What is the physical significance of the Dipole Transformation of Maxwell's Equations?

# The Question Given Maxwell's equations of the form \begin{align} \bar{\nabla}\times \bar{B} = \dfrac{4\pi}{c} \bar{J} \+ \partial_0 \bar{E} \\\ \bar{\nabla}\times \bar{E} = -\partial_0 \bar{B} \\\ \bar{\nabla} \cdot \bar{B}=0 \\\ \bar{\nabla} \cdot \bar{E} = 4 \pi \rho, \end{align} what is the physical significance of the following transformation of Maxwell's equations: \begin{align} -i \bar{\nabla} \times \bar{G} \+ \bar{\nabla} G_0 = \dfrac{4\pi}{c} \bar{R} \+ \partial_{0}\bar{G} \tag{Amp-Far Dipole} \\\ \bar{\nabla} \cdot \bar{G} \- \partial_{0}G_0 = 4\pi R_0 , \tag{Gauss Dipole} \end{align} where $\bar{G} =(i\bar{r} \times\bar{F}-x_0 \bar{F}) $, $G_0 = (\bar{r} \cdot \bar{F})$, $\bar{R} = (\rho c \bar{r}-x_0\bar{J}+i \bar{r} \times\bar{J})$, $R_0 = (\bar{r}\cdot\bar{J} - x_0\rho c)/c$ and $\bar{F} = \bar{E} - i \bar{B}$. # The Definitions In this post, I refer to (Amp-Far Dipole) and (Gauss Dipole) as the dipole equations, because I do not know what there actual names are or who first published these equations. I just stumbled upon them by accident on pencil and paper. $x_0 = ct$ is the time variable multiplied by the speed of light for condensed notation purposes. $\bar{R}$ is the complex combination of the electric dipole field density $\rho c \bar{r}-x_0\bar{J}$ and the magnetic dipole field density $\bar{r} \times\bar{J}$. $\bar{R}$ is interpreted as a fictitious current in the dipole equations (Amp- Far Dipole). The corresponding fictitious charge density of $\bar{R}$ is $R_0$, which is equal to the Minkowski inner product of the four-position and the four-current. # Physical Consequences Though $R_0$ and $\bar{R}$ are fictitious charge and current, they are conserved as a current when $G_0 = 0$. This implies that $G_0$ breaks charge conservation of the fictitious charge and current $R_0$ and $\bar{R}$. An interesting consequence of the dipole equations is they are identical to Maxwell's equations when $G_0 = 0$. # Complex Formulation of Maxwell's Equations I first write Ampere's law, Faraday's law and Gauss' law in complex form \begin{align} -i\bar{\nabla} \times \bar{F} = \dfrac{4\pi}{c} \bar{J} \+ \partial_{0} \bar{F} \tag{Amp-Far} \\\ \bar{\nabla} \cdot \bar{F} = 4\pi \rho \tag{Gauss} , \end{align} where $\bar{F} = \bar{E} + i \bar{B}$. # Formulation of Ampere-Faraday Law in Dipole form I use the following differential vector calculus identity \begin{align} \bar{r} \times (\bar{\nabla} \times) \+ \bar{r} (\bar{\nabla} \cdot) \+ x_0(\partial_{0}) = \bar{\nabla} \times (\bar{r} \times) \+ \bar{\nabla} (\bar{r} \cdot) \+ \partial_{0}(x_0) \end{align} to transform (Amp-Far) into the following: \begin{align} \bar{r} \times (\bar{\nabla} \times \bar{F}) \+ \bar{r} (\bar{\nabla} \cdot \bar{F}) \+ x_0(\partial_{0} \bar{F}) =\\\ \bar{r} \times \left( i\dfrac{4\pi}{c} \bar{J} \+ i\partial_{0} \bar{F} \right) \+ \bar{r} \left(4\pi \rho\right) \+ x_0\left(-i\bar{\nabla} \times \bar{F} \- \dfrac{4\pi}{c} \bar{J}\right) =\\\ \dfrac{4\pi}{c} (i\bar{r} \times\bar{J}) \+ \partial_{0} (i\bar{r} \times\bar{F}) \+ 4\pi \left( \rho \bar{r} \right) \- i\bar{\nabla} \times (x_0\bar{F}) \- \dfrac{4\pi}{c} (x_0\bar{J}) =\\\ \bar{\nabla} \times (\bar{r} \times \bar{F}) \+ \bar{\nabla} (\bar{r} \cdot \bar{F}) \+ \partial_{0}(x_0 \bar{F}) , \end{align} which reduces to the following expression \begin{align} -i \bar{\nabla} \times \left( i\bar{r} \times \bar{F} \- x_0\bar{F} \right) \+ \bar{\nabla} (\bar{r} \cdot \bar{F}) =\\\ \dfrac{4\pi}{c} \left( \rho c \bar{r} \- x_0\bar{J} \+ i \bar{r} \times\bar{J} \right) \+ \partial_{0} \left( i\bar{r} \times\bar{F} \- x_0 \bar{F} \right) . \end{align} One can perform the following substitutions $\bar{G} =(i\bar{r} \times\bar{F}-x_0 \bar{F}) $, $G_0 = (\bar{r} \cdot \bar{F})$, and $\bar{R} = (\rho c \bar{r}-x_0\bar{J}+i \bar{r} \times\bar{J})$ to obtain \begin{align} -i \bar{\nabla} \times \bar{G} \+ \bar{\nabla} G_0 = \dfrac{4\pi}{c} \bar{R} \+ \partial_{0}\bar{G} . \tag{Amp-Far Dipole} \end{align} # Formulation of Gauss' Law in Dipole form I use the following differential vector calculus identity \begin{align} -x_0 (\nabla\cdot) \+ \bar{r}\cdot (-i\bar{\nabla}\times) = \bar{\nabla} \cdot (i(\bar{r}\times)- x_0) \end{align} to transform (Gauss) into the following: \begin{align} -x_0 (\nabla\cdot\bar{F}) \+ \bar{r}\cdot (-i\bar{\nabla}\times\bar{F}) =\\\ -x_0 (4\pi \rho) \+ \bar{r}\cdot \left(\dfrac{4\pi}{c} \bar{J} \+ \partial_{0} \bar{F}\right) =\\\ \dfrac{4\pi}{c} (\bar{r}\cdot\bar{J} - x_0\rho c) \+ \partial_{0} (\bar{r}\cdot\bar{F}) =\\\ \bar{\nabla} \cdot (i\bar{r}\times\bar{F} - x_0\bar{F}) , \end{align} which reduces to the following expression \begin{align} \bar{\nabla} \cdot (i\bar{r}\times\bar{F} - x_0\bar{F}) \- \partial_{0} (\bar{r}\cdot\bar{F}) = \dfrac{4\pi}{c} (\bar{r}\cdot\bar{J} - x_0\rho c) . \end{align} One can perform the following substitutions $R_0 = (\bar{r}\cdot\bar{J} - x_0\rho c)/c$ to obtain \begin{align} \bar{\nabla} \cdot \bar{G} \- \partial_{0}G_0 = 4\pi R_0 . \tag{Gauss Dipole} \end{align}


Understanding the difference between co- and contra-variant vectors

I am looking at the 4-vector treatment of special relativity, but I have had no formal training in Tensor algebra and thus am having difficulty understanding some of the concepts which appear. One such concept is the notion of contra- and co-variant vectors. Wikipedia describes a contra-variant vector as varying with the co-ordinates, i.e. the inverse of the reference axes. It gives the example of a change of basis matrix $\mathbf{M}$, and a contra-variant vector $\vec{v}$ and states that in the new set of co-ordinates, we have: $$\vec{v}'=\mathbf{M}\vec{v}$$ It then states that a covariant vector varies with the co-ordinate axes. > Does this mean that for a covariant vector $\vec{u}$ and change of basis > matrix $\mathbf{M}$ we have: $$\vec{u}'=\mathbf{M}^{-1}\vec{u}?$$ I am also having problems understanding the concept of raising and lowering indices. For instance, Wikipedia states that for two four-vectors $\vec{A},\vec{B}$ and using Einstein summation we have: $$\left\langle \vec{A} \middle|\vec{B} \right\rangle=A^{\mu}B_{\mu}=A_{\nu}B^{\nu}$$ And I am having difficulty understanding how this relates to the other definition of the inner product: $$\left\langle \vec{A} \middle|\vec{B} \right\rangle =\mathbf{A}^{T}\mathbf{\eta}\mathbf{B}$$ If anyone could help give me a more intuitive understanding of these concepts I would be grateful.


Experimental information: is momentum conserved?

![enter image description here](http://i.stack.imgur.com/yGSEt.jpg) Could I say that it is conserved during the time, or instead there is a slight decrease?


Are there any Earth-Meteorites on Earth?

A follow-on from this question, if meteorites can come from Mars, it seems they could have come from Earth and ended up back here (though I'm guessing that to be less likely). Do we have any evidence of this having happened? Would there be a way of verifying that they left and came back, other than direct observation of them falling from space?


Why doesn't Ehrenfest's theorem work for a particle in a an infinite square well?

I'm reading Griffith's Intro to Quantum Mechanics, and he mentions that in an infinite potential well, a classical particle would simply bounce back and forth between the two walls indefinitely. He then proceeds to calculate the situation for a quantum particle, and gets a whole bunch of _stationary_ state solutions. My question is: Using Ehrenfest's theorem, can't we immediately say that the expectation values would mimic the classical particle and bounce back and forth as well? I know that this would give us the wrong answer(i.e. a non- stationary solution), but I'd like to know why this reasoning is wrong.


Is enthalpy also valid for liquids and solids?

The definition of enthalpy is $$H=E+PV,$$ is it only valid for (ideal) gas? As this naturally comes from the first law of thermodynamics, $$dE=dQ+PdV$$ for a enclosed system of gas. Can we talk about the enthalpy of (perfect) fluid and even solid (the pressure P seems uneasy to specify.)?


Would a small puddle of water evaporate faster if you spread a dry towel over it?

Let's say you spill 10ml of water on the kitchen counter. It forms a small puddle that would evaporate after a while (assuming room temperature and sane humidity). Would spreading a large, dry towel over the puddle cause the water to evaporate quicker? What I mean is a towel that's much larger than the diameter of the puddle. Say 4x as large. On the one hand, the water by itself comes into direct contact with the air which causes it to evaporate. On the other hand, assuming the towel absorbs the water almost entirely, it increases the wet surface area significantly, but also "locks" away some of the water in the fabric where it has less contact with air.


Why is earth on an axial tilt of 23.4 degrees?

What were the influences for this to occur? Did some things play a bigger part that others?


Explaining Newton's Laws of motion to a 6 year old

An old professor of mine once said that an effective means to get people interested in Physics is to get them started early. What would be an effective and meaningful (and fun) means to explain Newton's 3 laws of motion to a 6 year old?


Weak force: attractive or repulsive?

We are always told that there are the four fundamental forces or interactions of nature: gravitation, electromagnetism, and the weak and strong forces. We know that gravitation is attractive, that electromagnetism can be attractive or repulsive depending on the electric charge of the interacting particles, and that the strong force is attractive between quarks. But when the weak force is mentioned, the description is always something such as 'responsible for radioactive decay', but there is no mention of whether this force is attractive or repulsive. So my question is: is the weak force/interaction attractive or repulsive?


How to write units?

In a scientific article, should I write "3m", "3 m", "3 meters", or "3 [meters]" ?


How long would it take for electricity to flow from one terminal to other, via a 1 LY long wire?

Basically, how long does it take for electricity to determine there is a closed circuit and how does it know that the circuit exists? I'm curious to know how it knows there is a closed circuit at any length **Edit:** For clarification of the thought experiment, picture an electrical source of some type with a 1 light year wire coming from both terminals and connected in the middle. The wires are of zero resistance and capacitance. If electrons were to start flowing immediately then information (one end of the circuit knowing the circuit is now closed at the opposite end) has obviously traveled faster than light, which is impossible. **Edit:** It has been pointed out that at least some capacitance is required. So if required, consider that as a possibility in the hypothetical setup. I'm not quite sure how that affects this situation due to my lack of knowledge. I'm trying to not get to bogged down in the setup and trying to focus more on "How does the circuit know its complete & how long does it take to determine it (particularly when the distance is long)"


Energy of gravitation

EDIT: As some confusion has appeared, I want to make another clear question. If gravitational energy is meaningless in general relativity (since it is the geometry), how can one come up with the graviton, the quanta of the gravitational field? I would like to understand the concept of gravitation's energy (and maybe momentum) in term of Eintein's General Relativity and its non-quantum extension. Since the action of matter and metric is: $$S=\int d^4x(\sqrt{-g}R+\mathcal{L}_{matter})$$ it's understood that the energy momentum of matter is $$T_{\mu\nu}\sim\frac{\delta(\sqrt{-g}\mathcal{L}_\text{matter})}{\delta g^{\mu\nu}}$$ while $$G_{\mu\nu}\sim\frac{\delta(\sqrt{-g}R)}{\delta g^{\mu\nu}}$$ where $G_{\mu\nu}$ denotes Einstein's tensor. How can we come up with a energy term of the metric?


Harmonic oscillator modified by infinite well: are analytic solutions possible?

I'm trying to find solutions to a harmonic oscillator that sits within an infinite square well. I haven't spent too much time yet, and I've had no success so far. I'm wondering how possible or complex an analytic solution would be? The potential of a simple harmonic oscillator is: $$V_1(x)=1/2 m \omega_0^2 \, x^2$$ For simplicity, let us set $\omega_0=1$ and work in unitless time. The potential for the infinite well $V_2(x)$ is infinity outside of the box, $|x|>L/2$ and zero just about everywhere else. The potential for the modified problem is: $$V(x)=V_1+V_2 \,=\, 1/2 m x^2 + V_2(x)$$ I would like to find the energy eigenstates of this modified problem, and the eigenvalues. What does your intuition say about how energy eigenvalues of the SHO would be affected by an added infinite well (assume a well width $L$ much larger than the "wavelength" of the ground state), and how does this compare to the actual analytical or a numerical solution?


Masses and Pulley

I have a two mass and a pulley system as shown here: ![enter image description here](http://i.stack.imgur.com/leuTC.jpg) I have the following two questions: First, why does $\dfrac{d^2}{dt^2} y_p$ is not zero, i I mean I don't understand why does $ y_p$ is going to change over time (it's a constant, right?). Second, and why does $\dfrac{d^2}{dt^2} y_p = A$? (A is defined as the acceleration upward of the pulley)


Confused about SI and CGS Units For E/M Energy

I've been looking at some papers on wakefields of electron beams and I am a little confuded about the units. They do not specify whether or not they are in cgs or SI units. Here is a sample equation: ![enter image description here](http://i.stack.imgur.com/CbEpg.png) I am assuming that these are in cgs, am I correct? What units should electron charge and radius be in? What units is the change in energy in? Thank you!


How much energy does it take to simply run forward?

I'm interested in tracking as much data about my runs as I can in an effort to get faster, and while I can easily estimate energy expenditure during an uphill run due to the change in elevation, I can't estimate energy expenditure due to the requirements of just keeping in motion. I could estimate wind resistance, creating a simple cylindrical model and estimating drag, but obviously it would take energy to move forward on a flat surface in a vacuum as well. Where would I start for this? What would be my variables? I know for a bicycle it's pretty simple to account for rolling friction, but I'm not sure what I'd need to account for. The question: What makes running so much less energy- efficient than bicycling? seems to downplay the importance of vertical motion in energy dissipation. I guess to further boil down the question I'd ask: **where does the energy go when you run?**


Commutation of abstract $O(3)$ generators and vectors

I've been given the following problem, and I'm quite lost with it. Let $L_1$, $L_2$, and $L_3$ denote the abstract o(3) algebras. You are given that $\vec{A} = (A_1, A_2, A_3)$ and $\vec{B} = (B_1, B_2, B_3)$ transform as vector operators of o(3). Show that $[L_j, \vec{A} \cdot \vec{B}] = 0$ I know that $L_j = \varepsilon_{jlm} q_l p_m$, and I can obviously determine the dot product, but I'm not sure where to go from there. I do, however, know that $\vec{A} = \frac{1}{Ze^{2}\mu}(\vec{L} \times \vec{p}) + (\frac{1}{r})\vec{r}$, but I'm not sure how to integrate that into this problem.


Dark energy converted to electricity

> **Possible Duplicate:**   >  Dark Energy saving the world In theory, dark energy be used to generate electricity. Given its apparent abundance, this could provide rather a lot (perhaps too much) clean energy. Has any research been conducted on this or any other practical applications for dark energy? (other than repelling galaxies ;)


Equation describing the motion of an object attracted to magnet

Imagine there is a strong magnet at `(0, 0)`. There is a horizontal rail on `y=1` on which a ferromagnetic object rides. The object can only move on the rail. No friction anywhere. The object starts at `(0, 1)` with some positive velocity. How will the graph of its movement look like?


Phase space derivation of quantum harmonic oscillator partition function

I would like to derive the partition function for the quantum Harmonic oscillator from scratch: $$\tag{1} Z = \int dp \, dx\, e^{-\beta H}.$$ The free particle appears in many textbooks. $H = p^2$ so it is a Gaussian integral $$\tag{2} Z = \int dp \, dx \, e^{-\beta p^2} = L \int dp \, e^{-\beta p^2} = L \sqrt{2\pi/\beta}.$$ I wanted to do the same calculation for the Harmonic oscillator I get could take advantage that $$\tag{3} Z = \int dp \, dx \, e^{-\beta (p^2 + x^2)} = \int dx \, e^{-\beta x^2} \cdot \int dp \, e^{-\beta p^2 } = \sqrt{2\pi} \cdot \sqrt{2\pi} = 2\pi .$$ Or I could integrate over the circles $$\tag{4} H = p^2 + x^2 = E$$ take advantage the Hamiltonian is symmetric on rotation in phase space $$\tag{5} Z = \int dp \, dx \, e^{-\beta (p^2 + x^2)} = \int dE \, d\theta \, E \, e^{-\beta E} = 2\pi. $$ It seems I have forgotten that energy levels are quantized, so I should integrate over the circles $$\tag{6} H = E_n = \hbar \omega (n + \tfrac{1}{2}).$$ * * * **Where are the $E_n$ wavefunctions $\psi_n$ located in phase space?** By Heisenberg uncertainty, we can't specify both $(p,x)$ in the phase plane. Are they equidistributed on the circle $p^2 + x^2 = E$?


Minimal Extension of Wave Equation to Include Dispersion

Let's say you are modeling some process with the wave equation $\frac{1}{c^{2}}\frac{\partial^{2}\psi}{\partial t^{2}} = \nabla^{2}\psi$. You wish to improve your model by including dispersive effects, but you want your model to be as simple as possible for computational tractability. What is the minimal appropriate model for phase velocity, say $c \approx c_{0} + c_{2}\omega^{2}$? and how should the wave equation be altered?


Trying to understand the weak gravitational field metric (1)

In Carroll's online Lecture Notes on General Relativity (p112) he says, "the weakness of the gravitational field allows us to decompose the metric into the Minkowski form plus a small perturbation" ![enter image description here](http://i.stack.imgur.com/mnlOe.gif) He then says that, to first order in h ![enter image description here](http://i.stack.imgur.com/PjVle.gif) How does that happen? I've tried juggling round with various inverse metrics but I just can't see where that minus sign comes from. Nice and simple please. Thank you.


Why is water vapor released when an empty water bottle is squeezed?

I recently came across this tutorial and wondered why vapor will come out when the cap comes off. Heres the Tutorial ![enter image description here](http://i.stack.imgur.com/Pwbg1.jpg)


How is weight distributed when legs are astride?

Is there a simple and quick formula to find the weight (the force acting) on each leg, especially when legs are astride and they are not equally stretched? ![enter image description here](http://i.stack.imgur.com/lUesD.jpg) (or, better) not equally distant from the center of mass? Can you give a brief description of the principle and formulas involved? I couldn't find an appropriate picture, but suppose this man here below brings his back foot/leg forward nearer the Center of Mass. ![enter image description here](http://i.stack.imgur.com/25USG.jpg) Supposing he weighs 80Kg and is tall 170 cm, given that in a male the CoM is near the sternum, if the feet are equidistant from CM we would have two triangles   * 1-2) from CM to foot 1m, angle:45°, height (to CM) 71 cm, base 71 cm In this position can intuitively say that the weight is equally distributed (40-40) on the legs. (or are there other relevant factors to be considered?). But, if   * 2) the rear leg is moved further forward, then the triangles will not be equal anymore, how do you find out:   * the distribution of thw weight on each leg?   * the horizontal component of the force that must be counterbalanced by friction?   * can we apply the parallelogram to each triangle here? _Is it possible to find just the **percentage of the weight** distribution without examining the whole problem and doing a lot of operations?_ If you do not wish do use numbers just explain the principles. > You write down the equation that the total forces on all parts of the feet > from the floor are equal to the weight of the person; and the total torque > r⃗ i×F⃗ i relatively to the person's center of mass is zero Is this the _only or the simplest way to do it_? I asked if there is a **simple, quick formula** to get the weight on each leg, can you find one that i so simple that you can post it in a comment?


Gravitational and gauge-gravitational anomalies in N=1 D=4 supergravity coupled to a SUSY gauge theory with chiral matter

When people talk about the first superstring revolution they often mention the miraculous cancellation of anomalies via the Green Schwarz mechanism. My question is whether such a string-theoretic mechanism is also at work when the 4D gravitational and gauge-gravitational anomalies are tackled? In this context, would it be fair to say that a possible discovery of superpartners at the LHC, which automatically implies some version of N=1 D=4 supergravity, imply that stringy couplings (higher order in alpha') must be present in the corresponding lagrangian to cancel the anomalies? What type of coupling are those?


Can flux pinning be used for propulsion in Earth's magnetic field?

I was wondering if the following steps would result in a net displacement on a spacecraft operating in Earth's magnetic field:   1. Turn on a superconductor   2. Use a linear actuator to shift some mass   3. Turn off the superconductor   4. Shift the mass back to its original position   5. Repeat Would flux pinning allow pushing or pulling against the field?


Matrix Representations of Quantum States and Hamiltonians

I am a high school student trying to teach himself quantum mechanics just for fun, and I am a bit confused. As a fun test of my programming/quantum mechanics skill, I decided to create a computer program to model an atom. Initially, I learned about the matrix representations of quantum states (the first example I came across was that of a polarized photon, with the base states [0, 1] and [1,0]). This was all fine and dandy, but then I learned about the Schrodinger Equation, which replaces these matrices with functions. The way I understood it, both matrices and functions are just concrete representations of an element in a vector space; is this correct? It is much easier to work with matrices in code, so how do I transform operators and functions (Hamiltonian operator and state functions, specifically) into their corresponding matrix forms?


Why terminal velocity exists?

Why the acceleration of free-falling bodies become zero after some time? My only idea would be that the closer to the core of Earth we are, the value of "g" the lower becomes (since the mass below the body is lower and consequently, its gravitational force is also lower) but it works on large scales, like hundreds and thousands of kilometres. (and also, I'm not sure in it) I don't count with air resistance, since an experiment with checking terminal velocity would work in vacuum too. EDIT: I thought about enery aspects too, but obviously a body that falls with constantly increasing speed wouldn't be a perpetuum mobile, because once it's gonna reach with the larger body, and then speed becomes zero.


$U(N)$ gauged quantum mechanics

I'm studying the $U(N)$ gauge theory theory in 0+1 dimensions. The aim is to show that this is equivalent to a matrix model. Is there any literature on this topic? The action I am interested in is $$ S_E = \int d\tau \text{Tr} \left[ \left( D_\tau \Phi \right)^\dagger \left( D_\tau \Phi \right) + m^2 \Phi^\dagger \Phi\right] $$ where $\Phi$ is an adjoint matter field and $S_E$ is the Euclidean action. Also $$ D_\tau \Phi = \partial_\tau \Phi + \left[A_\tau, \Phi\right] $$ We gauge fix using $\partial_\tau A_\tau = 0$. I want to be able to describe this system in terms of a matrix model $U \in U(N)$.


If something weighs 25 kg, how do I find the mass of the object?

> An object is falling and it weighs 25 kg (on a scale, presumably). What is > its mass? I know that weight is measured in Newtons and mass in kilograms, but what if a problem states that something weighs in kilograms? Would I still use $\text w=mg$?


What does the velocity of a wave mean?

I know that the velocity of a wave is given by $v=\lambda f$ but what does this velocity represent in the physical sense. For instance, if I am told a car moves at a velocity of 5 $m/s$ I know that the car itself will cover 5 meters in displacement every second. What part of the wave is moving at velocity $v$?


Why group velocity represents energy or information transmission? What relation between phase velocity and special relativity

I have two question,   1. why group velocity represents energy or information transmission?   2. what is the relation between phase velocity and Special relativity: **why can it exceed C without violation of SR**? I didn't take any classes in special relativity or information theory or quantum mechanics, my background only contains   1. general physics (basic knowledge about Special relativity)   2. electromagnetism   3. signals and systems (basic knowledge about signal transmission), so please reply in a simple and easy way.


Why does no physical energy-momentum tensor exist for the gravitational field?

Starting with the Einstein-Hilbert Lagrangian $$ L_{EH} = -\frac{1}{2}(R + 2\Lambda)$$ one can formally calculate a gravitational energy-momentum tensor $$ T_{EH}^{\mu\nu} = -2 \frac{\delta L_{EH}}{\delta g_{\mu\nu}}$$ leading to $$ T_{EH}^{\mu\nu} = -G_{\mu\nu} + \Lambda g_{\mu\nu} = -(R_{\mu\nu} - \frac{1}{2}g_{\mu\nu}R) + \Lambda g_{\mu\nu} $$ But then, in the paragraph below Eq(228) on page 62 of this paper, it is said that this quantity is not a physical quantity and that it is well known that for the gravitational field no (physical) energy-momentum tensor exists. To me personally, this fact is rather surprising than well known. So can somebody explain to me (mathematically and/or "intuitively") why there is no energy-momentum tensor for the gravitational field?


Comparing predictions and reality for the gravitational attraction due to light beams

While doing some on-the-side reading, I stumbled across this question: Do two beams of light attract each other in general theory of relativity?. Great question and a great, easily understandable answer. In short, it explains that General Relativity allows photons/beams of light to gravitationally attract other things. It mentions a very important and well known phenomenon; that the deflection of light passing through the gravitational field of a massive particle is twice that predicted by Newtonian Gravitation. It also mentions and links to a wonderful article that shows that light beams travelling anti- parallel are deflected by each other's gravitation by four times what is predicted by Newtonian methods. The Newtonian predictions were able to be made because of the commonly accepted gravitational mass for a photon, which effectively uses Einstein's $E=mc^2$ and Planck's $E=h f$ to get $m=h f/c^2$. Not a bad strategy. My question is why we choose to equate the photon's gravitational mass with a hypothetical particle's rest mass? Given that the total energy of a photon (if you rescale the potential energy to 0) can be written as: $$Total~Energy=Kinetic~Energy+Rest~Energy$$ And given that it is nice to set the rest energy of our photon to $0$. Why then should we choose the mass on which to base the predictions using Newtonian Gravity to be a rest mass? Especially when Newtonian physics provides an adequate way of obtaining mass from kinetic energy (as long as that mass is used only in other Newtonian physics). I mean, why can we not say the following for purely Newtonian calculations: $$E=hf,~~K=\frac{1}{2}mv^2,~~Rest~Energy=E_o=0$$ $$\therefore hf=\frac{1}{2}mv^2\rightarrow m=2hf/v^2=2hf/c^2$$ This effectively doubles the gravitational mass of a light beam without altering the actual momentum of it. When predicting the deflection of a beam due to a massive particle, this would make the force of Newtonian gravitation twice as large and the fact that momentum didn't change means the deflection prediction would be twice as large. For the deflection of two antiparallel beams, since the gravitational masses of both are doubled, this would quadruple the force of attraction again without modifying each beam's momentum, making the Newtonian prediction four times that compared to using mass from the rest energy equation. Both of these new predictions are what is shown to actually happen. Understandably, if this were a good and valid idea, it would have been used or realized a long time ago. My question is centred around understanding why the rest mass equation must be used; I am not trying to say what has been done is wrong, just trying to understand why it makes sense.


Preons and 't Hooft condition

If some fundamental particles, like leptons or quarks are composite (e.g.in preonic models), or the same with gauge bosons or the higgs particles. **How could it be possible that preons were more massive than the "known" elementary particles?** Remark: I think to remember there are some conditions called 't Hooft's arguments or similar that are related to this question. Can someone help me? Any good reference about how to understand this hard issue? I am interested in compositeness since I think that BSM theories should be simpler in the same they should have less "elements"/"constituents"/"degrees of freedom" than the SM itself... Remark (II): I am interested in as simple as possible answers AND in technical responses related to effective theories and their operators with mass dimension X...


Calculate/estimate power of a fission bomb

I have some questions about the released energy and power of a nuclear fission bomb.   * What are the key dependencies of the power of a fission bomb?   * Is it true that the power of a fission bomb depends linearly on the mass of the bomb? If so, why?   * Is there a simple physical argument to estimate the magnitude of the total released energy of the bomb if you know the mass and the used isotope? Can you provide an example calculation?    * How can I estimate the power (released energy per time) from this information?   * Is there a closed formula to calculate the power of a fission bomb.


The Born-Oppenheimer approximation and muonic molecules

Does the Born-Oppenheimer approximation fail for muonic molecules (i.e. molecules where one or more electrons are replaced with muons)?


About interchange phase of identical particles in Weinberg's QFT book

In Weinberg's textbook on QFT(google book preview), he discussed the phase acquired after interchanging particle labels in the last paragraph of page 171 and the footnote of page 172. It seems he's suggesting interchanging particles of same species but different spin states will only bring a phase determined by convention, that is, the phase does not have to be $\pm1$. I'm having a hard time understanding this, because I was taught that interchange of identical particles must give a phase of $\pm 1$ and took it for granted. Is Weinberg actually treating particles of same species but different spin states as distinguishable particles? If so can I take it further and conclude particles of same species, same spin states but different momenta are also distinguishable, so that the interchange phase is also conventional? EDIT: I did a calculation of the commutation relation of $a'_m$ and $a'_n$ as Lubos suggested and verified his conclusion. Recently I did the same calculation again but could not reproduce the result I had, here is my recent calculation: $a'_ma'_n=(-1)^{N_{m',m'<m}} a_{m'}(-1)^{N_{n',n'<n}}a_{n'}=\pm (-1)^{N_{m',m'<m}}(-1)^{N_{n',n'<n}} a_{m'}a_{n'}$ where $+$ is taken when $m'\neq n'$, $-$ is taken when $m'=n'$. Also $a'_na'_m=(-1)^{N_{n',n'<n}}a_{n'}(-1)^{N_{m',m'<m}} a_{m'}=\pm (-1)^{N_{m',m'<m}}(-1)^{N_{n',n'<n}} a_{n'}a_{m'}$ again $+$ is taken when $m'\neq n'$, $-$ is taken when $m'=n'$. So $a'_m$ and $a'_n$ should obey the same commutation relations as those of $a_m$ and $a_n$. From another persepctive, it seems just from Pauli exclusion principle we can deduce the antisymmetry of fermion wavefunctions: Exclusion principle tells us the square of a fermion creation or annihilation operator must be zero, so $a^2_m=0,\ a^2_n=0,\ (a_m+a_n)^2=0$(assuming no superselection rule on $m,\ n$ so that the 3rd operator is well defined), and we can easily see this implies $a_ma_n+a_na_m=0$. So, I'm again puzzled by this question.


What does Metric Transitivity Mean?

Jaynes In his paper "Information theory and Statistical mechanics" says "Previously, one constructed a theory based on equations of motion, supplemented by additional hypothesis of ergodicity, Metric transitivity, or equal a priori probabilities, and the identification of entropy was made only at the end, by comparison of the resulting equation with the laws of phenomenological thermodynamics." In this context what does "Metric Transitivity" mean?


Is there is any difference between Electric Flux ($\Psi$) and Total number of Electric Field lines ($E\times$Area)? $\psi = \Sigma Q$, and $\phi$ = Electric field intensity $\times$ Area where Area = Total surface integral over Gaussian surface


de Broglie formula inconsistency

I recently stumbled across a small peculiarity I don't understand: According to de Broglie, the frequency of a matterwave can be written as $f=\frac{E}{h}$, and its wavelength as $\lambda = \frac{h}{p}$. However, relativistically for a particle with matter the following formulae are valid: $E = \gamma m_{0} c^{2}$, and $p = \gamma m_{0}v$, where $m_{0}$ is the particle's rest mass. So, with $v=\lambda f$ (with the particle's velocity v, unequal the speed of light!), using de Broglie we would end up with $E=v p$, which is clearly not valid when inserting the relativistic energy and momentum equations: $v*p = \gamma m_{0} v^{2} \neq E = \gamma m_{0} c^{2}$. Where am I mistaken?


Logistic map and attractors

Does the logistic map have a strange attractor for some "chaotic" values of the parameter?


How do virtual-photons curve in a magetic field?

From what I understand photons only move in straight lines unless reflected or refracted (other than influences from gravitational fields and their usual wavelike movement). And since they are a fundamental force carrier I am having difficulty understanding how they are "pulled" into a pole of a magnet after leaving the opposite end. How does this occur? ![Curves in magnetic field:](http://i.stack.imgur.com/vZmTY.png)


Do airlocks in space decompress violently as they do in movies?

The standard scene follows: The good guys have finally captured the enemies and have them in an airlock chamber on a spaceship. A button is pressed and out they go, violently propelled by the blast of air moving from a pressurized zone to a zone of much less pressure (space). Here's my question: If people don't immediately decompress, then why do airlocks do this in movies? If the air moves due to expansion to reduce pressure, then why wouldn't the human body do the same?


Calculating a spacecraft's deflection angle

> There is a spacecraft with mass $M_1$ moving in a certain trajectory with a > constant speed $V_r$. To modify its trajectory, a engine is turned on and > starts ejecting gas at a speed $V_g$, which is constant relative to the > spacecraft and perpendicular to the spacecraft's trajectory. The engine is > turned off when the mass of the rocket is $K$ times lower. What is the > trajectory's deflection angle, in radians, caused by the engine? I guess I should use the rocket equations to discover the rocket's thrust and the thrust caused by the engine. So I would have both vectors and then I could calculate the "total" thrust. But I don't know how could I get the angle with it, maybe using some projectile motion equations? Can anybody help me?


What is the world-sheet picture of high energy open string scattering?

Following Gross-Manes paper (which came right after Gross-Mende), the planar one-loop amplitude (the annulus) is dominated by the region in moduli space where the radius of the annulus shrinks to a point, i.e. it looks like the tree amplitude. Does anyone know if this generalizes for to all-loops?


What if we could give photons some mass?

I was reading an article and these paragraphs got me wondering... > Before I list the replies, here is some background. The Higgs mechanism > describes an invisible field that, it is argued, split one force into two > soon after the birth of the universe. Specifically, it divided an ancient > "electroweak" force into the electromagnetic and weak forces we see at work > today. The latter is seen in some radioactive decay processes, and is > involved in creating sunshine. > > The Higgs field splits the electroweak force by giving mass to the particles > that carry the weak force (the W & Z bosons) and leaving the particle that > carries the electromagnetic force (the photon) massless. The Higgs boson is > the quantum particle associated with the Higgs field. What I'm wondering is... can anyone hazard a guess as to the possibility at some point in the future of humans being able to manipulate the Higgs mechanism so as to give photons some kind of mass? I'm thinking they obviously couldn't go at the speed of light anymore, so could we make 'slow light'?


Double slit experiment

I can't understand the relation between Schrödinger's Cat and this amazing experiment of the double slit. It seems like in the double slit we know if we observe the particle the wave function collapse and the electron acts like a normal particle, so doesn't that solve the problem in Schrödinger's Cat? When we open the box the atom that should help release the poison acts like a particle,the poison is in the bottle so the cat stays alive, don't know really?!


Given eigenvalues of $\vec l^2$ and $\vec s^2$, calculate the eigenvalue for $\vec j^2$

There was an exam question that read approximatly: > _Let $\vec j = \vec l + \vec s$. Given eigenvalues of $\vec l^2$ and $\vec > s^2$, calculate the eigenvalue for $\vec j^2$._ We came up with $$\vec j^2 = (\vec l + \vec s)^2 = \vec l^2 + \vec s^2 + 2 \vec l \cdot\vec s$$ How can we get the eigenvalue for $\vec j^2$ only with this information? I think that we would need $m_l$ and $m_s$ or so.


Why does the gas cloud of an underwater gunshot pulse?

I have been watching some slow motion video and I was intrigued by the slow motion underwater gunshot. The first moments of the video go as expected. The gun fires and a cloud forms in front of the gun. Then the bullet rips through the water, leaving an open space (which I assume is a cavitation vacuum.) Soon after, the rip closes, and the gas bubble shrinks (I'm assuming this is due to the cooling and/or compression of the hot high pressure gasses.) Then something funny happens. The bubble pulses a few times and almost appears to emit light. Each pulse is accompanied by a noise. The video can be seen here. What is this pulse effect and/or what causes it? **EDIT:** Here is another great video for a possible explanation:


What are some of the best books on complex systems?

I'm rather interested in getting my feet wet at the interface of complex systems and emergence. Can anybody give me references to some good books on these topics? I'm looking for very introductory technical books.


Stellar Power(Luminosity) Flux

So I was applying some mathematical techniques I learned to physics, and one thing that captured my interest, is the power or luminosity flux of a star. So modeling the situation, taking the scalar field of Luminosity($L$), and the vector surface(stellar surface), $S$, I began by setting up a Surface (Flux, in this case) integral of $$ \iint_{S}L\,dS $$ Where, $$ S=\langle r\sin\theta \cos\phi,r\sin\theta \sin\phi,r\cos\theta \rangle \;,\; \theta \, \epsilon \, [0,\pi] \wedge \phi \, \epsilon \, [0,2\pi] $$ $$ L = 4\pi r^{2}\sigma T^{4} $$ $r$ in this model is the invariant stellar radius. Following logic, since $L$ has no explicit dependency on $ \theta $ or $\phi$, I rearranged as: $$ 4\pi r^{2} \sigma T^{4} \iint_{S} dS $$ Since the integral of dS over S is merely surface area, and our surface is a sphere of radius $ r $, this expression reduces to: $$ 16\pi^{2} r^{4} \sigma T^{4} $$ The dimensional analysis for this breaks down to $ W \cdot m^{2} $. Isn't flux supposed to be quantity per area?


Spin wave theory for 1D spin systems

In general, when one performs spin-wave theory, he has to assume some kind of long-range ordered state and perform spin-wave theory (using for instance Holstein-Primakoff bosons) around that state. Recall that in 1D there is no long-range order. Then it seems that spin-wave theory in 1D would be meaningless. **But yet, people still do spin-wave in 1D... why ?** (you can just type "spin-wave for spin chain" on google and you'll see plenty of examples)


Easy to do experiments that clearly show outstanding properties of graphene

By chance we received for free some monolayer graphene sheets (20 cm x 20 cm) and mixed coper-graphene wires at our University. I would like to prepare some very easy to do experiments for the chemistry students that show the awesome properties of graphene, and which could be carried out with very simple lab equipment or even home experiments, any ideas or references?


Question regarding Drift velocity in general?

The derivation of drift velocity in case of electrons is equivalent to the case of an charged ionic gas and therefore all the arguments also apply there. Now for an ideal "ionic" gas which interacts with each others only as perfect spheres would (for arguments sake), when we apply an external electric field, they would just show drift, that means that they "favour" one direction more than the others where drift velocity is given by $v_d=(eE/m)\tau$ where $\tau$ is the average relaxation time. Hence the net velocities of the electrons wont vectorially add to zero, and moreover the speed as given by kinetic theory of matter would not depend only on the temperature but also depend on electric field. Does this not contradict two fundamental postulates of kinetic theory? Moreover in case of electrons in a circuit with many series resistors, the relaxation time would be different. The resistivity as defined by $E=\rho J$ and $\rho=m/(ne^2\tau)$ where j=current density, n=number of electrons per unit volume and e=electronic charge, should be different for all the series resistors. But then will not the drift velocity be different for all the resistors if the top equation is correct since the resistivities might defer, causing the relaxation time to differ. Doesn't this contradict the postulation of a steady current which requires constant drift velocity. Moreover suppose we use the same material of different cross section, since there is no area dependence in the drift velocity, it shouldn't change but then again we don't get a steady current as J remains a constant and I is simply J times A.


Gravitation law paradox for very close objects?

We all know that gravitation force between two small (not heavenly) bodies is negligible. We give a reason that their mass is VERY small. But according to inverse square law, as $r\to 0$, then $F\to \infty$. But in real life we observe that even if we bring two objects very close, no such force is seen. Why is this so?


What is the meaning of the Fourier transform of Feynman propagator?

I know $K(a,b,t)$ is the probability amplitude of find a particle that starts at point a in b in a time t later. There is also an expression that sometimes is called green function:   $$G(a,b,E)=(i/\hbar)\int_{-\infty}^\infty\;\exp(iEt/\hbar)\;K(a,b,t)\;dt$$ or Fourier transform of Feynman propagator   See: Grosche. Handbook of Feynman path integrals page 149. Keller. the feynman integral page 461.   http://arxiv.org/abs/cond-mat/0304290v1 I want to know if $G(a,b,E)$ could be the amplitude that a particle of energy $E$ at the initial point $a$ will appear at some (arbitrary) time at $b$. It seems that Martin Schaden and Larry Spruch use this interpretation in http://arxiv.org/abs/cond-mat/0304290v1 but I have not found this in any book of quantum mechanics.


Temperature stability of permanent magnets

Hello physicists/engineers, I have a somehow very specific question I hope some expert can answer me: I'm working at university and try to build an experiment which requires comparable high magnetic field (2000Gauss or 0.2T if you prefer) at room temperature. This field is actually no problem using Neodymium iron boron magnets, the problem however is: I can see relatively slow drifts in the magnetic field in the order of several mG. Since this basically destroys my experiment, and definitely occurs due to temperature drifts of a few mK, i was wondering if there are permanent magnets that do not change their magnetization under such temperature drifts (SmCo?) and/or if there's a procedure (annealing) to stabilize the magnets I have right now Thanks


Laser Coherence Length/Time

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


"Inverted" quantum oscillator

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


Exact diagonalization of graphene's tight binding Hamiltonian

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


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

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


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

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


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

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


Why atomic bomb makes a mushroom?

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


Optical explanation of images of stars?

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


Is the normal force a conservative force?

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


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

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


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

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


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

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


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

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


Software to calculate forces between magnets

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


Cyclist's electrical tingling under power lines

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


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

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


How to get statics out of a dynamic force concept?

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


How to calculate the coefficient of restitution for 2 bodies?

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


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

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


Magnetic induction in Relativity

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


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

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


Parabolic motion (experiment)

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


Quantum Bit Commitment, restricting ourselves to pure states

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


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

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


What are virtual excitations in the context of superconductors?

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


Are quantum mechanics and quantum physics the same field?

What is the difference between _quantum mechanics_ and _quantum physics_?


Centripetal/Gravitational Force

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


Are Lifshitz and Berestetskii right in this case?

In the Quantum electrodynamics book (look at the problem) its authors Lifshitz and Berestetskei claim that operator of charge conjugation $\hat {C} = -\alpha_{2}$ in Majorana basis transforms as $\hat {C}^{M} = \alpha_{2}$. Here (they started from the standart (Dirac) representation of the gamma-matrices) $$ \alpha_{2} = \begin{pmatrix} 0 & \sigma_{y}\\\ \sigma_{y} & 0\end{pmatrix}, \quad \beta = \begin{pmatrix} 1 & 0\\\ 0 & -1\end{pmatrix}, $$ $$ \hat {C}^{M} = \hat {U}^{+} \hat {C} \hat {U}, \quad \hat {U} = \frac{1}{\sqrt{2}}(\alpha_{2} + \beta ) = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 & \sigma_{y}\\\ \sigma_{y} & -1\end{pmatrix} = \hat {U}^{+}. $$ I tried to get their result, but I only got $$ \hat {C}^{M} = \frac{1}{2}\begin{pmatrix} 1 & \sigma_{y}\\\ \sigma_{y} & -1\end{pmatrix}\begin{pmatrix} 0 & \sigma_{y}\\\ \sigma_{y} & 0\end{pmatrix}\begin{pmatrix} 1 & \sigma_{y}\\\ \sigma_{y} & -1\end{pmatrix} = \begin{pmatrix} 1 & 0\\\ 0 & -1\end{pmatrix}, $$ because $\sigma_{y}^{2} = 1$. Where is the mistake? **An edit**. I found the mistake. I used the wrong definition of transformation of charge conjugation operator under unitary spinor transformation. The correct one is $\hat {C}^{'} = \hat {U}\hat {C}\hat {U}^{T}$.


Where do flying gas molecules get their energy from?

If you heat up water in a tank to boiling point, some of the water will turn to steam. Gas bubbles are created in the water. According to Archimedes principle, the gas bubbles will have a force equal to the weight of what they are occupying. Let's denote this force $F_g$. $$F_g = \text{Bouyant force of a gas bubble}.$$ Let's denote the height of the tank $h$. We also assume the steam bubbles are created at the bottom of the tank and also that the viscosity of water is nearly zero. So the work done when a bubble reaches the surface is $$\text{Energy} = (F_g-g m)h$$ $g$ is the gravitational acceleration and $m$ the mass of the bubble. Where did this energy come from? From the surrounding water? From the boiler? From the bubble itself? Can someone in detail explain to me this phenomena?


Does the doppler shift have any effect on the energy that the atom absorbs?

Does an atom moving towards the light that is falling on it, absorb more energy than when it is stationary? The atom should a see blue shift due to Doppler effect so the atom will see a light of greater frequency, causing it to absorb a greater amount of energy (or, equivalently, it will absorb less energy when it is moving away from the light due to red shift). Or will it absorb the same energy in all the cases? Does the atoms motion have any affect on the energy it absorbs?


Why are high voltage lines “high voltage?”

If I have two spheres of the same size and one sphere has a small amount of charge compared to the other that has a lot more charge, then clearly the sphere with the larger charge has a larger voltage (relative to the ground). My question is do high voltage power lines have a lot more charge that is placed on them? Is that what gives them the high voltage? I think I have a grasp of the step up stations that use transformers to kick up the voltage from power plants. This question seems almost silly to me but I have been struggling with this for a long time. I’ve done several searches online and I am not able to find answers. If there is a link that someone can provide, I much appreciate it.


Does alternating current (AC) require a complete circuit?

This popular question about "whether an AC circuit with one end grounded to Earth and the other end grounded to Mars would work _(ignoring resistance/inductance of the wire)_ " was recently asked on the Electronics SE. ![Picture edited from the one in the above link](http://i.stack.imgur.com/fcz1i.png)   _(Picture edited from the one in the above link)_ Though I respect the AC/DC experts there, I think _(with the exception of the top answer)_ they are all wrong. * * * My issue is that they all assume that AC requires a complete circuit in order to function. However, my understanding is that a complete circuit is necessary for DC, but not AC. My intuitive understanding is that AC is similar to two gas-filled rooms with a pump between them - the pump couldn't indefinitely pump gas from one room to another without a complete circuit _(DC)_ , but it could pump the gas back and forth indefinitely _(AC)_. In the latter case, not having a complete circuit just offers more resistance to the pump _(with smaller rooms causing a larger resistance)_. Is my understanding correct - **can AC circuits really function without a complete loop?**   More importantly, **what are the equations that govern this**?   If larger isolated conductors really offer less AC-resistance than smaller AC conductors, how is this resistance computed/quantified? Would its "cause" be considered inductance, or something else?


Maybe photon energy is constant as the Universe expands?

This is a question following on from my previous post Time-like Killing vector in FRW metric? For simplicity I take the spatially flat FRW metric in cartesian co-ordinates given by: $$ds^2 = -dt^2 + a^2(t)(dx^2 + dy^2 + dz^2)$$ I transform the time co-ordinate $t$ to conformal time $\tau$ using the relationship: $$d\tau = \frac{dt}{a(t)}$$ so that the metric now becomes: $$ds^2 = a^2(t)(-d\tau^2 + dx^2 + dy^2 + dz^2)$$ We can introduce a function $\alpha(\tau)$ giving $$ds^2 = \alpha^2(\tau)(-d\tau^2 + dx^2 + dy^2 + dz^2)$$ One can now see that the FRW metric has a time-like conformal Killing vector field.. Now electromagnetic waves are conformally (or scale) invariant. If one makes the transformation: $$x \rightarrow \lambda x$$ $$t \rightarrow \lambda t$$ the electromagnetic wave equations are invariant. Thus it seems to me that there is a symmetry here that Nature could obey. If it did so then the implication would be that the energy of electromagnetic waves is constant as the Universe expands.Therefore wave lengths that scale with the Universe are associated with constant energy. However atoms have a fixed wavelength so that their energy would increase as the Universe expands. Thus the cosmological redshift would be explained not by electromagnetic waves losing energy in an expanding Universe but rather by atomic energies increasing. In conventional cosmology one tacitly assumes that atomic energies are constant as the Universe expands. But what is the basis for such an assumption? At least with photons one can offer a symmetry argument, consistent with general relativity, as to why their energy should stay constant.


Problem with ring-method of du Nouy for calculating surface tension

I need to calculate the surface tension of a liquid with the ring-method of du Nouy and the correct formula to use is: > $\gamma = \frac{F}{4\pi R}$ I understand that the surface tension is the force that acts on a certain length which, in this case, is the circumference of the ring. Since the circumference of the ring is $2\pi R$ where does this extra $\times2$ come from? Thanks!


Square bracket notation for dimensions and units: usage and conventions

One of the most useful tools in dimensional analysis is the use of square brackets around some physical quantity $q$ to denote its dimension as $$[q].$$ However, the precise meaning of this symbol varies from source to source; there are a few possible interpretations and few strict guidelines. What conventions are there, who uses them, and when am I obliged to follow them?


Is there a physical system that emulates mathematical cellular automata?

Theoretical cellular automata have been proposed as models for many physical phenomena from music to quantum mechanics. My question concerns the reverse: **Is there a _simple_ physical system that emulates a theoretical cellular automaton?** A computer running CA software such as Hashlife can be considered an (extremely complex) physical system that behaves according to the rules of the programmed CA. I am however looking for something much simpler, such as a set of dominoes that topple each other in a fashion conforming to the rules of a certain CA, wave interference patterns that model a CA etc. The point is that there have been enormous advances in CA theory lately (such as the discovery of the first self-replicating pattern in Conway's Game of Life in November 2013) that it would be interesting if these patterns could simply be "set up" in the physical world somehow and then they would evolve according to the CA rules without requiring a computer to simulate the rules. In the long run, this might even lead to the creation self-replicating physical systems.


Why was the conversion factor of the metric unit bar chosen the way it was?

The unit **bar** for pressure is clearly a metric unit, but its order of magnitude is a bit strange. In the centimeter–gram–second system of units we have:               1 bar = 1 000 000 baryes = 1 000 000 dyn/cm²      so the bar is not "coherent" with this system (the factor is not one). Also in the meter–kilogram–second (and SI) system we get:               1 bar = 100 000 pascals = 100 000 N/m²      while in the meter–tonne–second system:               1 bar = 100 pièzes = 100 sn/m²      So my question is simply, where does the conversion factor for bar come from, since it seems to not fit into usual systems? According to the Wikipedia article _bar_ this unit was created already in 1909 by British meteorologist Shaw, but not much detail is provided. Maybe the factor was simply chosen as the power of ten making the unit closest to the atmospherical pressure at sea level (which is 1.01325 bar by convention, and close to 1.01 bar on average)?


Is it possible to levitate a toy maglev train using only permanent magnets as long as the train stays in motion?

I am fascinated with magnets, and specifically the idea of magnetic levitation. I recently purchased a Levitron toy maglev train and have enjoyed finding the perfect sweet spot at which it levitates, and I understand that the Levitron does not violate Earnshaw's theorem because it is in rotating motion. I would like to build a toy maglev train that makes no contact with the track, or any guide rails, using only permanent magnets and their repulsive force. I understand that if I attempt to build a flat track with no guide rails, the magnets will not be able to stably levitate, and will shunt the train car to one side or the other. However, what if I built a trough, or a V-shaped track, with the train car sitting inside the track, and a corresponding v-shaped bottom, with repulsive magnets running the length of the track, inside the V? I realize that it wouldn't be stable while sitting still, but if the train were in motion, would that enable it to levitate stably? In my mind there should be a sweet spot where the train's motion overcomes any sideways shunting effects as it travels above the magnetic fields of the magnets in the track, while gravity's pull on the train car into the V would be negated by the repulsive force of the magnets. I haven't been able to find anyone who has tried this approach. Every toy maglev train that I've seen has either used guide rails or a superconductor. My gut tells me it's not possible, but I want to believe. :) Is a configuration like this possible? Is it possible to overcome the instability through one-directional motion?


Photon as the carrier of the electromagnetic force

My physics background goes as "far" as reading popsci books on QM, Particle Physics, and Cosmology so pardon my ignorance in the below questions. I've read that the photon is the particle (quanta in QFT) exchanged between electrons, and that is how the electromagnetic force is explained in QFT.   1. How do we know about these virtual photon exchanges given the fact that we can't observe them directly? How did we arrive at " ** _electrons exchange virtual photons and that's the cause of the electromagnetic force between them_** " from merely observing electrons absorbing or emitting photons?    2. If electrons throw photons at each other doesn't that mean that they should only scatter (repel)? If that is so why do magnets and opposite charged particles attract ?   3. I've read that because photons don't posses electric charge therefore they can't exchange between themselves other photons so they can't collide (scatter) in the usual way, unlike the W+, W-, Z0, and gluons which posses the charge of the force they mediate. The only collision processes between photons that I've heard of are Delbruck scattering and the phenomenon of conjuring matter (1 electron , 1 positron) out of 2 photons that collide at a close angle between their trajectories. My third question is: Do these two rare phenomena of photon "scattering" happen here on earth naturally (no LHC or other particle accelerators), in the upper atmosphere, or only in the deep dark space? Thanks!


A sphere, a simple object?

In this video, the woman says that a sphere is a _pretty simple object_. What intrigues me is the use of a sphere for such a calculation. First of all, the sphere wouldn't be perfect as a perfect sphere doesn't exist in nature (first source of imprecision). Second, one will have to use π for calculating the volume. As π is irrational, one will have to use an approximation for it (second source imprecision). So why not using a simple cube? The shape would be closer to a cube due it's harder to make a sphere and it's volume would be calculated more precisely. Why they choose to do this calculation using a sphere?


Kinetic energy of a gas molecule

I am working through the derivation of the equation $$pV=\frac13mNC_{rms}^2,$$ but relating to this formula my textbook states that the kinetic energy of an average molecule of gas is given by $$KE=\frac12mC_{rms}^2$$ but how is this formula right since the mean squared speed ($C_{rms}^2$) is not the same as the average speed of the molecule squared, $v^2$, so why do we use the mean squared speed for kinetic energy and not the means speed squared? Is it just that it is easier to calculate?


Magnetic Induction - How would different coil configurations affect the outcome?

This is a branch question (from this question) related to magnetic induction: These deal with how different configurations of coiled layers might affect the outcome:   * If one were to bundle multiple layers of coiling to up the coils, would it be more effective to have each layer connected in parallel and running in the same direction? (example - 2 ins and 2 outs)   * If one were to coil 'loosely' such that the wire is wrapping around the coil at a 45 degree angle - would it generate less induction than the same number of coils tighter coiled at nearly a 90 degree angle (pretty sure this is a yes - but wondering if there is any silver lining in terms of resistance - perhaps with a deluge of parallel-connected layers))?   * In the event that two or more layers are connected in parallel - would the energy created across each layer likely be significantly less? In the event of (say) five layers of coiling connecting, each with differing currents (or voltages), how might the resultant current be combined once reunited (presuming a similar wire is receiving the energy)?  I am keenly interested in the variables at play being defined - as well as any information on how such might play out. Thank you for any guidance.


Lippmann-Schwinger solution

What's wrong with this general solution of the Lippmann-Schwinger equation: $$ |\psi_k \rangle=|\phi_k \rangle+G_k V|\psi_k \rangle $$ Taking the inner product with $\langle\phi_{k'}|$ \begin{align} \langle \phi_{k'}|\psi_k \rangle&=\langle \phi_{k'} |\phi_k \rangle+\langle \phi_{k'} |G_k V|\psi_k \rangle \\\ &=\delta(k'-k)+g(E_k-E_{k'})\langle \phi_{k'} |V|\psi_k \rangle\\\ &=\delta(k'-k)+g(E_k-E_{k'})\langle \phi_{k'} |H-H_0|\psi_k \rangle\\\ &=\delta(k'-k)+g(E_k-E_{k'})\langle \phi_{k'} |E_k-E_{k'}|\psi_k \rangle \end{align} Basic algebra then gives us $$\langle \phi_{k'}|\psi_k \rangle[1-g(E_k-E_{k'})(E_k-E_{k'})]=\delta(k'-k)$$ and isolating the inner product on the left hand side gives us $$\langle \phi_{k'}|\psi_k \rangle=\delta(k'-k) [1-g(E_k-E_{k'})(E_k-E_{k'})]^{-1}$$ where $g(E_k- E_{k'})$ is the eigenvalue of $G_k$ resulting from hitting $\langle \phi_{k'} |$ on the left? This seems to suggest that the eigenstate $|\psi_k \rangle $ of the full Hamiltonian $H$ is the same as the eigenstate $|\phi_k \rangle$ of $H_0$, up to a proportionality factor, since $|\psi_k \rangle $ has no amplitude to be in $|\phi_{k'\neq k} \rangle$.


Are there any well-known theories successfully unifying the inertial and gravitational mass?

From what little I know of general relativity, the equality of inertial and gravitational mass is an axiom of the theory. I suspect that this precludes GR from unifying them in the same sense as electric, magnetic and weak interactions are unified as the electroweak force. But since the two appear to be equal under staggeringly precise measurements, there must be some interest in constructing a theory of mass and motion which fundamentally unifies the two, that is, makes the separation nonsensical / impossible. Are there any particularly outstanding attempts in this regard, even if the resulting theories are known to fail to correspond to our universe in some other important way?


Inverse square law in 2+1 dimensional universe from a Yukawa coupling?

There is a nice result that in 3+1 space time, a Yukawa coupling leads to an inverse square law force as the mass of the scalar field goes to zero. I was wondering what the corresponding force in a 2+1 dimensional Universe would be? Geometrically, I would guess that it should be $1/r$ rather than $1/r^2$, but my explicit calculation for a Yukawa coupling gives that it is $1/r^2$ in 2+1 space as well. I was wondering if that was correct?


Next crescent moon?

How do I find the date of next crescent moon on wolframalpha or any other site which tells the appearance date time of next crescent on a specific location? _I am searching for it to look for 1st of Ramadan_


Density of stars near the center of the Milky Way

At night, I can look up and see the Milky Way across the sky. My question is, supposing our solar system was, instead of way out on an 'arm' of the galaxy, if we were near the galactic center. Would the night sky be much brighter?? If so, how close would we have to be to never have darkness of night?? Thanks!


Does the "Andromeda Paradox" (Rietdijk–Putnam-Penrose) imply a completely deterministic universe?

**Wikipedia article:** http://en.wikipedia.org/wiki/Rietdijk–Putnam_argument **Abstract of 1966 Rietdijk paper:** > A proof is given that there does not exist an event, that is not already in > the past for some possible distant observer at the (our) moment that the > latter is "now" for us. Such event is as "legally" past for that distant > observer as is the moment five minutes ago on the sun for us (irrespective > of the circumstance that the light of the sun cannot reach us in a period of > five minutes). Only an extreme positivism: "that which cannot yet be > observed does not yet exist", can possibly withstand the conclusion > concerned. Therefore, there is determinism, also in micro-physics. **Question:** I realize that this is a very philosophically-heavy topic, but at the same time, it is physical in the strictest sense (in that it's not a "What if" question, but more of a thought experiment, like the Twin Paradox, that follows the logical implications of relativity to some extreme scenario). So, setting aside matters of "free will" and whatnot, which I'm not interested in... If I understand this article correctly: the content of these papers implies a 100% deterministic universe — one in which the future of any object is "set in stone"; **but,** at the same time, this future is completely inaccessible to any observer that is in its past (i.e: impossible to predict even given initial conditions, because of small-scale uncertainty, chaos, etc.). Are the proofs in these articles are well-accepted? (Meaning that this "everything has already happened depending on the observer" universe is the only conclusion one can arrive at?). I was not aware of the Andromeda Paradox before. I can't imagine it being very useful, but it's interesting enough as a mind-bender that I'd've thought one would hear about it more often.


Factorization of fermionic scattering integral in 2d momentum rep

the scattering integrals for fermions involves both momentum ($k$) and energy ($k^2$) conservation and a nonlinear phase space factor of a distribution function $f(k)$. $$\begin{multline}I(k) = \sum_{k_1, k_2, k_3} \delta(k^2+k_1^2+k_2^2+k_3^2) \delta(k+k_1+k_2+k_3)\times \\\ \Bigl[f(k)f(k_1)\bigl(1-f(k_2)\bigr)\bigl(1-f(k_3)\bigr) - f(k_2)f(k_3)\bigl(1-f(k)\bigr)\bigl(1-f(k_1)\bigr)\Bigr]\end{multline}$$ In energy space, energy conservation is linear and powerful factorizations are possible to compute the integrals fast. In momentum space, the nonlinear energy conservation constraint complicates factorizations. Trivially, in 2 dim one can work in polar coordinates separating off the radial part. But has anybody seen a more efficient reduction of numerical complexity (e.g. mapping to FFT,...) Thanks for any hint!


What libraries are available for solving problems at the college physics level?

Are there any libraries available for solving problems at the level of college physics? In particular, I'm interested in libraries for general purpose programming languages that can calculate results both numerically _and_ symbolically. Libraries which could be used for implementing something like this are SymPy (Python), GiNaC (C++), or SymbolicC++ (C++). I'm asking this question because I've started cooking up a C# library for symbolic computation and have used it to solve a few projectile motion problems, but would like to review other work out there. Here's an example of a program which solves such a problem. Here's what the program outputs when run: ![photo](http://i.stack.imgur.com/ZX14W.png) Any pointers or suggestions welcome. Thanks!


Does any particle ever reach any singularity inside the black hole?

I am not a professional physicist, so I may say something rubbish in here, but this question has always popped in my mind every time I read or hear anyone speak of particles hitting singularities and "weird things happen". Now to the question at hand, please follow my slow reasoning... As far as I've learned, to reach a black hole singularity, you must first cross an event horizon. The event horizon has this particular property of putting the external universe on an infinite speed to the falling observer. Now due to the Hawking radiation, and knowing that the cosmic background radiation is slowly dimming, sooner or later every black hole in this particular instance of inflation we are living in will evaporate, according to an external observer of said black holes. This means that every black hole has a finite timespan, as long as this universe survives that long. Now if we go back to the falling observer, we had already established that such an observer would see the outside universe "speed up" infinitely. This means that when the falling observer "hits" the event horizon, he will (or it if we speak about particles, which is clearer in this case), be immediately transported in time towards the final evaporation moment of the black hole. Either this or the particle gets some weird treatment. My point is, such a particle never gets to the singularity, because it has no time to get to it. The moment it crosses the event horizon, the black hole itself evaporates. Where am I wrong here?


Just a simple question. Does a muon detector on Earth's surface correctly measure the mean lifetime of a muon? I would think the answer is no because most muons detected are created about 15 km above Earth's surface, so on average they will have survived for some amount of time calculated using special relativity before reaching the detector. The detector I'm talking about is a scintillator, so it detects when the muon enters the detector with a flash, then if the muon decays it detects another flash. The time between flashes is said to be the muon's lifetime. Is this muon's lifetime the machine giving me the actual lifetime (2.2$\mu s$), or just the lifetime of it on Earth's surface? here is a plot of my data showing a histogram of decay times ![enter image description here](http://i.stack.imgur.com/Oc1yB.png)


Concentration of fluid pumped through a tube?

I have a bottle filled with a fluid A, and a tube of volume $V_0$ filled with a fluid B leading away from it. When I now start pumping fluid A out of the bottle, I am interested in the concentration $c(V)$ of fluid A in fluid B at the end of the tube after the volume $V$ exited the tube. In my second situation (I am afraid there are two), after having pumped $V_0$, a valve at position $V_1$ switches and the pump is reversed. The mixed fluids are now pumped into a second tube of volume $V_2$. Again, I'd like to know what drips out of there. **Situation 1**               |    A    |     |         |      V0     |      ====================    <- B     |         |     |         |     |_________|      **Situation 2**               |    A    |     |         |          V0     |      =========++==========    <- B     |         | V1  ||     |         |     || V2     |_________|     ||                           ^- B      I've tried using the Hagen-Poiseuille equation (or its proof as given in the Wikipedia), but that grows ugly fast. Is there a simple(r) way to approach this?


Algorithm to simulate impulse response of a solid?

There are well-documented methods of calculating the impulse response of a room (ex. image method, ray-tracing method); however, I have not been able to find anything similar for the impulse response of an arbitrary solid given its size, density, etc. Can anyone point me in the right direction?


Tachyon vertex operator (Polchinski's book)

* I would like to know how does Polchinski in his book "derive" what is the "tachyon vertex operator" (..as say stated in equation 3.6.25, 6.2.11..) I can't locate a "derivation" of the fact that $:e^{ikX}:$ is the tachyon vertex operator.  (..I understand that it follows from some application of the state-operator map but I can't put it together..)   * And then what is the meaning of the ``higher vertex operators" - which are of the form of arbitrary number of either operators of the above kind or the derivatives of $X$ w.r.t either $z$ or $\bar{z}$. (..like in equation 6.2.18..)


What does it mean that an inspiral is 'adiabatic'?

Binary systems emit gravitational radiation. This causes the system to lose energy, which results in a shrinking of the semi-major axis. I have read on countless occasions that this 'inspiral' is _adiabatic_ (here for example). What does this mean that the shrinking is adiabatic?