^2}{4\pi\epsilon_0r}u(r)+\frac{\hbar^2}{2m}\frac{\ell(\ell+1)}{r^2}u(r) = E_n u(r)$$ Ok. Then the exercise tells me to suppose $u(r) \overset{\underset{0}{}}{\sim} r^\lambda$ and to prove $\lambda > -\tfrac{1}{2}$. No problem, this follows from the fact that $R(r)$ must be a $L^2$ function. Then, as long as $\ell \neq 0$, we can solve the above equation for $r\to 0$, showing that $\lambda = \ell + 1$. The next part of the exercise says: > Apply Stokes's theorem on a sphere to show that > > $$\nabla^2 \left ( \frac{1}{r} \right ) = -4\pi\delta({\mathbf r}) $$ > > then use this result to prove that $\lambda = \ell + 1$ even when $\ell = > 0$. And I get completely stumped. I can prove the above formula, but I have no idea what this Laplacian has to do with anything.


title: Expansion of Helmholtz energy

text: To get an expansion of Helmholtz energy of a) an ideal gas b) a Van der waals gas we must integrate $\left ( \frac{\delta A }{\delta V} \right )_{T}=-P$ I saw the solution is : ![enter image description here](http://i.stack.imgur.com/Njbor.png) Can you explain why is that?


title: How to calculate the coefficient of restitution for 2 bodies?

text: I have 2 rigid bodies (from different materials) in a collision. As you know I should have the coefficient of restitution value to get the velocities after collision. What is the information/values should I know about each body to calculate the restitution? And how to calculate it?


title: Diffraction pattern when wavelength tends to zero

text: While studying about Fresnel and Fraunhofer diffraction, I came across a statement which says that the fringes disappears and the image would take on the limiting shape of the aperture when wavelength goes to zero, which is the same as predicted by geometrical optics. I don't understand the meaning of this, especially the usage 'wavelength goes to zero'. Can anyone help me?


title: Why is distance between two successive nodes equal to $\frac{1}{2} \lambda$ in standing wave

text: I'm reading my textbook and it says the distance between two successive nodes is equal to $\frac{1}{2} \lambda$ in standing wave. ![enter image description here](http://i.stack.imgur.com/86Y0R.jpg) If $\lambda$ here means the wavelength of the standing wave shouldn't it be: when there are two nodes: $L=\lambda$ when there are three nodes: $L=\frac{1}{2} \lambda + \frac{1}{2} \lambda$ when there are three nodes: $L=\frac{1}{3}\lambda + \frac{1}{3}\lambda + \frac{1}{3}\lambda$ Does $\lambda$ here mean anything?


title: Magnetic induction in Relativity

text: As we know magnetic phenomenon is a mere relativistic effect.My question is how to explain the magnetic induction in a relativistic manner?


title: Vector Summation

text: When two vectors are sketched from a single point, the angle between them is θ. Show that the size of their vector summation is given in the expression: $ \sqrt{A^2 + B^2 +2ABcosθ} $. Any suggestions on how one might tackle this one?


title: SInce heat is a vibration in solids, isn't it just small scale sound?

text: On an atomic scale I think of electrons hitting positive ions in an electric lattice. Most energy there is heat energy. If I think of ball bearings flying through a large lattice, I think sound is produced. The only difference between the two interactions, seemingly, is size. Am I wrong here? In retrospect, I didn't realize there was a similar thread: If the source of sound is vibration, why can't we "hear" a object whose molecules are vibrating?


title: Quantum state with zero standard deviation of position operator

text: Is any quantum state $|\psi\rangle$ possible such that the standard deviation $\sqrt{\langle\psi|(\Delta\hat{x})^2|\psi\rangle}$ of the position operator $\hat{x}$ is zero? If not, why?


title: Parabolic motion (experiment)

text: We performed a laboratory, performing six releases of a sphere with angles $15^\circ,30^\circ,45^\circ,60^\circ,75^\circ,40^\circ$ a parabolic movement, took five distances for each angle, the initial velocity was calculated $3.025~\text{m/s}$. Then doing 5 tosses of the sphere with an angle of $90^\circ$ times were recorded and calculated initial velocity was $3.33~\text{m/s}$, the question is: Why are these speeds almost the same?


title: Ratio of Size of Atom to Size of Nucleus

text: I have the following problem: > In nuclei, nucleons exists in nuclear energy levels and in atoms, electrons > exist in atomic energy levels. The order of magnitude of nuclear energy is > 1MeV whereas the energy of atomic energy levels is of the order 1eV. Use > this info and the particle in the box model to make an order of magnitude > estimate of the ratio $$\frac{\text{size of atom}}{\text{size of nucleus}}$$ So the way I approached this was to consider $$\frac{E_a}{E_n}=10^{-6}\Rightarrow \frac{\frac{h^2}{8m_aL_a^2}}{\frac{h^2}{8m_nL_n^2}}=\frac{m_nL_n^2}{m_aL_a^2}=10^{-6}\Rightarrow \frac{L_a}{L_n}=\sqrt{10^6\cdot m_n/m_a }.$$The mass of the nucleus we assume to be 1u but what about the mass of the atom? In the answers at the end of the book it says that we take $m_n = m_e$. But why is this so? Shouldn't the mass of a typical atom also include the mass of the nucleus, along with the mass of the electron? Please keep your answers simple since I am only doing A-Level physics. Thanks in advance.


title: Temperature of relic radiation and thermal equilibrium

text: The temperature or relic radiation is 2.7 K. Does it mean that any flying body sufficiently far from any stars will reach this temperature? As I understand it, you can warm up water with sufficient amount or radiation with 2.7K spectrum and you can freeze helium even in presence of a source of photons with 100000 K spectrum if the source is dim enough. So what temperature will reach a body placed in interstellar medium?


title: Finding The Resultant Vector

text: A plane is flying southeast at 215 km/h. Suddenly there is a wind from the north at 75 km/h. What is the plane's new velocity with respect to the ground in standard position?


title: Bi-vector in Minkowski space

text: I have a problem, I have a bi-vector that define like: $\omega^{\mu \nu}=a^{\mu}b^{\nu}-a^{\nu}b^{\mu}$ where, $a^{\mu}=(a^0,a^1,a^2,a^3)$ and $b^{\nu}=(b^0,b^1,b^2,b^3)$ I need show that $\Lambda^{\mu}{}_{\nu}=\exp(\alpha \omega^{\mu}{}_{\nu})$ and the explicit form like a matrix.


title: Is everything pre-decided?

text: "There is nowhere in the universe where the laws of physics are violated." Considering this general to be true,can i conclude that everything is pre- decided? I can explain this in the following manner: consider a photon or nerve impulse travelling in our brain at this exact moment,in accordance with the above theory, it will meet a definite fate determinable by sufficiently complex physical calculations, i.e strike a particular nerve ending and thus initiate a chain of thoughts which lead to a definite action on our part and thus determine future. Thus it can be essentially conclude that every particle or wave in the universe will necessarily follow the laws of Newtonian or quantum physics at every point of time and thus the future is definite and pre-decided. Please express your views.


title: Question on Pendulum

text: The maximum tension in the string of the pendulum is three time the minimum tension.If x(angle) is the amplitude of the pendulum then find cos(x)


title: Capacitance: How to increase total charge drawn from a battery?

text: I can't seem to find any information about how parallel capacitors or capacitors in a series affect how much a capacitor can charge, not in my textbook or online. Here is the question I am trying to answer: > You have a capacitor that you will connect across a battery. If you wish to > increase the total charge drawn from the battery, which of the following > options will work? Choose all of the correct answers. > > (a) Add a larger capacitor in series with the first. > > (b) Add a smaller capacitor in series with the first. > > (c) Add a larger capacitor in parallel with the first. > > (d) Add a smaller capacitor in parallel with the first. I would think that the charge $Q$ would increase as the capacitance $C$ increases according to $Q=CV$, and I know that $C$ increases with increased area, decreased separation, or a higher dialetric constant, but that has nothing to do with the problem. I think $C$ also is higher with parallel capacitors than capacitors in a series, but am unsure.


title: Quantum Bit Commitment, restricting ourselves to pure states

text: I have been going through the Preskill lecture notes on quantum computation, and there is a question on Quantum Bit Commitment: > Alice wants to make a prediction, either $0$ or $1$, before an event. She > doesn't want Bob to know her prediction beforehand, but wants to be able to > prove to him afterwards that she was right all along. To do this she > prepares one of two distinguishable states represented by density operators > $\rho_0$ or $\rho_1$ of a bipartite system $A\otimes B$, sending $B$ to Bob > and keeping $A$ for herself. Later to unveil her prediction she sends $A$ to > Bob who performs a measurement to determine if $A\otimes B$ is in $\rho_0$ > or $\rho_1$. We say the protocol is binding if after commitment Alice is > unable to change her prediction, and concealing if after commitment and > before unveiling Bob is unable to discern the value of the bit. Show that if > this protocol is concealing it is not binding. I am asked to show that we may without loss of generality restrict ourselves to $\rho_0$, $\rho_1$ pure states, which is what I am unsure about. My guess is that we may do this because we can always regard a mixed state $\rho$ on $A\otimes B$ as a pure state $\rho'$ on some $A\otimes B\otimes C$, where $\rho=\mathrm{Tr}_C\\{\rho'\\}$. The tricky bit would be to show we can do this simultaneously for $\rho_0$ and $\rho_1$. To do this, if: $$\rho_0=\sum_{i=1}^np_i|\psi_i\rangle\langle\psi_i|$$ $$\rho_1=\sum_{j=1}^mq_j|\phi_j\rangle\langle\phi_j|$$ We can then define $$p'_k=\begin{cases}p_k & k\le n \\\ 0 & k > n\end{cases}$$ $$|\psi'_k\rangle=\begin{cases}|\psi_k\rangle & k\le n \\\ 0 & k > n\end{cases}$$ And similarly for $q'_k$, $|\phi_k'\rangle$. Then: $$\rho_0=\sum_{k=1}^{m+n}p'_k|\psi'_k\rangle\langle\psi'_k|$$ $$\rho_1=\sum_{k=1}^{m+n}q'_k|\phi'_k\rangle\langle\phi'_k|$$ Then if we pick $C$ such that $|\alpha_k\rangle\in C$ are orthonormal for $k\in\\{1,\cdots,m+n\\}$ we have $\rho_0'=|\Psi\rangle\langle\Psi|$ and $\rho_1'=|\Phi\rangle\langle\Phi|$ pure for: $$|\Psi\rangle=\sum_{k=1}^{m+n}\sqrt{p'_k}|\psi_k'\rangle\otimes|\alpha_k\rangle$$ $$|\Phi\rangle=\sum_{k=1}^{m+n}\sqrt{q'_k}|\phi_k'\rangle\otimes|\alpha_k\rangle$$ Does this seem correct?


title: Dependence of the neutron-freeze-out temperature

text: What is the physical explanation for the dependence for the freeze-out temperature $T_\text{F0}$ of the neutrons as a function of the number of degrees of freedom $g^\star$? $$T_\text{F0}=\left(\frac{1.66}{G_F^2m_\text{PL}}\right)^{1/3} g_\star^{1/6} $$ Why does the temperature rise, implying that the decoupling happens _earlier_ in the history of the universe, with a large number of degrees of freedom? Reference: Grupen, Astroparticle Physics, Chapter 10.3


title: Why is the angle of a pendulum as a function of time a sine wave?

text: OK so I'm trying to understand why the angle of a pendulum as a function of time is a sine wave. I can't really find an explanation online and when I do find something partial there are certain symbols I don't understand. $$\frac{d^2\theta}{dt^2} + \frac{g}{l}\sin\theta = 0$$ This is the equation I found on Wikipedia.   1. What I don't understand here is the part $d^2\theta\over dt^2$, from what I know $d$ means instantaneous delta, instantaneous rate of change, why does the upper part of the function has the square sign right after the $d$ ($d^2\theta$) and the lower part of the fraction has the square sign after the $t$ and not after the $d$ ($dt^2$).   2. This part still doesn't really show the answer to my question cause the change of the angle and the time are squared so it doesn't mean much to, I'm hoping for an explanation that is simple as possible and intuitive as possible for why the angle as a function of time is a sine wave, hopefully as much mechanics as possible ans less math, a good reference is also good. EDIT: OK, I Understand the first part (the square sign notation), what I still don't understand is how we can get from the the equation I wrote above, or from the acceleration as function of the angle: $a = g\sin\ \theta$, to an equation that shows the angle as a function of time. It's confusing for me since the angle itself changes all the time, in both eqaution (the one at the top and $a = g\sin\ \theta$ we have $\theta$, but $\theta$ itself changes all the time! It seems it's not possible to use "little math" here, so use math where necessary. Thank you.


title: How much effect does the Bernoulli effect have on lift?

text: I understand that the Bernoulli effect is a flawed explanation for the cause of lift, and does not cause much at all, but _how much_? Is there any experimental data on the force caused by the Bernoulli effect? Maybe implicitly through data of the pressure difference between the top and underside of an aeroplane's wings. After that, I assume I could (crudely approximating the pressure to be acting perpendicularly to the flight direction) use $\Delta P A$ to work out the net force on the plane. Perhaps there is another way to _quantitatively_ analyse the extent to which the Bernoulli effect causes lift. **Edit:** see this short cartoon (content similar to Mike Dunlavey's answer).


title: Reflection of sound waves

text: I was doing a physics experiment, and i encountered a question which i couldnt answer. The experiment was about using a radar technique to measure the speed of sound. The apparatus was a plastic tube 90cm long and the sound source was a speaker connected to a function generator. The generator was sending a square wave with a low frequency(a couple of Hz)so we were creating sound pulses and we were observing their echoes through a microphone wich was connected ot an oscilloscope. When the other end of the tube was closed we could clearly see the reflected waves on the oscilloscope screen.However even with an open tube we could see that the waves reflected off the open end.(and they returned with an opposite pressure) Why did this happen?


title: What are virtual excitations in the context of superconductors?

text: In a superconductor I have read that the energy gap can be defined as the energy difference between the ground state and the virtual excitations of the system. Is this the energy required to form a cooper pair?


title: Are quantum mechanics and quantum physics the same field?

text: What is the difference between _quantum mechanics_ and _quantum physics_?


title: Is there a named unit that, when divided by 32, gives meters per second?

text: I am receiving unknown units of speed from another system. I must divide the value by 32 to get meters per second. What units do I use to refer to the values I'm receiving? Is there any such unit? Is this just random malice from the previous developers? Google give me nothing.


title: What is meant by the quantum integrity in a superconducting loop

text: I encountered this in the context of using squid microscopy to test for d-wave pairing in an iron arsenide superconductor where the goal of the experiment was to detect half integer flux quantum transitions to indirectly observe a change in the order parameter.


title: Expansion of Galaxy's Caused by Electromagnetic forces

text: From my limited physics knowledge i know that gravity when compared to Electromagnetic forces is very weak... So if every object is polarized (positive and negative separation), then on a bigger level shouldn't galaxy's have this too?   So then this could explain why galaxy's are repealing away from each other with their like polarity, seeing the force of gravity is no match for the electromagnetic forces acting on them. Just a thought though. Would be great to see if it is wrong and the real physics behind it.


title: Centripetal/Gravitational Force

text: Suppose a satellite is orbiting the Earth. The gravitational and centripetal force supposedly point towards the Earth. Therefore, the net force is towards the Earth. Since the satellite doesn't fall immediately towards the earth, what's the third force holding the satellite in orbit? First impression tells me its the elusive "centrifugal force", but I was told to use this with caution as it's often a misnomer. What's the correct line of reasoning behind this?


title: 2D Harmonic Oscillator Commutators

text: So I am given a 2-dimensional harmonic oscillator with $H=H_1+H_2$ where $$H_i=\frac{p_i^2}{2m}+\frac{1}{2}m\omega^2x_i^2$$ Additionally, $$L=x_1p_2-x_2p_1$$ If we define $$A=\frac{1}{2\omega}[H_1-H_2]$$ $$B=\frac{1}{2}L$$ $$C=\frac{-i}{\hbar}[A,B]$$ Where [A,B] is the commuatator of A with B. We are asked for the explicit form of C, but isnt it just $$[H_1-H_2,L] = [H_1,L]-[H_2,L]=0$$ Due to the isotropy of space. It just does not make sense that C would be 0, because then the three would not be closed under commutation (which I am supposed to show).