title: AM vs FM radio under bridges

text: I know what the difference between AM (Amplitude Modulation) and FM (Frequency Modulation) radios is. However, I noticed that when I drive under a bridge while listening to a station on AM, it just gets really quiet, while if I'm listening to an FM station is gets fuzzy and static-y. Does anyone know why this is? My guess it that AM radio waves can't reach their full amplitude under bridges? And maybe that FM radio waves will oscillate on a different range under a bridge, but I'm really not sure.


title: Paraxial wave equation solution?

text: I am looking for a worked solution to the paraxial wave equation, since most sources just say that _clearly_ Gaussians are solutions but they do not explicitly show so. This equation is equivalent to the TDSE for a free particle, if we replace _z_ by _t_. $$\frac{\partial^2 \Psi(x,y,t)}{\partial^2 x} + \frac{\partial^2 \Psi(x,y,t)}{\partial^2 y} = 2ik\frac{\partial \Psi(x,y,t)}{\partial t} $$


title: CM: Need to recover the Hamiltonian, knowing conserved quantities and information about the EOM, possibly via action-angle coordinates

text: **Statement of the problem:** I have a system with 2 degrees of freedom and I have found two independent conserved quantities, without knowledge of the Hamiltonian. I'm looking for a method to recover a Hamiltonian that describes the system. Action-angle variables seems like a promising approach, but most of the discussions that I've found on this assume knowledge of the Hamiltonian and at least two of the coordinates $(p,q) \rightarrow (I, \phi)$ that represent the canonical transformation, or the generating function, etc. I also recognize that some guess work may be necessary if no methodical procedure exists, but advice on "guided" guesses would also be helpful. Maybe simply treating the conserved quantities as the action variables and attempting to guess a generating function for the canonical transformation could work... **Further info** I have a coupled set of 4 first order nonlinear differential equations which describe the dynamics in 4D phase space. To be more specific, these equations take the form \begin{align} \dot{X} = f_1 (Y,S)\\\ \dot{Y} = f_2 (X,Z)\\\ \dot{Z} = f_3 (X,Y)\\\ \end{align} where $X, Y, Z$ are functions of time and I omit the equation for $\dot{S}$ because there is a hidden relationship $S^2 = X^2 + Y^2 + Z^2$ (these are the Stokes variables). I have no knowledge or "feel" for which coordinates might describe canonical positions or momenta at this point. As stated, I have independent conserved quantities \begin{align} \dot{J_1}(Y,Z) = 0\\\ \dot{J_2}(S,Y,Z) = 0 \end{align} A direct method for finding a Hamiltonian is of course desirable, but any ideas or direction towards relevant reference material would also be greatly appreciated.


title: In what limit do we *really* get Maxwell-Boltzmann statistics from Bose-Einstein and Fermi-Dirac?

text: Fermi-Dirac and Bose-Einstein energy occupation number $n(\epsilon)$ in natural units ($[T]=[\epsilon]$) read $$n(\epsilon) = \frac{D(\epsilon)}{e^{(\epsilon-\mu)/T}\pm 1},$$ where $D(\epsilon)$ is the density of states at the given energy $\epsilon$, $+$ is Fermi-Dirac and $-$ is Bose-Einstein. The usual answer to the question by which limit to arrive to Maxwell-Boltzmann statistics is: $$\frac{\epsilon-\mu}{T} \gg 1$$ or $$\frac{\epsilon_{min}-\mu}{T} \gg 1$$ This gives you purely formally Maxwell- Boltzmann distribution but is a make-it-work pseudo-argument as when $T \to \infty$, $\frac{\epsilon-\mu}{T} \to 0$ and on the contrary $T \to 0^+$, $\frac{\epsilon-\mu}{T} \to \infty$. Hence this rule would tell us to apply Boltzmann statistics to low temperatures and stick with the quantum statistics at high temperatures. This is obviously a fall into the fiery pit of slurry textbook mumbo jumbo. To resolve this, I believe we have to presume a growing density of states $D(\epsilon)$ and faithfully take the $T \to \infty$ limit showing that every macroscopic feature (i.e. every moment of the original distribution) is in the limit reproduced by Maxwell-Boltzmann up to $\mathcal O(T^{-2},\sqrt{N},...)$. The problem in the "textbook" argument as well as with the discussed limit is that $\epsilon-\mu$ actually always passes through values which are both greater and smaller than $T$ in the integration. The growing $T$ just "smears out" the distribution out into larger and larger regions in $\epsilon$, which motivated my guess that $D(\epsilon)$ must be growing so that the "smearing out" makes the high-energetic states dominate. So how is the limit rigorously done? And are there some extra assumptions which aren't usually mentioned? (My guess is that the junk with $\mu \approx \epsilon$ in Bose-Einstein also needs some handling.)


title: Self-study: resources for understanding Gamma spectroscopy

text: I work in applied mathematics and have no background in atomic physics. I need to have a good understanding of the principle and application of Gamma spectroscopy. I am ok to spend the needed time but would like not to spend too much time on avoidable subjects. I fell a bit lost into this subject and do not know what to begin with. If somebody can provide me a learning map (with online ressources if possible) I will greatly appreciate.


title: What's a good reference for this classical picture Feynman's talking about?

text: I have a mathematics background but am trying to educate myself a little about physics. At the beginning of Feynman's QED book (not the popular one) is the following: > Suppose all of the atoms in the universe are in a box. Classically the box > may be treated as having natural modes describable in terms of a > distribution of harmonic oscillators with coupling between the oscillators > and matter. I guess this is something that physicists learn, but I have never heard of it. What is Feynman talking about and where can I learn more about it? The Wikipedia article on harmonic oscillators gives no indication that physicists do this.


title: Why can the Schroedinger equation be used with a time-dependent Hamiltonian?

text: I have a puzzle about Schroedinger equation with time-dependent hamiltonian, which is usually used in time-dependent quantum systems. However, one of the axioms in quantum mechanics postulates that the hamiltonian as the generator of the one-parameter unitary group $U(t)$ is not time-dependent, and the evolution of a quantum state should obey the Schroedinger equation with time-independent hamiltonian. So why can Schroedinger equation with time-dependent hamiltonian be used without hesitation? Where does the time-dependent hamiltonian come from, except for the case of interaction picture? And should there be an additional axiom for it? * * * The key is to recognize which are from axiom and which are from model. I foolishly thought that every Shroedinger equation with a time-dependent Hamiltonian can be derived by starting from that strong-continuity axiom in quantum mechanics. Also, I do not think the above comments answered my question.


title: Noether's Theorem Notation Question

text: In Noether's theorem you want to be able to say that the functional $$J[x,y,z] = \smallint_{t_1}^{t_2} \mathcal{L}(t,x,y,z,x',y',z')dt$$ is invariant with respect to a continuous one-parameter group of transformations of the form $$T(t,x,y,z,\varepsilon) = (t^*(\varepsilon),x^*(\varepsilon),y^*(\varepsilon),z^*(\varepsilon)) = (t,x\cos(\varepsilon)+y\sin(\varepsilon),-x\cos(\varepsilon)+y\sin(\varepsilon),z)$$ (in the case of conservation of angular momentum). How would I say this in mathematical notation rigorously? Is there a nice way to talk about one-parameter groups acting on functionals? For clarity I see $J$ as a function on a function space, something like $$J : \mathcal{C}^1[a,b] \times \mathcal{C}^1[a,b] \times \mathcal{C}^1[a,b] \rightarrow \mathbb{R} \| \ (x,y,z) \mapsto J[x,y,z]$$ & the one-parameter group of transformations as a map of the form $$T : \mathbb{R}^4 \times \mathbb{R} \rightarrow \mathbb{R}^3 | (t,x,y,z,\varepsilon) \mapsto T(t,x,y,z,\varepsilon) = (t^*,x^*,y^*,z^*)$$ Thus as it stands I'm kind of going on intuition when I form $J[x^*,y^*,z^*]$, the notation has gaps which intuition plugs but I'm sure there's a nicer & more consistent/cogent way to do all this! I'm hoping there's something like $J[\vec{r}] = J[x,y,z]$ implying $J[T(\vec{r})] = J[\vec{r}]$, though obviously that is massively flawed, how would I say the next best thing rigorously? If something like group actions applies would you mind being a bit explicit as I see things like this being mentioned & no notation is offered to justify it & I get lost, thanks!


title: What is the massless limit of massive electromagnetism?

text: Consider electromagnetism, an _abelian_ gauge theory, with a massive photon. Is the massless limit equal to electromagnetism? What does it happen at the quantum level with the extra degree of freedom? And, what does it happen at the classical level? We ca not get an extra massless classical scalar field, do we?


title: Why are photons released from high-voltage electricity through a near vacuum?

text: I read an answer to what is inside a typical vacuum, and it suggested neutrinos from Space, and vapor from the container. So, maybe they add to this effect... But why are photons released from high-voltage electricity passing through a near vacuum?


title: Weightlessness by a parabolic flight

text: Do you actually achieve weightlessness during a parabolic flight? Because I believe I heard somewhere did you achieve 'near-weightlessness' and not 'weightlessness' (if this is true, why is this?) And, also, Is 'weightlessness' a correct term?


title: Relativistic speed/energy relation. Is this correct?

text: The relativistic energy-momentum equation is: $$E^2 = (pc)^2 + (mc^2)^2.$$ Also, we have $pc = Ev/c$, so we get: $$E = mc^2/(1-v^2/c^2)^{1/2}.$$ Now, accelerating a proton to near the speed if light, I get the following results for the energy of proton: 0.990000000000000 c = 0.0000000011 J = 0.01 TeV 0.999000000000000 c = 0.0000000034 J = 0.02 TeV 0.999900000000000 c = 0.0000000106 J = 0.07 TeV 0.999990000000000 c = 0.0000000336 J = 0.21 TeV 0.999999000000000 c = 0.0000001063 J = 0.66 TeV 0.999999900000000 c = 0.0000003361 J = 2.10 TeV **0.999999990000000 c = 0.0000010630 J = 6.64 TeV** 0.999999999000000 c = 0.0000033614 J = 20.98 TeV 0.999999999900000 c = 0.0000106298 J = 66.35 TeV 0.999999999990000 c = 0.0000336143 J = 209.83 TeV 0.999999999999000 c = 0.0001062989 J = 663.54 TeV 0.999999999999900 c = 0.0003360908 J = 2,097.94 TeV 0.999999999999990 c = 0.0010634026 J = 6,637.97 TeV 0.999999999999999 c = 0.0033627744 J = 20,991.10 TeV If the LHC is accelerating protons to 7TeV it means they're traveling with a speed of 0.99999999c. Is everything above correct?


title: Hamiltonian of oscillators quantized proof

text: https://docs.google.com/open?id=0BxrBcN1-BZWUOXNxR1l4S0l2MjQ http://www.2shared.com/complete/Qjy1_uzp/Quantum_Mechanics_in_Simple_Ma.html (I uploaded a pdf file that contains the parts of the textbook that I am having some difficulty understanding. Seven pages.) (two links point to the same pdf.) A matrix $N$ has all entries except diagnoal ones zero, and first row, first column also zero, other diagonal entries depending on its column, row place; for example, second row, second column has one as its entry, third row, third column has two as its entry and so on. It is known that $H = \hbar\omega (N + \frac{1}{2}) $. The question is, my textbook says that if there are a continuous range of possible values, then for some states, $\langle (N - \langle N \rangle )^2 \rangle \leq [\frac{1}{2} (n - \langle N \rangle)]^2$ , where $n = 0,1,2,3,...,$ then progresses to say that since $\langle (N - \langle N \rangle )^2 \rangle \geq (n - \langle N \rangle)^2 (\langle I_0 \rangle + \langle I_1\rangle + ...)$ then $\langle (N - \langle N \rangle )^2 \rangle \geq (n - \langle N \rangle)^2$. Then it says that this proves that there cannot be a continuous range of possible values, as $(0-\langle N \rangle)^2$, $(1 - \langle N \rangle)^2$ and so on cannot be smaller than $(n - \langle N \rangle)^2$. I am not sure how one gets $\langle (N - \langle N \rangle )^2 \rangle \geq (n - \langle N \rangle)^2 (\langle I_0 \rangle + \langle I_1\rangle + ...)$. Can anyone show me the proof? Also, how does one get the part - $(0-\langle N \rangle)^2$, $(1 - \langle N \rangle)^2$ and so on cannot be smaller than $(n - \langle N \rangle)^2$?


title: Carnot cycle: What happens to the rest of the energy?

text: In a Carnot cycle, the work potential is $W =Q_{in}( 1-\frac{T_0}{T_R})$ where $T_0$ is the temperature of the sink, $T_{R}$ is the temperature of the source heat reservoir, and $Q_{in}$ is the heat flow from the source to the sink. What happens to the rest of the energy that is left? That is, where does $Q_{in} - W $ go? What happens to it and why can't we use it?


title: Weight registered of a falling chain

text: Let's say I have a metallic chain. Suppose I hold one of its tip of in my hand let the chain hang freely downward such that the other tip touches a weighing scale. At this point, with me holding the chain the weight, registered on the machine will be zero. But once I let go of the chain, how will the reading on the machine change? Will it show the same reading all the way down (ie total weight of the chain) or will it only show the weight of the part of the chain that it is in contact with. I don't think this is one of those 'lift' problems since there, the weighing scale itself is NOT moving. Intuitively I think the weight registered will increase, but what is confusing me is that if I stand on a weighing scale, my head is not touching the scale, in the same way different parts of the chain are not touching the scale while falling.


title: Why Is Current $\propto$ Number of Loops?

text: When an additional loop is added to a generator the magnetic flux through the loops does not change yet the current increases. This seems counter-intuitive and has been described as 'non-conservative.' What causes this unexpected phenomena known as Faraday's Law?


title: Restriction of a Lagrangian

text: I'm wondering if anyone could help me with the following questions. Let $M$ be the Minkowski spacetime, given $f\in C^{\infty}(M) ; f(m)=x^{0}(m)$, with $\\{x^{\mu}\\}$ being a global Cartesian coordinates system, given the 3-dimensional submanifold $M\supset F_{t}=f^{-1}(t)$ relative to a regular value $t\in\mathbb{R}$ of $f$, and given the Lagrangian: $$ \mathcal{L}\in C^{\infty}(TM) $$ $$ \mathcal{L}(x^{\mu},\dot{x}^{\mu})=-\sqrt{\eta_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}} $$ where $\eta$ is the Minkowski metric and $\\{x^{\mu}\\}$ a global Cartesian coordinates system; what is the coordinates espression of the Lagrangian on $F_{t}$: $$ (T\iota_{t})^{*}\mathcal{L}\in C^{\infty}(F_{t}) $$ I "know" from other "sources" that I should find: $$ (T\iota_{t})^{*}\mathcal{L}(x^{i},\dot{x}^{i})=\mathcal{L}\circ T\iota_{t}(x^{i},\dot{x}^{i})=-\sqrt{1 - \delta_{ij}\dot{x}^{i}\dot{x}^{j}} $$ Is it totally wrong?


title: How to determine spatial color distribution?

text: Is there an easy way (i.e. free or open source tool) to determine the spatial color distribution (i.e "intensity") in a picture or in a region of interest of this picture? Let's say I used a laser or some lenses and mirrors and a light bulb to adjust a ray of light on a piece of paper in a dark room and photographed the spot and now I want to find out the rough intensity distribution, e.g. if it is a flat distribution or a gaussian or something else. With what tool is this possible?


title: How energy curves spacetime?

text: We know through General Relativity (GR) that matter curves spacetime (ST) like a "ball curves a trampoline" but then how energy curves spacetime? Is it just like matter curvature of ST?


title: Could a Class A Stellar Engine Use A Brown Dwarf

text: Could a class A stellar engine (or maybe a class C one) be built using a brown dwarf (for argument's sake, a T-dwarf like Gliese 229B)? Would it be capable of enough thrust to move itself any appreciable distance over a long enough period of time? (Let's say over a period of a billion years.) As a possible follow-up if the answer above is more or less "yes", would any appreciable additional thrust be possible to create by generating an asymmetrical solar-flare like eruption in a modified variation of a thermal- driven outflow "star lifting" technique?


title: Continuum model of the n-body problem?

text: Consider a collection of $n$ point masses evolving under the influence of gravity. For very large $n$, one might be interested in making a continuum approximation of this system (for cosmology, say). However, my naïve attempt to derive a continuum model directly from particle-particle interactions fails. In particular, consider a function $\rho: \Omega \rightarrow \mathbb{R}$ that describes the mass density over some domain $\Omega$. To compute the gravitational force at a point $x \in \Omega$, I might try to simply integrate the force contributed by every other point $y$: $$ f(x) = \int_\Omega \frac{\rho(x)\rho(y)}{|x-y|^2}\frac{y-x}{|y-x|} $$ Even if we exclude the point $x$ (i.e., we integrate over $\Omega \backslash x$ instead of all of $\Omega$) this integral diverges -- we effectively have particles arbitrarily close to $x$. So the question is: what's the right way to compute the gravitational force in this situation? It would also be nice to have some pointers to continuum models used in cosmology **_and their derivation from the particle model_** (many references simply start with a continuum model and do not explain its origin). By the way, I really only care about classical models -- let's forget about relativity, etc., for now. Thanks!


title: Do higher frequency EM waves travel longer than lower EM frequency waves?

text: In radio communication, the distance traveling seems to be an important parameter before we choose the frequency band of communication. There is also a path-loss model that describes a distance $\leftrightarrow$ frequency dependence of the attenuation with an isotropic antenna. However, I fail to find any specific physical law that demonstrates the frequency dependence of the attenuation. Some even say that higher the frequency, higher the energy content. What are the factors for propagation distance?


title: Why can't a stable star have radius 1 < r < 9/8 its Schwarzschild radius?

text: From http://www.spacetimetravel.org/ssm/ssm.html : > A mass of 1.78 [in geometric units] corresponds to a ratio of radius to > Schwarzschild radius of 9/8. Theory predicts that a smaller ratio is not > possible for a stable star. What causes this cutoff? Is this the TOV limit or something else?


title: Time-ordering vs normal-ordering and the two-point function/propagator

text: I don't understand how to calculate this generalized two-point function or propagator, used in some advanced topics in quantum field theory, a normal ordered product (denoted between $::$) is subtracted from the usual time ordered product (denoted $T$): $$\langle X^{\mu}(\sigma,\tau)X^{\nu}(\sigma',\tau')\rangle ~=~ T ( X^{\mu}(\sigma,\tau)X^{\nu}(\sigma',\tau')) ~-~ : X^{\mu}(\sigma,\tau)X^{\nu}(\sigma',\tau'):$$ My question is can the rhs of this propagator be derived or the meaning of the subtraction of the time ordered product explained and motivated in simple words?


title: Eigenfunctions of the Runge-Lenz vector

text: The hamiltonian for the hydrogen atom, $$ H = \frac{\mathbf{p}^2}{2m} - \frac{k}{r} $$ is spherically symmetric and it therefore commutes with the angular momentum $\mathbf{L}$; this causes all its eigenfunctions with equal angular momentum number $l$ but arbitrary magnetic quantum number $m$ to be degenerate in energy. The hydrogen atom also has a further degeneracy, in that given any angular momentum there are usually other $l$s with the same energy. This degeneracy is due to the existence of a second constant of motion, usually called the Laplace-Runge-Lenz vector, $$ \mathbf{A} = \frac{1}{2m} ( \mathbf{p} \times \mathbf{L} - \mathbf{L} \times \mathbf{p}) - k \frac{\mathbf{r}}{r}, $$ which is the generator of an even bigger symmetry, which is isomorphic for bound states to the group $\rm{SO}(4)$ of rotations in four dimensions, of the Kepler problem. The Runge-Lenz vector also has a rich geometrical interpretation. For a classical elliptical orbit, it points from the focus to the periapsis and its magnitude is proportional to the orbit's eccentricity. For circular orbits, it vanishes. ![enter image description here](http://i.stack.imgur.com/9Ejp0.png) Image source: Wikipedia The hydrogen atom is usually described in the common eigenbasis of the hamiltonian and the angular momentum, with the well-known and well-loved quantum numbers $|nlm\rangle$. However, the Runge-Lenz vector $\mathbf{A}$ is also a constant of the motion. ## What do its eigenfunctions look like? More concretely, I'm looking for the spatial structure of the common eigenfunctions of $H$ and at least one component of $\mathbf A$, and possibly also of $A^2$ (which, in analogy with the common eigenfunctions of $H$, $L^2$ and $L_z$, is the most one could expect), and if that's not possible then an explanation of why, and a description of suitable third quantum numbers to complete a CSCO. I would like to know what their corresponding eigenvalues are, and what the uncertainty of the other components is, whether one can assign a classical eccentricity to the orbital, and more generally in the relation to the corresponding classical geometry.


title: speed of an electromagnetic wave

text: I have this equation $$\dfrac{\partial E_z}{\partial y^2} - \mu_0 \varepsilon_0 \dfrac{\partial E_z}{\partial t^2} = 0$$ Why is $v^2 = 1 / (\mu_0 \varepsilon_0)$ ?


title: Trying to understand the weak gravitational field metric (2)

text: I'm still struggling with Carroll's discussion of the Newtonian Limit. I'm hoping no one will mind if I ask several questions here as they all relate to the same section (pages 105-106) of his “Lecture notes on general relativity” and I'm trying to get this thing straight in my mind. He states that “moving slowly” means $\frac{dx^{i}}{d\tau}\ll\frac{dt}{d\tau} $ , and therefore the geodesic equation becomes $\frac{d^{2}x^{\mu}}{d\tau^{2}}=\Gamma_{00}^{\mu}+\left(\frac{dt}{d\tau}\right)^{2}=0 $ Q1. Why does $\frac{dx^{i}}{d\tau}\ll\frac{dt}{d\tau}$ ? What's the justification in assuming that the rate of change of coordinate time wrt proper time is much larger than the rate of change of distance wrt proper time? If I can understand this I (I think I) can understand why he only considers the $\Gamma_{00}^{\mu}$ Christoffel symbols. Because the field is static, he changes the geodesic equation $\Gamma_{00}^{\mu}=-\frac{1}{2}g^{\mu\lambda}\partial_{\lambda}g_{00}$ to $\Gamma_{00}^{\mu}=-\frac{1}{2}\eta^{\mu\lambda}\partial_{\lambda}h_{00} $ Q2. So, he's gone from $g^{\mu\lambda}$ to $\eta^{\mu\lambda}$ . Is that because he ignores the $h^{\mu\lambda}$ bit of $g^{\mu\lambda}=\eta^{\mu\lambda} -h^{\mu\lambda}$ because it's negligible? Q3. He's also gone from $g_{00}$ to $h_{00}$ . Is that because the rate of change of $g_{00}=\eta_{00}+h_{00}$ wrt $\lambda$ (I know this is a clumsy way of stating it - sorry) equals $h_{00}$ because $\eta_{00}$ doesn't change wrt to anything? Thank you.


title: Does the speed of light vary in noninertial frames?

text: The speed of light is the same in all inertial frames. Does it change from a non-inertial frame to another? Can it be zero? If it is not constant in non-inertial frames, is it still bounded from above?


title: The appearance of volume $V$ in the Fourier series representation of a periodic cubic system

text: In the textbook _Understanding Molecular Simulation_ by Frenkel and Smit (Second Edition), the authors represent a function $f(\textbf{r})$ (which depends on the coordinates of a periodic system) as a Fourier series. I quote from page 295 of the text: > Let us consider a periodic system with a cubic box of length $L$ and volume > $V$. Any function $f(\textbf{r})$ that depends on the coordinates of our > system can be represented by a Fourier series: > > $$f(\textbf{r}) = \frac{1}{V} \sum_{\boldsymbol{\ell} = -\infty}^{\infty} > \tilde{f}(\textbf{k}) e^{i \textbf{k} \cdot \textbf{r}} \; \; \; \; > \textbf{(12.1.6)}$$ > > where $\textbf{k} = \frac{2\pi}{L}\boldsymbol{\ell}$ with $\boldsymbol{\ell} > = (\ell_x, \ell_y, \ell_z)$ are the lattice vectors in Fourier space. The > Fourier coefficients $\tilde{f}(\textbf{k})$ are calculated using > > $$\tilde{f}(\textbf{k}) = \int_V d\textbf{r} \; f(\textbf{r}) > e^{-i\textbf{k} \cdot \textbf{r}} \; \; \; \; \textbf{(12.1.7)}$$ Now, the authors use equation **(12.1.6)** to write the electric potential $\phi(\textbf{r})$ in Fourier space: $$\phi(\textbf{r}) = \frac{1}{V} \sum_{\textbf{k}} \tilde{\phi}(\textbf{k}) e^{i\textbf{k} \cdot \textbf{r}}$$ The authors write: > In Fourier space, Poisson's equation has a much simpler form. We can write > for the Poisson equation: > > $$-\nabla^2 \phi(\textbf{r}) = -\nabla^2 \left( \frac{1}{V} > \sum_{\textbf{k}} \tilde{\phi}(\textbf{k}) e^{i\textbf{k} \cdot \textbf{r}} > \right) = \frac{1}{V} \sum_{\textbf{k}} k^2 \tilde{\phi}(\textbf{k}) > e^{i\textbf{k} \cdot \textbf{r}} \; \; \; \; \textbf{(12.1.8)}$$ My question is, why is the $\frac{1}{V}$ factor present in equations **(12.1.6)** and **(12.1.8)**? What is the significance of the $\frac{1}{V}$ factor in $\phi(\textbf{r}) = \frac{1}{V} \sum_{\textbf{k}} \tilde{\phi}(\textbf{k}) e^{i\textbf{k} \cdot \textbf{r}}$? In contrast, the article on Wikipedia does not include this prefactor. I realize that that article is dealing with the general case, whereas here we are considering a system with a cubic box of volume $V$. But shouldn't the units of $\phi(\textbf{r})$ be the same as those of $\tilde{\phi}(\textbf{k})$? The $\frac{1}{V}$ seems to preclude $\phi(\textbf{r})$ and $\tilde{\phi}(\textbf{k})$ having the same units. Do you have any advice? Thanks.


title: How can one see that the Hydrogen atom has $SO(4)$ symmetry?

text: 1. For solving hydrogen atom energy level by $SO(4)$ symmetry, where does the symmetry come from?    2. How can one see it directly from the Hamiltonian?


title: The speed of light through space and its effects on black holes

text: I was curious about a couple of things that don't quite make sense to me: (and probably won't when I get the technical answer, but I'll ask it anywise.)   1. When light travels through space, is its speed effected by other bodies gravitational pull, specifically over great distances, so it LOOKS like the the universe is accelerating outward? Is the universes expansion even measured by light from distant stars?    2. Secondly, if what I've seen/heard is true about objects falling into a black hole, (that they would appear to freeze at the event horizon) wouldn't the heat and light falling into a black hole from an accretion disk make said black hole look bright?


title: What does imaginary number maps to physically?

text: I am taking undergraduate quantum mechanics currently, and the concept of an imaginary number had always troubled me. I always feel that complex numbers are more of a mathematical convenience, but apparently this is not true, it has occurred in way too many of my classes, Circuits, Control Theory and now Quantum Mechanics, and it seems that I always understand the math, but fail to grasp the concept in terms of its physical mapping. Hence my question, what does imaginary number maps to physically? Any help would be much appreciated


title: Particle inside a charged shell

text: Imagine that I have a particle of charge $q$ at the center of a spherical insulating shell of charge $Q$ and radius $R$. Both the particle and shell are initially at rest. Now I apply a force $\mathbf{F}$ to the particle which causes it to have an acceleration $d\mathbf{v}/dt$. The electric field due to the charge $q$ is given by: $$\mathbf{E} = - \mathbf{\nabla} \phi - \frac{\partial \mathbf{A}}{\partial t}.$$ Due to the symmetry of the situation the total static electric force on the shell due to the charge is zero. The $\mathbf{A}$-field due to the charge $q$ at the position of the shell is approximately given by: $$\mathbf{A} = \frac{q\mathbf{v}}{4 \pi \epsilon_0 c^2R}.$$ Therefore the charge $q$ induces an electric force $\mathbf{f_s}$ on the shell given by: $$\mathbf{f_s} = -\frac{qQ}{4 \pi \epsilon_0 c^2R}\frac{d\mathbf{v}}{dt}.$$ But the total force on the system as a whole should remain $\mathbf{F}$. Does this mean that the shell must induce a balancing electric force $\mathbf{f_s}$ back on the accelerating particle? P.S. As the particle acceleration is constant I believe that there is no radiation reaction force back on the particle (a controversial view I admit, see related link below). Thus there is no change of momentum in the EM field. Therefore the internal forces on the system of charges should balance.


title: Mysteries of the water meter

text: Recently, I got an exceptionally high bill for water consumption. I went to check my water meter, and saw that it rotates in a pace of about 200 cc per minute, even though all the faucets in my home are closed. So, I concluded that there is probably a hidden leakage somewhere in my home. But then I did the following experiment: I closed the main faucet (the one just next to the water meter). The water meter immediately stopped rotating. I waited for 1 minute, and then opened the main faucet again. At that second, the water meter rotated very fast - it made about 400 cc in a second! After that second, it returned to its previous pace of 200 cc per minute. Is there a physical explanation for this phoenomenon? (Some details that may be relevant: I live in the 2nd floor in a 3-floor house. We get our water from the urban pool, located at the top of the hill in the center of town).


title: Does antimatter curve spacetime in the opposite direction as matter?

text: According to the Dirac equation, antimatter is the negative energy solution to the following relation: $$E^2 = p^2 c^2 + m^2 c^4.$$ And according to general relativity, the Einstein tensor (which roughly represents the curvature of spacetime) is linearly dependent on (and I assume would then have the same mathematical sign as) the stress-energy tensor: $$G_{\mu \nu} = \frac{8 \pi G}{c^4}T_{\mu \nu}.$$ For antimatter, the sign of the stress-energy tensor would change, as the sign of the energy changes. Would this the sign of the Einstein tensor, causing spacetime to be curved in the opposite direction as it would be curved if normal matter with positive energy were in its place? Or does adding in the cosmological constant change things here?


title: Why does aperture size affect depth of field in photography?

text: Why do small and big holes (the aperture) in a lens change the depth of field in the image? (if you have big hole you have smaller depth of field). I wondered if I should ask this on photo.stackexchange.com, but then realize that I want optic based answer which I think will get here.


title: Effects of Electron/Ion Particle Spray on Antennae

text: I was wondering if an antenna could pick up some sort of signal if it was exposed to a spray of electrons or ions. For instance I have a patch antenna calibrated for 300 MHz. The patch antenna was placed in an environment where it could be getting sprayed by a wave of electrons followed by a wave of ions. In the antenna response I see a very strong signal voltage drop follow by a strong spike that is temporally correlated with when it would be getting sprayed, but the frequency of this signal is orders of magnitude below 300 MHz. Is this a possible effect of the particle spray, or is it more likely that something else is the cause of this low frequency signal showing up on a high frequency antenna. Any help would be GREATLY appreciated!


title: Calculate the change in the Fermi energy as the temperature is raised

text: Sodium has a volume expansion coefficient of $15 * 10^{-5} K^{-1}$. Calculate the percentage change in the fermi energy as the temperature is raised from $T = 0K$ to $T = 300K$. My attempt at the solution is below: Volume expansion coefficient is given by $\alpha _v = \frac{1}{V}\left( \frac{\partial V}{\partial T} \right)$. Then just using the definition of the fermi energy I get: $$E_f = \frac{ \hbar ^2 k_f^2}{2m} = \frac{ \hbar ^2 \left( 3 \pi^2 \frac{N}{V} \right)^{2/3}}{2m}$$ Where $N$ is the number of electrons that can contribute to conduction (They are in the conduction band I think, but their energy must be less than $E_f$ because they are inside the fermi surface so I am unsure about that.) And $V$ is some volume containing the electrons. Then it follows that: $$dE_f = \left( \frac{\partial E_f}{\partial N} \right) dN + \left( \frac{\partial E_f}{\partial V} \right) dV$$ $$=\frac{ \hbar ^2 \left( 3 \pi^2 \frac{1}{V} \right)^{2/3}}{2m}N^{-1/3}dN + \frac{ \hbar ^2 \left( 3 \pi^2 N \right)^{2/3}}{2m}V^{-5/2}dV$$ Now $dV$ I know from the thermal expansion coefficient, but am I supposed to evaluate this new term? I had the idea that each Na atom contributes 1 conduction electron, but from there I don't know how many atoms or what else. Also I find it strange that I am asked to evaluate the fermi energy at a non- zero temperature.


title: A few parity questions for simple harmonic oscillator

text: I think I understand that the solution to the Schrodinger equation for the SHO is based on the Hermite polynomials (and the Guassian function). The solution set of all _even_ Hermite polynomials are a solution to the Schrodinger equation, as well as the set of all _odd_ Hermite polynomials. Questions:   1. Is the set of all odd Hermite polynomials a solution because the potential $V(x)=1/2kx^{2}$ has even parity?    2. Does $\psi^{*}\psi$ have even or odd parity, or neither?   3. Why does $<x>=0$?  My main confusion is why the expectation value of the position results in an odd integrand, and why an odd integrand is equal to zero.


title: Why does a copper mug keep my Moscow Mule cold? (Or does it?)

text: The Moscow Mule is a delicious cocktail that is normally served ice cold in a copper mug. The general consensus among Moscow Mule drinkers is that the copper mug keeps the drink colder than a normal glass would, and my subjective experience seems to confirm this. But this doesn't make sense to me: it seems that the opposite should be true. That is, since copper is an excellent conductor of heat, shouldn't heat from my hand and from the air transfer to the beverage faster than normal, thereby quickly transforming my refreshing cold beverage into a room-temperature, ice-melt-diluted swill? What's going on here?


title: Angular momentum components as independent integrals of motion

text: I was told that in order to solve the Kepler problem (6 degrees of freedom in total) you have to proceed, step by step, to reduce those degrees of freedom using the integrals of motion. You do so until you have no one left. Now you can do so in "clever" ways (you see that the motion can happen only on a certain plane, then that the two bodies can be seen as a single body with a certain mass, and so on...) or in a more general and theoretical way (Hamilton-Jacobi). What matters is that you should find six integrals of motion. Now energy (1 scalar) plus angular momentum (3 components) plus linear momentum (3 components) make a total of 7 integrals of motion. I was told that one of the components of angular momentum is really dependent on the other components, so it cannot be considered as an independent integral of motion. My question is: what's the relationship between the components of angular momentum that makes them dependent?


title: Higgs field questions

text: The Wikipedia article about Higgs field poses some questions to me.   * The article says that the Higgs field is a "nonthermal" field, a field whose energy does not decrease as the universe expands, yet it claims that in the early universe Higgs field was stronger which led to inflation. How the field's strength could decrease then?   * If the Higgs field was stronger in early universe, why the article claims that elementary particles had no mass at early stages of the universe?   * The article claims that Higgs field is repulsive and is behind the inflation. Yet it does not interact with photons. Would not that make the photons not to follow the universe's expansion so that we would see distant stars not actually in the direction where they should be?   * The article claims that all electrons in neutron beta-decay are left-handed only for a stationary observer while for a moving observer this changes. Does it mean that one can measure his speed relative to the Higgs field by counting the right-handed electrons in neutron's decay?   * The article claims that in absence of Higgs field electrons would have no rest mass. But I cannot imagine a charged particle without rest mass because the electromagnetic field has potential energy. Does it mean that in absence of Higgs field electrons are chargeless?


title: How can electric displacement vector field is equivalent to flux density on free space

text: Wikipedia states that > "In free space, the electric displacement field is equivalent to flux > density, a concept that lends understanding to Gauss's law". Doesn't that meant that at free space $ \vec D $ is equal to $ \vec D = {\text{Total Flux} \over \text{Area} } = \frac{ \oint_s \vec E \cdot \hat n dS}{\oint_s dS} = \epsilon_0 \vec E$


title: Can I transfer vibration with hydraulics?

text: I need to transfer vibration to 10 meters. I can do this with solid metal rod, but its making mechanical design complex. So, I am thinking to do it hydraulically (tubes can easily be bend). But, I am not confident with hydraulics for vibration transfer. Vibration is an exclusive attribute of solids. Can pressurized liquid in hydraulic device hold and transfer vibration? What would happen if I apply vibration to one end of hydraulic pump? I want to see it at molecular level, too. Mathematical model is encouraged.


title: Calculation of Einstein Equation

text: I have a 3d system with Lagrangian $$e_3^{-1} L_3 = -\frac{1}{2} R_3 + \delta_{ab} \partial_\rho q^a \partial^\rho q^b + \frac{1}{2H} V(q)$$ From this I want to calculate the Einstein equation by performing the Euler- Lagrange procedure. First of all, I move the 3d dreibein to the RHS and then I apply the E-L eqns. Using that $$ \frac{\partial e_3}{\partial g^{\mu \nu}} = \frac{1}{2} e_3 g_{\mu \nu}$$, I see that $$ \frac{\partial}{\partial g^{\mu \nu}} (e_3 R_3) = e_3 R_{\mu \nu} + \frac{1}{2} e_3 R$$ Now, I don't want a Ricci scalar in the answer and I am unsure how to get rid of it. I tried looking at the trace of the Einstein equation $$ R_{\mu \nu} - \frac{1}{2} R g_{\mu \nu} = T_{\mu \nu} \Rightarrow R-\frac{3}{2} R = T \Rightarrow -\frac{1}{2} R =T $$ and so if I can work out $$T$$ then I can avoid a Ricci scalar being in the answer but I don't know how to calculate T let alone its trace.


title: Amount of free energy in the Earth-ionosphere waveguide

text: How much free energy is there in the Earth-ionosphere waveguide resulting from the constant bombardment of lightning strikes all over the Earth, and how do you go about calculating an estimate for it?


title: Quantum mechanics on a manifold

text: In quantum mechanics the state of a free particle in three dimensional space is $L^2(\mathbb R^3)$, more accurately the projective space of that Hilbert space. Here I am ignoring internal degrees of freedom otherwise it would be $L^2(\mathbb R^3)\otimes S$, but let's say it is not that time of the month. The observables are operators in that space and the dynamics is describe by the Schrodinger equation or in other equivalent ways. If we have more particles then it is $L^2(\mathbb R^{3N})$. My question is: are there any examples where one considers a system with configuration space a general manifold $M$, instead of $\mathbb R^3$, say a system of particles (a particle) with some restrictions, so that the state space is $L^2(M)$. There might be physical reasons why this is of no interest and I would be interested to here them. What I am interested in is seeing is specific (or general) examples worked out in detail. For example a system with a given Hamiltonian, where one can explicitly find the spectrum. Or if that is too much to ask for an example where the system has very different properties from the usual case. Say a particle living on the upper half plane with the Lobachevsky geometry, may be some connection to number theory! I am aware that there is quantum field theory on curved spacetime, I am interested in quantum mechanics. **Edit:** Just a small clarification. The examples I would like to see do not have to come from real physics, they can be toy models or completely unrealistic mathematical models. Something along the lines: take your favorite manifold $M$ and pretend this is the space we live in, what can we say about QM in it. The choice of $M$ doesn't have to do anything with general relativity. As I said the upper half plane is interesting or quotients of it by interesting discrete groups or generalizations $\Gamma\backslash G(\mathbb R)/K$ or anything at all. The answers so far are interesting. Hoping to see more.


title: Do senior physicists actually conduct research?

text: Senior physicists constantly complain they spend too much time on administration, teaching, getting grants, serving in committees, peer- reviewing articles, supervising, etc. . Do senior physicists conduct research by getting their post-docs and graduate students to do all the intensive work for them?


title: Observation of cosmological topological defects

text: Are there any projects/experiments running for the observation of topological defects (domain walls, monopoles, etc)? Are there any _past/canceled_ or _future_ such projects?


title: Quantum entanglement as practical method of superluminal communication

text: As I understand it (from a lay physics perspective), quantum entanglement has been experimentally demonstrated - it is a reality. As I understand it, you can measure something like the spin of an electron and know that its entangled pair will, in that same instant, no matter where in the universe it is, have the opposite spin. This would not seem to have any utility as the foundation of a superluminal communications device. Is this true, or has it been established that is there some aspect of quantum entanglement that can ultimately lead to the development of such a device. In other words: is superluminal communication via quantum entanglement an open scientific question, has it been settled as an impossibility, or is it currently more of an engineering problem than a scientific one?


title: CERN projects on gravity

text: Recently I was reading about CERN's upgrade to work on gravitational theories. But if most of the work has been done by General Theory of Relativity than with other theories are there that need to be tested. But gravity term is used in terms of heavenly bodies than how will they simulate it using atomic level particles as both follow entirely different postulates ?


title: Could metric expansion create holes, or cavities in the fabric of spacetime?

text: Is it possible for metric expansion to create holes, or cavities in the fabric of spacetime? According to the Schwarzschild metric, the metric expansion of space around a black hole goes to infinity at the event horizon. Normally, when something expands, it thins out, so, this metric stretching seems to imply that the energy density of the vacuum goes to zero at the event horizon. In other words the implication is that the spacetime manifold comes to an end there; making the black hole an actual hole, or cutout, in the spacetime manifold. I'm definitely not an expert in these matters but I can imagine the fabric of space being pulled apart by the explosive expansion of the early universe; empty bubbles opening up, imbedded regions devoid of manifold, devoid of vacuum energy, ... true voids, like what, presumably, lies beyond our universe.


title: If layers of insulation are continuously added to a heated object, will it continue to be better insulated?

text: If you were to keep adding layers of insulation to something, like blankets to a person, would each blanket continue to improve the insulation? Or do you reach some point at which the next blanket provides no marginal increase in insulation? Links to explanations of heat or insulation for the layperson are more than welcome.


title: Interesting Math Topics Useful for Physics

text: What are some interesting, but less popular, math topics that are useful for physics that can be self-studied? Specifically, topics that might ultimately be useful in high energy theory (even if it is a while down the road). My background is: real analysis, differential equations, advanced linear algebra, complex analysis, and some functional analysis.


title: Is it possible for a sound to be louder as you move away from it?

text: I was asked a puzzling question/thought experiment: Given the source of a sound in a wide open field so acoustics do not play a role, is it possible for a sound to be louder as you move away from it. My answer was instinctively no. As you move away from sound it dissipates, so it should not be louder. The response is that, if there is another, weaker source of sound closer to you, then by walking away the source closer to you will lose strength and the source farther will "shine" through better. This doesn't feel right to me. Surely when you move away from the sound, the farther one dissipates as well as the nearer one? Is this potentially because of the inverse square law? edit: This question probably applies in a similar way to light. Not quite sure what the right tags should be for this


title: Trying to rhyme Peskin and Schroeder with Weinberg

text: This is a follow up question of this one. In the Vol 1, Weinberg derives how a unitary operator $U(\Lambda)$ acts on one-particle states, which is given by equation (2.5.2): \begin{equation} U(\Lambda)|p,\sigma\rangle = \sum_{\sigma'} C_{\sigma' \sigma}(\Lambda,p)|\Lambda p, \sigma'\rangle \end{equation} Am I correct in thinking that Peskin and Schroeder use this result to derive the Lorentz transformation on a annihilation operator of the quantized Dirac field (given by equation (3.106)): \begin{equation} U(\Lambda) a^s_{\mathbf{p}}U^{-1}(\Lambda) = \sqrt{\frac{E_{\Lambda\mathbf{p}}}{E_{\mathbf{p}}}} a_{\Lambda \mathbf{p}}^s \end{equation} However, Peskin and Schroeder are _only_ considering Lorentz transformations that do not mix states with different spin indices! In other words, the first equation I wrote becomes: \begin{equation} U(\Lambda)|p,\sigma\rangle = C|\Lambda p, \sigma\rangle \end{equation} where $C$ is normalization constant which can be determined to be $C=1$. If this is not correct, can you please let me know how they (Peskin and Schroeder) have derived equation (3.109)? I hope this question is not too confusing. Edit: for you to see my reasoning. We can now write (for $C=1$): \begin{equation} U(\Lambda)|\mathbf{p},s\rangle = \sqrt{2 E_{\mathbf{p}}} U(\Lambda)a^{s\dagger}_\mathbf{p}U^{-1}(\Lambda)|0\rangle=|\Lambda \mathbf{p},s\rangle = \sqrt{2 E_{\Lambda \mathbf{p}}} a^{s\dagger}_{\Lambda \mathbf{p}}|0\rangle \end{equation} which implies: \begin{equation} U(\Lambda)a^{s\dagger}_\mathbf{p}U^{-1}(\Lambda) = \sqrt{\frac{E_{\Lambda\mathbf{p}}}{E_{\mathbf{p}}}} a^{s\dagger}_{\Lambda \mathbf{p}} \end{equation} Subsequently, using the unitarity of $U(\Lambda)$, we can take the adjoint of the above equation: \begin{equation} U(\Lambda)a^{s}_\mathbf{p}U^{-1}(\Lambda) = \sqrt{\frac{E_{\Lambda\mathbf{p}}}{E_{\mathbf{p}}}} a^{s}_{\Lambda \mathbf{p}} \end{equation} which is equation (3.109).


title: Wave packets and the derivation of Schrodinger's equation

text: I studied in my class, that a plane progressive wave cannot be used to represent the wave nature of a particle as it is not square integrable. Also, the phase velocity can get above the value of $c$, the speed of light. A simplistic argument can be- $$v_{phase}=\frac{\omega}{k}=\frac{E}{p}=\frac{mc^2}{mv_{particle}}=\frac{c^2}{v_{particle}}$$ (using de Broglie's formula and Planck's formula) Since $v_{particle}<c$, hence $v_{phase}>c$. Thus we use not a simple wave, but instead a superposition of multiple waves, which represents a wave packet. Wave packets, as far as I know, should have a varying amplitude. While deriving Schrodinger equation, why don't we treat the amplitude $A$ as variable quantity? Am I missing something?


title: Some hints for special case of metric tensor in GR

text: Let's have metric $$ ds^2 = dt^2 - dx^2 - dy^2 - dz^2 - 2f(t - z, x, y)(dt - dz)^2. $$ I need to prove that it is an exact solution for Einstein equations in vacuum for $\partial_{x}^{2}f + \partial_{y}^{2}f = 0$. The straightforward method is obvious, but does some method especially for this metric exist? Edit. As I see, first is replacing the variables as $$ u = (t - z), \quad v = (t + z). $$


title: Index Notation with Del Operators

text: I'm having trouble with some concepts of Index Notation. (Einstein notation) If I take the divergence of curl of a vector, $\nabla \cdot (\nabla \times \vec V)$ first I do the parenthesis: $\nabla_iV_j\epsilon_{ijk}\hat e_k$ and then I apply the outer $\nabla$... and get: $\nabla_l(\nabla_iV_j\epsilon_{ijk}\hat e_k)\delta_{lk}$ I am not sure if I applied the outer $\nabla$ correctly. If I did do it correctly, however, what is my next step? I guess I just don't know the rules of index notation well enough. Can I apply the index of $\delta$ to the $\hat e$ inside the parenthesis? Or is that illegal?


title: What tools do I need to track an orbit of a mystery object?

text: A while back I was observing Uranus with my 6" dob and a moving object tracked across my field of view. I was at about 90x at the time, and the object was pretty bright but slow moving. I dropped down to about 40x and it was still visible, if faint, which allowed better tracking. I followed it for a good 5 minutes until I lost it after a chair bumped my scope. It was clearly moving in the plane of the ecliptic and in the right direction. If that were to happen to me again, what do I need to have to get a reasonable track on it? Is it just a time plot vs. background stars and my exact location, or is there something more I need?


title: How do I find the acceleration of an object at a certain velocity if I know it's mass and terminal velocity?

text: I have a mass of 85kg and a terminal velocity of 46.7m/s. I think I correctly calculated the coefficient of air resistance( _b_ ) as 17.84 because I know that the Resistance force at terminal velocity is 833.13 = b($v_t$). I need to figure the acceleration of the object when it is traveling at 30m/s and what the drag force would be at that point as well. R=bv doesn't seem to work at that point, so my _b_ must be off which doesn't really make sense to me.


title: Lagrangian of Schrodinger field

text: Usual Schrodinger lagrangian is $ i(\psi^{*}\partial_{t}\psi ) + \frac{1}{2m} \psi^{*}(\nabla^2)\psi $. It gives correct equation of motion, with conjugate momentum for $\psi^{*}$ vanishing. This lagrangian density is not real but differs from real Lagrangian density $ \frac{i}{2}(\psi^{*}\partial_{t}\psi -\psi \partial_{t}\psi^{*} ) + \frac{1}{2m} \psi^{*}(\nabla^2)\psi $ by a total derivative. My trouble is that these two lagrangian densities lead to different conjugate momenta and hence when setting equal time commutation relations, I am getting different results (a factor of 2 is causing the problem). I can rescale the fields but then my hamiltonian also changes. Then applying Heisenberg equation of motion, I don't get the operator schrodinger equation. Is it possible to work with real lagrangian density and somehow get the correct commutation relations? I would have expected two lagrangians differing by total derivative terms to give identical commutation relations (since canonical transformations preserve them). But perhaps I am making some very simple error. Unless all conjugate momenta are equivalent for two lagrangians differing by total derivatives, how does one choose the correct one? I guess same thing happens for other first order systems like Dirac lagrangian also. I am learning QFT for the first time and so I would appreciate any help! thanks a lot Ankit *Thanks a million to Qmechanic for very detailed answer. It makes perfect sense.


title: Dirac paper quoted at Wikipedia

text: From Wikipedia's http://en.wikipedia.org/wiki/Uncertainty_principle: > In 1936 Dirac offered a precise definition and derivation of the time-energy > uncertainty relation in a relativistic quantum theory of "events". The citation to the paper is missing. Can anyone provide the title of this paper and where can I find it? Thanks.


title: Do strong and weak interactions have classical force fields as their limits?

text: Electromagnetic interaction has classical electromagnetism as its classical limit. Is it possible to similarly describe strong and weak interactions classically?


title: Observation of violation of the uncertainty principle?

text: I stumbled upon this piece of news in the BBC's website http://www.bbc.co.uk/news/science-environment-19489385, discussing this paper http://prl.aps.org/abstract/PRL/v109/i10/e100404, which reports the "Violation of Heisenberg’s Measurement-Disturbance Relationship by Weak Measurements"   * What does this mean? The uncertainty principle is wrong!?   * What are weak measurements?    * All in all, what do we learn from this experiment? I don't have access to the journal, so it would be great if someone who saw and read the paper gave us some answers.


title: Do intergalactic magnetic fields imply an Open Universe?

text: According to a recent paper on the arXiv, they do. How credible is this result? The abstract says: > The detection of magnetic fields at high redshifts, and in empty > intergalactic space, support the idea that cosmic magnetism has a primordial > origin. Assuming that Maxwellian electromagnetism and general relativity > hold, and without introducing any `new' physics, we show how the observed > magnetic fields can easily survive cosmological evolution from the > inflationary era in a marginally open Friedmann universe but fail to do so, > by a very wide margin, in a flat or a marginally closed universe. Magnetic > fields evolve very differently in open and closed Friedmann models. The > existence of significant magnetic fields in the universe today, that require > primordial seeding, may therefore provide strong evidence that the universe > is marginally open and not marginally closed.


title: Does anyone know the difference and relation between $k\cdot p$ method and tight binding (TB) method?

text: Among the methods of calculating energy bands for crystals, first-principles method is the most accurate. Besides first principles, two commonly used modeling methods are the $k\cdot p$ method and tight binding (TB) method. They can both give a Hamiltonina matrix of wave vector $k$, i.e. $H(k)$. I want to know the detaild difference and relation between $k\cdot p$ and TB method, especiall their relations. Does anyone know? Are there any books or literatures to cover it? I know TB can be used to calculate the energy bands in the full Brillouin zone (BZ), while $k\cdot p$ generally used for neighbourhood of band edges. However, I know an article which uses $k\cdot p$ to calculate the bands in full BZ [Phys. Rev. 142, 530 (1966)]. Is $k\cdot p$ fully equivalent to TB method?


title: Explanation of equation that shows a failed approach to relativize Schrodinger equation

text: I'm reading the Wikipedia page for the Dirac equation: > $\rho=\phi^*\phi\,$ > > ...... > > $J = -\frac{i\hbar}{2m}(\phi^*\nabla\phi - \phi\nabla\phi^*)$ > > with the conservation of probability current and density following from the > Schrödinger equation: > > $\nabla\cdot J + \frac{\partial\rho}{\partial t} = 0.$ > > The fact that the density is positive definite and convected according to > this continuity equation, implies that we may integrate the density over a > certain domain and set the total to 1, and this condition will be maintained > by the conservation law. A proper relativistic theory with a probability > density current must also share this feature. Now, if we wish to maintain > the notion of a convected density, then we must generalize the Schrödinger > expression of the density and current so that the space and time derivatives > again enter symmetrically in relation to the scalar wave function. We are > allowed to keep the Schrödinger expression for the current, but must replace > by probability density by the symmetrically formed expression > > $\rho = \frac{i\hbar}{2m}(\psi^*\partial_t\psi - \psi\partial_t\psi^*).$ > > which now becomes the 4th component of a space-time vector, and the entire > 4-current density has the relativistically covariant expression > > $J^\mu = \frac{i\hbar}{2m}(\psi^*\partial^\mu\psi - \psi\partial^\mu\psi^*)$ > > The continuity equation is as before. Everything is compatible with > relativity now, but we see immediately that the expression for the density > is no longer positive definite - the initial values of both ψ and ∂tψ may be > freely chosen, and the density may thus become negative, something that is > impossible for a legitimate probability density. Thus we cannot get a simple > generalization of the Schrödinger equation under the naive assumption that > the wave function is a relativistic scalar, and the equation it satisfies, > second order in time. I am not sure how one gets a new $\rho$ and $J^\mu$. How does one do to derive these two? And can anyone show me why the expression for density not positive definite?


title: Lagrangian dynamics with initial conditions: motion of free particle

text: I am very new to Lagrangian dynamics so I am trying to get my head around the practical usage. So far on here all I could find were proofs and they did not make much sense to me, especially when time was involved in a problem. Let's take a simple example. I have a free particle, and it can be represented in Cartesian coordinates at time $t=0$: \begin{align} \mathbf{r}(0) & = x(0)\mathbf{e_x} + y(0)\mathbf{e_y}, \\\ \mathbf{v}(0) & = v_x(0)\mathbf{e_x} + v_y(0)\mathbf{e_y}. \end{align} A standard thing to want to know is the displacement $\mathbf{r}$ at a later time $t$. This is simple enough with Newtonian mechanics. How does Lagrangian mechanics do this? I know I need to find the kinetic energy at some point - is it just the kinetic energy at time $\small{t=0}$? How does a formula for energy get me the path of the particle for all times? I am not so much interested in the solution to this particular problem as seeing how Lagrangian mechanics can replace Newtonian mechanics and give me the same concrete results.


title: Gauss' law giving zero field where field is not zero?

text: Two plastic sheets with charged densities as shown: ![enter image description here](http://i.stack.imgur.com/DlpDA.jpg) I'm trying to find the field at $B$. I obtained the correct answer by adding up the fields created by each charge density. But I realized that since the field is uniform in the region between the two sheets, I should be able to make a Gaussian surface between the sheets with the shape of a box and with one edge at $B$. Thus, the flux would be $$\int{E \cdot dA} ~=~ EA ~=~ \frac{q_{encl}}{\epsilon_{0}}~=~0.$$ Because there would be no charged enclosed inside the surface. However, that means that $EA$ is $0$ (Note the integral reduced to $EA$ because $E$ is uniform). Since $A$ is not zero (it is the area of two sides of the box), this means that $E$ must be zero. However, $E$ is not zero there, as you can see by adding $\frac{\sigma}{2\epsilon_0}$ for each charge density. What am I doing wrong when using Gauss' law?


title: Small confusion related to leaving of electrons from atoms

text: Picking up a circuit board from a table, or our clothing rubbing as we walk, sit and work, are all examples of movement that can create static charge. One object, or surface, picks up additional electrons from the surface that it is separated from, and becomes negatively charged. The surface that had given up the electrons then becomes positively charged. As electrons transfer, the absence or surplus of electrons creates an electrical field known as static electricity. _**Okay so the thing giving up electrons should get destroyed in the long run? I mean from where the lost electrons come back from?_** I know this sounds insane but really? if all electrons leave the atom, then atom positive charge, then slowly other atoms take way the protons also, then the nucleus is ASHAMED is attempts sucide! please help really confused.


title: Creation of Nuclear Isomers

text: As I understand it, if a nucleus is excited with energy exceeding its ground state, it releases energy via gamma radiation. An example would be technitium 99m, a medical tracer with a 6 hour half life and 140 keV gamma radiation. What are the characteristics of ground state nuclei allowing them the ability to be converted to metastable isomers? Can any element be converted to a metastable isomer?


title: Mixed state after measurement

text: I'm looking at Section 2.4.1 of Nielsen and Chuang's _Quantum Computation and Quantum Information_ were they derive the density operator versions of the evolution and measurement postulates of quantum mechanics and something is bugging me. Let $$\\{(p(i), | x_i \rangle) \colon i=1,2,...,n\\}$$ be an ensemble and suppose you perform a measurement on this ensemble that results in outcome $m$. Then the post-measurement ensemble is $$\\{(p(i|m), | x'_i \rangle) \colon i=1,2,...,n\\}$$ where $p(i|m)$ is the probability that the state of the system was originally $| x_i \rangle$ given that the measurement outcome was $m$. It seems natural enough to use $p(i|m)$ for the probabilities of the post-measurement ensemble. However, mathematically, I do not see why that should be the case. My question then is this: Given only the definition of what an ensemble is and the state vector version of the postulates of QM, is there a way to derive the rules to compute the post-measurement probabilities of an ensemble?


title: Conservation of Hamiltonian vs Conservation of Energy

text: What is the difference between conservation of the Hamiltonian and conservation of energy?


title: Entropy of two expanding and mixing ideal gases

text: I am currently stuck with the following problem:                _________________________         _________________________     |                 |       |    \  |                         |      |        V1       |   V2  | ====> |        V1+V2            |     |                 |       |    /  |                         |           |_________________|_______|       |_________________________|       In a box there are two different gases in two compartments seperated by a solid wall. The Temperature $T$ of both gases is the same and both gases consist of $1mol$ particles. Therefore on the left side of the box, you have a volume of $V_1$ with a pressure of $p_1$ and on the right hand side we have volume $V_2$ and pressure $p_2$. No we remove the separating wall. Lets first assume that we have to different gases on the left and on the right side. My goal is to calculate the change of entropy which occurs due to the removal of the wall. My first Ansatz was: $\Delta S_1 = nR\ln(\frac{V_1+V_2}{V_1})$ $\Delta S_2 = nR\ln(\frac{V_1+V_2}{V_2})$ Where $\Delta S$ is the change of entropy of the system, $n$ the number of mols and $R$ the universal gas constant. The total change of entropy would now be the sum of both: $\Delta S_{tot} = S_1 + S_2 = nR\ln(\frac{(V_1+V_2)^2}{V_1V_2})$ After a while I thought about the problem in a different way: First I move the wall isothermally so that $V_1=V_2$. The entropy change due to the movement would be: $\Delta S_{mov1} = nR\ln(\frac{V_1+V_2}{2V_1})$ $\Delta S_{mov2} = nR\ln(\frac{V_1+V_2}{2V_2})$ $\Delta S_{mov} = S_{mov1} + S_{mov2} = nR\ln(\frac{(V_1+V_2)^2}{4V_1V_2})$ Now i remove the wall and get a entropy change due to the mixing of the gasses: $\Delta S_{rem} = R(n_1 ln(\frac{n_1+n_2}{n_1}) + n_2 \ln(\frac{n_1+n_2}{n_2})) = 2nR\ln(2)$ so $\Delta S_{tot} = \Delta S_{rem} + \Delta S_{mov} = nR\ln(\frac{(V_1+V_2)^2}{4V_1V_2}) + 2nR\ln(2)$ which is clearly different from the result I got from my approach in the first Ansatz. However since entropy is a state variable this should not happen. No I am stuck. I tend to the second result, but I am very far from sure. It would be great if you could explain which ansatz is correct (if any^^). Moreover if I had identical gases in $V_1$ and $V_2$, how would the entropy behave? Since the pressure changes after the removal of the wall, I would say that the entropy also changes. Just like in my second Ansatz, however without the mixing term. Is this correct? ||~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ EDIT: Oh, dear. I just realized that $2nRln(2)$ can be written as $nRln(4)$. Therefore $nR\ln(\frac{(V_1+V_2)^2}{4V_1V_2}) + 2nR\ln(2) = nR\ln(\frac{(V_1+V_2)^2}{V_1V_2})$ so both ansatzes give the same result. Since this is cleared out, my second question remains. In case of two identical gases does the entropy change according to my second ansatz (without the mixing entropy)? Thanks in advance ftiaronsem


title: How does Hubble's constant affect the Earth's orbit

text: If Hubble's constant is $2.33 \times 10^{-18} \text{ s}^{-1}$ and the earth orbits the sun with average distance of 150 million kilometers; Does that mean the earth's orbital radius increases approximately $11\text{ m}/\text{year}$? Does the earth's angular momentum change? If so, where does the torque come from? If the angular momentum doesn't change, does the earth's orbital velocity (length of a year) change? If so, where does the lost kinetic energy go? * * * Aside: the 11 meters per year figure comes from Hubble expansion of space the distance of the earth's orbital radius integrated over an entire year. $$(2.33 \times 10^{-18}\text{ s}^{-1}) (1.5 \times 10^{11} \text{ m}) (3.15 \times 10^7 \text{ s}/\text{year}) = 11 \text{ m}/\text{year}$$


title: Why do power lines buzz?

text: When near high tension power lines, particularly after a good rain, the lines themselves emit a buzzing noise. A similar noise can be heard coming out of the electric meters attached to my apartment. I've heard before that this is supposedly from the 60Hz AC current that's running through the lines -- namely, that the buzz is the same 60Hz which is in the lines. I'm skeptical of this though for a couple of reasons:   1. I don't see any reason the change in electricity would somehow be audible.   2. The noise subjectively sounds relatively high pitch. 60Hz would sound extremely low pitched -- it's near the base of human hearing of 20Hz (typical). What is the actual cause of that buzzing? EDIT: I just spent some time playing with a tone generator and the noise I hear from these things sounds closest to 120Hz using a square or triangle wave. (Oddly, not a sine wave, as I would have expected) Perhaps that helps?


title: Complete description of two electrons with spin

text: We have two electrons described by the wave function $\phi(\vec{x}_1,\vec{x}_2,s_1,s_2;t)$ where $\left| \phi(t)\right\rangle$ is the state vector and $(\vec{x}_1,\vec{x}_2,s_1,s_2)$ is the index of each dimension in the vector. $\vec{x}_1,\vec{x}_2\in\mathbb{R}^3$. $s_1,s_2\in\\{-{1\over2}\hbar,{1\over2}\hbar\\}$ are the spins of each electron measured along some axis in space (but lets just say both are measured along the $z$ axis). What are the first-quantization equation(s) that evolve the system? I am expecting the Schrodinger equation where the Hamiltonian $\hat{H} = \hat{T} + \hat{V}_{Coulomb} + \hat{V}_{External}$, $\hat{V}_{Coulomb}$ is the classical (first quantization) Coulomb potential, and $\hat{V}_{External}$ is potential due to external forces. But I do not see how spin fits into this picture. How are spin and position states are affected when electrons interact via electromagnetism? I do not want to get into quantum field theory so I am hoping that a complete description of quantum particles in classical fields exists. Please include the equations (probably Schrodinger) in your answer and define anything that I have have not already so that there is no ambiguity.


title: Would a closed field of gravity neccesarily lead to paradoxes?

text: I've asked wether artificial gravity, as seen in some SF-Movies, would violate known laws of physics. To recap, my idea of an Artificial Gravity (AG) system was like this:   * A Device that creates an 'artificial gravity' (AG) inside a Box, with these properties:   * the system does not use inertial forces (like centrifugal force) or a huge mass to create AG   * the AG 'feels' just like gravity/inertia over most of the volume of the box   * the Device, the Box and its content are still subject to any gravitational forces that may be present, but the AG force adds to other gravity/inertia forces, net force may be zero   * Matter, Energy and Information can leave and enter the box there is a smooth but strong gradient in the AG at the walls of the box -   * conservation of energy still holds: if the AG does work, this energy has to be supplied to the device in some way, vice versa for work done against the AG   * conservation of momentum still holds: for every force due to the AG on an object, there is a corresponding counteracting force on the device Basiclly, I tried to rule out all the obvious ways such an AG would violate laws of physics (or could be used to costruct another device that does so), and see if one can find a less obvious way und thus rule out the possibility of AG. If you want to talk about the original question, please head over (and please do!). I may have thought of a way a device like this could create an impossible situation, but I don't understand enough relativity to know wether I'm right. Suppose you take a number of boxes as describet, and cut open two sides. Then you arrange the boxes in a rough circle or polygon. The AG in each box is adjusted so that the net 'gravity' (AG + gravity at the place the box is) result in a sort of circular, homogenous field of gravity. The devices are anchored somwhere to account for any momentum. An observer placed inside the sytem would not 'fall' freely around the circle, but would bump into the wall after the first bend since there would be no centripetal force present. However, we can build a circular rail through the boxes and attch our oberver to that, then he could travel round and round while constantly getting faster. Whee! Now, I think there would be no meaningful way to define gravitational potential in such a situation, so we could not tell how fast the clock of the observer on our merry go round is going. So the device would create a prardoxical situation is thus impossible. On the other hand, either our observer is in free fall or we know the accelration she is subject to (either from the rail, or from bumping into the walls of the boxes). So her watch will behave like any other wach subject to the same acceleration. I tend to believe the second interpretation is correct, our AG would not create an impossible situation, but since I don't realy undersant gravity, relativity and all the rest I thought I ask. p.s. I've thought about posting this thought experiment as an answer to my first question, but since I'm totally unsure I want to exp.ose my little thought experiment to more scrutiny. p.p.s Why is there no 'thought experiment' tag?


title: Can a hierarchy of fixed points potentially be used to describe a kinetic energy spectrum which is composed of multiple scale invariant subranges?

text: Making use of a nonequilibrium functional renormalization group (Berges and Mesterhazy, 2012) are able to investigate a whole hierarchy of fixed points that explain the successive evolution of a system from a nonequilibrium state to equilibrium. Can such an appoach be applied to investigate the transition between two inertial subranges in atmospheric turbulence too? I am asking about this, since the illustration of a dual cascade in fugure 1 of the paper ![enter image description here](http://i.stack.imgur.com/i8AAT.png) Looks quite similar to the TKE spectra one observes in atmospheric turbulence. ![enter image description here](http://i.stack.imgur.com/w5g44.jpg) In both cases, the different regimes are associated with with a stationary transport of conserved quanitities. So, is this similarity I percieve only superficial and misguided, or should it in principle be possible for example to describe the kinks in the Nastrom-Gage spectra by such dual cascades too? And if so, has there already some work on this be done or is under way?


title: Helicity Representation of Massive Spinor

text: For massless spinors case we can decompose momentum into Weyl sub-parts as $$p = \lambda_{a}\tilde \lambda_{\dot a}.$$ But for the case of massive fermions can I do something like this? Decompose them into Weyl subparts with some additional terms? If so, how? Why do I need it? I am performing a twistor transform for the equation of the process $q\bar q \to gg$ so I have to write the amplitude and the 4d delta function of the momentum and then Fourier transform the $\lambda$ and $\tilde \lambda$ separately.


title: Gravity Interaction versus Gravity Well

text: This question may be misguided, but bear with me. Is there such a thing as gravitational 'invisibility', and if so, is it comparable to optical invisibility (photons/gravitons 'continue' in their initial direction). What should I expect from such an object when it exist in a gravity well (like on Earth)? Weightlessness? Deformation?


What determines how much electrical charge an object can hold? Does increase voltage force more electrical charge to be store in an object (Van de Graaff generator), since electric field increase as voltage increase. I don't think it is about relative permittivity in dielectric material because it just creates a bipolar.


title: Power of Shockwaves

text: My question is about shockwaves and their power when they are created/how do they lose their power? Let's say that we have ground 0 with 10 grams of TATP on it. The detonation velocity of TATP is 5,300m/s. How can we calculate the pressure it will make and the shockwave's power when it is detonated. Also,how can we calculate how fast the shockwave will lose power?


title: Accuracy of various optical instruments

text: _I understand that this may not be the type of question allowed here, but I'm not sure. Feel free to close this if you feel that it shouldn't be here_ I'm planning on carrying out a certain set of experiments regarding quantum mechanics and double-slit setups in half a year or so. The experiments will be similar to the quantum eraser. The issue is, I don't know what sort of experiments are feasible. I'd like someone who has experience with such optical instruments to explain the avaialiblity, cost, and feasability of the following instruments. Also if they can be found in a lab of any institution that deals with QM. _Whenever I ask for a value, I only want a ballpark figure or an order of magnitude_   1. Single-photon detectors: I've seen these referred to as "click-detectors". How accurate are they? Do they measure every single photon that eneters or only a significant fraction of them?   2. One-photon-at-a-time emitters: How accurate?   3. Fiber optics:      * If I send a single photon through a fiber optic of some length, how much percentage of the time will it be lost? If there are different types of fiber optics, please let me know which one is the best for this.     * If I have a pair of parallel-running fiber optics, how coherent can they keep two coherent sources? For example, if set up a normal YDSE, feed fiber optics into the slits, and feed the source into the fiber optics some $s$ meters away, will I observe a fringe pattern?    4. Screen:What sort of screens do we get nowadays for YDSEs? Can they detect single photons with xyz accuracy? How many photons are required on a screen to see a distinct fringe pattern? In the quantum eraser, they used a detector on a tractor. Any reason for doing this instead of using a screen? Are there any screens which can detect single photons, pinpoint their location with xyz accuracy, and feed the data to a computer?   5. Environment: While doing a single-photon YDSE, what sort of environment is required? After all, we need to block all external photons. Do we need to sheath the YDSE with something? If so, how will we view the results without tampering with them (taking a photographic screen out to view it can easily damage it)?   6. Mirrors/lenses: How good are mirrors for directing single photons? Do they eat up xyz% of the photons? Do they tamper with coherence?   7. Beam splitters: Same situation as mirrors.   8. Beta-barium borate crystal (used in the quantum eraser): Same situation as splitters and mirrors.  An example setup would be where I use a beam splitter to create two possible paths for a single photon. These paths are directed to a YDSE, in such a way that they cannot interfere till they reach the slits. One path corresponds to one slit. I've though of fiber optics so that the paths are kept from interfering with each other. A thin opaque barrier might also do the trick. For this experiment to work, basically, it should be able to build up an interference pattern from one photon emitted at a time, with the interference happening ONLY in the YDSE (the reason for this is that I want to do something else involving the BBO crystal in between).


title: How to initiate fusion explosion without a fission trigger

text: I'd like to know what theoretical methods for initiating a fusion detonation without a fission trigger exist, and which would be the most practical in the near future. I'd also like to know how small A fusion bomb might be practical under those circumstances. For instance I think you could probably initiate a fusion detonation if you collide 2 suitable masses into each other at a couple hundred miles per second, but that wouldn't be practical from a weapons standpoint. I'm thinking maybe laser initiated fusion or some kind of explosive magnetic compression or combinations or a host of other things I probably haven't even considered.


title: Young's double-slit experiment with detectors

text: _Related:Accuracy of various optical instruments_ In many books, it's written that knowing which slit a photon passes through (by placing a detector before the slit) in a Young's double-slit setup beforehand destroys the interference (producing two bright fringes). My question is, does this require there to be only one photon in the double slit experiment at a time? The one-photon interference is explained by there being "half a photon" coming from each slit, and these interfere. If we know which slit it comes through, then it becomes a full photon in one slit and it can't interfere with itself. But can _other_ photons interfere with it? I'm quite sure that light can be either a particle or a wave in such experiments, never both. Aside from this, how was the photon detected without absorbing it? If I keep a detector behind a slit, yes, it will detect all the photons, but it won't let any past it to the main apparatus. Unless the detector was something else.. UPDATE: A clearer version of this question:   1. If you know which slit a photon went through, its wavefunction is collapsed. One can say that it ceases to be wave like. This collapsed photon cannot interfere with itself   2. Can it interfere with others? It is, after all, not really a wave anymore (it is a wave, but collapsed to a point).   3. In a normal double slit experiment (beam of light, no doodads), are photons interfering with themselves, each other, or both? (See comment: Is Time Significant in the Double Slit Experiment)


title: Calculating Kramers-Kronig using Mathematica

text: First of all, I know there is a Mathematica group in beta, but I don't think the problem of the following is directly a Mathematica issue. I am trying to calculate the change of the refractive index dnF from the change of absorption daF. Note that l,li,ln is in nm. The form of the resulting curve is fine (which is a sign for me that there is no Problem with e.g. PrecionGoals), but the order of magnitude is lower by 13 orders of magnitude than the expected one. The MathLab script von Lucarini et. al. "KK Relation in Optical Materials Research" gives the correct values. The formula used here is from the same book and matches the one in German Wikipedia. Does anyone see my mistake?               c = 300000000;          daF[l_] = 500 * 0.28 Exp[-((l - 500)/90)^2];          dnFpoints = Table[         {             ln,             c/Pi NIntegrate[                 daF[li] / ((2 Pi c 10^9 /li)^2 - (2 Pi c 10^9 / ln)^2),                 {li, 200, 800},                 Method -> {"PrincipalValue"},                 Exclusions -> ((2 Pi c 10^9 /li)^2 - (2 Pi c 10^9 / ln)^2) == 0             ]         },         {ln, 300, 600}     ];          ListPlot[dnFpoints, Filling -> Axis]


title: Electrical machines: electrical and mechanical properties of steels and cast iron

text: I would like to know a good source of information for the common metals (steels and cast iron). Information that I need are basically the mass density and the electrical conductivity or resistivity. There are just too much source of information on the web and one doesn't know which source is accurate.


title: Is resonating valence bond (RVB) states long-range entangled?

text: Quantum liquid is at the core of condensed matter theory study, examples include superfluid in Bose Hubbard model, quantum spin liquid around the RK point of a quantum dimer model, string-net condensation in a quantum many- string system, etc. Recently it was realized that quantum entanglement may be the key to differentiate these liquids. For example, the Bose superfluid is simply a direct product state with no entanglement, while string-net condensate is claimed to be long-range entangled. To me their differences would be that the former is a condensation of point objects, and the latter is a condensation of string-like objects. I would conclude from this observation that the condensation of extended objects leads to long-range entanglement. It seems to me that it is the extended objects that pass on the quantum entanglement in the system. So when I come to the RVB state in the quantum dimer model, I think of condensing the dimers -- stick-like objects. Sticks are more extended than points but less than strings. So I guess the dimer condensation (or the RVB state) would have an entanglement between the superfluid and the string-net condensation. I can see that the RVB state can not be written as a direct product state, so there is definitely some entanglement there. My question is: is RVB state long-range or short-range entangled?


title: Solving Schrödinger's equation with Crank-Nicolson method

text: I am trying to numerically solve Schrödinger's equation with Cayley's expansion ($\hbar=1$) $$\psi(x,t+\Delta t)=e^{-i H\Delta t}\psi(x,t)\approx\frac{1-\frac{1}{2}i H\Delta t}{1+\frac{1}{2}i H\Delta t}\psi(x,t)$$ and second order finite difference approximation for space derivative $\psi''(x)\approx\frac{\psi_{j+1}^n-2\psi_j^n+\psi_{j-1}^n}{\Delta x^2}$, as described in Numerical Recipes. The Hamiltonian is that of one-dimensional harmonic oscillator ($m=1$): $H=\frac{\partial^2}{\partial x^2}+\frac{1}{2}kx^2$. What i end up computing is this system of linear equations with a 3-band matrix at every time step: $$(1+\frac{1}{2}iH\Delta t)\psi_j^{n+1}=(1+\frac{1}{2}iH\Delta t)^*\psi_j^n$$ or explicitly $${\boldsymbol A}{\boldsymbol \Psi}^{n+1}={\boldsymbol A}^*{\boldsymbol \Psi}^n\;,$$ where the elements on the main diagonal of ${\boldsymbol A}$ are $d_j=1+\frac{i\Delta t}{2m\Delta x^2}+\frac{i\Delta t}{4}x_j^2$ and the elements on the upper and lower diagonals are $a=-\frac{i\Delta t}{4m\Delta x^2}$. For the initial condition i choose the coherent state, for which there is a known analytical solution. While trying to propagate the wavefunction over few periods of oscillation, the wavefunction gets distorted. When timestep is smaller, the distortions appear later. I am wondering what is the reason for this process, if it has to do with the Courant condition and if there is a known relation between the size of timestep and start of these distortions. * * * Here is a video of propagation.


title: Get an intuition for pressure values

text: I am currently working with pressure differences in the milibar range. However I do not have a good feeling for pressure differences. In every day life:   * Where does a pressure of say 100 mbar over atmospheric pressure occur?   * When I blow up my mouth, what is the pressure I get?   * When I blow up an air balloon, what is the pressure inside?


title: People eating and gaining energy

text: Let's say we have 100 grams of chocolate which has 571 kilocalories (so it's about 2.4 megajoules) and we have barbell which weights 100 kilograms and need to lift it to 0.5 meters height. So how many times you can lift it eating only 100 grams of chocolate? Let's use potential formula $$E=mgh$$ n – how many times you lift $$E=nmgh$$ $$n=\frac{E}{mgh}$$ $$n=\frac{2.4\cdot 10^6J}{100kg\cdot10\frac{m}{s^2}\cdot0.5m}=4800$$ So I get 4800 times, but I doubt it's true. So where and how many (approximately) our gained energy disappears, I mean becomes other kind of energy?


title: Tunnelling through a Dirac potential barrier

text: I am reading a QM book by Griffiths, which says it is possible for wave particle to tunnel through a barrier formulated by a Dirac function. This function is known to peak at infinity and also infinitesimally narrow. Is there anything special about the Dirac function for this to happen? If I change the function to a infinitely tall rectangular function with a finite width, would it still be able to tunnel through this barrier If we try to solve the Time independent Schrodinger equation, I thought the exponential would fall off very quickly since it has a large potential. Note that gamma is a function of Energy and potential. If we use exponential gamma as trial function.


title: Two formulas for a particle's acceleration

text: While on a class my teacher was taking about particle's motion in space. At some point she said the following: Consider that the particle's path is described by a curve in space defined by the parametric equations $x^i=x^i(s)$, where $x^i$ are the coordinates in space and $s$ the length along the curve. The particles velocity is then given by $$\tag{1} v^i=\frac{dx^i}{dt}=\frac{ds}{dt}\frac{dx^i}{ds}=v~e^i,$$ where $v=ds/dt$ is the particle's speed and $e^i=dx^i/ds$ is a unit vector tangent to the curve. Then she wrote that the particle's acceleration would be given by: $$\tag{2} a^i=\frac{dv^i}{dt}=\frac{ds}{dt}\frac{d(v~e^i)}{ds}=v^2 \left(\frac{de^i}{ds}\right)+e^iv\frac{dv}{ds}.$$ Then I started wandering, what is the relation between this formula for the acceleration and then one using the covariant derivative: $$\tag{3} a^i=v^j~\nabla_j~v^i=v^j\left(\partial_j v^i+\Gamma^i_{k j}v^k \right)=\dot{v}^i+\Gamma^i_{k j}~v^k v^j~,$$ where $\Gamma^i_{k j}$ are the Chritoffel symbols of second kind? I know I should have asked her right away but she left pretty quickly from class. Can anyone help me? I can't seem to prove that the formulas are equivalent.


title: In Path Integrals, lagrangian or hamiltonian are fundamental?

text: When studying path-integrals one question arose to my mind... Which presentation is more fundamental to calculate the propagator? The one based on the Hamiltonian (phase space)? $$K(B|A) = \int \mathcal{D}[p]\mathcal{D}[q] \exp \\{ \frac{i}{\hbar} \int dt [ p \dot q - H(p,q) ] \\} $$ or the one based on the lagrangian (configurational space)? $$K(B|A) = \int \mathcal{D}[q] \exp \\{ \frac{i}{\hbar} \int dt L \\} $$ Reading Feynman thesis we see he affirming that "[...] a method of formulating a quantum analog of systems for which no Hamiltonian, but rather a principle of least action, exists has been worked out. It is a description of this method which constitutes this thesis." He seems to take lagrangian form as more fundamental. Other authors, like Hatfield or Swanson, seems to take the phase space form as more fundamental. They see the other form as a special case where the $p$ dependence is quadratic. So, this is my question.   Which one is more trustful? There is any example where one view is privileged?


title: In Schrödinger equation, can we get $\Psi$ if we know $\varphi(n)$?

text: Let $A$ be a mechanical quantity and we know its eigenfunction $\varphi(n)$. Can we get its wave function $\Psi$, if we measure $A$ for many (maybe infinite) times at the same time?


title: Why do non-stick frying pans work?

text: Modern non-stick frying pans use a mixture of titanium and ceramic that is sandblasted onto the pan surface, and then fired to 2,000 °C (according to Wikipedia). Can anyone explain (at the molecular level), Why cooking materials do not stick to these surfaces?, a hydrophobic effect would work if the food was mainly water but these surfaces are not affected by oily materials either.


title: Radio communication and antennas

text: I'm trying to understand how an antenna is able to understand, but have a few quesions:   1. I understand that for an antenna to distinguish between 2 (or more) signals, they have to be in different frequencies. This means that, for example, every mobile phone transmits in a unique frequency? otherwise the signals would mix in the antenna..   2. So if I want to disrupt a signal that an antenna receives, I need to transmit some sort of noise in the same frequency (preferably the inverse of the received wave), and then the antenna couldn't understand that signal?    3. Also, a third question (not related to the last two): what is the tolerance of an antenna? How sensitive it can get? for example, how far away from a transmitter an antenna could receive the sent message (e.g. in communication between earth and a satellite).


title: Determine when a light is going into the object

text: Please look at this image: (http://mypages.iit.edu/~smart/acadyear/refract.gif) ![enter image description here](http://i.stack.imgur.com/zcuBA.gif) How can I determine when light is going into the object? Actually, if there is, what is the simplest way to determine whether light is going into the object or not? NOTE: I could not find any meaningful title. Sorry.


title: Translations of field operators in QFT

text: A question in the book _QFT_ of Srednicki: This concerns the relativistic QFT generalization $$\tag{2.21} {{e}^{-i\hat{P}x/\hbar}}\psi (0){{e}^{i\hat{P}x/\hbar}}~=~\psi (x)$$ of the formula $$\tag{2.20} {{e}^{i\hat{H}t/\hbar}}\psi (\vec{x},0){{e}^{-i\hat{H}t/\hbar}}~=~\psi (\vec{x},t)$$ from non-relativistic QM. How can we prove this two formula just use Poincare algebra? $\psi (x)$ is an field operator. About the second formula we can use Schrodinger equation to prove. I think there is a proof just using the Poincare algebra. I just have no idea about this.


title: What's the relation between output voltage and time to boil water given the same kettle?

text: > An electric kettle rated 220V, 2000W needed 10 minutes to boil water when it > is half filled with water in Singapore where the output voltage is 220V. > Estimate the amount of time needed to do the same task if the kettle was > brought to the USA where the output voltage is 110V. Options:               1. 5 minutes     2. 10 minutes     3. 20 minutes     4. 40 minutes      My answer to this question is Option **3** (20 minutes) because I thought that since output voltage is the USA is 2 times less than that of Singapore, it would take twice as long to do the same task of boiling water half filled. However, the answer key was Option **4** (40 minutes). Can anyone explain how 40 minutes was derived in the question above? P.S. I don't quite understand what it is meant by " An electric kettle rated 220V, 2000W" ??? I get that 2000W means the kettle uses 2000J/s and V means work done by kettle/charge. But what is the difference between the 2 (power and charge of the kettle)? And how does the V of kettle differ from output voltage? Thank you very much!


title: Parity of a decay

text: If a particle of unknown intrinsic parity decays into 2 particles each with negative intrinsic parity, does that necessarily imply that the original particle also has negative parity?


title: Simple home pressure experiment: how to understand the dependence on the glass opening?

text: Suppose you put a drinking glass under water such that it is completely filled with water. Then hold it upside down and slowly raise the drinking glass out of the water. The water in the drinking glass will stay in the glass until the glass is almost completely above the water. This is because in the glass there is only water and no air such that atmospheric pressure causes the water to stay in the glass. However why isn't his the case anymore if the class is completely above the water surface? If you do this with thin a straw (with is closed on one side) instead of a drinking glass, the water will remain inside. So it seems to depend on the area of the glass opening. Why is that the case? How can one describe this difference with formulas?


title: How does a stronger magnet affect the MRI image quality?

text: In which ways is a stronger magnet better for magnetic resonance imaging? I read that: > The field strength of the magnet will influence the quality of the MR image > regarding chemical shift artifacts, the signal to noise ratio (SNR), motion > sensitivity and susceptibility artifacts. but I don't understand why. I suppose the dephasing is quicker, reducing the transverse net magnetization more quickly. Does this go for the longitudinal net magnetization too? How would this reduce motion sensitivity and SNR?


title: Classical mechanics and the speed of a train-mosquito collision, when perfectly rigid bodies

text: This is all under the assumption that they are perfectly rigid bodies: A train is moving at 300m/s. A mosquito is moving directly towards it, head- on, at 4m/s. When the mosquito and the train collide, the mosquito is at 0m/s, and then changes direction, moving at 300m/s in the direction of the train (because it's stuck to the windscreen). Surely this says that, in theory, because there is a period of time where the mosquito is at 0m/s, the train must also be at 0m/s. But the train surely doesn't stop moving, and a mosquito certainly couldn't stop a train, even for the smallest amount of time.   1. Is this really what theory predicts? If not, where am I going wrong?   2. Is the theory flawed with this almost paradoxical situation?


title: A foundational question about entanglement

text: The recent Nature article http://www.nature.com/news/data-teleportation-the- quantum-space-race-1.11958 prompts me to ask this question, which is of the same tenor as that asked at the recent Perimeter Institute meeting attended by Zeilinger et al., i.e. "Do the pairs somehow communicate though some still- unknown information channel?" It seems to me that entanglement could be described as behaving as if in some sense the coordinate separation is described by an extra degree of freedom (dimension) in which either a) the two entangled components are co-located in the extra dimension, allowing instantaneous communication; or b) in the extra dimension, the speed of light is no longer a limitation, and so co-location is not a requirement. ... or some mixture of the two. This could all be rephrased in terms of the concepts of "brane" and "bulk", of course. Are there many other such "unconventional" interpretations of the nature of entanglement out there?


title: What is the difference between a vortex ring and a circular line vortex?

text: Can someone explain the difference between a vortex ring and a circular line vortex? Or are they the same?


title: Matter waves - DeBroglie's relations

text: I am currently studying from _Modern Physics for Scientists and Engineers_ by Taylor et al. They derive the DeBroglie relation $p=h/\lambda$ from setting mass $m=0$ in the energy-momentum relation $E^2=m^2c^4+p^2c^2$, resulting in $E=pc$, then $p=E/c$. This was originally done to show particle-wave duality for photons. However, later they apply $\lambda=h/p$ to find the wavelength of electron waves. My question is how can this equation be used if electrons have mass? In fact in an example demonstrating the short wavelengths of electron waves, they give an energy $K$ and use the equation $\lambda=h/\sqrt{2mK}$. Is using the DeBroglie relation for matter waves pertinent or misleading?


title: Is it reasonable for a heavy door to "open by itself" (ie from differences in air pressure) if it had already been slightly ajar?

text: If you consider a basically uniform massive door (say, 300 N) where there is some coefficient $\mu_{s,k}$ of static and kinetic friction between the thing on the inside of the door, and where the door is 1 meter long (1 meter from pivot point to end of door), is it reasonable that differences in air pressures inside and outside of the room could cause the door to open slightly, seemingly by itself, if it started where the metal thing was not snugly in the hole?