title: How do I derive the critical temperature for bose condensation in two dimensions?

text: In class we derived the 3D case, but there's a step I don't understand: $$ N = g \cdot {V \over (2 \pi \hbar)^3} \cdot \int\limits_{0}^{\infty}{1 \over{e^{\left( E_p \over{K_B T}\right)}-1}} d^3 p = g \cdot {V \over (2 \pi \hbar)^3} \cdot 4 \pi \cdot \int\limits_{0}^{\infty}{p^2 \over{e^{\left( E_p \over{K_B T}\right)}-1}} dp $$ ... I feel like if I knew why that step made sense, I could figure out how to do the equivalent thing for the 2D case, but I'm stuck on that.


title: topological twisting by introducing bosonized operator

text: In this paper http://arxiv.org/abs/hep-th/9309140 on page 125, the authors claim that one can twist the $N=2$ theory by introducing a term in the action $\frac{1}{2}\int R \phi$, where $\phi$ is the bosonized version of the field we want to shift the spin of, and $R$ is the scalar curvature. I don't see how this could work at all. Further, which twist is being used here? Presumably the $A$-twist from the context the claim appears in in the paper, but where in this term is a choice of $U(1)$ action to couple to the worldsheet holonomy? As an additional question, is there any way to think about twisting this way in 4 or other dimensions?


title: why is the transition $3p^53d^2 \to 3p^63d^1$ (hydrogen atom) forbidden?

text: What I was thinking is that in 3d subshell (l=2) we have two electrons with $$m_l=-2$$ (spin up and down) and if we move to 3p we will fill the last vacant position - that is $$m_l=1$$ with spin down position. and that doesn't obey the selection rule of $$ \triangle m_l=0,-1,1$$ but I saw in my notes that the transition $$3p^53d^1 \to3p^63d^0$$ is allowed so I don't understand something fundamental in there...


title: Choosing the right point when calculating moments

text: A body is composed of two straight pins that are joined at a right angle. They have lengths $a$ and $b$ and the mass per unit length is $\rho$. When the body is balanced on a flat surface, as shown, how large is the normal force against the ground in the right point of contact? 4 options as can be seen in the picture. ![picture](http://i.imgur.com/Hr1RBRF.jpg) Let me point out that this is a conceptual question. When I first tried to solve this problem I decided to choose to calculate the moment around the left contact-point in order to reduce one term(left normal force). This seems like a natural way but gives a false answer($N_2= \rho g$. If I instead calculate about the vertex of the triangle and use Newton's second law I get the correct solution.(answer is D). So how should I choose the "correct" point? **Around left contact point:** By dropping an altitude **h** at the right angle we get that: Notice that: $\cos\alpha=\frac{a}{\sqrt{a^2+b^2}}$, $\cos\beta=\frac{b}{\sqrt{a^2+b^2}}$ and $m_ag=\rho ag$ and $m_b=pbg$ $-\frac{\rho a^2g}{\sqrt{a^2+b^2}}-\frac{\rho b^2g}{\sqrt{a^2+b^2}}+N_2\sqrt{a^2+b^2}=0$. Simplifying gives $N_2=\rho g$


title: Quasi-particle and quasi-hole excitations of Laughlin states and generalization of Laughlin states

text: The Laughlin wave function at filling fraction $\nu=\frac{1}{m}$ is \begin{equation} \Psi_m=\prod_{i<j}(z_i-z_j)^m e^{-\sum|z_i|^2/4l_B^2} \end{equation} It is claimed in section 7.2.3 of Wen's book that the wave function of a quasi-hole excitation on the top of this state is \begin{equation} \Psi^h(\xi,\xi^*)=\sqrt{C(\xi,\xi^*)}\prod_i(\xi-z_i)\Psi_m \end{equation} I am wondering why this is a wave function of a quasi-hole, it looks more like the wave function of a quasi-particle to me. If this is indeed the wave function for a quasi-hole instead of a quasi-particle, what is the wave function of a quasi-particle? Later in section 7.2.4, the author discusses a way to construct generalization of Laughlin states by adding quasi-holes or quasi-particles on the top of Laughlin states, and when the density of quasi-holes or quasi-particles reaches certain value, they form a Laughlin state by itself. And there are two examples: by condensing quasi-holes of $\nu=1/3$ Laughlin state, we get a $\nu=2/7$ FQH state with wave function: \begin{equation} \Psi=\int\prod_id^2\xi_i\Psi_3\prod(\xi_i- z_j)(\xi_i^*-\xi_j^*)e^{-\frac{1}{4l_B^2}\frac{1}{3}|\xi_i|^2} \end{equation} and by condensing quasi-particles of $\nu=1/3$ Laughlin state, we get a $\nu=2/5$ FQH state with wave function: \begin{equation} \Psi=\int\prod_id^2\xi_i\prod_{i<j}(\xi_i-\xi_j)^2(\xi_i^*-2\partial_{z_i})\Psi_3 \end{equation} My understanding is, the original electrons and added excitations form a Laughlin state with a common filling fraction, then by charge conservation, we should have \begin{equation} {\rm original\ filling}+{\rm resulting\ filling}\times{\rm charge\ of\ excitation}={\rm resulting\ filling} \end{equation} However, these two examples are telling me (notice the factor $\frac{1}{2}$): \begin{equation} {\rm original\ filling}+{\rm resulting\ filling}\times{\rm charge\ of\ excitation}\times\frac{1}{2}={\rm resulting\ filling} \end{equation} (The first example fits as \begin{equation} \frac{1}{3}-\frac{1}{3}\times\frac{2}{7}\times\frac{1}{2}=\frac{2}{7} \end{equation} and the second example fits as \begin{equation} \frac{1}{3}+\frac{1}{3}\times\frac{2}{5}\times\frac{1}{2}=\frac{2}{5}.) \end{equation} My questions are: 1) Why should there by a $\frac{1}{2}$ factor? 2) Is it related to the power of $(\xi_i-\xi_j)^2$, namely $2$? 3) If it is related to that $2$, can I change that $2$ into another integer to get another FQH state?


title: Tadpole diagram and vacuum

text: This statement may be well-known. For many **massless** theories, these **tadpole diagram graphs vanish in dimensional regularization (by dimensional analysis and the absence of any inherent mass scale in the loop integral).** ![enter image description here](http://i.stack.imgur.com/aWjxr.png) How can one show this is QFT statement is true(tadpole diagram graphs vanish) regardless the coupling, say by Yukawa-Higgs via $\lambda \bar{\Psi} \phi \Psi$ or by gauge field $g \bar{\Psi} \gamma^\mu A_\mu \Psi$, etc? i.e. does not depend on $\lambda,g$ or an energy cutoff $\Lambda$?


title: Poseuille number for flat channel?

text: For a flat channel with fully developed flow, the Poseuille number when using the hydraulic diameter is $$Po=C_fRe_{D_{H}}=24$$ But when using 2b as the length scale, the Poseuille number is $$Po=C_fRe_{2b}=12$$ Why is this? I thought that the hydraulic diameter of a flat channel is 2b. Shouldn't they be the same? In the derivation used, b is half the channel width


title: How to measure vector potential such as that of magnetic field?

text: If we want to measure gravitational potential energy of a body-earth system, we can simply do it by measuring the body's height from ground. How could we can measure vector potential such as that of magnetic field?


title: Does the opening angle of the cone matter?

text: When discussing orbital mechanics, you learn that all orbits roughly follow an ellipse which is obtained as the intersection of a cone with an inclined plane, creating conic sections. ![enter image description here](http://i.stack.imgur.com/v4rml.gif) Below is a plot of different mathematical variables of a cone. I am trying to figure out if the ellipses that orbits follow are a special rule where the opening angle (at the top of the diagram) of the cone is 90 degrees, or if it is irrelevant what that angle is. ![enter image description here](http://i.stack.imgur.com/TL4rA.gif)


title: What is the shortest controllable time?

text: I was reading http://en.wikipedia.org/wiki/Planck_time and noticed a reference to: http://phys.org/news192909576.html where it is stated that: **12 attoseconds is the world record for shortest controllable time** This article is now almost 4 years old, is this still the record or is in the meantime this record broken?


title: $E,H,\rho,\vec{j}$ intensive? $B,D,\Phi,\vec{A}$ extensive? Why not $ DdE,BdH ,\Phi\delta\rho,\vec{A}\cdot \delta \vec{j}$ as infinitesimal work?

text: $u$ is the e&m field energy $\frac{\partial u}{\partial t}=H\cdot \frac{\partial B}{\partial t}+E\cdot \frac{\partial D}{\partial t}$ Compared with thermo physics: $PdV,TdS,\mu dN$ ${H,E,P,T,\mu}$ are intensive variables ${B,D,V,S,N}$ are extensive variables But in microscopic level, there is no $H$ field. $B$ should be more meaningful. My question is, why are $H$ and $E$ intensive variable. Why are $D$ and $B$ extensive variables? **I think all of them $E,D,H,B$ should be intensive,** because energy density $u$ itself is an intensive variable.


title: What happens if you let a cable roll slip into a black hole?

text: 1. Does the cable roll spin faster the more cable goes into the black hole in reference of a observer standing next to it?   2. Can gravity pull the cable that it exceed the speed of light inside a black hole in reference to the center of the black hole?


title: Translation invariance in crystals

text: **Context** : solid state physics - dynamics of atoms in crystals. Coupling constants are defined as: \begin{align*} \frac{\partial^2\Phi}{\partial r_{nai}\partial r_{mbj}} = \Phi_{nai}^{mbj} \end{align*} where $\Phi$ is the total energy of the crystal expressed as a function of nuclei coordinates. $r_{nai}$ is the position vector of the $n$-th cell, $a$-th atom at $i$-th spatial direction. My book says that coupling constants have some properties. One of them is the "translation invariance", which is stated as: \begin{align} \Phi_{nai}^{mbj} = \Phi_{0ai}^{(m-n)bj} \end{align} I don't understand the zero index in RHS. Since the $n$ or $m$ represent the cell, what does it mean to refer to zero-th cell ?


title: Reflection of an evanescent matter wave within a finite barrier?

text: To my understanding if I have a _finite_ barrier with potential $V(x)>E$, then to the left of the barrier, the wavefunction can be represented as two exponentials: $$\psi= e^{(ik_{left} x)} + e^{-(ik_{left} x)}$$ Where the negative exponent represents a wave travelling in the opposite direction. Firstly is this correct, or me making things up? So in the barrier, as $V>E$, $k$ is imaginary so we get a solution of the form: $$\psi = e^{-k x}$$ i.e. exponential decay. However I can't understand why it shouldn't be reflected upon leaving the finite barrier. If this is so, then it would produce a similar wave but of negative exponent to the incident wave, i.e. a wave which is exponentially growing from left to right? Is this correct? Is this allowed?


title: Can Allan variance be generalized such that the "Oscillator model" is not presumed?

text: The definition of Allan variance, $\sigma^2[ \tau ]$, which relates to "stability of clocks" is described on the Wikipedia page as being derived in terms of an " _Oscillator model_ ": > " _The oscillator being analysed is assumed to follow the basic [wave form] > model of_ $V_{osc}[~t~] := V_0 ~ \text{Sin}[ ~ \Phi_{osc}[~t~] ~ ]$" where $V_0$ is constant (i.e. independent of the parameter $t$) and the " _total phase_ " is also expressed as $$ \Phi_{osc}[~t~] := \omega_{osc}~t + \phi_{osc}[~t~] $$ in terms of a (constant) " _nominal angular frequency_ , $\omega_{osc}$" and a " _fluctuating component_ , $\phi_{osc}[~t~]$", along with the parameter $t$. **My question:**   Are there "generalizations of Allan variance" based on more general "wave form models"? Such as perhaps based on   * "smooth amplitude modulated oscillation": $V_{AM}[~t~] := | V_{AM}[~t~] | ~ \text{Sin}[ ~ \Phi_{AM}[~t~] ~ ]$,   where $| V_{AM}[~t~] | \gt 0$, and $\exists ~ \frac{d}{d~t}\\!\\!\\!\left[~V_{AM}[~t~]~\right]$, or even   * "any sequence which changes sign often enough": $V_{alt}[~t~] := | V_{alt}[~t~] | ~ \text{Sin}[ ~ \Phi_{alt}[~t~] ~ ]$,   where $| V_{alt}[~t~] | \gt 0$, but where differentiability (wrt. parameter $t$) or even continuity (wrt. parameter $t$) are not required of $V_{alt}[~t~]$, or even   * "any suitably wiggled sequence": $V_w[~t~]$,   where $\small \forall t_k : \left( \exists t_j, t_p : (t_k - t_j) ~ (t_p - t_k) > 0 \text{ and } (V_w[~t_k~] - V_w[~t_j~]) ~ (V_w[~t_p~] - V_w[~t_k~]) > 0 \right) \implies $ $\small \left(\exists t_q : (t_p - t_j) ~ (t_q - t_p) > 0 \text{ and } (V_w[~t_q~] - V_w[~t_j~]) ~ (V_w[~t_p~] - V_w[~t_q~]) > 0 \right)$, or even   * "any strictly monotonous function": $V_m[~t~]$,   where $\small \forall t_k : \left( \exists t_j, t_p : (t_k - t_j) ~ (t_p - t_k) > 0 \right) \implies \left( (V_m[~t_k~] - V_m[~t_j~]) ~ (V_m[~t_p~] - V_m[~t_k~]) > 0 \right)$ ? And if so, how are the corresponding "generalizations of Allan variance" called?


title: Where do our 4 macroscopic spacetime dimensions reside in multidimensional models of the universe?

text: In models such as M-theory with 7 'higher dimensions' plus the 4 macroscopic spacetime dimensions, where do our 4 macroscopic spacetime dimensions reside _ordinally_? My reason for asking is TV shows such as the 'Fabric of the Cosmos' that alludes to 'lower' dimensions in string theory. I can understand if our 4 are, say the 7th, 8th, 9th and 10th so there are dimensions lower than us but that we cannot readily detect.


title: Are we so sure about superposition?

text: Apparently particles can be anywhere when not observed. How strong is this theory really? Okay the wave-function can be collapsed through observation but how are we so sure that when an object is not being observed that it is in potential states? How are we so sure whether or not the wave function is collapsed by a conscious observing force other than us? And is superposition an assumption due to the fact that when we observe waves they act as particles, so we assume that everything has different potential until measurement; when really we don't know for sure whether it is that an object is in potential states or a conscious observance by a higher power already puts things in order already eliminating this potentiality?


title: Fugacity of the fermi gas

text: It can be shown that in the high temperature exploration of the Fermi gas, the Fermi function may be expanded to second order in $e^{\beta \mu}$, where $\beta = 1/kT$ and $\mu$ is the chemical potential like so $$f_+ \approx e^{-\beta(\epsilon_i - \mu)} - e^{-2\beta(\epsilon_i - \mu)}$$ Using the following results $$\sum_i \epsilon_i e^{-\beta \mu} = \frac{3}{2} \frac{Z(1)}{\beta}\,\,\,\,\,\,\,\sum_i \epsilon_i e^{-2\beta\epsilon_i} = \frac{3}{2} \frac{Z(1)}{2^{5/2}\beta}$$ we obtain $$e^{\beta \mu} \approx \frac{N}{Z(1)} + \frac{1}{2^{3/2}} \left(\frac{N}{Z(1)}\right)^2$$ My question is what is the physical interpretation of the fugacity $e^{\beta \mu}$ in the small regime? Many thanks.


title: matrix elements of the electronic molecular Hamiltonian between a hartree product and a Slater determinant

text: This may belong in Chemistry, but I thought I might try my luck here first. In Szabo's book, an exercise requires a proof that = (N!)^(1/2) * given that |K(HP)> is the Hartree product wave function corresponding to the N-electron Slater determinant |K>. I've been trying for a couple days now, but the only idea I have is to evaluate both integrals explicitly by throwing away all terms arising from the Slater determinant that would vanish by orthonormality (all but one for the one) then evaluating the integrals over the sums of one and two electron operators as a sum over allowed orbital permutations. This would defeat the point of having the theorem to simplify the derivation. Does anyone have a better more clever way or a hint to point me in the right direction? Thanks


title: How to measure rainfall? And how is it determined for a specific location?

text: How is rainfall measure in a pinpoint location? This is what I think:   There are automatic rainfall measuring stations at: A, B, C. But say I want to know point D, located between A and B, its rainfall level. How do I figure that out? Do I take the average? Do I go to the nearest neighbor on the map? Thanks in advance!


title: Time reversal symmetry in the presence of friction

text: I was reading a paper on time reversal symmetry, and came across an example of a pendulum swinging in the presence of friction: > When we consider the more realistic physical situation of a swinging > pendulum in the presence of friction, we can tell the difference between a > forward and a reverse film of this pendulum. Namely, the original (forward) > film will show the swinging pendulum losing amplitude until it comes to a > standstill. However, the film in reverse direction shows a swinging pendulum > whose amplitude is increasing in time. The latter film is clearly unphysical > as it does not satisfy the natural laws of motion anymore (assuming there is > no hidden source of energy feeding the pendulum). The presence of friction > breaks the time-reversal symmetry of the ideal pendulum. I don't understand this. If I am not wrong, we are only reversing the direction of time, not that of the frictional force (the nature and cause of this force is not relevant either). In the movie played backwards, the frictional force will be along the direction in which the bob swings. This force would be proportional to $v^2$, and would give the bob more momentum consistent with the observation. So what am I missing? How does the presence of friction break time reversal symmetry?


title: Status of Teleparallel Quantum Gravity

text: There are several approches to incorporate torsion into a theory of gravity. According to this article, requiring a consistent coupling to Dirac fields leaves us with the Einstein-Cartan approach or the rules of general relativity. The latter can be motivated by the principle of general covariance (see section IV of this review article), but does _not_ restrict us to torsion-free connections: Rather, there's a sort of gauge symmetry (see this article) between curvature and torsion, ie a whole class of connections that can be used to reproduce the equations (if not the geometry) of general relativity, with the torsion-free Levi-Civita connection and the curvature-free Weitzenböck connection as limiting cases. Einstein-Cartan theory comes with new physics and is not without its own problems (see this opinion piece (PDF)), so teleparallel gravity would be the conservative choice. As teleparallel gravity is a gauge theory and actually does not depend on the equivalence of inertial and gravitational mass, it is conjectured to be more friendly to quantization. However, I've yet to see a theory of quantum gravity based on teleparallelism. Assuming this is not just the case because everyone qualified to work on it is busy playing with yarn (strings and loops, you know ;)), what are the problems that one runs into with a teleparallel theory of quantum gravity? Are they the same ones (renormalizability) that prevent us from quantizing general relativity? For bonus points, is there any relation of the Teleparallelism/GR equivalence and Gauge/Gravity duality of string theory besides the superficial?


title: Energies and numbers of bound states in finite potential well

text: Hello I understand how to approach finite potential well (I learned a lot in my other topic here). However i am disturbed by equation which describes number of states $N$ for a finite potential well ( $d$ is a width of a well and $W_p$ is potential ): $$ N \approx \dfrac{\sqrt{2m W_p}d}{\hbar \pi} $$ I am sure it has something to do with one of the constants $\mathcal L$ or $\mathcal K$ defined this way: \begin{align} \mathcal L &\equiv \sqrt{\tfrac{2mW}{\hbar^2}} & \mathcal{K}&\equiv \sqrt{ \tfrac{ 2m(W_p-W) }{ \hbar^2 }} \end{align} and the transcendental equations for ODD and EVEN solutions: \begin{align} &\frac{\mathcal K}{\mathcal L} = \tan \left(\mathcal L \tfrac{d}{2}\right) &&-\frac{\mathcal L}{\mathcal K} = \tan \left(\mathcal L \tfrac{d}{2}\right)\\\ &\scriptsize{\text{transc. eq. - EVEN}} &&\scriptsize{\text{transc. eq. - ODD}} \end{align} **QUESTION:** Could anyoe tell me where does 1st equation come from?


title: Partial derivative of the classical action with respect to time

text: Does anyone know how to derive the general identity: $$\frac{\partial S}{\partial t}=-E$$ where $S$ is the classical action defined as $$S=\int_0^t\left[\frac{1}{2}m\dot x-V(x))\right]d\tau$$ and $E$ is the total energy? Any help will be really appreciated.


title: Travel time for accelerating object

text: I have an object which starts at velocity = 0. Accelerates to a known max speed and continues thereon at constant velocity (max speed). > _My problem: How long does it take to reach a known destination?_ I've visually described it like this: ![visual description of problem](http://i.stack.imgur.com/f6Rhe.png) Where we have velocity on vertical-axis, and time on horizontal-axis. Up to where the lines intersect (let's call this time $t_M$, the blue line is the current velocity, after that, the red line. Equation for blue line is $v = a \cdot t$ where $a$ is acceleration, $v$ is velocity (vertical-axis) and $t$ is time (horizontal- axis). Equation for the red line it $v = M$, where $M$ is the max speed (known) and $v$ velocity. Since the distance to travel is know, it should be as easy as adding up the area under the blue line, up until it equals the requested distance. Is there any way to do this without handling the areas before and after $t_M$ individually? My current solution is: $a$ (acceleration, known), $d$ (requested distance, known), $M$ (max speed, known) $t_M = M / a$ $distanceDuringAccel = t_M * M / 2$ (the area under the blue line)               if distanceDuringAccel > d          We will not reach our destination during acceleration, add the area under the red line aswell          else          we will reach destination during acceleration. t = sqrt(2 * d / a)      The above works, but feels very clumsy. Is there a better way to address this?


title: Update velocity or position first in computation?

text: I am trying to make a simulation of a vibrating string. The string is divided into $n$ points, and each point along the string is acted upon by a force due to the positioning of its neighbors. I eventually have to update both velocity due to acceleration, and position due to velocity at each point, but the two are dependent on each other. Is there a _physically correct_ order in which to update position and velocity?


title: Is there an orthogonal time dimension in Minkowski spacetime?

text: Space dimensions are orthogonal one to each other. But what about time in the Minkowski diagram? At first sight, time seems to be orthogonal to space. But we have to consider that each Minkowski diagram is an observer's diagram. Thus, an object that is not moving with regard to the observer will describe a vertical worldline in the Minkowski diagram of the observer. However, from the point of view of another observer (inertial frame) the worldline described by the object might be inclined. Even the worldline of the observer himself which is the y-axis might be sloped from the point of view of other observers. The conclusion seems to be that time is not orthogonal to space. Or is there an error?


title: Could non-locality imply a holographic universe?

text: I am just a physics groupie, so please excuse me if this is stupid. But wouldn't non-locality possibly imply a holographic universe? An analogy would be a projector and a hologram-- if a change occurs at the projector regarding two "entangled" objects that are actually not separate, but appear to be so in the projected hologram, wouldn't that change appear to have happened instantaneously at two separate points in the hologram?


title: How can a spiral galaxy exist?

text: A spiral arm orbiting a central mass should be dispersed quite quickly as the outer elements would move more slowly than the inner ones. The Milky Way, is about 59 Galactic Years old, which, one would have thought, would be enough rotations to disperse a spiral structure entirely. Is there, then, something keeping the spiral arms in existence, and if so what could it be? Or are the spiral galaxies monstrous co-incidences?


title: How to determine what size telescope to buy

text: A couple of years ago my son showed an interest in astronomy and we bought a 6" reflector telescope. We use it pretty regularly and have enjoyed it immensely. Lately we've both been wishing we had something bigger to be able see more things and to see what we can see now with more detail. How do you determine the size of telescope needed to view a certain object? I understand that there are a lot of other factors that come into play when talking about what you can see and how well you can see it. Ideally I suppose what I'm looking for is some sort of chart/table that gives a general guideline of the scope size and some of the objects that should be viewable (with an average setup). Are there any such resources?


title: Optimal Angular Field of View (AFOV)

text: Given the rather huge price differences between eye pieces at the same focal length. How exactly does the AFOV affect the view seen through the eyepiece? Are higher / lower AFOV better for certain situations? or is higher always better?


title: How exactly can we describe the normal force on a static person standing on earth's surface using general theory of relativity?

text: For planetary motion I can understand that the planets move along the geodesics e.g. the warped space-time geometry. Imagine that the moon gets suddenly stopped by some external force and comes to rest momentarily, which force will bring it to motion towards earth. And how can we describe "Normal force" as mentioned in the subject. Please excuse me for my ignorance in advance but an excessively mathematical explanation would be very difficult for me to understand.


title: Original Paper on Slow Neutron Capture by Fermi?

text: I was wondering what the title of his original publication on Neutron capture was. I've found his Nobel lecture but I want to read his initial publication. A title would be most appreciated. It's currently hard to find it since searching his name brings up a lot autobiographical material, telescopes, Stanford publication archives, you name it. A link to the original wins you a big e-hug =)


title: Calculate the Riemann tensor and Ricci tensor

text: Given a metric tensor $\gamma_{ij}$ (where $i, j = 1, 2, 3$; the metric tensor of 3- dimensional space is denoted by $\gamma_{ij}$ to distinguish it from the metric tensor $g_{\mu\nu}$ of 4-dimensional space-time), show that the Riemann tensor is equal to \begin{equation} ^{(3)}R_{ijkl}=\frac{\xi}{R^2}\left(\gamma_{ik}\gamma_{jl}-\gamma_{il}\gamma_{jk}\right),\end{equation} and \begin{equation}^{(3)}R_{ij}=2\frac{\xi}{R^2}\gamma_{ij}\end{equation} where $\xi=0,\pm 1$, distinguishes 3-plane, 3-sphere and 3-hyperboloid. The first equation above looks quite simple, but I have no idea how to derive it. Can you please suggest me how to start deriving the above expressions?


title: Does inducing charge on a grounded conductor do work?

text: Imagine a grounded conductor as an plane of infinite area. If I bring a charge to a close proximity to the plane, it induces some charge distribution on the conductor. The question is: is any work being done on the (induced) charge during this process?


title: Space distance between observers and Lorentz transformation

text: I'm not a specialist of Lorentz transformation. I would like to know if the Lorentz transformation from one observer to another preserves the space distance between these two observers. Example (edited): Two spaceships are flying through space at arbitrary, different velocities. Two signal flashes are shot simultaneously (Earth frame) from Earth to the spaceships, which are equidistant to Earth (Earth frame) at the moment of the simultaneous (Earth frame) two explosions at the arrival of the two flashes. Each space ship is measuring the space distance between the two flashes. Will their result be the same? And, will a measurement from Earth have a different result because of the length contraction effect? (No gravity must be taken into account. In case of need you may replace Earth by a third spaceship).


title: Simulation of broken object

text: I was thinking and the following question came out: how an object that is falling is simulated once it hits the ground? Specifically, I would like to understand how one would be able to simulate the cracking of the object, and its consequent breakdown in several pieces. As far as I understood, this topic is quite important in computer games, but I was wondering how this things are studied in physics. One approach I recognized was the use of meshes and continuum physics. On the other hand, a N body problem where the particles were "glued" could also work. Does anyone knows what are the principles of crack formation and object breakdown in this sense?


title: Ways of achieving parallel rays of light?

text: I am thinking of ways of achieving parallel rays of light. Let's say I have an object that emits light, for instance the display of my mobile phone. The display emits rays of light in every direction. Is there a "filter" I can put in front of the screen that only lets rays of light traveling in a certain direction through - e.g. only rays of light traveling orthogonal to the screen? Or what other ways are there to achieve parallel rays of light?


title: Twin Paradox - different approaches

text: What was the difference between Langevin's approach to the twin paradox and Max Von Laue's? I don't understand how Langevin tried to use the idea of absolute acceleration to explain the distinction in time difference? (Also not sure if this is physics or philosophically relevant?)


title: Per unit of energy, what radiation is the most penetrating?

text: This question about nuclear reactor shielding got me wondering, what forms of radiation are the hardest to shield? I suppose Neutrinos will have to be excluded since they're the obvious winner. So assuming an equal kinetic energy, what's the hardest radiation to shield? I think I've read Muons penetrate to much greater depths in accelerators but surely there are other things that are hard to attenuate. I'm not really looking for a list so much as an explanation for what makes a particular particle (or groups of particles in the case of Alpha particles) difficult to shield.


title: Solving 2nd-order ODEs

text: I was reading this PDF REF per request:               Title: Fourier Series: The origin of all we’ll learn     Link: http://www.math.binghamton.edu/paul/506-S11/CpxFn2.pdf     Author: Paul Loya       which I found very good to explain the origin of the Fourier series. However, I have a problem with 2 key points of the paper which are left unexplained. I was wondering if someone could put me on the right track: In page 12, he goes on writing: $$ F''(x) = -m^2 F(x)$$ And then "we see that" $$F(x) = A \cos(mx) + B \sin(mx)$$ This is the part I have a problem with. What's the method to get to the second equation. Why would $F(x)$ equals to such thing? Where is that coming from? Later for the second part of the equation he uses he states: $$G''(y) = m^2 G(y)$$ And "From elementary differential equations, we know that" $$G(y) = Ae^{my} + Be^{-my}.$$ I didn't really go that far at school, so probably miss the bit where this was explained. I don't necessarily ask someone to explain me in detail something that might be taking a long time, but maybe just pointers to documents, wiki articles, name of these identities which I could lookup on the web myself... having that would already put me on the right path.


title: What happens to light after it enters an eye

text: What happens to the light [energy] after it enters an eye and hits the rods and cones? I presume the energy becomes electrical, and it must be near 100% perfect, else our eyes would heat up? Or am I missing something? The motivation of this question is solar panel technology.


title: Why does a car engine not do work if the wheels don't slip?

text: I saw this mind boggling result that if the tires don't slip then the work done by an engine to move a car is zero. Why is this true? Moreover, what does this truly mean? **Update:** Sorry about not being clear, but I was talking about an idealised case where air resistance is negligible and the tires are perfectly rigid and there is no internal friction at play.


title: Riemann normal chart and special relativity

text: When you pick Riemann normal coordinates at a point, then the Christoffel symbols vanish and the metric is flat, but the Riemann curvature tensor does not necessarily vanish. Since Einstein said that locally the laws of special relativity hold, how does that reconcile with the Riemann curvature tensor not being zero (which means that spacetime doesn't appear to be flat in that local neighborhood)?


title: Does any object placed in an electric field change the electric field?

text: Lets say I have a point charge of magnitude $+q$, All around it I would have a symmetric radial electric field. Now if I place a neutral object lets say a sphere (doesn't matter insulating or conducting) in this field some distane away from the point charge. A negative charge will be induced on the object near the point charge and a positive charge on the opposite side. No matter how small this induced charge is, due to the radial distance of the two (positive and negative) there must be an increase/decrease in net electric field on either side of the object and mostly everywhere else too ! I hope that what I am thinking is wrong, because we have not been taught that anything placed in electric field would affect the field itself regardless of it's nature. But I can't figure out what am I thinking wrong, how to solve this dilemma ?


title: Does a HDD ( Hard Disk Drive) emit radio waves?

text: While I was reading the "guide" of a HDD I had bought a while ago, I found that it might emit radio waves. I was wondering if that is true and why that happens?


title: Does the covariant derivative of a derivative of the metric vanish?

text: The title holds the main question. For a little more backgroud: In a calculation I keep coming across terms of the form $$ D_M \partial_N e_P^A \overset{?}{=} 0$$ where $e_P^A$ is the vielbein and $D_M$ is the covariant derivative _including_ both the spin _and_ the affine connection, i.e. the vielbein postulate reads $D_M e_P^A = 0$. Generally, I would not expect the expression to vanish, but the again I know little about the more involved machinery of differential geometry.


title: Phase Space dimension of Lorenz Strange Attractor

text: It is often discussed in 3 spatial dimensions and the need for third dimension to prevent self intersection is mentioned. But shouldn't the phase space of the Lorenz system be 6 dimensional, i.e., the 3 momenta also?


title: What does the sum of two qubits tell about their correlations?

text: > How much can I learn about correlations between two quits by measuring the > sum of their values? What is the best way to formalize such a question? Below is my original, longer formulation of the same question in spin language. I have an ensemble of pairs of spin-$1/2$ particles. All I can do is to measure total $S_z$ for each pair, each time learning one of the three standard outcomes: $+1$, $0$ or $-1$. Based on the knowledge of this probability distribution $P(S_z)$, what can be said about the density matrix of my ensemble? In particular, what is best measure for correlations between the two particles in these circumstances? I understand that I won't be able to distinguish whether the correlations are classical or necessarily quantum-mechanical, but struggle to come up with a measure better than a classical correlation coefficient $$r=\frac{\langle s_1 s_2 \rangle-\langle s_1 \rangle \langle s_2 \rangle}{\sqrt{\langle (s_1- \langle s_1 \rangle)^2 \rangle \langle (s_2- \langle s_2 \rangle)^2 \rangle }}$$ where $s_1$ and $s_2$ are the $\pm 1/2$ spin projections (or $0$, $1$ qubit values) of the 1st and the 2nd particle, respectively, so that my only observable can be expressed as $S_z=s_1+s_2$. I then minimize $r$ over all possible $\langle s_1 \rangle$ and $\langle s_1 \rangle$ consistent with measured $P(S_z)$ and declare that the outcome is the minimal possible correlation in the pair. Is there a more elegant way? **UPDATE:** Classically, if we take the definition of zero correlations as $\langle s_1 s_2 \rangle =\langle s_1 \rangle \langle s_2 \rangle$, then it implies that $s_1 = +1/2$ and $s_2=+1/2$ with some _independent_ probabilities $p_1$ and $p_2$, so that the sum is distributed bimomially $P(+1)=p_1 p_2$ and $P(-1)=(1-p_1)(1-p_2)$. These two equations have a solution with $0\leq p_1, p_2 \leq 1$ only if $$P(+1) < 1-2 \sqrt{P(-1)}+P(-1)$$ Graphically, this condition is satisfied within the region bounded by a black line in the plot below. The rest of the available probability space is the region of excess correlations ($r_{\text{min}}>0$, depicted in red). ![enter image description here](http://i.stack.imgur.com/BjYIe.png) **Updated question:** is there a quantum loophole in such analysis? I suspect not but would really like to come up with a shorter/cleaner argument. **UPDATE-2** My selected spin/qubit degree of freedom is one of the many variables characterizing the state of the two particles. Can we be sure there exists no pure product state that can lead to $r_{\text{min}}>0$? See a related active question on fermionic entanglement of two particles.


title: Atmospheric hydrogen in Titan

text: In 2005, some astrobiologists proposed that methanogenic life on the surface of Titan would consume both hydrogen and acetylene from the atmosphere, causing a downward gradient in the concentrations of the gases My naive thinking would be that, the most natural reason for a downward gradient for molecular hydrogen on any atmosphere would be that, being the lightest gas that is out there, it will tend to accumulate in the upper layers of the atmosphere. Why isn't natural buoyancy of molecular hydrogen enough to explain the observed downward gradient? Nasa - What is consuming hydrogen and acetylene in Titan?


title: $\frac{dt}{d\tau}=\gamma$ in special relativity

text: I hope this is not too silly a question: We often see $$\frac{dt}{d\tau}=\gamma=\frac{1}{\sqrt{1-v^2}},$$ taking $c=1$. Problem: I don't understand why... In the Minkowski metric, using the $(-+++)$ signature and taking $c=1$, $$ds^2=-dt^2+d\vec x^2\\\ d\tau^2=-ds^2\\\ \implies d\tau^2=dt^2-d\vec x^2\\\ \implies 1=\left(\frac{dt}{d\tau}\right)^2-v^2\\\ \implies \frac{dt}{d\tau}=\sqrt{1+v^2}\neq \frac{1}{\sqrt{1-v^2}}$$ What has gone wrong? with my reasoning?


title: Has Voyager 1 entered a solar radiation belt?

text: The Voyager 1 probe was sent out in 1977 to go where no man made object has gone before, after more than 35 years it’s still going strong. It’s now 124 AU away from Earth and many are wondering when it’s going to leave our heliossphere and set off into unknown interstellar space. The readings from the probe have been more or less as expected until a dramatic change appeared. In August 2012. Voyager 1 entered a strange region where the solar wind came to a complete stop while the amount of cosmic rays set new records: ![enter image description here](http://i.stack.imgur.com/bXiXt.png) As Voyager 1 entered the new region, it seems like the region boundary where fluctuating back and forth. The magnetic field also strengthened as Voyager 1 entered the region, but the magnetic field did not change direction and that is the reason why the Voyager team has concluded that Voyager 1 is still in our heliossphere. Voyager 1 is now in this new unknown region within our heliossphere and we don’t know what this region is. Here is a suggestion; the new region is a solar radiation belt. Most scientists currently believe that the Sun has no radiation belts because the magnetic field, which turns every 11 years, is not stable enough to sustain a solar radiation belt. But recent observations from Earths outer radiation belts show that the belts can be drained and refilled with particles within weeks, so it may be time to take a second look at the possible existence of stellar radiation belts. But why is the possible solar radiation belt so far from the Sun? Earths outer solar radiation belt is compressed by the solar wind, so a solar radiation belt may experience an opposite effect and get blown far away from the Sun, the unknown interstellar wind could on the other side push it against the sun. **The readings we got when Voyager 1 entered the new region may be explained by Voyager 1 leaving the high density solarplasmasphere and entering the low density energetic solar outer radiation belt.** ![enter image description here](http://i.stack.imgur.com/mDtVV.gif) -The observed disappearance of the solar wind could be Voyager 1 leaving the solar plasmasphere. Earths plasmasphere is filled with an outflow of ions from the ionosphere from low and mid-latitudes, which could be similar to the solar wind filling the solar plasmasphere. What happens to the solar wind ions on the threshold to this new region seems to be unclear, are they slowing down and piling up, or are they accelerated out of there? -The observed sharp increase in cosmic rays could be energetic ions and electrons in the solar radiation belt itself. Ions with low cosmic ray velocity have recently been found in Earths radiation belts. We know that Jupiter has much stronger and more energetic radiation belts, so size amplify radiation belts and create cosmic ray velocity particles. We could then expect cosmic ray velocities to be normal in solar radiation belts which are located in a huge magnetic heliossphere. This also gives possible solutions to other cosmic ray problems, like why cosmic rays are decreasing during solar maximum, why cosmic rays can come in showers and temporary vary in particle type. It could also explain why we have found unstable isotopes in cosmic rays which cannot survive the long journey from the stars. Collisions with interstellar medium causing spallation are a proposed explanation of the unstable isotopes and the observed anti-matter, but collisions by the radiation belt with the solar wind will also give spallation and isotopes and anti-matter.  We currently don’t know what creates cosmic rays, supernovas is one proposal, but there is not enough supernovas in the universe to count for all the cosmic rays, so an additional source is needed. Cosmic rays from the radiation belts of stars could be such a source. If we have a solar outer radiation belt which is a Cosmic Ray source, CR-particles could also gyrate along solar magnetic field flux lines and bombard the Sun, with a much higher CR intensity than Earth experience in its magnetic field shielded quiet zone. -Voyager 1 observed an increase in the magnetic field as it entered the new region. Earths outer radiation belt is also the home of Earths ring current which induces a magnetic field. So the solar radiation belt itself could harbor a solar ring current which induces a magnetic field which can explain the observed increase in the magnetic field: ![enter image description here](http://i.stack.imgur.com/ECGmg.png) -The observed fluctuations of the region boundary could be caused by irregular solar wind causing expansion and compression of the solar radiation belt, similar to how solar wind causes Earths radiation belt to fluctuate. The fluctuations could also be caused by variations in the solar magnetic field as the Sun spins. -The number of cosmic rays hitting Earth is inverse proportional to the number of sunspots. An explanation to this could be that a varying solar magnetic field during solar maximum cannot sustain or confine a big stable outer radiation belt which may produce cosmic rays that hit Earth. It’s also possible that solar storms deplete the solar radiation belt of particles like solar storms deplete Earth’s outer radiation belts of particles. ![enter image description here](http://i.stack.imgur.com/1G6Rv.jpg) The Voyager team states that Voyager 1 has not left the heliossphere, the observed region then have another explanation. A solar radiation belt may be a solution, and may also explain how the majority of cosmic rays may be created in stellar radiation belts. It’s a proposal which so far seems to fit well with the observations and it could give us some great answers. Any help in the evaluation process or comment is most appreciated. If the Sun and Earth have more in common than we think, this realization may lead us to further insights in how these great machinery's works.


title: How does QED explain magnetism?

text: I understand how QED explains, for example, Coulomb's law. But how does it explain the behavior of two bar magnets whose N and S poles attract one another?


title: Is there is a specific original source where the "quantum operator ordering issue" is stated?

text: During my research, when the quantum operator ordering ambiguity is mentioned is deemed usually in the likes of "the well-known problem of ordering in quantum mechanics". However, could anybody point me in the right direction as where to find an article or book where is first considered?


title: Is phosphorene a topological insulator?

text: All of the other "-ene" materials (silicene, germanene, stanene, and even graphene) are topological insulators. Phosphorene and these other materials all have honeycomb lattices and would appear to have similar properties. However, I don't see any papers referring to phosphorene as a topological insulator, so is phosphorene just a semiconductor? Thanks.


How did the Bardeen-Brattain point-contact transistor work? What are the conditions under which a Schottky barrier injects minority carriers? In a bipolar junction transistor the forward-biased emitter injects minority carriers into the base, which are then swept across the reverse-biased base- collector depletion layer. At first glance a point-contact transistor might work the same way: the rectifying Schottky barrier might inject minority carriers from the metal emitter into the doped silicon base which would then be swept across the reverse-biased base to metal-collector depletion layer. Except that all the textbooks I can find say things about Schottky barriers like, > Under normal operating conditions, the minority carrier current is orders of > magnitude smaller than the majority carrier current. (Sze, _Semiconductor > Devices_ , 1985). Or, > the forward current ... is due to the injection of majority carriers from > the semiconductor into the metal. The absence of minority carrier injection > and the associated storage delay time is an important feature of Schottky > barrier diodes. Although some minority carrier injection occurs at high > current levels, these are essentially majority carrier devices." (Streetman, > _Solid State Electronic Devices_ , 1980). So what gives? The point contact transistor could only possibly work if a gold to n-type germanium point contact is injecting. Is it perhaps the case that at high current _densities_ minority carrier injection is actually quite common? (But point-contact diodes should have very high current densities, and thus if this is the explanation then we don't have a good explanation for their good capacitance.) There is a lot of erroneous information about how point-contact transistors work floating around on the web. In the "maker" community the suggestion (probably from the article Henderson, PB; Home made transistors, _Wireless World_ , Jan 1954, pp. 20-23) is that somehow melting phosphor-bronze point contacts into N-type germanium does a P-type doping, so that point contact transistors are actually just simple PNP junction transistors. But phosphor is an N-type dopant, so this seems an unlikely explanation. Jeri Ellsworth repeats this assertion in her Youtube video. The Wikipedia page, the PBS page and the Nobel prize page(see frames 19-26 of the animation) all claim that the point contacts somehow magically create a hole based inversion-layer _across the entire surface of the crystal_. (To be fair to all of them, Bardeen/Brattain's original papers in Physical Review (74(2):230-232, July 1948) made extensive arguments about surface states that may have confused the issue). In Bardeen's Nobel acceptance speech, (as published in Science, 126:105-112, Jul 19, 1957 (paywall), however, he gives the traditional explanation of a Schottky rectifying contact and then says (page 109) > _If $\chi$ (the energy barrier) is sufficiently large, the Fermi level at > the interface may be close to the valence band, implying an inversion from > n-type conductivity in the bulk to p-type near the contact. The region of > hole conduction is called, following Schottky, an "inversion layer." An > appreciable part of the current flow to the contact may then consist of > minority carriers in this case, holes. An important result of the research > program at Bell Laboratories after the war was to point out the significance > of minority carrier flow._ So I've got a Nobel laureate saying one thing, and two eminent IEEE fellows saying something that seems contradictory.


title: Ground state of a quantum mechanical system

text: When looking back at my courses of quantum mechanics, I noticed that assumptions about the ground state of a quantum mechanical system where rather vague and unprecise. It is always assumed that a ground state exists and that it had a finite energy. So my questions are the following:   * Should every quantum mechanical system have a ground state? And how can we be sure of this ?   * Should this ground-state have a finite energy ? Or is an energy of $-\infty$ also allowed ? > A neat proof for the ground state is given in "Introduction to Quantum > Mechanics" by D. J. Griffiths for the simplest system (problem 2.2). There > it is shown that if you have an energy-eigenstate $\psi(x)$ (working in the > position space for simplicity) with energy $E$, so: > $$\hat{H}\psi(x)=E\psi(x),$$ where we consider a simple non-relativistic > point-particle, so $$\hat{H}=\hat{T}+\hat{V}.$$By applying this on this on > the equation for the eigenstate $\psi(x)$ this > yields:$$\left[-\frac{\hbar^2}{2m}\frac{\partial^2}{\partial > x^2}+V(x)\right]\psi(x)=E\psi(x),$$ or (after some simple rewriting) > as:$$\frac{\partial^2\psi(x)}{\partial > x^2}=\frac{2m}{\hbar^2}\left[V(x)-E\right]\psi(x).$$In the case that we > would have an energy which is lower than the minimum of $V(x)$ (assuming > that the potential has a minimum), the wave-function would be non-normizable > since $\psi(x)$ and $\partial_{xx}\psi(x)$ whould have the same sign. This > is because of the fact that $\psi(x)$ can only have a minimum if it's > positive and a maximum if it's negative, which yields the non-normalizable > character. So this still leaves me with the question on how they know it for sure for potentials of the Coulomb-kind and how they do this in quantum field theory (and, by extension, classical physics)?


title: Theories with non-vanishing commutators outside the lightcone

text: I'm reading Weinberg's new book on Quantum Mechanics, and in Chapter 8.7 "Time-Dependent Perturbation Theory" he derives the usual Dyson series for the $S$ matrix when the interaction Hamiltonian $V_I(t)$ (interaction picture) is the integral of a local density $V_I(t) = \int \mathrm{d}^3x\ \mathcal{H}(x,t)$: $$ S_{\beta\alpha} = \sum_{n=0}^\infty \left[ -\frac{i}{\hbar} \right]^n \int \mathrm{d}^4 x_1\cdots\int \mathrm{d}^4 x_n\left(\Phi_\beta,\ T\left\\{\mathcal{H}(x_1)\cdots\mathcal{H}(x_n)\right\\}\Phi_\alpha\right) $$ with his notation $\left(u,v\right)$ for the Hilbert space inner product. He then discusses when this formula is Lorentz invariant. There is no problem defining the time ordering when the $x_i$s are inside the light-cone, but the time ordering is ambiguous outside the light-cone. So the usual argument leads to the condition: $$\left[\mathcal{H}(x,t),\mathcal{H}(x',t')\right]=0$$ if $ (x'-x)^2 \geq c^2 (t'-t)^2 $. So far so good - I've seen all this before. But then he gives this parenthetical: > (This is a sufficient, but not a necessary condition, for there are > important theories in which non-vanishing terms in the commutators of > $\mathcal{H}(x,t)$ with $\mathcal{H}(x',t')$ for $ (x'-x)^2 \geq c^2 > (t'-t)^2 $ are canceled by terms in the Hamiltonian that can not be written > as the integrals of scalars.) There are no references for this, and as far as I can tell it is not clarified anywhere else in the book. If this is true it seems to contradict some of the arguments for local quantum field theories as being somehow a unique (apart from string theories) set of consistent relativistic quantum theories. Does anyone know the theories Weinberg is making reference to here? (If it's string theory I guess I'll go listen to the Derpy song.)


title: relevant 4-dimensional theory with interacting vector field

text: A simple langragian that gives the simplest interaction is $\mathcal{L}=(\partial\phi)^2+(m\phi)^2$ where $m$ is some constant. Does anyone know of theory in four dimensions which is physically interesting and couples three vector fields over a 4-dim'l manifold.


title: Does the Earth rotate the same encased in ice during the height of an Ice Age as it does when the bulk of it's water is liquid and always in motion?

text: Ice Age vs. Now. Does the Earth rotate at the same rate when encased in ice during the height of an Ice Age as it does when the bulk of it's water is liquid and always in motion?


title: Lifetimes of stable particles

text: What are possible lifetimes of up/down quarks, electronic/muonic/tau neutrinos, photon, gluon?   I understand they are said to be stable, but, as I saw on wikipedia, the lower bound for the "stable" electron's mean lifetime is $4.6×10^{26}$years. Are there any bounds for the mentioned particles?


title: Why is water clear?

text: Water appears transparent to visible light, yet most other objects are opaque. Why is that? Is there an explanation why water appears transparent? Is water transparent at all wavelengths, or are the visible wavelengths somehow special? If it is not transparent at all wavelengths, is there some evolutionary explanation why we would expect water to have low absorption at the wavelengths that we can see with our eyes? Is there some explanation along the lines of "because we evolved in an environment where water plays (some important) role, therefore it's not surprising that our eyes are sensitive to wavelengths where water has low absorption"?


title: How do the particles in a standing wave vibrate in phase within half a wavelength?

text: I would like to have a better understanding of phase difference when talking about stationary waves. How exactly are particles in phase within half a wavelength in a stationary wave?


If U238 ions were circulating in a particle accelerator would their neutron absorption cross section spectra change?


title: Can we project a 4D world using 3D video technology?

text: Traditional movies, TV, etc, faithfully show our 3-dimensional world using 2 dimensions. So can we have a movie that shows a 4-dimensional world using 3D technology?


title: Pressure at the front and back of a fan

text: I have a fan which is inside a duct. When the fan is off, the pressure at the front and back of the blade remains the same. Now when it is running, the velocity of the air below will be more because fan blades induce momentum to the air. What happens to the pressures at the back and front of the blade? Will it be different?


title: Does weather obstruct microwaves and radiowaves?

text: I read an article ('How airplanes use radar' at Bright Hub.com) that stated that airplanes use radio waves for communication because they aren't obstructed by weather. Is this true?


title: multibody problem and determinism

text: Given that there is no exact general solution to the $N$-body problem, can it be concluded that the Universe is non-deterministic, even for the Newtonian case (ignoring relativistic and quantum effects)?


title: Is it cheaper to boil water on high gas heat or low gas heat?

text: Assume my time value is \$0. High gas heat adds more energy per second to the water but the pot may absorb the heat less efficiently, or maybe the shape of the flame is less good, or for other reasons diminishing returns are encountered. Low enough gas heat might never boil the water. Is there an efficient optimum in the middle that uses the least total gas to get the water to 100℃? More generally: what are the factors in a real-life situation of warming something up that differ from the simplest model of a knob that makes "temperature" rise (characterises the entire fluid by one scalar), forgetting that something physical has to heat up something else physical?


title: Does nature tetrate?

text: We see addition, multiplication and exponentiation in the natural formulae that make up physics. However, do we ever see tetration (repeated exponentiation) or higher hyper- operators in nature? (Not counting cases where the tetration step could be trivially eliminated.)


title: A general wavefunction in a square lattice

text: Suppose we have a square lattice with periodic condition in both $x$ and $y$ direction with four atoms per unit cell, the configuration of the four atoms has $C_4$ symmetry. What will be a general wavefunction in this lattice look like? How will it different from the usual Bloch wavefunction?


title: Recently publicized experiment on destructive interference between two laser beams

text: Recently I've had several non-physicist friends ask me, independently of each other, about an experiment where two collinear laser beams destructively interfere along a certain length. Everybody wants to know "where does the energy go?" Answering that question is not the problem, but I would be more convincing if I knew what experiment they were talking about! None of them can recall where they read it, but it seems to have made the rounds of popular science websites sometime in the last year. I am especially surprised that this has been published _now_ , since similar experiments have been done for decades, so I'm guessing there must be some other twist to the experiment that didn't register in my friends' memories. Can anybody point me towards a published paper or a popular science article?


title: How many digits of Pi are required in physics?

text: In other words: which physics experiment requires to know Pi with the highest precision?


title: Mechanical energy problem - please help

text: Quite new to all this - simple question. "A ball starting from rest rolls down a hill and reaches 10m/s at the bottom. If the same ball starts rolling down the hill from an initial velocity of 5m/s, will it's velocity at the bottom be: (A) Less than 15m/s; (B) Equal to 15m/s; (C) Greater than 15m/s." The answer I have marked in the book this question comes from says (A). However, I am unable to reason to that conclusion from the way the question is stated. Here is my reasoning, please point out where it is wrong. If the same ball (the same mass) traveses the same distance down this same hill (the same change in height), the gravitational potential energy must be the same at the point on top of the hill, regardless of its initial velocity. Therefore, on rolling down the hill, by the time it reaches the bottom, this potential energy would have transferred to kinetic energy - indicating (B)? Thanks for your time. Tim.


title: Calculating the rate at which a car turns

text: This is going to be difficult to describe, as I have no real concept of this sort of physics, so please bear with me. If this is too much or beyond the scope of this website, I apologize. I am making a simple-ish driving simulator and I need a formula that I can use to determine the rotation of the vehicle for every frame. If the car's rotation is 90.0 degrees and the wheels are pointing 20.0 degrees to the right, that should mean I can use a simple polar offset to send it a distance of _a_ toward 70.0 (90.0 - 20.0). This does not work when the car is not moving, or when the air resistance is large, like when the car is turning at high speeds, and I am not sure how I should be dealing with such factors. What I have works for now, but I need a more accurate solution. **What I have** What I have barely turns the car at low speeds, and allows it to turn on a dime at high speeds. This is the algorithm that I am currently using in pseudo-code.                   if (vehicle.engineCurrentRPM > vehicle.engineMaxRPM) vehicle.engineCurrentRPM = vehicle.engineMaxRPM;         if (vehicle.engineCurrentRPM < -vehicle.engineMaxRPM) vehicle.engineCurrentRPM = -vehicle.engineMaxRPM; // This allows it to go full speed in reverse, which I can fix myself.          // The engineMaxRPM is from a Bugatti Veyron and is equal to 6000.0.     // That is 600.0 lower than the redline, and I did that on purpose.          vehicle.engineCurrentTorque = (vehicle.engineCurrentRPM * (922.0 / 2200.0));     // A simple formula that approximates the torque-rpm relationship. I only include this because torque _might_ be relevant.          differentialRatio = 4.0;     drag = 0.39;     gearRatio = 3.64;     grip = 1.0;     wheelRadius = (2.0 * 0.0254);     // 20 inch wheels converted to meters and set to 10%. I don't know why this is necessary, but it seems to be in all of the formulas I have seen     weight = 1995.806; // 4400 pounds * 0.453592 to get kilograms.     power = (vehicle.engineCurrentTorque * vehicle.engineCurrentRPM) / 5252.0; // HP = Torque x RPM ÷ 5252          // The variable "power" is unused, but there if you need it.          airResistance = 0.05 * (1.0 - drag);          tWheels = vehicle.engineCurrentTorque * gearRatio * differentialRatio;     f = (tWheels / wheelRadius);     vehicle.acceleration = (f / weight) * grip;          vehicle.object.rotation.z -= vehicle.wheelAngle * vehicle.acceleration * time;     float angle = vehicle.rotation.z;     angle *= M_PI; // The pi constant.     angle /= 180.0;          vehicle.velocity.x -= vehicle->acceleration * sin(angle) * time;     // The world is upside-down for some reason, so subtracting is necessary. This will be fixed!     vehicle.velocity.y += vehicle->acceleration * cos(angle) * time;     vehicle.velocity.z += 0.0;          vehicle.position.x += vehicle.velocity.x;     vehicle.position.y += vehicle.velocity.y;     vehicle.position.z += vehicle.velocity.z;          vehicle.velocity.x *= (1.0 - airResistance);     vehicle.velocity.y *= (1.0 - airResistance);     vehicle.velocity.z *= (1.0 - airResistance);


title: Electric Field Homework Question

text: Can a charged particle move through an electric field that exerts no force on it?


title: Zero magnetization of spin model without external magnetic field

text: For a given Hamiltonian with spin interaction, say Ising model $$H=-J\sum_{i,j} s_i s_j$$ in which there are no external magnetic field. The Hamiltonian is invariant under transformation $s_i \rightarrow -s_i$, so there are always two spin states with exactly same energy. For the magnetization $M = \sum_i s_i$, we can take the ensemble average $$\left\langle M\right\rangle =\sum_{\\{s_{i}\\}}M\exp(-\beta E)$$ and the result should be $\left\langle M\right\rangle = 0$ because the two states with all spin flipped will exactly cancel each other. There is an argument for this in the wiki. So the question: how is this situation handled for finite and infinite lattice? How can they obtain the non-zero magnetization for the 2D Ising model? $$M=\left(1-\left[\sinh\left(\log(1+\sqrt{2})\frac{T_{c}}{T}\right)\right]^{-4}\right)^{1/8}$$ * * * **Some information** : for a 1D Ising model with external magnetic field, one can solve the Hamiltonian $$H=-J\sum_{i,j} s_i s_j - \mu B \sum_i s_i$$ and obtains the magnetization as $$\left\langle M\right\rangle =\frac{N\mu\sinh(\beta\mu B)}{\left[\exp(-4\beta J)+\sinh^{2}(\beta\mu B)\right]^{1/2}}$$ It gives the result $\left\langle M\right\rangle \rightarrow 0$ when $B \rightarrow 0$ for any temperature and this match the definition of the magnetization above. However, it gives $\left\langle M\right\rangle \rightarrow N\mu$ when we take the limit of $T \rightarrow 0$. It suggests the ordering of limit is important, but we still get $\left\langle M\right\rangle = 0$ when there is no external magnetic field. * * * **Reminder** : There are some precautions needed to care when you run computer simulation using the definition of magnetization above directly, otherwise, you will always get 0. These methods are similar to create a spontaneous symmetry breaking manually, for the Ising model, the following may be used:   * Use the $\left\langle |M|\right\rangle$ instead   * Fix the state of one spin so that the system will close to either $M=+1$ or $M=-1$ at low temperature. In general, the finite size scaling should be used because we are likely interested in the thermodynamic limit of the system. * * * **Update** : Visualization should explain the problem better. Here is the figure of the canonical distribution as a function of magnetization $M$ and temperature $T$ for 3D Ising model ($L = 10$). ![alt text](http://i.stack.imgur.com/jsf5R.png) At a fixed temperature $T \lesssim T_c$, there are two symmetric peaks with opposite magnetization. If we blindly use the $\left\langle M\right\rangle$ defined before, we will get $\left\langle M\right\rangle = 0$. So how do we deal with this situation? For a finite system, there is a finite probability that the transition between two peaks can occur. However, the "valley" between two peaks will become deeper and deeper when the size of the system $L$ increase. When $L \rightarrow \infty$, the transition probability tends to zero and those two configuration space should be separated. Note that the "flat mountain" in the figure at $L = 10$ will also become a very very sharp peak when $L \rightarrow \infty$. One method discussed in the answers below is to consider the average of $M$ for separate configuration spaces. This seems reasonable for infinite system, but becomes a problem for finite system. Another problem raised here is that how to find each separated configuration spaces? * * * Thanks people try to give the answers to this question. In the following discussion, Kostya gives the typical treatment of spontaneous symmetry breaking. Marek discusses the ensemble average below and above the critical temperature. Greg Graviton gives an analogue for the real space spontaneous symmetry breaking. If anyone can explain better to the problem of configuration space and how to take the average for Ising model, other spin models or general case, you are welcome to leave answer here.


title: Lorentz transformations of fields evaluated at a point

text: I'm am sure that I must be missing something very simple, so apologies in advance. Considering the Lorentz transformation $\Lambda$ of a spinor fields, for the plane-wave solution $u(p)$, I cannot for the life of me agree why (1) $$ u^s(\Lambda^{-1} {p'}) = \Lambda_{\frac{1}{2}} u^s(p') $$ where $$ p' = \Lambda p $$ This is in Peskin & Schroeder, pg 59, just above equation (3.110). I have tried to get this a dozen times, to no avail. I know that, for a scalar field, under a Lorentz transformation $\Lambda$ we get, as per Peskin & Schroeder, pg 36, equation (3.2) $$ \phi(x) \rightarrow \Lambda \phi(x) = \phi'(x) = \phi(\Lambda^{-1} x) $$ This makes sense to me as "the transformed field at the transformed point in spacetime should be the same as the un-transformed field at the untransformed- transformed point in spacetime". So trying to do that with inverse transformations, now using $\Lambda_{\frac{1}{2}}$ for a spinor plane-wave solution, I get $$ \Lambda u(p) = u (\Lambda^{-1} p) $$ and applying an inverse transformation would give $$ \Lambda^{-1} \Lambda u(p) = \Lambda^{-1} u (\Lambda^{-1} p) $$ or $$ u(\Lambda^{-1} \Lambda p) = u' (\Lambda^{-1} p) $$ so $$ u(\Lambda^{-1} p') = u( [\Lambda^{-1}]^{-1} \Lambda^{-1} p) $$ whence $$ u(\Lambda^{-1} p') = u(p) $$ that is, $$ u(p) = u(p) $$ So it's consistent alright, but not of much use! Can anyone show me what I'm missing to derive equation (1) above. Thank you in advance!


title: Zero photon energy in moving frame

text: Suppose I have a photon with energy $E$ (in its rest frame) moving along the $z$ direction with momentum $\mathbf{p} = p_z\mathbf{\hat{z}}$: The Lorentz transform in the laboratory frame should be: $$ E'=\gamma(E-p_zv) $$ so using $p_z = E / c$ and $v=c$ for light, I get that $E'=0$. How is this possible? Is there a physical meaning?


title: Bulk Modulus, How are these formulas equivalent?

text: Bulk Modulus is defined as $$ B = \frac{VdP}{-dV}$$ Where $V$ is volume and $P$ is pressure. It is also defined as, $$ B = \frac{\rho dP}{d\rho}$$ Where $\rho$ is density. Question: How are these equivalent? If $V = \frac{m}{\rho}$, $dV = \frac{m}{d\rho}$ Substituting into the first equation of the bulk modulus, I get $$ B = \frac{\frac{m}{\rho}dP}{\frac{m}{d\rho}}$$ or $$ B = \frac{d\rho dP}{\rho} $$


title: Energy carried by photon not conserved?

text: In an imaginary frame of reference traveling with a photon, the length of the path traveled is 0. If the length of the path is 0, isn't it similar to say that the photon is either at the source or at the destination and nowhere in between? In that case, in the frame of reference of the destination, the photon is planned to warp toward us. It's destiny is unstoppably toward us and will pop up in the coming future. The energy transported by the photon can not be lost in spacetime and must be conserved. A way to explain this is that there is a time offset between the two points in space. The quantum of energy warps through space from one object through another and there is no such thing as a wave or particle traveling. Does this make any sense at all?


title: Stress calculations in a perforated paper

text: ![Unperforated Paper](http://i.stack.imgur.com/y1H0E.jpg) You have a sheet of paper (torn out of a good quality foolscap notebook) as shown above, and you start pulling it apart with both your hands (forces indicating by the blue arrows). Its difficult to tear the paper apart this way --- it has a very high tensile strength, I assume. If you try this at home, you are bound to fail. ![Perforated paper](http://i.stack.imgur.com/75tzW.jpg) Now if you make tiny perforations in the paper (indicated by red circles), pull it apart the same way, you'll notice that its very easy to tear apart the paper. And the line of tearing will definitely have a few circles on them. Is this happening because of the high stresses that are formed around the circle? Can someone give a good mathematical as well as verbal explanation of this phenomenon.


title: Definition of vacuum in field theory; Connection between the classical definition and the connection to QFT

text: I am a bit confused by what is defined to be a vacuum in field theory. Classically a vaccum state is defined to be the state where the field sits at some minima of the potential $\frac{\partial V}{\partial \phi}=0$. My main confusion is that when we go to quantum framework the vacuum state is defined to be a unique state in the Fock space with no-particles $|0\rangle$? I would be grateful if anyone can clarify the correspondence between these various definitions.


title: Is crystal momentum an operator?

text: My teacher has for Bloch waves the notation $\langle \vec{r}|\vec{k} \rangle = e^{i\vec{k}\cdot \vec{r}}u_{\vec{k}}(r)$ and _uses it consistently_. However, does this not assume that there is an operator that has eigenstates $|\vec{k} \rangle$? If so, how would such an operator be defined?


title: Would sea water moderate a nuclear submarine meltdown?

text: My son asked me: > If nuclear submarines get sunk or blown up, what is to stop them going into > nuclear meltdown? I thought about it and came to the conclusion that because they're in the sea the water itself would act as a coolant, stopping nuclear meltdown. Is this correct, as I wouldn't want to teach him nonsense?


title: What is Getting in the Way of Testing D-Wave?

text: I know there are other questions i.e. Do quantum computers manufactured by D-Wave Systems, Inc. work? , What can the D-Wave quantum computer do? , etc. But I can't seem to find my answer. What is getting in the way of determining if D-Wave is a quantum computer? Why can't we just analyze the way they built, and see if it's quantum? Is there a problem with patents getting in the way? Why can we _still_ not tell?


title: What's the avarage temperature of the Universe?

text: I'm sorry if this was asked before but with all the stars and dark matter and all the other stuff, curently in the Universe, what's the avarage temperature of the Universe? Is it like extremely high or extremely low?


title: If a window in a plane cabin breaks will a bunch of water vapor form?

text: If a plane has some windows broken mid-flight (ie open) will the plane suddenly be full of water vapor because of the difference of pressure?


title: About Earth Gravity and its characteristics in different scenarios

text: 1. Earth's gravitational field gives rise to a typical acceleration of 9.80 m/s². If you say gravity changes with altitude, then what will be the gravity at per say 8900 meters. Is there any specific formula?   2. Does the moon have any noticeable effects on the gravitational acceleration? If so, what will be the change?   3. If gravity changes during the above 2 scenarios, then is it possible for an aircraft to save fuel during take-off at high altitude and when flying relatively close to the moon?


title: Newton's second law invariant under law of addition of velocities

text: I'm currently reading Schutz' first course in general relativity, and on the second page (already) I've encountered a problem: We have the Galilean law of addition of velocities: $ v(t) = v'(t) = v(t) - V$ where $V$ is a constant velocity and $v'(t)$ is the velocity relative to the object moving with $V$ (relative to..). He goes on to show that Newton's law are invariant under the above law, but I don't understand how he shows that the second law is invariant: $$a'=\dfrac{dv'}{dt}=\dfrac{d(v-V)}{dt}=\dfrac{dv}{dt}=a$$ I think what mainly confuses me is the fact that it looks like he's using 2 different notations for derivatives at the same time, but even then this is what confuses me: I'm not sure, but from what I know $a'$ should mean the relative acceleration, but $\dfrac{dv'}{dt}$ to me seems to mean the derivative of the relative velocity = relative acceleration. Of course those would cancel out, so my view is incorrect. Can someone clear this up?


title: attractive and repusive potential barrier

text: In the paper of quantum mechanics we have a topic "attractive and repulsive potential barriers" included alongwith the potential step problem.I wonder what does attractive and repulsive potential mean?Is it about the magnitude or something else?


title: Zero probability of finding an electron in the nucleus

text: One and the same electron in a p orbital and taking part in a common π (pi) bond has two lobes visualized as connecting through the nucleus. There is however zero probability of finding an electron at the plane through the nucleus at right angles to the lobes. Where has the electron gone? Is the electron tunneling through the nucleus as a virtual photon? Ed./ Without altering the meaning of the original question it is more clearly stated as: How does the electron cross the nucleus.


title: What is time teleportation?

text: I read this article about time teleportation. Can someone explain the concept better?


title: How do Dirac fermions arise in graphene, and, what significance (if any) does this have for high-energy physics?

text: Graphene has a honeycomb lattice (in the absence of defects and impurities). By considering the low-energy limit of the half-filled Hubbard model used to model the strongly interacting electron gas we find that the low-energy quasiparticles obey the dispersion relation for massless fermions. These details are all covered very nicely in a paper by Gonzalez, Guniea and Vozmediano (reference) among others. It might seem like I'm answering the question. I'm following this line of exposition because I don't want to assume that this is a topic something commonly known or understood outside the condensed matter community. Any answers which elaborated on these basics would be very useful as they would help make the discussion more broadly accessible. My _primary_ question is more about the implications this fact has for high- energy physics, in particular the question of emergent-matter in theories of quantum gravity. The case of graphene is a canonical example in that regard where one obtains _relativistic_ , _massless_ excitations in the low-energy corner of an otherwise non-relativistic system - the 2D electron gas (2DEG). Obviously I have my own beliefs in this regard and I will try to outline them in an answer. But I also want to solicit the communities views in this regard.


title: Electronic configuration for singlet and triplet states

text: Is there a difference in the electronic configuration for singlet and triplet states? For example, He atom has 1s2 configuration in its ground state (singlet state) But what about when the He atom is excited. It can come in both states (Signlet and Triplet) Can I tell from the excited electronic configuration whether the atom is in a singlet or a triplet state? Thanks!


title: How is the Plane of the Solar System oriented to the Sun's motion through space?

text: How is the Plane of the Solar System oriented to the Sun's motion through space: parallel, perpendicular, or some other angle?


title: Visualizing Electromagnetic Waves in 3D Space

text: I did one module of physics for my GCSE one year ago which taught me about transverse EM waves & the EM spectrum, but since then, I do not understand how a wave would move in 3D space. Can someone show me some animation or something? I can understand it in 2D space (ie on a graph) but not 3D. I also read somewhere that they do not oscilate in space, but in electromagnetic field strength and direction? Is this true?


title: How does the Ocean polarize light?

text: I know that the light gets polarized in the plane of the ocean, but why is that? It seems to me that the molecules are free to be in any position or location, so why is it that the 'vertical' polarization gets ignored?


title: Electric fields problem

text: Why is this the case.. ![enter image description here](http://i.stack.imgur.com/ItA9v.jpg) I would have thought it would have followed the E field?


title: Commutator of Lorentz boost generators : visual interpretation

text: I have always struggled to visualize the correctness of the commutation relation for the generators of the boost in the Lorentz group. We have $$[K_i,K_j] = i \epsilon_{ijk} L_k$$ I fail to picture this. For definiteness' sake, let's take a point $\vec{x}$ in my coordinate system, lying in the $O_{xy}$ plane. The difference between boosting first in the $x$ and then in the $y$ direction, and boosting first in the $y$ and then in the $x$ direction ammounts simply to a rotation in the $Oxy$ plane, since $$[K_1,K_2] = iL_3$$ I must be doing something wrong when I draw this on a paper. I realize that this holds at the infinitesimal level, but a simple drawing should be able to realize this principle 'at first order' shouldn't it ? My problem is that when I draw it, I end up with the same situation regardless of which direction I boost into first.


what is expectation values of this anti-commutator? $$\langle \\{ \Delta \hat x,\Delta \hat p\\} \rangle$$ where the $\Delta \hat p=\hat p-\langle \hat p \rangle$ and $\hat p$ is momentum operator and so $\Delta \hat x=\hat x-\langle \hat x \rangle$ and $\hat x$ is position operator operator


title: The difference between an apparent horizon and event horizon?

text: I'm currently writing a project on minimal surfaces and general relativity - however I don't understand the difference between the apparent and event horizon? They ultimately both seemed to be defined as observer-dependent horizons which act as a black hole threshold of which light can't escape once it has passed the horizon? Can anyone clear the difference up?


title: Is the U(1) gauge theory in 2+1D dual to a U(1) or an integer XY model?

text: The compact U(1) lattice gauge theory is described by the action $$S_0=-\frac{1}{g^2}\sum_\square \cos\left(\sum_{l\in\partial \square}A_l\right),$$ where the gauge connection $A_l\in$U(1) is defined on the link $l$. I was told that this theory in 2+1D spacetime is dual to a U(1) XY model on the dual lattice, described by the following action $$S_1=-\chi\sum_l \cos\left(\sum_{i\in\partial l}\theta_i\right)-K\sum_i\cos(\theta_i),$$ where the XY variable $\theta_i\in$U(1) is defined on the dual site $i$. It was said that the K term in the action is to take into account the instanton effect in the compact U(1) lattice gauge theory (which I don't understand). However when I tried to derive the the dual theory, I arrived at the following integer XY model (or height model?) $$S_2=-\chi\sum_l \cos\left(\sum_{i\in\partial l}m_i\right),$$ with the integer variable $m_i\in\mathbb{Z}$ defined on the dual site $i$. Because the Pontryagin dual group of U(1) is simply $\mathbb{Z}$ but not U(1), so I believe that the U(1) gauge theory $S_0$ should dual to an integer XY model $S_2$, and this duality is exact. But every book or paper that I have encountered did not mention anything about $S_2$, instead they all point to the U(1) XY model $S_1$. Therefore I was forced to conjecture that the integer XY model is equivalent to the U(1) XY model with additional K term. Can anyone tell me if my conjecture is correct or not? How to go from $S_2$ to $S_1$ (or maybe directly from $S_0$ to $S_1$)? How is the K term being added? What is the physical meaning of the K term?


title: How could scientists know how far a star or galaxy is from us?

text: How do astronomers measure how far a star (or galaxy) is away from the earth? How would they know that it has taken 13 million years for light to travel in space before it reaches us?


title: How do you calculate the radiant for Kreutz Sungrazer comets? where in the sky do I look?

text: According to wikipedia: > All known members of the [Kreutz Sungrazer] group up until 1965 had almost > identical orbital inclinations at about 144°, as well as very similar values > for the longitude of perihelion at 280–282°, with a couple of outlying > points probably due to uncertain orbital calculations. This longitude of 280 deg seems to point in the direction of the Orion constellation. However, the radiant would seem to be 180 degrees away (opposite) since I expect the radiant to point to the location of the apehelion. I'm also wondering if there is any reason behind the apparent mirror image similarity (or otherwise some connection) between the sungrazers and the orionids, I found this image on pinterest http://www.pinterest.com/pin/129197083031621626/ ![enter image description here](http://i.stack.imgur.com/Z6JZr.jpg) and this picture is actually what peeked my interest in wanting to know where in the sky the radiant for the sungrazers is. FWIW the approximate RA of the orionids radiant seems to be roughly that of Saiph's, or 05h 47m 45 seems to translate to a longitude of -93 deg = 267 deg which, as I noted above, lies close to the perihelion of the sungrazers (http://math.mercyhurst.edu/~griff/courses/webmapping/RA_convert.php). [--edit: oops it appears the orientation of the pic is "upside" down, the red star I mistook for Saiph there in Orion constellation actually seems to be Betelguese whose longitude is -91 deg = 269 deg. Comparing with the stars shown in Stellarium, the radiant seems to lie right in the middle part of Gemini, at RA of 7 hours = longitude of -75 deg = 285 ] [--edit 2: http://ssd.jpl.nasa.gov/sbdb.cgi?ID=dK11W030;orb=1;cov=0;log=0;cad=0#orb contains a jpl orbit diagram for comet lovejoy--an example of a Kreutz sungrazer; the orbit diagram shows that the RA of the APHELION--not perihelion-- of lovejoy is aligned roughly with gemini (or jupiter currently), so now I am completely confused and am wondering if there is a typo in the wiki sungrazer page, or what the misunderstanding on my part is. ] If anyone can point me to where in the sky the sungrazer radiant (such as nearest star/s) is it would be really appreciated from me.


title: What does spin 0 mean exactly?

text: I heard two definitions: (1) Spin 0 means that the particle has spherical symmetry, without any preferred axis. (2) The spin value tells after which angle of rotation the wave function returns to itself: 2 pi / spin = angle. Therefore, spin 1/2 returns to itself after 4 pi, spin 1 after 2 pi, and spin 0 after an infinite rotation angle. These seem to contradict each other: a sphere returns to itself even after infinitesimal rotations. Can somebody clarify?


title: $\pi/2$ phase shift in RC circuit

text: I just recently learnt Alternating Current and RCL circuits and I also learnt about phasers and various graphs. Now I also know that current lags in an LR circuit. My intuitive understanding of this is according to Lenz Law, the inductor will oppose the current and hence slow it down. So the current lags. But I wanted to know if there is any similar logic for a RC circuit. The capacitor doesn't seem to make the current lead by a phase difference of $\pi/2$. I have already studied the mathematical proof and everything works, but I wanted to have an intuitive understanding of both.


title: Is such an orbit possible that allows a satellite on Earth and another on Mars to always be in direct line-of-sight?

text: If not always, how much "most of the time" could it get? Or would a multi-satellites setup be more suited?


title: Where does the "borrowed energy" come from in Alpha decay?

text: I was also thinking about the uncertainty principle in regards with energy & time. The question of something like: Alpha tunneling out of the nucleus is where this can be invoked, but having an uncertainty in energy does not imply that energy is not conserved - it just means that energy got borrowed from somewhere and then can be returned such that the product of uncertainties remains above the minimum and it looks like a fluctuation. So I got to wondering where the energy is borrowed from and the only thing I can really come up with right now is that it gets borrowed from another field that it interacts with. Perhaps because of Coulomb repulsion in the nucleus, whose interaction field would extend far outside the nucleus, there is a temporary drop in the Coulomb field energy? That was my guess, because I don't consider borrowing from other matter fields very realistic, since several would all have to contribute something, rather than just one. Anyways, just thinking "out loud" here.


title: What makes water heavier than air?

text: I apologize in advance if this is too basic for this site. Considering that hydrogen is lighter then oxygen, and water is 2/3 hydrogen, and only 1/3 oxygen, and our "air" is comprised mainly of oxygen, why isn't water lighter then air?


title: Probability, quantum physics, and why (can't it/does it) apply to macroscale events?

text: Quantum physics dictates that there are probabilities that determine the outcome of an event, ie: the probability of a quark passing through a wall is X, due to the size of the quark in comparison to the wall), but couldn't macroscale events be predicted this same way, assuming all variables were accounted for (lets hypothetically say we have a computer powerful enough to take into account all factors). My understanding is that as the scale of an object increases, the probability of it doing anything other than what classical physics dictates is almost 0%, but could conceivably do something not predicted (Example: a human being cannot pass through a wall, but given an infinite timescale, eventually s/he would because of probability not being 0). Is this accurate? If so, then shouldn't we, (except as a tool for conceptualization, like how we round 9.8 m/s^2 or pi to 3.14), use only quantum mechanics to explain events in any scale (for more accuracy)?


title: Why is wave function so important?

text: I am almost sure that the wave function is the most important figures in modern physics book. On the other hand I know that wave function even do not have a physical meaning it self alone!   * Why is wave function so important?   * How many types of wave function exist?   * What means time and space coordinates inside of wave function? (I know how to find wave vector and angular frequency but I don't know how to find time and space coordinates.) $$\Psi=\exp i({\bf {k}\cdot \bf {x}} -\omega t)$$


title: Evolution of Gibbs energy change of reaction with temperature

text: The Gibbs energy change of reaction is given by $$\Delta G^0_{rx}=\Delta H^0_{rx}-T\Delta S^0_{rx}$$ From the look of it seems the direction of variation of equlibrium with $T$ is given by the sign of the entropy change $\Delta S^0_{rx}$. However the Van't Hoff equation has $$\frac{d Ln K_a}{dT}=\frac{\Delta H^0_{rx}}{RT^2}$$where it now seems that the sign of the change in enthalpy is the relevant parameter. How are the two reconciled?


title: Why does a tail of a balloon poodle not collapse?

text: I was just wondering. How can it not collapse? There is pressure in the tail top and in tail itself. How can that little "bridge" (with that I mean the part that in not inflated, see image 2) and tail top sustain and not collapse?      ![enter image description here](http://i.stack.imgur.com/xj6kj.jpg)   **1\. Image of balloon poodle** ![enter image description here](http://i.stack.imgur.com/hj9lY.png)   **2\. Scheme of the balloon tail with a schematic "pressure"** As you can see in _Image 2_ I guess that the pressure is something like that. My best guess is that the surface tension or strenght of the "bridge" is strong enough to sustain. But I would like to know what you guys think!


title: EMF in an Open circuit?

text: If a wire that is connected to a circuit passes a magnetic field, it would induce $EMF$ that would allow current to flow, and generate an opposing force. Now what if the wire is not connected to a circuit? EMF will be induced and $I$ = $0$? What will happen to the induced $EMF$ while the conductor passes the magnetic field? And how fast would it dissipate?


title: Calculate trajectory of projectile with gravity from disk planets in 2D?

text: I'm making a 2D game where you control a spaceship and fly between disk-formed planets of differing size (mass/pull). The planets are small and you can fly near several at once. I need to calculate the gravity produced by each of these planets pulling on the ship including the velocity of the ship. I figured out a way to approximate it:   1. Calculate the position of the rocket in the next frame based on the angle and velocity   2. From this position, calculate the position of a point pulled towards the planet where the larger the planet, the further this point is towards said planet   3. Do this for each planet then average each of these points to find the new location for the rocket this frame   4. Rotation doesn't matter in my case, but I would calculate it by comparing this point's location and the previous point's location.   5. Repeat for every frame. This may be a good approximation, and accuracy isn't a huge deal in my case, but I am wondering if there is a better way to do this with physics formulas. I haven't taken a physics class yet. The other main problem with this solution is that it must calculate it step by step or frame by frame. If there is a formula to do this the path could be calculated from the very start before.


title: Can scientists tell the energy levels of the atom?

text: In the hydrogen spectral series how did the scientists know the number of the energy level which the electron is moving from or to?


title: Ultracold atoms and General theory of Relativty

text: I am looking for good reviews for the subject of Ultracold atoms and it's application in test of General theory of relativity. I am planning to pick up this topic as a semester project, can somebody guide me ? I am basically looking forward to some good books and papers which gives me a flavor of theoretical aspects of this subject. I came across an website which says about Quantum gauge theories and Ultracold atoms and that has interested me a lot. http://cordis.europa.eu/search/index.cfm?fuseaction=proj.document&PJ_RCN=10276613


title: Are they the same thing: Wigner distribution in quantum Boltzmann equation and Wigner function in quantum optics?

text: We know that quantum Boltzmann equation (QBE) is an equation of motion for the interacting Green's function $G^<(\vec{x}_1,t_1;\vec{x}_2,t_2)\equiv\mathrm{i}\langle \psi^\dagger(x_2)\psi(x_1)\rangle$ when transformed to center of mass and relative coordinates, also coined as Wigner distribution. Besides, in quantum optics and quantum information, we have another Wigner function for phase space description, defined as $W(x,p) = \frac{1}{2\pi\hbar} \int_{-\infty}^{\infty}{\mathrm{d} \xi \mathrm{e}^{\mathrm{i} p \xi/\hbar} \langle x-\frac{\xi}{2} \vert\psi\rangle \langle \psi\lvert x+\frac{\xi}{2}\rangle} $. (Certainly, we have more complete definition for more modes and density matrices.) Unluckily, I happen to be that kind of person who cares about nomenclature. Both of these two Wigner distribution functions are quasiprobability distributions, i.e., not necessarily nonnegative. And I even remember that QBE shares some similar form with certain equation for Wigner function in quantum optics. However, these are no more than faint traces. I guess they are different things, but there must be some plausible relation. Can anyone shed light on this? **Update** For anyone who are interested in this question, besides the answer underneath, it may also be helpful to refer to Chap. 4 & 16 of Schieve and Horwitz's book _Quantum Statistical Mechanics_.


title: General covariance and global Poincaré algebras

text: Reading an article (page 7) I read this: > _Just as ordinary general covariance may be regarded as the local gauge > symmetry corresponding to the global Poincare algebra and local gauge > invariance as the gauge symmetry corresponding to the global abelian > algebra_ I would like if somebody could tell me with as much detail as possible the relation between general covariance and a global Poincaré algebra.


title: Understanding how clouds form: warm air near the earth's surface?

text: I'm trying to understand how clouds are formed. I've read that it begins with air near the earth's surface heating up. But I'm hearing two different reasons for how this happens:   1. Air pressure near the earth's surface is high and this causes water vapor near the earth's surface to heat up, since the high pressure causes the kinetic energy of the water vapor molecules to increase.   2. The sun heats up the earth's surface, and this heats up the water vapor near the earth's surface. So what is the truth?


title: Which is more efficient cooling? Cooling yourself from cold water from Referigerator or Airconditioning?

text: Case a: You chill a glass of water in refrigerator to a certain temperature and drink it.. it lowers your body temperature by X degrees. Case b: You switch on the a/c for a certain duration.. it lowers your body temperature by X degrees. Which is more efficient method of cooling? (in terms of energy used and money spent respectively)


title: Nonlinear combination of velocities implies no absolute time?

text: Landau 1961 begins with a brief presentation of special relativity. This question is about the validity of a certain argument that they use in building up the foundations of the subject from scratch, so I think I should first sketch what they consider foundational. They define an inertial frame as one in which Newton's first law holds. They telescope Einstein's two 1905 postulates into a single principle of relativity, which is that the laws of physics are form-invariant w.r.t. changes of inertial frame. They assert (based on unspecified experiments) that forces are propagated at a finite speed. They also seem to assume (presumably also based on experiment) that this speed is a maximum and universal for all interactions. Given a maximum velocity, they argue that time is not absolute. The argument is that in Galilean relativity, velocities combine linearly, which is incompatible with a maximum velocity and with the Michelson-Morley experiment. Also, Galilean relativity posits absolute time. Therefore absolute time must be wrong. This seems like a clear logical fallacy to me. (A implies B, and A also implies C. B is false, so C must be false.) [EDIT] Oops, crucial mistake in my original statement of the question. The parenthetical originally ended with "C must be true." Should have ended with "C must be false." Is there any way of salvaging this argument? Am I misunderstanding it? Landau and Lifshitz, The classical theory of fields, 2nd ed., 1961


title: Interaction Between Two Bar Magnets

text: How would one describe the interaction between two bar magnets? In the E&M course I took we described quite a few very basic electric interactions, but things that I would have thought of as being basic magnetic interactions didn't end up being discussed. What's the best way we have of described the force between two bar magnets, perhaps parallel, and some nontrivial distance apart?


title: Does electrical current have a measurable momentum?

text: If I were to build a levitating super-cooled superconducting loop of wire with a electrically charged circuit that would discharge its stored energy to produce a large circulation of current around the superconducting loop levitating off the ground, would the loop of wire experience a counter torque, forcing the loop to rotate in the opposite direction that the electrical current rotates to have a total torque and angular momentum equal to zero? Is this experiment possible?


title: What is the infinitesimal work done when the force is given by the gradient of a scalar function that depends both on position AND time?

text: The title is slightly confusing but I didn't know how else to phrase my question. Basically, this is the situation: When the force applied to a particle is given by the gradient of a scalar function that depends explicitly on JUST the position of the particle, that force is conservative and $T+V$ is conserved.   However, if it depends explicitly on both position AND time, then $T+V$ is still defined but not conservative. In my notes, this is proven: $$F(r(t),r)\cdot \dot{r} dt = -\nabla V(r(t),t)\cdot \dot{r}dt = \left( -\frac{dV(r(t),t)}{dt}+\frac{\partial V(r(t),t)}{\partial t}\right)dt$$ What I don't understand is the second line, working it out a few times but I never get the correct result.


title: When gravity pulls on light it blueshifts or redshifts it, which way around does it go?

text: when light is propagating away from a mass does it get blue shifted or red shifted? And if its going towards a mass whats the effect?