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<p>In PADMANABHAN, Gravitation (Foundations and Frontiers), Cambridge, p
$304$, exercice $7.6$, an example of the Schwarzschild metric in a
different coordinate system is given : </p>
<p>$$\mbox{d}s^2= -c^2\mbox{d}T^2 +
\dfrac{4}{9}\left(\frac{9GM}{2(R-cT)}\right)^ {\large \frac{2}{3}}
\mbox{d} R^2+\left(\frac{9GM}{2}(R-cT)^2\right)^ {\large
\frac{2}{3}} \mbox{d}\Omega^2 \tag{7.52}$$</p>
<p>The second part of the exercise is : </p>
<blockquote>
<p>"Consider observers located at fixed spatial coordinates. Show that
they are on a free fall trajectory starting with zero velocity at
infinite distances and that $T$ represents their proper time. This
provides a simple interpretation of this coordinate system"</p>
</blockquote>
<p>I am afraid I don't really know how precisely to see this...</p>
| 3,999 |
<p>From my limited classical interpretation of the universe, i've always found it convenient to think of energy as what happens when an object isnt in its equilibrium position in the universe, and thus has some combination of the 4 fundamental forces acting on it to cause it to move to an equilibrium state.</p>
<p>So in terms of the subatomic world, when people say for example that an electron can be at different energy levels, does that mean that either the electron or its internal components are farther from an equilibrium position and thus more energy will be used when the forces act on the electron? Or does my interpretation of energy fall apart at this level?</p>
<p>My apologies if my lack of knowledge has caused me to pose the question in an ill formed manner.</p>
<p>I was just really wondering if the concept of energy works along the same principles at the subatomic level</p>
| 4,000 |
<p>In the gravity well-like 2d surfaces that are used in documentaries to illustrate the fact Earth orbits the Sun, I don't seem to find any kind of geodesic that will at least resemble an ellipse... What am I doing wrong? Is spacetime curved entirely different from what the animators show us? Any mathematical argument is apreciated but not neccesary... :)</p>
| 4,001 |
<p>For generators of the Lorentz group we have the following algebra:
$$
[\hat {R}_{i}, \hat {R}_{j} ] = -\varepsilon_{ijk}\hat {R}_{k}, \quad [\hat {R}_{i}, \hat {L}_{j} ] = -\varepsilon_{ijk}\hat {L}_{k}, \quad [\hat {L}_{i}, \hat {L}_{j} ] = \varepsilon_{ijk}\hat {R}_{k}.
$$
For the splitting of algebra, we can introduce operators
$$
\hat {J}_{k} = \hat {R}_{k} + i\hat {L}_{k}, \quad \hat {K}_{k} = \hat {R}_{k} - i\hat {L}_{k}.
$$
So
$$
[\hat {J}_{i}, \hat {J}_{j} ] = -\varepsilon_{ijk}\hat {J}_{k}, \quad [\hat {K}_{i}, \hat {K}_{j} ] = -\varepsilon_{ijk}\hat {K}_{k}, \quad [\hat {J}_{i}, \hat {K}_{j}] = 0.
$$
So, each irreducible representation of Lie algebra is characterized by $(j_{1}, j_{2})$, where $j_{1}$ is max eigenvalue of $\hat {J}_{3}$ and $j_{2}$ is max eigenvalue of $\hat {K}_{3}$.</p>
<p>Then I can classify objects that transform through the matrices of the irreducible representations,
$$
\Psi_{\mu \nu}' = S^{j_{2}}_{\mu \alpha }S^{j_{2}}_{\nu \beta}\Psi_{\alpha \beta},
$$
where $S^{j_{i}}_{\gamma \delta}: (2j_{i} + 1)\times (2j_{i} + 1)$.</p>
<p>For $(0, 0)$ I have scalar field, for $\left(\frac{1}{2}, 0\right); \left(0; \frac{1}{2}\right)$ I have spinor, for $(1, 0); (0, 1)$ I have 3-vectors $\mathbf a, \mathbf b -> \mathbf a + i\mathbf b$ creating antisymmetrical tensor etc. </p>
<p>Also, for scalar $j_{1} + j_{2} = 0$, for spinor - $\frac{1}{2}$, for tensor - $1$. So, the question: is sum $j_{1} + j_{2}$ experimentally observed? Is it connected with a spin?</p>
| 4,002 |
<p>It just occurred to me that almost all images I've seen of the (in)famous mushroom cloud show a vertical column rising perpendicular to the ground and a horizontal planar ring parallel to the ground. </p>
<p><img src="http://i.stack.imgur.com/yqTdB.jpg" alt="enter image description here"></p>
<p>Not that I'm an expert (that's why this question) but I have rarely seen anything go in the 45 degree angle. Or for that matter anything other than the 'special' 0 degree horizontal plane and 90 degree vertical column. Shouldn't there be radial vectors at all angles between 0 and 90 degrees giving rise to a hemispherical explosion envelope? Why is it a vertical cylinder? </p>
<p>PS: I understand the top expands eventually on cooling and lowered air pressure giving the mushroom look but my questions is for the previous stage - the vertical column.</p>
| 4,003 |
<p>Can anyone explain to me the meaning of this picture? I Know that the argument is Quantum Physics and that cat is the Schrödinger's cat, but I don't know how to interpret the bra–ket notation and also the two numerical coefficients...</p>
<p><img src="http://i.stack.imgur.com/Of40B.jpg" alt="enter image description here"></p>
| 4,004 |
<p>I was watching a show on discovery and according to it, in a nebula the dust and gases slowly come together and as the gravity increases and the pressure rises in the core the gases fuse together and a star is born and the rest of the left over dust and gases come together and form planets and moons.</p>
<p>So my question is that isn't all of the dust and gases used in the formation of the star? Why is some of it left and used to form planets, or why it is not sucked by the newly born star due to it's gravity. Why does the left over turn into planets and not other stars?</p>
| 4,005 |
<p>We all know (or maybe know) that to move, we need to spend energy. If you want to drive a car, you gotta spend gasoline.</p>
<p>We also know that energy can't be created (<a href="http://en.wikipedia.org/wiki/First_law_of_thermodynamics" rel="nofollow">first law of thermodynamics</a>, and <a href="http://en.wikipedia.org/wiki/Perpetual_motion" rel="nofollow">perpetual motion</a>).</p>
<p>Also, we know that in energy transformation, in real-world almost some part of it is converted into heat produced because of the friction between motion bodies. (for example, part of the gasoline of the car in burned to overcome car's friction with air, and thus would be converted into heat, and won't serve any movement at all).</p>
<p>Now a question has obsessed my mind. <strong>How electrons circulate around nucleus for ever</strong>?
Where does electron get its energy from? </p>
| 202 |
<p>I noticed an unusually fast moving cloud this morning.</p>
<p>My questions:</p>
<ul>
<li><p>What is the average velocity of a cloud on Earth?</p></li>
<li><p>What is the greatest ever recorded cloud velocity?</p></li>
<li><p>What factors affect the velocity of a cloud? (e.g. do they experience inertia?)</p></li>
</ul>
| 4,006 |
<p>Let $H= \frac{p^2}{2m}$, then I am supposed to calculate $[x,e^{-iHt}]$.</p>
<p>My idea was to use $[x,p^n]=i \hbar n p^{n-1}$ and so I ended up by using the series for the exponential function with $-\frac{t \hbar}{m} e^{-iHt}$.</p>
<p>Could anybody tell me, whether this result is correct? </p>
| 4,007 |
<p>Let's consider a single-particle(boson or fermion) with $n$ states $\phi_1,\cdots,\phi_n$(normalized orthogonal basis of the single-particle Hilbert space), and let $h$ be the single-particle Hamiltonian. As we all know, the second quantization Hamiltonian $H=\sum\left \langle \phi_i \mid h \phi_j \right \rangle c_i^\dagger c_j$ of $h$ should <em>not</em> depend on the single-particle basis we choose(where $c_i,c_i^\dagger$ are the bosonic or fermionic operators.), and this can be easily proved as follows:</p>
<p>Choose a new basis, say $(\widetilde{\phi}_1,\cdots,\widetilde{\phi}_n)=(\phi_1,\cdots,\phi_n)U$, where $U$ is a $n\times n$ unitary matrix. Further, from the math viewpoint, an inner product can has two alternative definitions, say $\left \langle \lambda_1\psi_1\mid \lambda_2 \psi_2 \right \rangle=\lambda_1^*\lambda_2 \left \langle \psi_1\mid \psi_2 \right \rangle(1)$ or $\lambda_1\lambda_2^* \left \langle \psi_1\mid \psi_2 \right \rangle(2)$.</p>
<p>Now, if we think $(\widetilde{c}_1^\dagger,\cdots,\widetilde{c}_n^\dagger)=(c_1^\dagger,\cdots,c_n^\dagger)U$ combined with the definition (1) for inner product, then it's easy to show that $\sum\left \langle \widetilde{\phi}_i \mid h \widetilde{\phi}_j \right \rangle \widetilde{c}_i^\dagger \widetilde{c}_j=\sum\left\langle \phi_i \mid h \phi_j \right \rangle c_i^\dagger c_j$; On the other hand, if we think $(\widetilde{c}_1,\cdots,\widetilde{c}_n)=(c_1,\cdots,c_n)U$ combined with the definition (2) for inner product, one can also show that $\sum\left \langle \widetilde{\phi}_i \mid h \widetilde{\phi}_j \right \rangle \widetilde{c}_i^\dagger \widetilde{c}_j=\sum\left\langle \phi_i \mid h \phi_j \right \rangle c_i^\dagger c_j$. </p>
<p>Which <strong>combination</strong> of transformation for operators and definition for inner product is more reasonable? I myself prefer to the former one. </p>
| 4,008 |
<p>Previously, on <a href="http://physics.stackexchange.com/q/86509/33806">Save us from swallowing baby universes, please!</a>, I pointed out the dangers associated with false vacuum decay.</p>
<p>I wish to be more specific here. Suppose the Standard Model remains a good description of our universe even at higher energies. <a href="http://physics.stackexchange.com/q/31291/33806">The observed value of the Higgs mass predicts that our current phase is metastable.</a> If our phase decays in a bubble, its interior will have a different much higher value for the Higgs field. </p>
<p>What happens to matter as it encounters the expanding bubble wall? Coleman-de Luccia assumed the exterior is initially in the vacuum state. They did not take the presence of matter into account. A changing value for the Higgs field is unlikely to affect the electromagnetic field much. But we expect the electromagnetic coupling strength to change inside the bubble due to a different running of the coupling constants under the renormalization group in the new phase. How much will this affect photons, and what is its frequency dependence? The leptonic and quark masses due to the Yukawa Higgs coupling will jump up dramatically in the new phase. Will this lead to a partial reflection and a partial transmission of leptons as the bubble passes through? What about confined bound states of quarks in hadrons? How will hadrons be affected? Leptons and quarks also happen to be dressed states, and the dressing will differ between the two phases. Might this cause a particle shower if these particles are transmitted? The masses of the W and Z bosons will also jump up. What will happen to them? Will neutrinos be unaffected? What if neutrino oscillations are also taken into account?</p>
<p>This leads us to the ultimate question. Suppose we have some lump of matter, like a cyborg, which is a complicated bound state of leptons, hadrons and photons. What will happen to this cyborg as the bubble expands and hits it?</p>
| 4,009 |
<p>How do you derive the solution to Poisson's equation with a point charge source? Without using Coulomb's law or the electric field! To be more explicit, we have a point charge at $(0,0)$ of charge $q$ and we want to solve Poisson's equation to find the potential. </p>
| 4,010 |
<p>I'm familiar with gravitational lensing but still I'm wondering if there is experiments (conducted here on Earth) which show that light bends due to gravity. For example mirrors setup to hold the light or something like that.</p>
<p>My question is inspired by this <a href="http://physics.stackexchange.com/questions/121514/slowing-of-time-under-gravity">bounty question</a>.</p>
| 4,011 |
<p>Is it true that a Mach-Zehnder interferometer with two polarizing beam splitters (PBS) is nothing but a bit flip for the polarisation degree of freedom?</p>
<p>Say the PBSs reflect vertical polarized light and transmit horizontal polarized light. If we send in a state like $|+\rangle = \frac{1}{\sqrt 2}(|H\rangle+|V\rangle)$ the horizontal part gets reflected only once (by the mirror in the interferometer) but the vertical part gets reflected three times (first PBS, mirror, second PBS). At the end we have a relative Phase of $i^2=-1$ between $|H\rangle$ and $|V\rangle$ which means that we end up with the state $|-\rangle = \frac{1}{\sqrt 2}(|H\rangle-|V\rangle)$.</p>
<p>Thus it is a bit flip of the polarisation. Right?</p>
| 4,012 |
<p>Would it be possible for a neutron to lose a positron and become an antiproton? Or would would it need to be the decay of a antineutron to antiproton instead? </p>
| 4,013 |
<p>What factors determine whether or not wind resistance will have an
important effect on the trajectory of a projectile? </p>
| 4,014 |
<p>I'm trying to understand precession for a gyroscope or top.</p>
<p>I do understand why precession occurs using the vectors for the weight force and torque and angular momentum. But what I don't understand is why precession only occurs at high angular speeds. Looking at the different vectors, there should still be a resulting change in the top's angular momentum.</p>
<p>I've tried to find an answer in my favorite physics textbook, but it only says that at lower angular speeds, the situation becomes much more complex. Which is not very helpful.</p>
<p>Can anyone explain the basic idea why precession only occurs at high angular velocities. I don't need this for an exam or anything, so I don't require a detailed mathematical explanation. I just want to get the basic idea.</p>
| 4,015 |
<p>In scattering theory, P wave means $l=1$, where $l$ is the azimuthal quantum number. However, what does P wave mean when referring to particle states? For example, in <a href="http://arxiv.org/abs/hep-ph/9208254" rel="nofollow">this paper</a> (arXiv link), the authors are talking about <em>P-wave charmonia states</em>. What does that mean?</p>
<p>More specifically, I understand that in some sort of potential model, solved using Schroedinger equation for example, there will be states that may be labeled by $n$=something, $l=1$. But here, the article says P-''wave'' charmonia! What is this wave?</p>
| 4,016 |
<p>Following Kepler's publication of his <a href="http://en.wikipedia.org/wiki/Kepler%27s_laws_of_planetary_motion#Third_law">3rd law of planetary motion</a><sup>1</sup>, </p>
<p>$$p^2 / r^3 = 1$$</p>
<p>in 1619, it would have been possible to use telescopic observations to arrive at an estimate of the orbital radii the <a href="http://en.wikipedia.org/wiki/Moons_of_Jupiter#Discovery">Jovian moons observed</a> by Galileo 1610, expressed in Earth orbital radii.<sup>2</sup> Along with the observed periods of the moons, these radii could have been used in a Jovian version Kepler's formula to discover that (at least for Io, Europa, and Callisto<sup>3</sup>)</p>
<p>$$p^2 / r^3 \approx 1.053 \times 10^3$$</p>
<p>or as close as telescopic technology of the day would allow.</p>
<p>Did anyone at the time take this, or a similar approach, to arrive at the conclusion that there is an attribute the Sun and Jupiter in which they differ by a factor of approximately a thousand? That attribute, of course, turns out to be mass.</p>
<p>Did Kepler, Galileo, or any of their contemporaries perform this calculation in the early 1600s? If not, why?</p>
<hr>
<p><sup>1. For $p$ in Earth years and $r$ in Earth orbital radii (today's AU).<br/>
2. Applying trigonometry to the observed angular extent of the moons' orbits and the distance to Jupiter.<br/>
3. Ganymede produces a slightly different value.</sup></p>
| 4,017 |
<p>I don't know how to solve question like this: </p>
<p>A transverse wave is propagated in a string stretched along the x-axis. The equation of the wave, in SI units, is given by:y = 0.006 cos π(46t - 12x). The frequency of the wave, in SI units, is closest to ...</p>
<p>Will appreciate your help!</p>
| 4,018 |
<p>Here is one task below.
How to solve equation
$$
m\ddot {x} + ax = F(t), x(0) = \dot x (0) = 0
$$
in quadratures by using two methods?</p>
<p>I tried to create a system of equations</p>
<p>$$
\begin{matrix} \dot v = F(t) - w_{0}^{2}x \\ \dot x = v \\ \end{matrix},
$$</p>
<p>but I don't know, what to do next without using some vector $\varepsilon = v + \alpha x$. So, i know only one way. Can you help me with the second way?</p>
| 4,019 |
<p>So I was wondering, with all this Higgs talk going on, they just detected a particle with a mass of 125 GeV (CMS) or 126.5 GeV (ATLAS). But they still don't know what it is, since there is tons of data to go through.</p>
<p>How do you determine spin of a particle from such experiments? Measuring the magnetic moments of a particle or?</p>
<p>If they find out that it has spin 0, and the other decay channels are in agreement with this, then they can say: it's Higgs, right?</p>
<p>So how do you do it? What do you need to analyse?</p>
| 211 |
<p>What's a physical meaning of, for example, complex part of the solution for coordinate change of the anharmonic oscillator?
Why after substitute (for diff. equation solve) for real x we can earn $x = Re(z) + iIm(z)$? It's because of substitute?</p>
| 4,020 |
<p>As Wikipedia explains, one photon passing through a crystal sometimes down-converts to two
photons. Wikipedia says total energy and momentum are conserved by just considering the
three photon states; is Wikipedia wrong here?</p>
<p>It seems a phonon (or something else) is needed too. If Wikipedia is right, can you
provide 3 example (non-parallel) momenta vectors so that I can see my logic mistake?</p>
| 4,021 |
<p>My sister asked me this question and I keep thinking that water would conduct heat much faster than sand. Hence the energy transfer of heat across the lake does not allow it to heat up soon. Sand on the other hand is probably a bad conductor of heat and hence more heat energy is held up by each grain of sand. </p>
<p>But then with water there is also convection which perhaps plays a role. </p>
<p>I would like to see what the experts say. My thinking is quite informal. </p>
| 4,022 |
<p>Consider the following action with a fermionic field $\psi$ and a scalar field $\sigma$, </p>
<p>$S = \int d^dx \{ -\bar{\psi}(\gamma^\mu \partial_\mu +\sigma )\psi + \Lambda^{d-4}[ \frac{(\partial_\mu \sigma )^2 + m^2\sigma^2 }{2g^2 } + \frac{\lambda \sigma^4 }{4!g^4 } ] \} - (N'-1)Trln(\gamma^\mu \partial_\mu + \sigma )$</p>
<p>Assuming that this has a large-N saddle with uniform $\sigma$ one gets the large-N free energy density as, </p>
<p>$E(\sigma) = \Lambda^{d-4}(\frac{m^2\sigma^2}{2g^2} + \frac{\lambda \sigma^4 }{4!g^4 } ) - \frac{N}{2}\int^\Lambda \frac{d^dq}{(2\pi)^d}ln [\frac{q^2 + \sigma^2 }{q^2 } ]$</p>
<p>And the large-N saddle value of $\sigma$ is determined by the large-N gap equation, $E'(\sigma)=0$</p>
<p>Now from here how do the following conclusions come? </p>
<ul>
<li>Firstly that a non-trivial solution to the gap equation exists only when, $\frac{m^2}{g^2} < N\Lambda^{4-d}(\frac{1}{(2\pi)^d} \int^\Lambda \frac{d^dk }{ k^2 } ) $ </li>
</ul>
<p>How does this one come? </p>
<ul>
<li>Secondly from this apparently follows that the inverse $\sigma$ propagator in the massive phase is, </li>
</ul>
<p>$\Delta_\sigma^{-1}(p) = \Lambda^{d-4}(\frac{p^2}{g^2} + \frac{\lambda \sigma^2}{3g^4} ) + \frac{N(p^2+4\sigma^2) } {2(2\pi)^d}\int^\Lambda \frac{d^dq }{(q^2+\sigma^2)((p+q)^2 + \sigma^2)} $</p>
<p>How does this equation come? </p>
<ul>
<li><p>Now from this one can show that $\Delta_\sigma \sim \frac{2}{N b(d) p^{d-2} }$</p>
<p>From the above it follows that the canonical dimension of $\sigma$ is 1. How does one understand that the mass dimension of the field $\sigma$ does not depend on the space-time dimension? </p></li>
<li><p>Now I don't understand this argument which says that now since $[\sigma] =1$, both the terms $(\partial_\mu \sigma)^2$ and $\sigma^4$ are of dimension $4$ and hence for $2\leq d \leq 4$ these terms vanish in the IR critical theory? For this argument to work was it necessary that the IR theory was critical? </p></li>
</ul>
| 4,023 |
<p>Suppose we have a quantum state $\rho$ and let's denote the photon number operator $\hat{n}=\hat{a}^\dagger\hat{a}$ where $\hat{a}$ is the annihilation operator. Let mean photon number $\bar{n}=\operatorname{Tr}(\rho\hat{n})<\infty$ obviously be finite. Can $\rho$ have infinite photon number <em>variance</em>, which we define as follows:</p>
<p>$$\operatorname{Var}(\rho)=\operatorname{Tr}(\rho\hat{a}^\dagger\hat{a}^\dagger\hat{a}\hat{a})-\bar{n}^2$$</p>
<p>I ask this because it came up in a quantum information theory research problem that I am working on. Mathematically, it's easy to define $\rho$ with infinite photon number variance: let $\rho=|\psi\rangle\langle\psi|$ where $|\psi\rangle=\sum_{n=0}^\infty a_n|n\rangle$ such that $|a_n|^2=(n+1)^{-3}/\zeta(3)$ with $\zeta(3)=\sum_{k=1}^\infty1/k^3$ being the <a href="http://en.wikipedia.org/wiki/Riemann_zeta_function" rel="nofollow">Reimann zeta function</a>. That is, $|\psi\rangle$ is a superposition of number states which is prepared such that the photon number of the state is distributed according to the <a href="http://en.wikipedia.org/wiki/Zeta_distribution" rel="nofollow">Zipf distribution</a>. </p>
<p>However, I am wondering if such state (or one like it) can actually exist in nature. If yes, how would one construct it? If not, is there a concrete reason as to why? (ideally, I am looking for a mathematical proof, but would accept a generally accepted well-cited explanation.) </p>
<p>I am confused because it seems to me that the requirement if one applies a measurement $\sum_{m=k}^\infty|m\rangle\langle m|$ with a large $k$ to a coherent state there is a chance of it returning a positive answer, however small, as coherent states are not "peak photon number" limited (if they were, that chance would be zero starting at some fixed $k$). Here the difference is the heavy tail on the photon number.</p>
<p>This is my first question to this forum. I am not a physicist, I study information theory and have been looking at quantum information theory lately. I know a lot of math, but not necessarily how things work in the "universe we touch". I only have a rudimentary knowledge of quantum mechanics. Please forgive my ignorance and feel free to refer this question somewhere else if it's not suitable for this forum.</p>
| 4,024 |
<p>Assuming there is an incident beam(i.e. $p$ or $\alpha$) and a target. How can I be sure if a rutherford backscattering will take place?</p>
<p>I know that for high $Z$ it more likely to happen as well as for higher beam energies. Is there a boundary/formula/criterion to tell me that at that beam, for that energy, for this $Z$ target I will have rutherford backscattering.</p>
<p>To rephrase it, for a certain beam(i.e. $p$) and a certain target(i.e. $SbF_3$) at what energies I am going to have Rutherford Backscattering?</p>
| 4,025 |
<p>I'm trying to develop an inertial navigation system.</p>
<p>I can access data from an accelerometer sensor (acceleration on three axes) and gyroscope sensor (angular velocity on three axes).</p>
<p>First of all, I integrate my angular velocity data with respect of time, and get angles on all three axes at every moment ($x \to \phi, y \to \theta, z \to \psi$).</p>
<p>Then I feed the angles to this rotation matrix</p>
<p><img src="http://i.stack.imgur.com/USq1f.png" alt="Rotation matrix"></p>
<p>and I use it to rotate my acceleration vector at that time, thus taking all my acceleration values on the same reference frame.</p>
<p>Finally, I integrate acceleration to get space travelled, using the simple formula
$$s(t) = \frac12 a(t) t^2 + v(t - 1)t + s(t - 1)$$</p>
<p>My method seems to work fine with fake data, but performs really bad when I plug in the real data, the output is almost meaningless.</p>
<p>Am I doing something wrong with the math or I have to search the problem in my implementation?</p>
| 4,026 |
<p>Consider a liquid solid interface $z =\zeta(x,t)$ moving at constant speed $v$, for a two dimensional problem. Due to solidification interface
is changing it position. For simplicity heat conductivities, heat capacity of liquid and solid are assumed to be the same. Heat conduction equation in the liquid and solid region is
\begin{eqnarray}
C\frac{\partial T}{\partial t} &=& k\nabla^2 T \nonumber \\
\end{eqnarray}
Here, $k$, $C$ are heat conductivity and heat capacity.</p>
<p>Heat balance equation at interface is
\begin{eqnarray}
Lv &=& k(\nabla T_s - \nabla T_l)\cdot \hat n \nonumber \\
\end{eqnarray}</p>
<p>$L$ latent heat of solidification. $\hat n$ is unit normal to the interface and $v_n$ is velocity of interface in the normal direction. $T_l$ and $T_s$ are temperatures in the liquid and solid domains.</p>
<p>Boundary conditions are:
\begin{eqnarray}
T &=& T_{\infty} \hspace{1cm}z \to \infty \\
T &=& T_m \hspace{1cm} z = \zeta(x,t)
\end{eqnarray}</p>
<p>I need to find Green's function for this problem. Green's function for
heat diffusion of point source is</p>
<p>\begin{eqnarray}
G = \frac{1}{{4\pi D(t-t_1)}}\exp\left(-\frac{(x-x_1)^2 +(z-z_1)^2}{4\pi D (t-t_1)}\right)
\end{eqnarray}
Where $D= k/C$</p>
<p>Then I proceeded as follows:</p>
<p>Since all the heat generated is at the interface $\zeta(x,t)$,
\begin{eqnarray}
T - T_{\infty} = \int^{t}_{0}dt_1 \int^{}_{}dx_1\frac{1}{{4\pi D(t-t_1)}}\exp\left(-\frac{(x-x_1)^2 +(z-z_1)^2}{4\pi D (t-t_1)}\right) \zeta(x,t)
\end{eqnarray}</p>
<p>Research <a href="http://journals.aps.org/rmp/abstract/10.1103/RevModPhys.52.1" rel="nofollow">paper</a> gives following equation,
\begin{eqnarray}
\frac{T-T_\infty}{L/C} &=& \int^t_0 dt_1 \int^{\infty}_{-\infty}dx_1 G[x-x_1, z-\zeta(x_1,t_1)+ v(t-t_1), t-t_1] \left( v + \frac{\partial \zeta}{\partial t}\right)
\end{eqnarray}</p>
<p>where,\begin{equation}
G(x,t) = \frac{1}{{4\pi D(t)}}\exp\left(-\frac{(x)^2 +(z+vt)^2}{4\pi D (t)}\right)
\end{equation}</p>
<p>which is clearly different from what I assume. How can I arrive at this equation?</p>
| 4,027 |
<p>Nuclei spontaneously decay according to a certain decay rate. There are however different kinds of decay, alpha, beta, gamma... What causes then the nuclei, when they decay, to do so in one way of another? Is there a different decay rate for each kind of decay? </p>
| 4,028 |
<blockquote>
<p>Is there any reason the universe has matter not being able to exceed the speed of light, or why there is a speed limit in the first place?</p>
</blockquote>
<p>I know why it can't, meaning the basic physics of it. I am just wondering if the reason why the universe is like this is or the benefit of it known.</p>
<p>I know this might be too philosophical, but I am just wondering.</p>
| 122 |
<p>Given an electromagnetic wave in resonance mode in a vacuum cavity inside a perfect conductor, on the boundary, the parallel component of $E$ field vanishes, and the perpendicular of component of $B$ field vanishes. How does one derive, through solving Maxwell equation boundary value problem, say by the way of stress-energy tensor or other by products of the Maxwell equation, $\frac{\partial \int_{\text{cavity}}\mathbf S dV}{\partial t} = 0$ or $\frac{\partial \int_{\text{cavity}} \mathbf<S>dV}{\partial t} = 0$ where $\mathbf S=E\times B$ is the Poynting vector, the integral is over the space of the cavity, $<\cdot>$ denotes time average? In other words, how does one show the total momentum of the field inside the cavity vanishes from Maxwell equation computation?</p>
| 4,029 |
<p>I am an undergrad intern at a national lab currently working with a basic proton NMR device. The device consists of two big coils which provide the static magnetic field, and a smaller coil, which sends both the "excitation" signal and receives the NMR signal. A week or so ago, my supervisor asked me to calculate the magnitude of the magnetic field of the inner coil. </p>
<p>Since I knew the maxiumum voltage of my signal, I chose to use Faraday's law of induction for a tightly wound coil of wire: $\mathcal{E}=-N\frac{d\phi_B}{dt}$.
Knowing that the excited net magnetization vector $\vec{M}$ has a torque exerted on it (thanks to the static field), I reasoned that the magnetic flux through the smaller coil would be:
$\phi_{B}=BA\cos\omega t$. Taking the derivative of this, I reasoned that the maximum voltage would be equivalent to the maximum of $-N\frac{d\phi_B}{dt}$:</p>
<p>$V_{max}=NBA\omega$.</p>
<p>I figured that from this equation; knowing the area, precession frequency, and number of coils; I would be able to find $B$ pretty easily. </p>
<p>But we measured B another way, using $\theta=\gamma B_1t_p$, and got a result that was half the size of the "Faraday way". At first, even my supervisor was confused, but then he quickly remembered that we forgot to remember that we were working in a "rotating frame". </p>
<p>For this reason, our supervisor said that the correct relation between max voltage and max flux was really $\frac{V_{max}}{2}=NBA\omega$.</p>
<p>Well, this baffled me. It still baffles me. He tried to explain to me that we can think of our signal as two arrows which oscillate in opposite directions in a circle, each arrow having a magnitude half the size of the actual thing. I know I'm using vague language right now, but that's because I don't get it. Why do I cut my max voltage in half. Even if we did use this rotating frame, wouldn't the max voltage be when the arrows are in phase, and add up together. </p>
<p>If you guys could help me out, that would be awesome. Let me know if you need any clarification on the setup of the experiment; I'll be happy to elaborate.</p>
| 4,030 |
<blockquote>
<p>A closed calorimeter of negligible water equivalent contains 1kg ice at 0 degree Celsius. 1kg of steam at 100 degrees Celsius is pumped into it. Find the resultant temperature of the mixture. </p>
</blockquote>
<p>I did the standard approach. Heat absorbed by ice = Heat lost by steam. I took the change in temperature as $\Delta\theta_{i}$ and $\Delta\theta_{s}$. I will use sub $s$ for steam and sub $i$ for ice.</p>
<p>$$Q_\text{ice} = Q_\text{steam}$$</p>
<p>$$m_is_f + m_is_w\Delta\theta_i = m_ss_v + m_ss_w\Delta\theta_s$$</p>
<p>Where $s_f$ is specific heat of fusion of water, $s_w$ is specific heat of water and $s_v$ is specific heat of vaporization of water. </p>
<p>Since $\Delta\theta_s = 100 - \Delta\theta_i$,</p>
<p>$$m_is_f + m_is_w\Delta\theta_i = m_ss_v + m_ss_w(100 - \Delta\theta_i)$$</p>
<p>$$m_is_f + m_is_w\Delta\theta_i = m_ss_v + 100m_ss_w - m_ss_w\Delta\theta_i$$</p>
<p>$$\Delta\theta_i(m_is_w + m_ss_w) = m_ss_v + 100m_ss_w - m_is_f$$
and thus
\begin{align}\Delta\theta_i&=\frac{m_ss_v + 100m_ss_w - m_is_f}{m_is_w + m_ss_w}\\&= \frac{1000g\times2230\,\mathrm{\frac{J}{g}} + 100\,\mathrm{K}\times1000\,\mathrm{g}\times4.18\,\mathrm{\frac{J}{gK}} - 1000g\times334\,\mathrm{\frac{J}{g}}}{1000\,\mathrm{g}\times4.18\,\mathrm{\frac{J}{gK}} + 1000\,\mathrm{g}\times4.18\,\mathrm{\frac{J}{gK}}}\\&= 276.79\,\mathrm{K}\end{align}</p>
<p>Which makes no sense. I know I am making some naive error. But I have gone through this problem at least ten times but to no avail. </p>
| 4,031 |
<p>This question is concerned with a thermodynamic paradox for radiating bodies and radiation in a cavity of a specific shape.</p>
<p>Consider two nested shells that are axisymmetric ellipsoids with the same two foci, A and B, as shown in the figure (line AB is the axis of symmetry). Cut the system along the vertical plane of symmetry and remove the right side of the outer shell, and remove the left half of the inner shell. Then connect the two halves with vertical surface, as shown in the figure, to make it a continuous enclosure. The result is a figure of rotation shown by the thick black line in the figure.</p>
<p>Next, make the inner surface of it a perfect mirror. The property of such a cavity is that each ray emitted from point B comes to point A; but not each ray emitted from point A comes to point B - some rays emitted from A (shown in blue) come back to A.</p>
<p>Now, put two small black bodies (say, two spheres of some small radius) at points A and B. Thermodynamic equilibrium requires that eventually the temperatures of the two spheres equilibrate. However, according to the geometric properties of this cavity, all energy emitted from B comes to A but only a fraction of energy emitted from A comes to B; so the equality of temperatures is not consistent with balance of emitted and absorbed power. How to resolve this paradox?</p>
<p><img src="http://i.stack.imgur.com/igGAT.jpg" alt="enter image description here"></p>
| 4,032 |
<p>We all know that when we say A it sounds different than when we say B. I was wondering what exactly can be the difference between saying A and B in terms of physics. I first thought that it may due to difference in the combination of frequencies. Then I realized I can say any alphabet in many different tones. </p>
<p>So what exactly in terms of physics is the difference between saying A and B ?</p>
| 4,033 |
<p>Is it correct that the number of lines originating from vertices on Feynman diagrams is equal to the order of phi in interaction lagrangian for scalar field?</p>
| 4,034 |
<p>Apparently to create wormholes you need negative energy/matter. Say you had negative matter/energy, how would it be applied towards making a wormhole?</p>
| 4,035 |
<p>I have a pot of vigorously boiling water on a gas stove. There's some steam, but not alot. When I turn off the gas, the boiling immediately subsides, and a huge waft of steam comes out. This is followed by a steady output of steam that's greater than the amount of steam it was producing while it was actually boiling. </p>
<p>Why is there more steam after boiling than during boiling? Also, what's with the burst of steam when it stops boiling?</p>
| 4,036 |
<p>I recently have read about interception of wireless information, however this mentions that people can intercept the information, and then somehow the recipient also gets the information. Regardless of this context, what happens to the actual photon if it is absorbed by one antenna how can another person receive the same signal? Is it that when the photon is absorbed exciting the electron, the electron will then leap back to the lower energy state causing it to emit another photon? so the antenna acts as a receiver & transmitter? to be honest I'm confused overall in how antennas work.</p>
| 4,037 |
<p>Can anybody suggest me a good algorithm for the time evolution of the reduced density matrix using Linbald equation. My Hamiltonian is time dependent. I am aware about Qotoolbox and Qutip. I have checked both things but I don't have any clues about the algorithms they are using. I have to find the entanglement dynamics in a bipartite system under dissipation and forcing.</p>
| 4,038 |
<p>In the Metropolis algorithm, the change in the energy given by the hamiltonian is compared for flipping a spin. This is not the free energy, but for systems above absolute zero you are trying to minimize the free energy, not the energy. So how is free energy built into this kind of simulation? How is entropy?</p>
| 4,039 |
<p>I've got a homework question. </p>
<blockquote>
<p>Consider a 2 dimensional space with metric $$ ds^{2} = \frac{dr^{2}}{1 -\frac{2}{r} } + r^{2}d\theta^{2} .$$ </p>
<p>I need to show that this is the induced metric (not sure if im translating this correctly) from $\Bbb R ^{3}$ and I need to find the parametrization of the surface. </p>
</blockquote>
<p>I really don't know how to start with the parametrization. Any help getting me started would be greatly appreciated. </p>
<p>Edit: since the question got put on hold I will try to clarify my question. I need some intuition behind this metric, specifically the $g_{rr}$ coefficient. I tried different things involving trig functions of $\theta$ and $r$ to be the usual polar transformation, but I'm stuck. I'm pretty sure that is I can manage to get the parametrization right I can prove that the given metric is the induced metric from $\Bbb R ^{3}$. Hope this clarifies the question.</p>
| 4,040 |
<p>So, there are all kinds of 'standard' explanations that people just 'know' about how things work which are wrong. The sort of explanations a dad may give to their son when he asks the infamous "why" word to explain how the world works. Not all of these answers are entirely true. </p>
<p>I thought, for the sake of helping us all be a little better informed, we could think up some of the most commonly misunderstood phenomenon and give a better or more accurate explanation to them. Think of it as something like Wikipedia list of common misconceptions, but specifically targeted to physical explanations and more of a focus on explaining the <em>true</em> reasons. I'll even include a bounty for the best answer</p>
<p>For example, the #1 misconception that comes to me is "how airplanes fly", which even Google itself will simplify to Bernoulli's principle, while the real answer is...actually a bit too complicated for me to explain well lol.</p>
<p>To make this sporting here are some 'rules' for the sort of answers I'm looking for</p>
<p>1) erroneous explanation should be about something common place, in other words if your couldn't see someone under the age of 13 asking about a phenomenon it's probably too complicated
2) corrected explanations given should be simple enough that an intelligent laymen would have a chance of understanding it. I'm looking for answers that could actually be used to correct a misconception when it occurred.</p>
<p>I'll generally select whatever answer has the highest vote count as the winner. However, I reserve the write to pick something else as the winner if I feel that the top voted answer does not fit the two categories, or that it only won because it was posted early and had more time to get up voted.</p>
| 4,041 |
<p>First of all I beg your forgiveness as I am not a physicist and the question I am going to ask may sound silly.</p>
<p>I am aware that beyond a certain threshold in the hierarchy of building blocks of matter (electrons, atoms, etc.) the 'standard' laws of physics (e.g. <a href="http://en.wikipedia.org/wiki/Classical_mechanics" rel="nofollow">Newtonian physics</a>) do not apply and we enter a totally different environment where the so called <a href="http://en.wikipedia.org/wiki/Quantum_mechanics" rel="nofollow">quantum mechanics</a> apply. </p>
<ul>
<li>Where is this threshold located in relation to types of particles? </li>
<li>Are there any other similar thresholds in physics indicating completely new environments? If yes, what are they? (other than classical mechanics, quantum mechanics, ...maybe <a href="http://en.wikipedia.org/wiki/String_theory" rel="nofollow">string theory</a>?).</li>
</ul>
| 4,042 |
<p>I have this problem: They give me, from a satellite that is in orbit in earth, a value for the period, and the closest height to earth surface, the ask me what the eccentricty of the orbit is. I have no idea how to do this. I've tried using Binet's equation, and the equation that comes for these movements ($p/r=1+e\cos(\theta-\theta_0)$) or the conservation of angular momentum to try to get some relations between quantities, but I can't get anything.</p>
<p>For example, the angular momentum is conserved, so:
$$l=mr^2\dot \phi$$
I can get from here:
$$\int_0^T\frac{l}{mr^2}dt=2\pi$$
Being T the period, but I don't know $r(t)$ Or the other way:
$$T\frac{l}{m}=\int_0^{2\pi}r(\phi)d\phi$$
Now I now the function $r(\phi)=\frac{p}{1+e\cos\phi}$, where $p=\frac{l^2}{\mu k}$, being $l$ the angular momentum and $k$ the constant of the gravity potential: $U=-k/r$. I don't know how to integrate that if it's possible or how to use the known value of the closest point.
Some help?</p>
| 4,043 |
<p>And what happens with the magnetic field of a star that goes supernova? The magnetic radiation is scattered through the cosmos? Each particle will go away with its own magnetic radiation?</p>
| 4,044 |
<p>The assumptions are: </p>
<ul>
<li>Alice and Bob have perfectly synchronized clocks</li>
<li>Alice and Bob have successfully exchanged a pair of entangled photons</li>
</ul>
<p>The idea is simply to have Alice and Bob perform the Quantum Eraser Experiment (doesn't need to be the delayed choice). </p>
<p>Alice and Bob agree on a specific time when Bob's photon will be between the "path marker" (which is usually just after the slits) and the detector.</p>
<p>If Alice acts using the eraser on her photon, the interference pattern will appear. If not it won't.</p>
<p>Alice and Bob can be spatially separated...</p>
<p>What am i misunderstanding? </p>
<p>The only meaningful difference from this spatially separated quantum eraser experiment to one done on tabletop is that you won't be able to use a coincidence detector, but that is not impeditive to identifying the interference pattern, just will make errors more probable. Which we should be able to deal with a appropriate protocol...</p>
<p>There is a experimental paper with a small amount of citations pointing out to the breaking of complementarity in a very similar setup:
<a href="http://www.pnas.org/content/early/2012/05/23/1201271109" rel="nofollow">http://www.pnas.org/content/early/2012/05/23/1201271109</a></p>
| 4,045 |
<p>my problem is that I can’t find a way to calculate the power required to drive the shaft and thus decide on a motor, I have tried several times using different methods and calculations but can’t seem to get a power requirement that would seem accurate, most recently I tried using a fluid equation of force on an immersed body travelling through a fluid using integration I got formula for torque, on each of the paddles rotating in the hopper of 1/8*Coeff drag*density*angular velocity^2*width of paddle*radius of paddle^4 and adding the inertia of the system, this gave me a power requirement of a 150 watts which seems ridiculous, I was wondering how i should go about calculating the power requirement.
Any help would be greatly appreciated.
Thanks
ollie</p>
| 4,046 |
<p>It is well-known that some exotic phases in condensed matter physics are described by <a href="http://en.wikipedia.org/wiki/Topological_quantum_field_theory#Schwarz-type_TQFTs">Schwarz-type TQFT</a>s, such as Chern-Simons theory of quantum Hall states. My question is whether there are condensed matter systems that can realize <a href="http://en.wikipedia.org/wiki/Topological_quantum_field_theory#Witten-type_TQFTs">Witten-type TQFT</a>s?</p>
| 4,047 |
<p>Ryder in his QFT book writes in eqn (2.20):</p>
<p>Probability density, $\rho = \frac{i\hbar}{2m}(\phi^*\frac{\partial \phi}{\partial t} - \phi \frac{\partial \phi^*}{\partial t})$</p>
<p>Then in the next paragraph he writes: Since the Klein-Gordon equation is second order, $\phi$ and $\frac{\partial \phi}{\partial t}$ can be fixed arbitrarily a given time, so $\rho$ may take on negative values,..</p>
<p>What does he mean by this line?</p>
| 4,048 |
<p>I have a question about Eq. (4.3.3) in Polchinski's string theory book volume I, p. 131. It is said</p>
<blockquote>
<p>Replacing the $X^{\mu}$ with a general matter CFT, the BRST transformation of the matter fields is a conformal transformation with $v(z)=c(z)$, while $T^m$ replaces $T^X$ in the transformation of $b$.
Noether's theorem gives the BRST current
$$ j_B = c T^m + \frac{1}{2} : cT^g : + \frac{3}{2} \partial^2 c, $$
$$ = c T^m + : bc \partial c : + \frac{3}{2} \partial^2 c, \tag{4.3.3}$$</p>
</blockquote>
<p>My question is, what is the explicit expression of $T^m$?</p>
<p>According to <a href="http://www.science.uva.nl/onderwijs/thesis/centraal/files/f1989820784.pdf" rel="nofollow">this thesis</a>, p 29,
$$-\frac{1}{\alpha'}: c \partial X \cdot \partial X = :c T_X:$$</p>
<p>Suppose this expression is correct, I cannot use it to vertify Eq. (4.3.11)
$$T(z) j_B(0) \sim \frac{ c^m - 26}{2z^4} c(0) + \frac{1}{z^2} j_B(0) + \frac{1}{z} \partial j_B(0) \tag{4.3.11}$$</p>
<p>if in (4.3.11), $T(z)= -\frac{1}{\alpha'} : \partial X^{\mu} \partial X_{\mu} : \tag{2.4.4}$ and I applied contraction Eq. (2.2.11).</p>
| 4,049 |
<p>I'm quite familiar with SR, but I have very limited understanding in GR, singularities, and black holes. My friend, which is well-read and is interested in general physics, said that we can "jump" into another universe by entering a black hole.</p>
<p><em>Suppose that we and our equipments can withstand the tidal forces near black holes. We jumped from our spaceship into a black hole. As we had passed the event horizon, we couldn't send any information to the outside anymore.</em></p>
<ol>
<li><p>Can this situation be interpreted as that we were in another universe
separate from our previous universe?</p></li>
<li><p>Is there corrections or anything else to
be added to above statement?</p></li>
</ol>
| 4,050 |
<p>If, as I asked in <a href="http://physics.stackexchange.com/questions/19631/question-about-the-perturbative-renormalization-group">this question</a>, a relevant operator in a renormalization group transformation can't be used in a perturbative expansion since it becomes large as the transformations are applied, does this mean that the operator can't be used in 'normal' perturbation theory? </p>
<p>I.e. Is using the renormalization group a way to determine whether or not perturbation theory can be used at all? Or is it only relevant within renormalization groups since without it, the operator remains small?</p>
| 4,051 |
<p>I'm currently learning about the renormalization group (RG) in condensed matter physics and just want to clarify a couple of things:</p>
<p>When doing the RG transformation, there's a flow to a fixed point. A coupling constant is a relevant operator (or relevant coupling, depending on which book you look at) if it gets larger as the transformation continues, and flows towards the fixed point. </p>
<p>Have I understood that correctly?</p>
<p>If doing perturbative renormalization, is it right that if you have a relevant operator it is not possible to use that operator in a perturbative expansion because it's large once the transformations are done (even if it was small to start with - before the RG transformations were done)?</p>
| 4,052 |
<p>Since electromagnetic waves have both electric and magnetic field components, which oscillate in phase perpendicular to each other and perpendicular to the direction of energy propagation. How much is that electric charge? Can you possibly get a shock in some way?</p>
| 485 |
<p>I'm supposed to derive a relation for the range of wavelengths that's being transmitted by a spectrometer (bandpass) in terms of the dispersion, focal length and exit slit width.</p>
<p>Given grating is $100mm$ wide, slit separation $d=5.6\times 10^{-7} m$, focal length $f=1\space m$, exit slit width $w=100\mu m$ and spectrometer working in first order regime.</p>
<p>Then I'm supposed to find value of slit width that corresponds to a bandpass of theoretical resolving power of $\lambda \approx 500 nm$</p>
<p><strong>Attempt</strong></p>
<p>I found dispersion $\frac{d\theta}{d\lambda} = \frac{p}{d cos \theta} $ and resolving power $Np$.</p>
<p>Thus The amount of wavelengths contained in the dispersion is simply $\delta \theta \times \frac{d\lambda}{d\theta} $</p>
<p>But what is the angle of spread? Is it simply $\delta \theta = \frac{w}{f}$.</p>
<p>If so, then I find the bandpass $\Delta \lambda = \frac{w}{f} \times \frac{d}{p} = 5.6 \times 10^{-11} $.</p>
<p>Last part of the question doesn't make sense, as resolving power is only dependent on $N$.</p>
| 4,053 |
<p>I've heard a couple of scary stories from experienced accellerator physiscists about something called neutron clouds. Apparently, if you have an experiment like a fixed-target experiment that produces a lot of neutrons with the correct energy, they don't just dissipate or get caught in surrounding matter. Instead, they hang around due to their large half-life (~15 minutes). The rumor goes that they actually form clouds, that wander around the facility, and that in the early days of some CERN experiment, people didn't think about the effect, and got a nasty (although not accute) dose when they entered the collision hall just after shutting down the beam.</p>
<p>The description of the behavior of these clouds varies in different accounts. Sometimes they just pass through everything, but sometimes they're supposed to behave like a real gas, being held back by walls (but creeping through small openings).</p>
<ul>
<li>I can imagine this phenomenon is real, but how much of an issue is it in real experiments / nuclear facilities?</li>
<li>Do the clouds really behave like a gas (I'd think the n-n cross section is not big enough to create pressure)? How do they behave wrt. walls? </li>
<li>And in light of the recent nuclear waste transports in France and Germany: The waste emits a lot of gamma and neutron radiation, could it leave a temporary trail of low-energy neutron clouds behind?</li>
</ul>
| 4,054 |
<p>In an experiment we were given non-homogenous dielectric substances described by functions of coordinate. How can capacitance be determined from this?</p>
| 4,055 |
<p>If we have a spherical capacitor with inner radius or r1 and outer radius of r2, with charges (+/-)q on them and there is a dielectric material (with constant e) in between them with. </p>
<p>What kind of a potential would this create outside the entire capacitor? in the region with the dielectric? and inside the entire thing?</p>
| 4,056 |
<p>This question is based on the description of Longair in his book "Theoretical Concepts in Physics".</p>
<p>He starts by giving some provisions:</p>
<ul>
<li>Conservative force field</li>
<li>Fixed times $t_1$ and $t_2$</li>
<li>Object moves from fixed point at $t_1$ to fixed point at $t_2$</li>
</ul>
<p>Then he defines:</p>
<ul>
<li>Lagrangian: $L = K-U$</li>
<li>Action: $S = \int_{t_1}^{t_2}Ldt$</li>
</ul>
<p>He goes on to explain, that the principle of least action means, that an object moves on a path so that $S$ is minimized.</p>
<p>Then he claims that this priciple is equal to Newton's 2nd law of motion, following through with a proof which is beyond my comprehension (which of course is my fault).</p>
<p>After I calculated $S$ for a few examples, I am convinced, that this claim is correct only adding one additional provision (which Longair clearly does not state directly or indirectly):</p>
<ul>
<li>The object moves on a path fixed in space. (Just the speeds at the points is allowed to differ.)</li>
</ul>
<p>My argument for why this is necessary follows from a counterexample:</p>
<ul>
<li>Assume a central force field with constant force. Setup the object so that its trajectory is a circle. Take time $t_1$ and $t_2$ so that the object is at opposing ends of the circle, describing a half circle. Now change the force field, so that there is no force inside this circle. (This is still a conservative force field and the object moves still in the same circle.) Compare the $S$ of this half circle to the $S$ of the object moving with constant lower speed along the diameter of the circle. For both trajectories the $U$ is the same but the $K$ is lower for the shortcut along the diameter (lower speed). So the shortcut along the diameter has a lower action. Still, with the correct initial speed the object will move the half circle, fully in accordance to Newton's second law of motion. </li>
</ul>
<p>Since I cannot assume, that I found an error in Longair's standard book, can anyone please explain, what I got wrong.</p>
| 4,057 |
<p>We use ceiling and table fans in home which are can be set to low or high speeds using regulators. I want to ask that does it uses or consumes same amount of electricity at different speeds? Here, "same amount of electricity" means exactly I want to know that, my electricity bill will be different for different speeds or will it be same. I know that, when it has less speed, less electricity flows through it, but my friend told me that, when it is at less speed, the remaining electricity get wasted at regulator, so total electricity consumed by fan + regulator is same, so same electricity bill will be generated regardless on which speed we use it. So is my fried correct or wrong?</p>
<p><img src="http://i.stack.imgur.com/KB3k8.jpg" alt="image of ceiling fan with regulator"> </p>
| 4,058 |
<p>I am experimenting and playing around with some data, and I'm having trouble seeing how to generate invariant mass plots.</p>
<p>The data I have has a bunch of events, and variables such as $P,P_T,\eta,\phi$ etc.. but no energy. There is a histogram which generates the invariant mass, but no macro provided for this. For example, I know there are tracks for electrons and protons (TPC), and I want to see if there are any pairs as a result of $\gamma\rightarrow e^+ e^-$. Conceptually I need to loop over all positive negative pairs, and plot the histogram. I should get a peak around zero for $\gamma$ pair production. The invariant mass formula is $$m_\gamma^2 = (E_{e^+}+E_{e^-})^2 - (\vec{p}_{e^+}+\vec{p}_{e^-})^2$$</p>
<p>But there is no data for the energy, how do the high energy physicists do it? Do they guess some value for the energy and make plots with that guess?</p>
| 4,059 |
<p>Suppose we are given two conducting, cocentric spheres of radius $a_1$ and $a_2$ respectively. The inner sphere with charge $q$, the outer sphere with charge $-q$.</p>
<p>I can calculate the capacitance of this system by calculating the potential difference $U$ between the plates and then use the definition $C = q / U$. This is easy enough and indeed gives me the right result</p>
<p>$$C = \dfrac{4\pi \epsilon_0}{a_1^{-1} - a_2^{-1}}$$</p>
<p>But now I tried to caluclate $C$ by finding out the energy stored in the system in two different ways:
First we have the expression
$$W_{el} = \frac12 \frac {q^2}C$$
for the energy stored in any capacitor. Now I wanted to compare this to the energy stored in the electric field (since that's where the energy is, right?!) $$E(r) = \begin{cases}\frac1{4\pi \epsilon_0} \frac q{r^2} & a_1 < r < a_2 \\ 0 & \text{otherwise}\end{cases}$$
to derive the above formula for $C$ again. The energy density is $w_{el} = \frac {\epsilon_0} 2 E^2$, therefore the energy stored in the electric field is
\begin{eqnarray*}
W_{el} &=& \int w_{el} \;\mathrm{d}V \\
&=& 4\pi \int_{a_1}^{a_2} \frac{\epsilon_0}2 \left(\frac1{4\pi \epsilon_0} \frac q{r^2}\right)^2 r \; \mathrm dr \\
&=& \frac1{4\pi \epsilon_0}\frac{q^2}{2} \int_{a_1}^{a_2} \frac{\mathrm d r}{r^3} \\
&=& \frac1{4\pi \epsilon_0}\frac{q^2}{2} \frac14\left(\frac 1{a_1^4} - \frac1{a_2^4} \right)
\end{eqnarray*}</p>
<p>Comparing the two expressions gives </p>
<p>$$C = \dfrac{16\pi \epsilon_0}{a_1^{-4} - a_2^{-4}}$$</p>
<p>So my question is: Why is this wrong? Where is the energy in this capacitor stored, if not in the electric field (as it doesn't seem to be - unless I have made a mistake in deriving the result somewhere...)?</p>
<p><strong>Edit:</strong> I also noticed, that the second result can be rewritten as</p>
<p>$$C = \dfrac{4\pi \epsilon_0}{a_1^{-1} - a_2^{-1}}\frac 4{a_1^{-3} + a_1^{-2}a_2^{-1} + a_1^{-1}a_2^{-2} + a_2^{-3}}$$</p>
<p>but I don't know whether this has any significance.</p>
<p>Thanks for reading and any help will be greatly appreciated! :)</p>
| 4,060 |
<p>I am trying to do a crude particle identification, using a Bethe Bloch tenchnique. Here is a plot I made from the data that I have
<img src="http://i.stack.imgur.com/SeyLv.png" alt="enter image description here"></p>
<p>From what I've read, the standard method to identify charged particles is by measuring the ionization for a given momentum, in what is known as the Bethe-Bloch Equation</p>
<p>$$\langle \frac{dE}{dx}\rangle = \frac{P_1}{\beta^{P_4}}\left( P_2 - \beta^{P_4} - \log \left(P_3+\frac{1}{(\beta\gamma)^{P_5}}\right)\right)$$</p>
<p>So I tried to fit it and found out the parameters that are used by the detector people (ALICE, in their documentation, they write that the parameters were found using some simulation program). So these parameters become given constants. Now the entire Bethe-Bloch curves are normalized to the values of the minimum ionizing particle. In the literature, this is referred to as MIP value. So I tried a one-parameter fit for the above curve, leaving ROOT to find out the value of the normalization constant. </p>
<p>However, I get absurd results. </p>
<p><img src="http://i.stack.imgur.com/u0tk6.jpg" alt="enter image description here"></p>
<p>The three "fits" in the figure are supposed to be for pions, electrons and protons. However, they are not really accurate, as you can compare this to any of the standard plots released by CERN. </p>
<p>For instance, this is a plot from STAR experiment, which obviously has much lower multiplicity.</p>
<p><img src="http://i.stack.imgur.com/CYv56.jpg" alt="enter image description here"></p>
<p>What are the techniques used by practitioners to find this normalization constant. I am using the same platform but the fitting routine is clearly inadequate for such magnitude of data. </p>
<p>PS: I do not have access to any online data or grid, etc. Most of the analysis code on the web seem to be based on somethings called ESDs and AODs and not for local standalone analysis (which is understandable but undfortunate for me)</p>
| 4,061 |
<p>I've been having trouble with my physics homework. The problem is: </p>
<blockquote>
<p><em>You may have measured the properties of a simple spring-mass system in the lab. Suppose you found ks = 0.9 N/m and m = 0.01 kg, and you observed an oscillation with an amplitude of 0.5 m. What is the approximate value of N, the "quantum number" for this oscillator? (That is, how many levels above the ground state is this oscillator?)</em></p>
</blockquote>
| 4,062 |
<p>What is the advantage/purpose of using $\psi$ for wavefunctions and getting the probability with $|\psi|^2$ as opposed to just defining and using the probability function?</p>
| 4,063 |
<p>I am trying to derive the <a href="http://en.wikipedia.org/wiki/Fermi%E2%80%93Dirac_statistics" rel="nofollow">Fermi-Dirac statistics</a> using density matrix formalism.
I know that </p>
<p>$$<A>= Tr \rho A.$$</p>
<p>So I started from</p>
<p>$$<n(\epsilon_i)>= Tr \rho n(\epsilon_i)=\frac {1}{Z} \sum e^{-\beta \epsilon_i n_i}n_i=\frac {1}{Z} e^{-\beta \epsilon_i}. $$</p>
<p>In the last passage I used the pauli principle ($n_i=0,1$). Now to derive the correct Fermi-Dirac distribution I have to use for $Z=1 +e^{-\beta \epsilon_i}$.
Why I have not to use the general form of </p>
<p>$$Z=\prod_i (1 +e^{-\beta \epsilon_i})~?$$</p>
<p>Can anybody give me a good explanation?</p>
| 4,064 |
<p><img src="http://i.stack.imgur.com/hbp7B.jpg" alt="enter image description here"></p>
<p>The mass is released at height $h$ above the spring, how far will the spring move?
$E_i=mgh, E_f= kx^2/2+mgx$...why the second equation isn't $ E_f= kx^2/2-mgx$? Since it is below the "zero".</p>
| 4,065 |
<p>In the first term the energies are given by the Wentzel–Kramers–Brillouin (WKB) formula</p>
<p>$$ \oint p dq = 2\pi \left( n+\frac{1}{2} \right) $$</p>
<p>However, can this formula be improved to include further corrections? For example the wave function in the WKB approach can be evaluated to several orders of magnitude in $ \hbar $</p>
| 4,066 |
<p>There is a large metal container in form of a cube made of stainless steel. It is used for storing water in it for technical uses. The problem is that all joints at the bottom of the container have micro cavities and water leaks through them very slowly. I am thinking of a method to close these micro cavities from inside. And the most reasonable method I can think of is to make them cover by a layer of limescale, as any electrical teapot or heater does. The only problem is that this is a big container, and it isn't easy to heat that amount of water, or to boil it. </p>
<p>So does anybody know any other way to cover the bottom or just micro cavities in the joints by limescale without heating during long time? Maybe some use of electrolysis?</p>
| 4,067 |
<p>I have a differential operator $L$,</p>
<p>$\displaystyle L = i (t\frac{\partial}{\partial z} - z\frac{\partial}{\partial t})$</p>
<p>I can trivially hit this operator to $x,y,z$ and $t$ as $L x$, $L t$, $L y$, $L z$.</p>
<p>But I have a problem with <em>exponential of that operator</em>. I want to hit this operator to $x,y,z$ and $t$ as well.</p>
<p>$\exp(i\eta L)\,\,x$</p>
<p>($\eta$ is rapidity in this case)</p>
<p>The first thing that comes to my mind is to use definition of exponential of an operator:</p>
<p>$\displaystyle \exp(A) = I + A + \frac{A^2}{2!} + \frac{A^3}{3!} ...$ </p>
<p>But, I don't know why, I don't want to use this infinite sum. There should be a smart way of doing this..</p>
<p>Do you have any suggestions for me?</p>
| 4,068 |
<p>I'm trying to derive the equation for the cosmological fluid:</p>
<p>$$\dot \rho + 3 \frac{\dot a}{a}(\rho +P)=0$$</p>
<p>by starting from the conservation of the stress-energy tensor:</p>
<p>$$\nabla^\mu T_{\mu \nu} = 0$$</p>
<p>with the stress-energy for a perfect fluid in its own frame being:</p>
<p>$$ T_{\mu \nu} = \text{diag} (\rho, a(t)^2 P,a(t)^2 P,a(t)^2 P) $$</p>
<p>in a spatially flat FLRW metric:</p>
<p>$$g_{\mu \nu} = \text{diag}(1,-a(t)^2,-a(t)^2,-a(t)^2)$$</p>
<p>But I keep getting a bogus answer! Consider the equation you get from $ \nabla^\mu T_{\mu \nu} = 0$ when $\nu =0$:</p>
<p>$$
\begin{align*}
\nabla^\mu T_{\mu 0} &= 0 \\
g^{\mu \alpha}\nabla_\alpha T_{\mu 0} &= 0
\end{align*}
$$</p>
<p>$T$ is diagonal, so $\mu$ must be zero, but $g$ is diagonal as well, so if $\mu$ is zero, then so is $\alpha$. This gives:</p>
<p>$$
\begin{align*}
g^{0 0}\nabla_0 T_{0 0} &= 0 \\
\nabla_0 \rho &= 0 \\
\dot \rho &= 0
\end{align*}
$$</p>
<p>Because $\rho$ is just a scalar, so the covariant derivative is the partial derivative. Except this answer is wrong.</p>
| 4,069 |
<p>I am working through Franz Schwabl's book on Statistical Mechanics, and he has a number of derivations of thermodynamic quantities that are different than those I have seen before. I am also having difficulty finding them repeated elsewhere.</p>
<p>In particular, he has a method for calculating $\Omega(E)$, the number of states with a given energy $E$, of a series of $N$ independent Quantum Harmonic Oscillators ($\mathcal{H} = \sum_{j=1}^N\hbar\omega(n_j+\frac{1}{2})$) that I hadn't seen before. Proceeding from the result</p>
<p>$$\Omega(E) = \mathrm{Tr}\,\delta(\mathcal{H}-E)=\sum_{n_1=0}^{\infty}\cdots\sum_{n_N=0}^{\infty}\delta\left(E - \hbar\omega\sum_{j=1}^N\left(n_j+\frac{1}{2}\right)\right),$$</p>
<p>my strategy would be combinatoric: the delta-function turns the unrestricted sums over $n_j$ to a constraint on the total number of quanta. Calculating the number of ways you can partition $n=\sum_{j=1}^Nn_j$ quanta of energy among $N$ oscillators gives you $\Omega(E)$. This is the way we did it in undergraduate stat mech.</p>
<p>Schwabl's approach proceeds differently: by taking the Fourier Transform of the delta function, one obtains</p>
<p>$$\Omega(E) = \int \frac{dk}{2\pi}e^{ikE}\prod_{j=1}^N\left(e^{-ik\hbar\omega/2}\sum_{n_j=0}^{\infty}e^{-ik\hbar\omega n_j}\right)=\int\frac{dk}{2\pi}e^{ikE}\left(\frac{e^{-ik\hbar\omega/2}}{1-e^{-ik\hbar\omega}}\right)^N,$$</p>
<p>where this last step involves <strong>summing a divergent geometric series,</strong> declaring
$$\sum_{\ell=0}^{\infty}e^{-i\alpha\ell} = \frac{1}{1-e^{-i\alpha}}$$
and ignoring the fact that this series doesn't converge in a conventional sense. </p>
<p>This simplifies to
$$\Omega(E) = \int\frac{dk}{2\pi}e^{N(ik(E/N) - \log(2i\sin(k\hbar\omega/2)))}$$</p>
<p>which is solved using the saddle-point approximation. The maximum of the argument of the exponential occurs at a value</p>
<p>$$k_0 = \frac{1}{\hbar\omega i}\log\frac{\frac{E}{N}+\frac{\hbar\omega}{2}}{\frac{E}{N}-\frac{\hbar\omega}{2}}$$</p>
<p>Which is clearly imaginary, <strong>despite the fact that in a Fourier Transform $k$ is supposed to be a real number!</strong></p>
<p><em>In spite of all this</em>, if you evaluate the integral using the saddle point approximation at $k=k_0$, you get the same form for $\Omega(E)$ that one derives through the garden-variety combinatorial argument!</p>
<p>My question(s) are</p>
<blockquote>
<p>Why does this work? Specifically:</p>
<ul>
<li><p>Why does it make sense to write the convergence of a divergent geometric series in the form given here (is this relying on some sense of convergence other than the typical one, and if so, what? And what does that imply about convergence in stat mech?), and </p></li>
<li><p>Why can you use the saddle point approximation when the maximum value does not occur in the space over which you are integrating?</p></li>
</ul>
</blockquote>
<p>Answers to this question might rely on appeals to other situations in which this math occurs and has been rationalized, physically if not mathematically. </p>
| 4,070 |
<p>This is just a question about terminology that is used in the beginning of a chapter about phonons.</p>
<p>In a simple cubic crystal, we can consider elastic oscillations in f.i. the [100] direction. In this direction there is one longitudinal and two transversal oscillation modes. Now my textbook tells that these two transversal oscillation modes can be degenerate. I wonder in which sense the word "degenerate" is used here and I hope that someone can clarify this.</p>
| 4,071 |
<p>I'm currently a third year undergrad writing about Minimal Surfaces. In particular, trapped surfaces and black holes.</p>
<p>What does the <a href="http://www.google.com/search?as_q=Positive+Mass+Theorem" rel="nofollow">Positive Mass Theorem</a> have to do with this? And does the theorem directly predict the existence of black holes?</p>
| 256 |
<p>I have a throlabs half- and quarter-wave plate with rotation plate. There shown the angle scales and a line denotes the fast axis. But what does the angle mean? I do some research and someone said the angle reading tells the angle away from the transmission direction and other said that's the angle away from the fast axis. So here is the transmission direction same as the fast axis? </p>
<p>If I know the incoming light is linear polarized but don't know the orientation.
Is that possible to tell the orientation by using the half-wave plate?</p>
<p>Also, I am studying the same optics element. When I read the material of the polarized beam splitter, I know that the incident beam will be split by the splitter with the two perpendicular outgoing beams. </p>
<p>What really confusing is if the incident light is linear polarized, what can I tell about the outgoing beams after a splitter?</p>
<p>Can I say they are bother linear polarized and what about the polarized orientation?</p>
<p>The last question is pretty odd to me. In the text, it said we can use a quarter-wave plate to elliptical polarize a linear-polarized light. In some special case, the outgoing light could be circularly polarized. But how can I verify that? I tried the following: I let the linear-polarized light go through a half-wave plate so to control the orientation of the polarization, then let that light pass a quarter-wave plate. If I measure the power out of the quarter-wave plate, it is pretty constant. I think it doesn't tell if the beam is elliptical or circular because the power I measure is the average one, right? So I put a polarized beam splitter after the quarter-wave plate to observe the power of the split light. I think if the light is elliptical, the power should change with time, but again it is pretty constant. Doesn't matter how I rotate the quarter-wave or half-wave plate, the power is pretty constant. </p>
<p>Is that anything wrong with this testing way?</p>
| 4,072 |
<p>Suppose a warm body moving in an empty space with high speed. </p>
<p>The body emits radiation based on its temperature. The protons emitted forwards of the body will have higher energy due to Doppler shift than those emitted backwards. Thus they will care greater momentum. The body should slowdown due to emitted radiation.</p>
<p>Is there a mistake in this reasoning? Possibly the number of photons emitted forwards will be smaller?</p>
| 4,073 |
<p>If one looks into a mirror, he can see a certain field of view.</p>
<p>If he places a convex lens that magnifies (or a concave lens that does the opposite) in front of the mirror, but so that he can still see the entire mirror, will it affect the field of view shown in the mirror?</p>
| 4,074 |
<p>The Problem:</p>
<blockquote>
<p><em>An 8-m-long beam, weighing 14,700 N, is hinged to a wall and supported by a
light rope that is attached 2.0 m from the free end of the beam. If the beam is
supported at an angle of 30.0° above the horizontal, (a) find the tension in the
rope and (b) the reaction force of the wall on the beam.</em></p>
</blockquote>
<p><img src="http://i.stack.imgur.com/jhAhT.png" alt="enter image description here"></p>
<hr>
<p>My question: I have found part (a), the tension in the rope appears to be 13203.5 N. To solve part b can I move away from torques as use Newton's second law to sum the forces in the $x$ to find the normal force since the beam is not accelerating? Otherwise, how can I solve part (b)?</p>
| 4,075 |
<p>In David J. Griffiths's <em>Introduction to Electrodynamics,</em> the author gave the following problem in an exercise.</p>
<blockquote>
<p><em>Sketch the vector function</em>
$$ \vec{v} ~=~ \frac{\hat{r}}{r^2}, $$
<em>and compute its divergence, where</em>
$$\hat{r}~:=~ \frac{\vec{r}}{r} , \qquad r~:=~|\vec{r}|.$$
<em>The answer may surprise you. Can you explain it?</em></p>
</blockquote>
<p>I found the divergence of this function as
$$
\frac{1}{x^2+y^2+z^2}
$$
Please tell me what is the surprising thing here.</p>
| 4,076 |
<p>I read that </p>
<blockquote>
<p><em>Halo nuclei could be seen as special Efimov states, depending on the subtle definitions.</em> (The last sentence in the second to last paragraph of <a href="http://en.wikipedia.org/wiki/Efimov_state" rel="nofollow">this</a> Wikipedia article.) </p>
</blockquote>
<p>This does not seem trivial to me in the least. Can someone shed some light on this for me? </p>
| 4,077 |
<p>This question is more about trying to feel the waters in our current abilities to compute (or roughly estimate) the refraction index of vacuum, specifically when high numbers of electromagnetic quanta of a given frequency $\omega$ occupy a spherical symmetric incoming wavefront modes
$$\frac{e^{i k r}}{r^2}$$</p>
<p>I'm interested in intensities above the <a href="http://en.wikipedia.org/wiki/Schwinger_limit" rel="nofollow">Schwinger limit</a>. Do exist analytical or lattice QED estimates?</p>
<p><strong>Why this is interesting?</strong></p>
<p>Usually i try to make my questions as self-contained as possible, but i believe it might be interesting to others why i'm interested in nonlinear vacuum refraction indices, so here it goes:</p>
<p>Let's review what classical theory says about our ability to create micro black holes with electromagnetic radiation. Our sun produces about $10^{26}$ watts of power, so in principle in the future we could harness all that power. The question trying to be answered here is: is the energy output from Sol enough for a sufficiently technically advanced humanity to create micro black holes?</p>
<p>Let's suppose we focus a spherically symmetric beam of light in a single focal point with the purpose of condense enough energy to create a black hole, well, the Schwarzchild radius of a given flow of energy is</p>
<p>$$ R = \frac{ G E }{c^4}$$</p>
<p>substituing constants,</p>
<p>$$ R = 10^{-45} E $$</p>
<p>Now, since this energy propagates electromagnetically as radiation, it needs to stay long enough inside the critical radius so that the black hole forms. If the (radial) refractive index inside a small enough region is $n$, then the light will stay inside the radius a time $T$</p>
<p>$$ R = \frac{cT}{n}$$</p>
<p>equating both terms we are left with an expression for the power that needs to be delivered in a spherical region in order to create a black hole</p>
<p>$$ \frac{cT}{n} = 10^{-45} E $$</p>
<p>$$ \frac{10^{53}}{n} = \frac{E}{T} = P $$ </p>
<p>So, assuming a refractive index of vacuum of 1, it means that we require $10^{53}$ watts focused in a spherical region in order to create a black hole. This is $10^{27}$ more power than what would be available for a humanity that managed to create a Dyson shell around our sun!</p>
<p>But if the refractive index could be managed to be above $10^{30}$ before reaching the focus region, then that future humanity would have a chance to create such micro black holes</p>
<p>Even a less generous increase of the refractive index could be useful, if they could store the energy of the sun for a few years and then zapping it into an extremely brief $10^{-20}$ wide pulse, in a similar fashion as to how the National Ignition Facility achieves 500 TeraWatt pulses (by chirp-pulse compression)</p>
| 4,078 |
<p>What is Curie-Weiss temperature? What is the difference between Curie-Weiss temperature and <a href="http://en.wikipedia.org/wiki/Curie_temperature" rel="nofollow">Curie temperature</a>?</p>
| 4,079 |
<p>This is homework and I need some guidance.
The question I struggle with is:</p>
<p>Place a particle with $m=2$ , $q=3$ in a constant electric field $\vec{E}=(5,0,0)$
Choose $\vec{r}(0)=(0,0,0)$ and $\vec{v}(0)=(0,0,0)$ and time steps $dt=1*10^{-4}$</p>
<p>I've nummerically found a solution using Euler-Cromer method in python and I set the acceleration vector $a=\frac{q\vec{E}}{m}$ Furthermore i must plot the motion in x direction with t, with this solution and in the same plot I must plot the analytic solution of x and t. The problem is what analytic solution. </p>
<p><code></p>
<p>from pylab import *</p>
<p>from mpl_toolkits.mplot3d import Axes3D</p>
<p>from mpl_toolkits import *</p>
<p>q = 3.0</p>
<p>m = 2.0</p>
<p>dt = 1e-4</p>
<p>t0 = 0</p>
<p>t1 = 1 </p>
<p>E = array([5.0, 0.0, 0.0])</p>
<p>t = linspace(t0, t1,(t1-t0)/dt)</p>
<p>n=len(t)</p>
<p>r = zeros((n,3))</p>
<p>v = zeros((n,3))</p>
<p>r[0] = array([0.0, 0.0, 0.0]) </p>
<p>v[0] = array([0.0, 0.0, 0.0])</p>
<p>a=q*E/m</p>
<p>for i in range(len(t)-1):</p>
<pre><code>v[i+1] = v[i] + a*dt
r[i+1] = r[i] + v[i+1]*dt
</code></pre>
<p>figure("Exercize 1a og b", figsize=(12,10))</p>
<p>plot(t, r[:,0], label="$x_{direction}$")</p>
<p>title("Exercize 1a og b")</p>
<p>xlabel("time")</p>
<p>ylabel("position")</p>
<p>legend()</p>
<p>E=array([1.0,2.0,-5.0])</p>
<p>a=q*E/m</p>
<p>for i in range(len(t)-1):</p>
<pre><code>v[i+1] = v[i] + a*dt
r[i+1] = r[i] + v[i+1]*dt
</code></pre>
<p>figure("Exercize 1c",figsize=(12,10))</p>
<p>plot(t, r[:,0], label="$x_{direction}$")</p>
<p>plot(t, r[:,1], label="$y_{direction}$")</p>
<p>plot(t, r[:,2], label="$z_{direction}$")</p>
<p>title("Exercize 1c")</p>
<p>xlabel("time")</p>
<p>ylabel("position")</p>
<p>legend()</p>
<p>show()
</code></p>
<p>Both velocity ad position can be solved analytic. (And is what the motion is.)</p>
| 4,080 |
<p>I understand the nature of light can be complex and has extensive theories/experimental data. We hear light can be both a wave and particle, so why can't it be both, a wave of particles?</p>
| 4,081 |
<p>It is well known that the speed of light $c$ can be interpreted as the speed limit for information propagation. Similarly, the Planck's constant $h$ is interpreted as the minimum quantum package of action/entropy. Is there a similar interpretation for the Newton's constant $G$?</p>
| 4,082 |
<p>I'm reading <a href="http://www.tcm.phy.cam.ac.uk/~bds10/aqp/handout_operator.pdf" rel="nofollow">this tutorial</a> by Ben Simons entitled <em>Operator methods in
quantum mechanics</em> in connection with his course in advanced QM, and I'm a bit puzzled by an identity in page 25, a bit above relation (3.3):</p>
<p>With the momentum operator $\textbf{p}=-i\nabla$ and the vector $\textbf{a}$ we have</p>
<p>$e^{-i\textbf{a}\cdot\textbf{p}}=e^{\ \textbf{a}\cdot\nabla}=\displaystyle\sum_{n=0}^{\infty}\frac{1}{n!}a_{i_1}\cdots a_{i_n}\nabla_{i_1}\cdots\nabla_{i_n}\ ,$</p>
<p>where repeated indices are summed over. What confuses me here is that (in 3D) $\textbf{a}$ and $\nabla$ have 3 components while this expression seems to refer to component $i_n$ where $n\rightarrow\infty$, and I don't recognize the usual expansion of the exponential.</p>
<p>What am I missing here?</p>
| 4,083 |
<p>I am in a serious doubt about it.</p>
<p>Consider a battery of emf E and we connect it to an inductor. Initially the switch is open, now we close the switch. My question is: What mechanism happens just after closing the switch?</p>
<p>When we close the switch, the electric field produced in the conductor by the battery causes the electron to flow in the inductor. As the electrons flow inside the inductor, the flux changes and an emf is induced, my question is that how is this induced emf Ldi/dt is equal to the external emf e, not simply by saying Kirchoff voltage law but by the mechanism happening that it should be E only?</p>
| 4,084 |
<p>I got given this question and im just lost.</p>
<blockquote>
<p>An object that weighs 25 kg experiences two perpendicular
forces of 80N and 50N respectively. </p>
<p>What's the acceleration that object gets?</p>
</blockquote>
<p>What i don't understand is how i would use that information to work out the answer.</p>
<p>Any Thought, I would like to know how you worked it out and what the answer is..</p>
<p><em>I believe that it would be 30N as gravity is not in play here, nor would be mass of the object affect the acceleration due to unknown factors.</em></p>
| 4,085 |
<p>Is the flow of time regular? How would we come to know if the <strong>our</strong> galaxy along with everything in it stops for a while(may be a century) <strong>w.r.t to the galaxies far beyond our reach</strong>. Is there a way to know if flow of time is smooth,or irregular?</p>
<p>PS I would describe myself as an illiterate physics enthusiast, so I hope you'll forgive me if my ignorance is borderline offensive.</p>
| 4,086 |
<p>In this <a href="http://physics.stackexchange.com/questions/26617/how-to-determine-day-night-based-on-latitude-longitude-and-a-date-time">question about how to calculate sunrise</a>,
there is a link to a page that describes a <a href="http://williams.best.vwh.net/sunrise_sunset_algorithm.htm">algorithm to calculate sunrise sunset</a>.</p>
<p>But when I start to follow the instructions I find that there is a lot of magical constants in that instruction and it is hard to understand what is going on.</p>
<p>Can someone help me a little with the magic numbers.</p>
<p>For example this part:</p>
<pre><code>3. calculate the Sun's mean anomaly
M = (0.9856 * t) - 3.289
</code></pre>
<p>Where did he find 0.9856 and 3.289?</p>
<p>And in the next part</p>
<pre><code>4. calculate the Sun's true longitude
L = M + (1.916 * sin(M)) + (0.020 * sin(2 * M)) + 282.634
NOTE: L potentially needs to be adjusted into the range [0,360) by adding/subtracting 360
</code></pre>
<p>Where did he find 1.916, 0.020 and 282.634? </p>
| 4,087 |
<p>The all-electron code Wien2K will optionally calculate the character tables for a specified list of $k$-points. I'd like to know the parity eigenvalue for a given $k$-point and band index. Is there some way to compute the parity eigenvalue from the irreducible representation of the group given that $\{i\}$ is a symmetry class? This would be handy to compute the value of $Z_2$. </p>
| 4,088 |
<p>Consider a block sliding down an incline plane at an angle $\theta$ with the horizontal. For the acceleration as a function of $\theta$ I find $$\ddot{x}=g \ \sin\theta $$ My text then claims we can find the block's velocity after it moves a distance $x_0$ from rest by multiplying both sides by $2\dot{x}$ and doing the following:</p>
<p>$$2\dot{x}\ddot{x}=2\dot{x}g \ \sin \theta$$
$$\frac{d}{dt}(\dot{x}^2)=2g \sin\theta\frac{dx}{dt}$$
$$\int_0^{v_0^2}d(\dot{x}^2)=2g\sin\theta\int_0^{x_0}dx$$
$${v_0}^2=2g\sin\theta \ x_0$$
$$v_0=\sqrt{2g\sin\theta \ x_0}$$</p>
<p>I think I understand up until the 3rd line. The $dt$'s disappear because both sides are exact differentials, yes? Then in the next step, why does $\dot{x}$ (the velocity) vary from 0 to ${v_0}^2$? Thanks in advance.</p>
<p>Also, is there another way to do this?</p>
| 4,089 |
<p>As far as my knowledge goes, Higgs field is only currently formulated in term of classical gauge theory. What is the importance of Higgs field being formulated in term of quantum gauge theory? In other words, why do we need to formulate Higgs field in term of quantum gauge theory?</p>
<p><a href="http://en.wikipedia.org/w/index.php?title=Higgs_field_%28classical%29&diff=618826334&oldid=555240452" rel="nofollow">Reference</a>: "However, no adequate mathematical model of this Higgs vacuum has been suggested in the framework of quantum gauge theory, though somebody treats it as <em>sui generis</em> a condensate by analogy with that of Cooper pairs in condensed matter physics." </p>
| 4,090 |
<p>My layman understanding of the Uncertainty Principle is that you can't determine the both the position and momentum of a particle at the same point in time, because measuring one variable changes the other, and both cannot be measured at once.</p>
<p>But what happens if I measure a charged particle with any number of detectors over a period of time? Can I use a multitude of measurements to infer these properties for some point in the past? If not, how close can we get? That is, how precise can our estimate be?</p>
| 4,091 |
<p>I am trying to figure out if Snell's Law for refraction can be derived from <a href="http://www.google.com/search?q=scalar+diffraction+theory" rel="nofollow">Scalar Diffraction Theory</a>.</p>
<p>The setup is this: light (plane wave, with wave vector $\vec k_i = (k_x, k_y, k_z)$ ) falls on a flat interface which is taken to be x-y plane; the incident side has refractive index $n_i$ . I want to figure out the refracted ray wave vector $\vec k$, on the otherside of the interface, which has refractive index $n$. Of course, I assume that the magnitutes of wave vectors are proportional to the refractive indices of respective media; i.e. $ |k|/|k_i| = n/n_i$.</p>
<p>I start with the Fresnel Diffraction Integral, with plane wave light $$U(\xi,\eta,z=0) = \exp( i \left[k_x \xi + k_y \eta + 0 \right] )$$ and got to the point of showing that $$U(x,y,z) = \exp( i \left[ k_x x + k_y y + k_z^{\prime} z \right] ),$$ where $$ k_z^{\prime} = k - \frac{k_x^2 + k_y^2}{2k}. $$</p>
<p>I was happy to see that this field has the same $k_x$ and $k_y$ values, but no matter how I play with $k_z^{\prime}$, I am unable to show that it is consistent with Snell's Law.</p>
<p>Any pointers? </p>
| 4,092 |
<p>There are a number of factors that I can think of:</p>
<ul>
<li><strong>The larger mass of the hammer</strong> would cause more force to be applied due to
F=MA.</li>
<li><strong>The larger surface area of the pillow</strong> would spread out the force of the impact, applying less pressure at any given point.</li>
<li><strong>Air resistance due to the pillow's larger surface area</strong> seems like it would also reduce the acceleration of the pillow, which would affect the force due to F=MA. Or perhaps it would just reduce the speed the pillow could be swung at, which would lower the pillow's momentum (does that matter?)</li>
<li><strong>The pillow is more compressible</strong>, which I believe should absorb some of the energy of the impact.</li>
</ul>
<p>So the question is two-fold:</p>
<ol>
<li>Are there other factors which I've missed?</li>
<li>What is the relative importance of the factors in determining the overall harm that a pillow versus a hammer would cause?</li>
</ol>
<p>Note: Don't hit people with hammers. Question for fun only ;)</p>
| 4,093 |
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