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<p>Starting with Euclidean space, suppose I make a number of copies of a coordinate system all coincident at the origin, together with copies of a standard unit length.</p>
<p>For any space interval common to all the coordinate systems, all the observers will count the same minimum number of times their unit length can be laid end to end to one another between this interval. And from this invariant counting process we derive the Euclidean metric in terms of its orthogonal components.</p>
<p>What invariant counting process do we carry out to derive the Minkowski metric in terms of its orthogonal components?</p>
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<p>(c.f Di Francesco, Conformal Field Theory chapters 2 and 4).</p>
<p>The expression for the full generator, $G_a$, of a transformation is $$iG_a \Phi = \frac{\delta x^{\mu}}{\delta \omega_{a}} \partial_{\mu} \Phi - \frac{\delta F}{\delta \omega_a}$$
For an infinitesimal special conformal transformation (SCT), the coordinates transform like $$x'^{\mu} = x^{\mu} + 2(x \cdot b)x^{\mu} - b^{\mu}x^2$$</p>
<p>If we now suppose the field transforms trivially under a SCT across the entire space, then $\delta F/\delta \omega_a = 0$.</p>
<p>Geometrically, a SCT comprises of a inversion, translation and then a further inversion.
An inversion of a point in space just looks like a translation of the point. So the constant vector $b^{\mu}$ parametrises the SCT. Then $$\frac{\delta x^{\mu}}{\delta b^{\nu}} = \frac{\delta x^{\mu}}{\delta (x^{\rho}b_{\rho})} \frac{\delta (x^{\gamma}b_{\gamma})}{\delta b^{\nu}} = 2 x^{\mu}x_{\nu} - x^2 \delta_{\nu}^{\mu}.$$
Now moving on to my question: Di Francesco makes a point of not showing how the finite transformation of the SCT comes from but just states it. $$x'^{\mu} = \frac{x^{\mu} - b^{\mu}x^2}{1-2x\cdot b + b^2 x^2}$$ I was wondering if somebody could point me to a link or explain the derivation. Is the reason for its non appearance due to complication or by being tedious?</p>
<p>I am also wondering how, from either of the infinitesimal or finite forms, we may express the SCT as $$\frac{x'^{\mu}}{x'^2} = \frac{x^{\mu}}{x^2} - b^{\mu},$$ which is to say the SCT is an inversion $(1/x^2)$ a translation $-b^{\mu}$ and then a further inversion $(1/x'^2)$ which then gives $x'^{\mu}$, i.e the transformed coordinate.</p>
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<p>Has subatomic particles ever been seen in a state of superposition or do we just detect information like qubits about the state of the particle? So is actual matter in superposition or is it just information about matter that's in a state of superposition?</p>
| 6,845 |
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<p>I have a homework problem:</p>
<blockquote>
<p>The Sun emits $ \sim5 x 10^{23}$ photons per second with $hν > 13.6$
$eV$. If the density of hydrogen atoms in interplanetary space is $n =$
$109 m^{-3}$, what is the size of the <a href="http://en.wikipedia.org/wiki/Str%C3%B6mgren_sphere" rel="nofollow">Stromgren sphere</a>? Assume a
recombination coefficient $α = 2.6 x 10^{-19} m^3s^{-1}$.</p>
</blockquote>
<p>From Wikipedia, I was able to get to</p>
<p>$$
R_S = \left( \frac{3S_*}{4\pi n^2 \beta_2} \right)^\frac{1}{3}
$$</p>
<p>And I know $\beta_2 = \frac{\alpha}{T}$, but I have no idea what the $S_*$ is. Wikipedia describes it as a source of flux, which is obviously the sun, but I cannot figure out anything else about it. I'm actually quite sure that the professor would have given us a different equation, but he never went over it, so I don't know. Anyone know how to solve this?</p>
| 6,846 |
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<p>In the Weyl basis we can separate the spinor field into 2 components: the right-chiral spinor and the left-chiral spinor. Each of these fields has again 2 components which are coupled. What is the physical interpretation of these 2 components that make up the left-chiral (or right-chiral) field?</p>
<p>In the Dirac basis the interpretation of the 4 components is:<br>
1. Electron spin-up<br>
2. Electron spin-down<br>
3. Positron spin-up<br>
4. Positron spin-down<br></p>
<p>So my question is what is the corresponding interpretation in the Weyl basis (in the massless case). Is it like this?<br>
1. Left-chiral electron $\psi_{4}$<br>
2. Left-chiral positron $\psi_{3}$<br>
3. Right-chiral electron $\psi_{2}$<br>
4. Righ-chiral positron $\psi_{1}$<br></p>
<p>If this is the case than I don't understand why the left-chiral electron $\psi_{4}$ couples to left-chiral positron $\psi_{3}$ as can be seen in the equations:</p>
<p>$$ \partial_{t} \psi_{4} + \partial_{x} \psi_{4} - i\partial_{y} \psi_{4} + \partial_{z} \psi_{3} = 0 $$
$$ \partial_{t} \psi_{3} + \partial_{x} \psi_{3} + i\partial_{y} \psi_{3} - \partial_{z} \psi_{4} = 0 $$
$$ \partial_{t} \psi_{2} - \partial_{x} \psi_{2} + i\partial_{y} \psi_{2} - \partial_{z} \psi_{1} = 0 $$
$$ \partial_{t} \psi_{1} - \partial_{x} \psi_{1} - i\partial_{y} \psi_{1} + \partial_{z} \psi_{2} = 0 $$</p>
| 6,847 |
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<p>If an ferromagnetic object is heated and reaches Tc the magnetization gradually drops as we get closer to Tc or it's a instant drop?
Can I assume as I heat the object, the magnetization is weakening gradually? Likewas as it cools?</p>
| 6,848 |
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<p>I am trying to model a pressure sensor in COMSOL. The basic work is that there is current flowing inside it, if the pressure of the gas around it drops down, the temperature goes down as well (ideal gas law) , hence there is a variation in the conductivity (described by the temperature coefficient of resistivity) so variation of current.
I am figuring out that i have a vacuum chamber with a specific amount of gas (let 's say air) and then fixed volume. I then start to pumping out the gas, which means reduction of pressure and reduction of the number of molecules.
I tried to use the built-in COMSOL modules (version 4.3a) but i was not able to properly describe the physics. Then I tried to use the gas law to "manually" calculate the temperature of the gas in the chamber and i have:</p>
<p>$PV=nRT\Rightarrow T=\frac{PV}{nR}$</p>
<p>but then i have two unknown parameters. I mean, i can decide to sweep the pressure (for instance simulating a drop of pressure from 760 torr up to 1E-4 torr) but i will have problem with the n (how many molecules have been pumped out from my vacuum chamber?) and viceversa. Is there a way to describe the pressure inside the chamber as a function of the number of molecules?</p>
| 6,849 |
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<p>Please help. My example:</p>
<p>You take a sheet of steel and suspend it in the air.</p>
<p>Then you take a permanent magnet (i.e. a neodymium magnet) and attach it to the sheet of steel and from that you suspend a 1kg weight. The weight and magnet will stay put providing that the room doesn’t change and that gravity is still acting on the objects.</p>
<p>Now you take a separate piece of steel (which is not magnetized) and try attach that and the 1kg weight to the suspended sheet of steel. You notice that without an external magnetic force you are unable to attach the two pieces together.</p>
<p>Now here is where my confusion lies. You then attach a electromagnet to the 2nd piece of steel and notice that it stick but the battery you have used to power the device drains. But all this time the permanent magnet with the same load and in the same conditions hasn’t why??</p>
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<p>I'm slightly confused with the following situation:</p>
<p>Suppose you have an electron in a tight-binding model, and let's say we are in one dimension with $N$ lattice sites. </p>
<p>Add to this a single impurity at a specific site $i$. </p>
<p>The pure system is translationally invariant and has as eigenfunctions the plain waves. As a consequence, the probability to find the (free) electron on the impurity site should scale as $1/N$ and I'd imagine that the full forward-scattering Green function for the problem has some form </p>
<p>$$G(k,\omega) = G_0(k,\omega) + \frac{1}{N} T(k,\omega) G_0(k,\omega)^2$$</p>
<p>where $T(k,\omega)$ would be some function independent of $N$.</p>
<p>As a consequence, in the limit of an infinite system $N \rightarrow \infty$ I would expect to not see the impurity effect at all as a consequence of that $1/N$ factor. </p>
<p>However, if I imagine that the impurity is a temperature effect (an example would be a flipped local spin in a ferromagnetic ground state, or an excited phonon) then the impurity could occur on any of the $N$ sites of the system. If I confine myself to very low temperatures where I assume that the entire system contains at most one impurity, the temperature averaged propagator would then look like</p>
<p>$$G_T(k,\omega) = 1/Z \times \left[ \left\langle \phi_0 | \hat G(k,\omega) |\phi_0 \right\rangle + \sum_i e^{-\beta E_I} \left\langle i | \hat G(k,\omega + E_I) | i \right\rangle \right]$$</p>
<p>where $|\phi_0\rangle$ is the free system's ground state and $| i \rangle$ is the state with an impurity at site $i$ and $E_I$ is the energy cost of the impurity.</p>
<p>$Z$ is the partition function and in our simple example just gives $1 + Ne^{-\beta E_I}$.</p>
<p>If that's all correct, then the finite temperature Green's function should just be
$$G(T,k,\omega) = \frac{G_0(k,\omega) + Ne^{-\beta E_I} \left( G_0 + G_0^2 1/N T_{kk}(\omega)\right)}{1 + Ne^{-\beta E_I}}$$
which can be simplified to
$$G_0(k,\omega) + \frac{e^{-\beta E_I} G_0^2 T_{kk}}{1 + Ne^{-\beta E_I}}$$</p>
<p>What confuses me here is that if we let $N$ go to infinity, the impurity term will again vanish at small temperature and thus seems to play no role at all in determining the Green function or the spectral function, but clearly that cannot be correct, since we know that impurities can, for example, lead to bound states.</p>
<p>Am I missing something really obvious here?</p>
<p>EDIT:
With your answers so far, let me just add what I think I could do right now. </p>
<p>At low temperature, I'd assume a low density of thermally excited impurities, so I will do an expansion to first order in that density. The partition function is
$$Z = (1 + e^{-\beta E_I})^N = \sum_{n}^N \begin{pmatrix}N\\n\end{pmatrix} e^{-\beta n E_I}.$$</p>
<p>The numerator for $\langle G \rangle$ becomes approximately, only accounting for interactions with <em>one</em> impurity at a time, justified by the low density:
$$\sum_n e^{-\beta n E_I} \begin{pmatrix}N \\ n \end{pmatrix} \left[ G_0(\omega) + n T(\omega) G_0^2(\omega)\right]$$
Now the $G_0$ don't depend on anything we're summing over, so we can pull them out of the sum and obtain</p>
<p>$$\langle G \rangle = G_0 + \langle n \rangle T G_0^2$$
where I omit the momentum and frequency arguments because they're the same everywhere anyway.</p>
<p>I like this answer, it makes intuitive sense: The extra term due to impurity scattering is, to first order, linear in the expected number of thermally excited impurities. I understand that this gets the physics of disordered systems wrong since it can't account for interference due to multi-impurity-scattering and thus for example won't ever see Anderson localization. But just to get started on this, I think it's at least the correct way to start thinking about things.</p>
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<p>Arguably the most well-known and used system of units is the SI-system. It assigns seven units to seven ‘fundamental’ quantities (or dimensions). However, there are other possible options, such as Gaussian units or Planck units. Until recently, I thought that these different systems differed only in <em>scale</em>, e.g. inches and metres are different units, but they both measure <em>length</em>. Recently though, I discovered that it is not simply a matter of scale. In the Gaussian system for example, charge has dimensions of $[mass]^{1/2} [length]^{3/2} [time]^{−1}$, whereas in the SI-system it has dimensions of $[current] [time]$. Also, I have always found it a bit strange that mass and energy have different units even though they are equivalent, but I find it hard to grasp that a quantity can be ‘fundamental’ in one system, and not in an other system.</p>
<p>Does this mean that all ‘fundamental’ quantities are in fact arbitrary? Would it be possible to declare a derived SI-unit fundamental, and build a consistent system with more base units? What is the physical meaning of this?</p>
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] |
<p>Given for example, Hydrogen electron in ground state. What is probability to find that electron at certain distance (not interval of distances) from center of nucleus, for example at radial coordinate $r=0.5 \cdot 10^{-10}$ (and any angle)? </p>
| 6,853 |
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<p>Is possible estimate the needed size of an geodesic dome (like in the Eden project) for creating an real hydrosphere - especially clouds (and rain)?</p>
<p>With other words, under what circumstances can happen clouds formation in an closed space -isolated from the outer atmosphere?</p>
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<p>The de Donder gauge is often used to simplify the linearised equations of motion of general relativity. If the metric is linearised as $g_{ab} = \bar g_{ab} + \gamma_{ab}$, then the de Donder gauge reads<br>
$\nabla^a(\gamma_{ab} - \frac{1}{2}\bar g_{ab}\gamma) = 0$.</p>
<p>The partial differential equation for the gauge transformation vector $v^a$ is
$ \nabla^b\nabla_b v_a + R_a^b v_b = \nabla^a(\gamma_{ab} - \frac{1}{2}\bar g_{ab}\gamma)$. </p>
<p>In chapter 7.5 of Wald, I read that this equation can always be solved because it is of the form
$g^{ab}\nabla_a\nabla_b \phi_i + \sum_j (A_{ij})^a\nabla_a \phi_j + \sum_j B_{ij}\phi_j + C_i$.<br>
Theorem 10.1.2 of Wald says that in a globally hyperbolic spacetime this equation has a well posed initial value formulation on any spacelike Cauchy surface.</p>
<p>In stead of de Donder gauge, I want to use a similar gauge:<br>
$\nabla^a(\gamma_{ab} - n \bar g_{ab}\gamma) = 0$.<br>
The partial differential equation changes to<br>
$ \nabla^b\nabla_b v_a + (1 -2n)\nabla_a\nabla_b v^b + R_a^b v_b = \nabla^a(\gamma_{ab} - n\bar g_{ab}\gamma)$.</p>
<p>This equation is not covered by theorem 10.1.2 of Wald. My question is: is the existence of a solution for this equation guaranteed in an AdS background when $n=1$?</p>
| 6,855 |
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<p>Obviously electrons annihilate with positrons, but can a muon annihilate with an positron, or can an anti-taon cancel with a muon? similarly for quarks of different species, e.g. u and anti-strange.</p>
<p>I think this is possible as long as quantum numbers like charge and spin are conserved, with the excess energy being given off in kinetic energy, but has it ever been observed?</p>
| 6,856 |
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<p>I've been trying to get my head around the formalisms used in the Standard Model. From what i've gathered Dirac Spinors are 4 component objects designed to be operated on by Lorentz Transformations much like 4-Vectors are in Special Relativity. However they also incorporate additional information: Spin and "Handedness". Due to the nature of Spin, Spinors also transform differently then vectors.</p>
<p>This leaves me with the impression that the 4 components can be classified as: Left Handed and Spin Up, Left Handed and Spin Down, Right Handed and Spin up, Right Handed and Spin Down.</p>
<p>My question is if this impression is the right general idea or not?</p>
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<p>My question is what is the relation between N=2 super Yang-Mills and its twisted version topological field theory? After twisting N=2 super Yang-Mills, i.e. diagonally embedding $SU(2)'_R$ into $SU(2)_R \times SU(2)_I$, we get a topological field theory. My question is since N=2 SYM and TQFT are different i.e. one is physical and the other is topological. Why can we use TQFT to calculate partition of N=2 SYM? What are the same for these two different theories?</p>
<p><strong>Update</strong>: From the second paper of Trimok, the authors claim that SYM under twist are just redefination. How to understand it?</p>
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<p>I had a question on a dimensional regularization identity. A reference or a quick sort of derivation will be greatly appreciated. I looked at some textbooks of QFT, but couldn't find the one I was looking for.</p>
<p>I found in <a href="http://www.maths.tcd.ie/~cblair/notes/list.pdf" rel="nofollow">http://www.maths.tcd.ie/~cblair/notes/list.pdf</a>, a result for $\int\frac{d^dp(p^2)^a}{(p^2+D)^b}$ (see eq. 3.2 of the above link). I wanted something which is $\int\frac{d^dp(p^2)^a}{(p^2+2pq+D)^b}$ i.e the integrand has a linear power of $p$ too. May be a derivation of the previous equation will help. But anyway, some light on $\int\frac{d^dp(p^2)^a}{(p^2+2pq+D)^b}$ or $\int\frac{d^dp(p_\mu p_\nu..p_\lambda)}{(p^2+2pq+D)^b}$ is what I need. Thanks in advance.</p>
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<p>Water (and other substances) can exist in many distinct solid phases (with different crystallic micro-structure), but only in two fluid phases - liquid and gaseous, in which the molecules are oriented randomly (they is no long range order). Is there an explanation in the molecular theory, why there are just two "disordered" phases? Why isn't there just one? Or more than two?</p>
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<p>Let's have generating functional $Z(J)$:
$$
Z(J) = \langle 0|\hat {T}e^{i \int d^{4}x (L_{Int}(\varphi (x)) + J(x) \varphi (x))}|0 \rangle , \qquad (1)
$$
where $J(x)$ is the functional argument (source), $\hat {T}$ is the chronological operator, $\varphi (x)$ - some field. </p>
<p>I want to understand the reasons for its introduction for the summands of expansion of S-matrix. As I read in the books, it helps to consider only the vacuum expectation values, forgetting about in- and out-states. But in $(1)$ appear summands like $\int \frac{J(p)dp}{p^2 - m^2 + i0}$ instead of the contributions from external lines. It may refer to the internal lines. So what to do with them and are there some other reasons to introducing $(1)$ except written by me?</p>
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] |
<blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="http://physics.stackexchange.com/questions/1775/why-is-there-no-absolute-maximum-temperature">Why is there no absolute maximum temperature?</a> </p>
</blockquote>
<p>Is there any upper limit of the temperature that can be achieved? Is the speed of light a kind of barrier?</p>
| 237 |
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<p>I have recently been looking into the <a href="http://en.wikipedia.org/wiki/Multiple_time_dimensions">two-time theories</a> and the implied concepts.</p>
<p>For me this seems slightly hard to grasp.</p>
<p>How can I see the basic concept in this theory in a fundamental way based on its implied interaction with normal 3+1 dimension?</p>
<p>I am interested specifically in how <a href="http://en.wikipedia.org/wiki/Gauge_symmetry">gauge symmetries</a> that effectively reduce 2T-physics in 4+2 dimensions to 1T-physics in 3+1 dimensions without any <a href="http://en.wikipedia.org/wiki/Kaluza-Klein_theory">Kaluza-Klein</a> remnants.</p>
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<p>This may be a stupid question but I am having trouble getting the same result in CGS units as I would if I used SI units for a unitless calculation. </p>
<p>I have to calculate $E_c=\frac{m_e c\, \omega_c}{eB}$ I have in CGS units that $\frac{e}{m_ec}$ is $(1.76\times10^{7}s^{-1}G^{-1})$ and $B=10^{-4}G=10^{-8}T$ and $\omega_c$ is just some frequency. Essentially, when I do this calculation in SI units and in CGS units I get very different results. From this equation it seems clear to me that $\frac{e}{m_ec}$ in CGS and SI should only be off by a few factors of 10 (namely 4) since $B$ is scaled by $10^{-4}$ from CGS to SI, but this is not the case... What am I doing wrong? </p>
<p>Edit: Here are my calculations.</p>
<p>(i) CGS: $\frac{(1.52\times10^{18}s^{-1})}{(1.76\times10^7s^{-1}\,G^{-1})(3\times10^{-4}G)}$</p>
<p>(ii) SI: $\frac{(9.11\times10^{-31}\mathrm{kg})(3\times10^8\mathrm{m \, s^{-1}})(1.52\times10^{18}s^{-1})}{(1.6\times10^{-19}C)(3\times10^{-8}T)}$</p>
<p>Unless the value I'm using for $\frac{e}{m_ec}$ in CGS is wrong...</p>
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<p>This posting is directly related to the issue in <a href="http://physics.stackexchange.com/questions/32385/the-system-and-the-measuring-gadget">The System and the Measuring Gadget</a>.</p>
<p>The QM expectation is given by:</p>
<p>$$\langle\sigma_{1}.\vec{a}{\;}\sigma_{2}.\vec b\rangle=-\vec a.\vec b$$</p>
<p>In the above relation we are considering <em>the measured value</em> of spin which is the outcome between the value of some property of the system itself and the measuring gadget</p>
<p>The "classical" formula for evaluating the expectation with the hidden variable is as follows:</p>
<p>$$P(\vec a.\vec b)=\int d\lambda\,\rho (\lambda) A(\vec a)B(\vec b)$$</p>
<p>Now some property of the system may depend on the value of $\lambda$ and the probability distribution $\rho (\lambda)$. Is the effect of measurement being fully accounted for by the the hidden variable $\lambda$ and the <a href="http://en.wikipedia.org/wiki/Probability_density_function" rel="nofollow">pdf</a> $\rho (\lambda)$, especially in view of the fact
that the process of measurement modifies the wave function itself.</p>
<p>Would it be possible remove the contradiction between QM and commonsense intuition,expressed through Bell's Inequality, by considering the above factors?</p>
<p><a href="http://www.drchinese.com/David/Bell.pdf" rel="nofollow">Reference for Bell's Original Paper</a>. </p>
| 6,864 |
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<p>I've already searched Physics StackExchange for some similar question but I didn't find anything about this.</p>
<p>Assumptions:</p>
<ul>
<li>Earth is a perfect sphere with it's core (X,Y,Z) -> (0,0,0) as a reference-frame center</li>
<li>Air resistance can be ignored</li>
<li>Earth rotation can be ignored</li>
<li>Moon gravity-effect can be ignored</li>
</ul>
<p>And if I know (projectile starting properties):</p>
<ul>
<li>Current Earth GPS-coordinates</li>
<li>Starting angle</li>
<li>Starting direction (relative to (0,0,0))</li>
<li>Starting velocity (can be larger than Earth-escaping speed)</li>
<li>(mass of the projectile is irrelevant (I guess) when we know starting velocity of the projectile)</li>
</ul>
<p>How could I calculate aprox. coordinates of a projectile landing somewhere around the Earth globe (OR detect that projectile will "leave" the Earth)?</p>
<p>EDIT: I've found tons of links on the net about projectile trajectories when starting velocities are quite small (& Earth can be considered like a flat plane), but non about upper situation. </p>
| 6,865 |
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<p>Perhaps an elementary questions. Given a time limited measurement situation, would it be better for one to measure more averages or more data points?</p>
<p>More averages will increase the SNR by $$\sqrt{n}$$ , i.e., making the data point more reliable, but more data points may make the fitting better. </p>
<p>Consider the model is A*exp(-Bt) + C*exp(-Dt) which is a difficult model to fit when noise is introduced.</p>
<p>Assume due to time limit, one could only do 100 measurements in total. Should one measure 20 data points with 5 averages each or 100 data points, or 5 data points with 20 averages each?</p>
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<p>What is the capture cross-section of a black hole region for ultra-relativistic particles? I have read that it is </p>
<p>$$\sigma ~=~ \frac{27}{4}\pi R^{2}_{s}$$ </p>
<p>for a Schwarzschild BH in the geometric optics limit. Where does the coefficient come from?</p>
<p>Edit - Sources:</p>
<ol>
<li><a href="http://prd.aps.org/abstract/PRD/v18/i4/p1030_1">Absorption and emission spectra of a Schwarzschild black hole.</a></li>
<li><a href="http://arxiv.org/abs/gr-qc/0503019">Fermion absorption cross section of a Schwarzschild black hole.</a></li>
</ol>
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<p>At the introduction to quantum mechanic <a href="http://en.wikipedia.org/wiki/Phase_velocity" rel="nofollow">phase</a> $v_p$ and <a href="http://en.wikipedia.org/wiki/Group_velocity" rel="nofollow">group</a> $v_g$ velocities are often presented. I know how to derive $v_p$ and get equation:</p>
<p>$$
\scriptsize
v_p=\frac{\omega}{k}
$$</p>
<p>What i dont know is how to explain a derivation of a group velocity $v_g$ to myself. Our professor did derive it, but i am having some difficulties with it.</p>
<hr>
<p>1st he did a superposition of 2 waves with the same amplitude $s_0$:</p>
<p>$$
\scriptsize
\begin{split}
s &= s_0 \sin(\omega_1t-k_1x) + s_0 \sin(\omega_2t - k_2 x)\\
s &= s_0 \left[ \sin(\omega_1t-k_1x) + \sin(\omega_2t - k_2 x) \right]\\
s &= 2s_0 \left[ \sin\left(\frac{(\omega_1t-k_1x)+(\omega_2 t -k_2 x)}{2}\right) \cdot \cos\left(\frac{(\omega_1t-k_1x)-(\omega_2t - k_2 x)}{2}\right) \right]\\
s &= 2s_0 \left[ \sin\left(\frac{\omega_1 + \omega_2}{2} t - \frac{k_1 + k_2}{2} x\right) \cdot \cos\left(\frac{\omega_1 - \omega_2}{2} t - \frac{k_1 - k_2}{2} x\right) \right]\\
s &= 2s_0 \left[ \sin\left(\overline \omega t - \overline{k} x\right) \cdot \cos\left(\frac{\Delta \omega}{2} t - \frac{\Delta k}{2} x\right) \right]\\
\end{split}
$$</p>
<hr>
<p>Here $\overline \omega$ is larger than $\Delta \omega$ and this is why:</p>
<ul>
<li>$\scriptsize\sin \left(\overline{\omega}t - \overline k x\right)$ is a part which declares an <em>envelope</em> and</li>
<li>$\scriptsize\cos \left(\frac{\Delta{\omega}}{2}t - \frac{\Delta k}{2} x\right)$ is a part which declares <em>phases inside an envelope</em>.</li>
</ul>
<p><img src="http://i.stack.imgur.com/GgLq5.png" alt="enter image description here"></p>
<hr>
<p>Than professor takes only a part which declares an envelopa and says that phase of this part must be <strong>constant</strong> like this:</p>
<p>$$
\scriptsize\frac{\Delta{\omega}}{2}t - \frac{\Delta k}{2} x = const.
$$</p>
<p><strong>QUESTION:</strong>
What does this mean? Does a constant phase mean to only look at one point which is allways at the same distance from $x$ axis? Please someone explain this a bit.</p>
<p>Well then he derives the group velocity easily from now on like this:</p>
<p>$$
\scriptsize
\begin{split}
\frac{\Delta{\omega}}{2}t - \frac{\Delta k}{2} x &= const.\\
\frac{\Delta k}{2} x &= \frac{\Delta{\omega}}{2}t - const.\\
x &= \frac{\Delta{\omega}}{ \Delta k} t - \frac{2}{\Delta k}const.\\
\end{split}
$$</p>
<p>If i partially diferentiate $x$ i finally get group velocity:</p>
<p>$$
\scriptsize
\begin{split}
v_g &= \frac{\partial x}{\partial t} \\
v_g&= \frac{\Delta{\omega}}{ \Delta k}\\
v_g&= \frac{\textrm d{\omega}}{ \textrm d k}
\end{split}
$$</p>
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<p>Im designing a model for Kelvin Method. Some of my calculation results are as follows: </p>
<ol>
<li>Radius of the membrane : 50 micron </li>
<li>thickness of the membrane : 3.25 micron </li>
<li>resonate frequency : 1.32MHz</li>
<li>spring constant : 1.81*10^4 N/m</li>
<li>mass : 1.045*10^-8 Kg</li>
</ol>
<p><img src="http://i.stack.imgur.com/B6rjl.png" alt="enter image description here"></p>
<p>Vdc is applied, so that we can get a continuous vibrations of the membrane.
Amplitude of vibration is same as the amplitude of Vac when there is no damping. i want to calculate the amplitude when damping(air and support) is included </p>
<p>I want to introduce damping into my design. Im not sure of the calculation. How can i calculate the following </p>
<ol>
<li>Damping coefficient (considering the system is surrounded by air)</li>
<li>damping ratio </li>
<li>Final amplitude with damping </li>
</ol>
| 6,869 |
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<p>A spring is a mechanical flexible element and has to absorb energy from outside gradually. We are manufacturing springs in helical shapes on a large scale. What is the reason behind this?</p>
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<p>What is the difference between implicit and explicit time dependence e.g. $\frac{\partial \rho}{\partial t}$ and $\frac{d \rho} {dt}$?</p>
<p>I know one is a partial derivative and the other is a total derivative. But physically I cannot distinguish between them. I have a clue that my doubt really might be understanding the difference between explicit and implicit time dependence. </p>
| 362 |
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<p>You have the following energy maps for 14 different structures so about 4-5 pixels for one unit. The color density represents energy.</p>
<p><img src="http://i.stack.imgur.com/4VNj3.png" alt="enter image description here"></p>
<p>I am thinking that you can set multivariate time series models from energy maps by <code>vgxqual</code> and <code>vgxset</code> for instance in Matlab.</p>
<p><strong>How are energy maps related to the measure of stability in quantum physics a) by wave models and b) non-commute dynamics?</strong></p>
<p>I think the latter measure is the right one here.
There may be different models too to measure stability however.</p>
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<p>Let's say the objects are marble size or even single atoms or quarks. They are placed in an otherwise empty universe(expanding or non-expanding) at opposite ends of the universe with an arbitrarily large distance between them. With a combination of great enough distance and small enough mass will the gravitational pull between the two objects ever equal zero or merely approach it? Given an infinite amount of time would they ever meet?</p>
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<p>There are many applications for orbital space mirrors in astronomy (better telescopes) and space propulsion (solar power for deep space probes), but this is limited by the minimum beam divergence achievable with current technology</p>
<p>So, i'm trying to understand what physical and technological limits exists in our capacity to build mirrors that can keep as small beam divergence as possible. For instance, a sail probe to Saturn would require that the beam doesn't significantly diverge above 300m-600m (the biggest sail we can conceive of building in the inmediate future) at distances as big as 5-6 AU ($10^{11} - 10^{13}$ meters)</p>
<p>What is the best beam focusing divergence we can achieve for solar light with mirrors <em>right now</em>, and what limits the improvement of this? technological limits? fundamental physical limits?</p>
<p><strong>Edit</strong> let's assume the concrete case of a wavelength of $10^{-6}$ meters, and a distance of $10^{12}$ meters (Neptune orbit). Can't i, for instance, build a focusing element with a focal length of $10^{12}$ meters that would push the far field beam divergence at farther distances from the focal point? Is this a manufacturing limitation of focusing element engineering (not enough precision to build lens made of atoms with the required focal length), or something more instrinsic, say, a focal point cannot be farther than some finite distance that depends on the wavelength?</p>
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<p>I have a problem in which the tell me that you drop a bag of 50 kg of sand from 10 meters high, and you have to caltulate the entropy difference of the sand, asuming that the speific heat of the sand is so high that its temperature (298K) doesn't change. My result is 16.5 J/K, and my problem is with the sign. The book doensn't give an explanation but just an answer and it's positive, but I'm thinking that in the impact the heat must flow out of the sand, as it's loosing its kinetic energy, so the entropy would decrease, and the surroundings entropy would increase; what am I missing, or what is really going on here?</p>
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<p>What force do the arms have to generate to do a pushup? Let us look a this simplified model:</p>
<p><img src="http://i.stack.imgur.com/ueWHS.jpg" alt="pushup physics"></p>
<p>The body can be represented by the green plank of mass B. Its angle to the ground is $\theta$. </p>
<p><em>This question was asked on <a href="http://fitness.stackexchange.com/questions/1717/what-is-the-effective-weight-when-doing-push-ups">fitness.stackexchange.com</a>, although not phrased with physics in mind.</em></p>
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<p>Person A in reference frame A watches person B travel from Star 1 to Star 2 (a distance of d). Of course, from person B's reference frame, he is at rest and is watching Star 2 traveling to him. </p>
<p>Now we know from the principle of relativity, each one will measure the other one’s clock as running slower than his own.</p>
<p>Let’s say that Person A measures Person B’s speed to be v, and that Person A measures 10 years for person B to make it to Star 2. Let’s also say that person B is moving at the speed so that the Gamma Factor is 2. This means person A observes person’s B’s clock to have elapsed a time of 5 years. </p>
<p>Now let’s look at this from Person B’s perspective:</p>
<p>Person B observes Star 2 approaching (and Star 1 receding from) him also at speed v. Since the two stars are moving, the distance between them is length contracted (after all, if there were a ruler in between the stars, the moving ruler would be contracted) by a factor of 2. Since person B measures the initial distance to Star 2 to be d/2 and its speed v, he calculates the time to Star 2’s arrival to be 5 years. Since he observes person A’s clock as running slow (since Person A is moving also at speed v), when Star 2 arrives, he measures Person A’s clock to have elapsed a time of 2.5 years. </p>
<p>Do you see why I’m confused? Person A measures Person B’s elapsed time to be the same as Person B measures Person B’s elapsed time (both 5 years), but Person B does not measure Person A’s elapsed time to be the same as Person A measures Person A’s elapsed time (Person B get’s a measurement of 2.5 years while Person A measured 10 years). This is asymmetrical, which probably means it is wrong. But I’m not sure what the error is.</p>
<p>I suspect if I had done this correctly, each person should measure his own elapsed time to be 10 years and measure the other’s elapsed time to be 5 years. This would be symmetrical and would make the most sense, but again, I can’t seem to justify how person B wouldn’t measure his trip time to be 5 years. </p>
<p>What's my mistake?</p>
| 6,875 |
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<p>Is it possible to predict what the final temperature will be by taking temperature samples. For example, an object is 0ºC and moved to a room above 0ºC. I'm taking temperature of the object using a thermometer every second. Can I predict (approximation) on what the final temperature would be after a few samples? I guess the more samples the more accurate it would be. Can I calculate when the final temperature might occur based the rate of the temperature change?</p>
<p>Is there any formulas for these kind of calculations?</p>
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<p>Most derivations I have seen of the Born-Oppenheimer approximation are made using wave-functions. To understand it better, I was trying to write a derivation using Dirac notation, but I am stuck. I am going to post what I have done so far so you guys can help me out.</p>
<p>The hamiltonian of the molecule can be written as the sum of two parts, $H_\text{mol} = H_\text{el} + H_\text{nuc}$, where $H_\text{nuc}$ is the hamiltonian of the nuclei by themselves, and $H_\text{el}$ is the hamiltonian of the electrons interacting with the nuclei:</p>
<p>$$H_\text{el} = T_\text{el} + V_\text{el-el} + V_\text{el-nuc}$$
$$H_\text{nuc} = T_\text{nuc} + V_\text{nuc-nuc}$$</p>
<p>We want to find the energy levels of the molecule. That is, we want to solve $H_{\text{mol}} |\mathcal{E}\rangle =\mathcal{E} |\mathcal{E}\rangle$, where $\mathcal{E}$ and $|\mathcal{E}\rangle$ are the eigenenergy and corresponding eigenket of the molecule.</p>
<p>The state space of the molecule can be separated into electronic and nuclear parts: $\mathcal{S}_\text{mol} = \mathcal{S}_\text{el} \otimes \mathcal{S}_\text{nuc}$. Let $|R\rangle \in \mathcal{S}_\text{nuc}$ be the position basis of the nuclei, where $R$ denotes the coordinates of all the nuclei. </p>
<p>Define $H_{\text{el}}(R) = \langle R| H_{\text{el}} |R\rangle$, which is an operator in $\mathcal{S}_{\text{el}}$. Let $E_a(R)$ and $|E_a(R)\rangle$ be the eigenvalues and corresponding eigenkets of $H_{\text{el}}(R)$ in $\mathcal{S}_{\text{el}}$, so that $H_{\text{el}}(R) |E_a(R)\rangle = E_a(R) |E_a(R)\rangle$. For each $R$, the kets $|E_a(R)\rangle$ make a basis for $\mathcal{S}_{\text{el}}$.</p>
<p>The set of kets $|E_a(R)\rangle |R\rangle \in \mathcal{S}_{\text{mol}}$ for all $R$ and $a$ then make a basis for $\mathcal{S}_{\text{mol}}$. Using this basis, the state of the molecule can be written:
$$|\psi \rangle =\sum _a \int \chi_a(R) |E_a(R)\rangle |R\rangle dR$$</p>
<p>where $\chi_a(R)$ is an amplitude function.</p>
<p>Note that:</p>
<p>$$H_\text{el} |E_a(R)\rangle |R\rangle = [ H_\text{el}(R) |E_a(R)\rangle ] |R\rangle = E_a(R) |E_a(R)\rangle |R\rangle $$</p>
<p>Therefore, the molecular eigenproblem $H_{\text{mol}} |\psi \rangle =\mathcal{E} |\psi \rangle$ can be written:</p>
<p>$$\sum_a \int (E_a + T_\text{nuc} + V_\text{nuc-nuc} - \mathcal{E}) \chi_a(R) |E_a(R)\rangle |R\rangle dR = 0$$</p>
<p>Multiplying by $\langle E_a(R)|$ on the left:</p>
<p>$$\int (E_a + T_\text{nuc} + V_\text{nuc-nuc} - \mathcal{E}) \chi_a(R) |R\rangle dR = 0$$</p>
<p>At last, we define a ket $|\chi_a\rangle \in \mathcal{S}_\text{nuc}$ such that $\chi_a(R) = \langle R | \chi_a \rangle$:</p>
<p>$$|\chi_a\rangle := \int \chi_a(R) |R\rangle dR$$</p>
<p>Then we can write:</p>
<p>$$(T_\text{nuc} + V_\text{nuc-nuc} + E_a - \mathcal{E}) | \chi_a \rangle = 0$$</p>
<p><strong>HERE ENDS MY DERIVATION SO FAR.</strong></p>
<p>I must have done something wrong somewhere, because the final equation that I obtain is, as far as I can tell, the Born-Oppenheimer <em>approximation</em>, but I am obtaining it here as an <em>exact</em> equation. What did I do wrong?</p>
<p>Also, if anyone knows of some reference, textbook or paper, that deals with the Born-Oppenheimer approximation in Dirac notation, please post it.</p>
| 6,877 |
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] |
<p>The many-worlds interpretation of quantum physics is built around a configuration space, where the position of a particle is three components of the position of that universe.</p>
<p>What happens with particle-antiparticle creation or annihilation? It can't just change the number of dimensions, can it?</p>
| 6,878 |
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<p>According to an answer in <a href="http://skeptics.stackexchange.com/questions/1664/it-is-not-the-voltage-that-kills-you-it-is-the-current/1667#1667">this thread</a> on Skeptics:</p>
<blockquote>
<p>If you take one of the little 12V
garage door opener batteries and short
out (directly connect) the two
terminals with a piece of wire or
something else. You'll get a light
current flow through the wire or
metal. It may get a little warm. This
battery is only capable of supplying a
small amount of current.</p>
<p>If you take a 12V car battery and
short out the two terminals (don't do
it, it's not fun), you will be met
with a huge current arc that will
likely leave a burn mark on whatever
was used to short it. This is because
the car battery is capable of
discharging a large amount of current
in a very short period of time.</p>
</blockquote>
<p>I'm not sure how this could work given Ohm's law V=IR. If we assume the resistance of toucher is constant, then we'd expect the current to be the same as well.</p>
<ol>
<li>Could it be that a car battery has
less internal resistance than a
garage battery?</li>
<li>Does it have anything to do with contact area? If it does, then how would you model it? Generally resistances add in series, but if I only half touch a contact, then neither the battery nor I change, so you'd expect our resistances to stay the same, but somehow our total resistance changes. So which objects resistance would change - mine or the batteries?</li>
</ol>
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<p><strike>Science and science fiction alike</strike>Science fiction describes black holes as these amazingly different entities in space that don't behave according to the same laws of physics that the rest of the universe is bound to. I've heard them described as wormholes to other universes, singularities, or tears in space-time.</p>
<p>However, are they really that special? </p>
<p>My thought is that all matter has event horizons. However, most of it isn't dense enough for that event horizon to be large enough to affect the way the matter interacts with the rest of the universe. So, a black hole is simply just a dense star that is simply dark. </p>
<p>And, if black holes <em>must</em> have a singularity at their center because of general relativity, can't we conclude from that that all matter creates singularities, since the gravitational field intensifies into infinity the closer you get to the point-mass? What is the difference being really close to a point-mass and being very close to a black hole?</p>
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<p>I know that there are a lot theses being published on lives of physicists. Is there a history/non-fiction book that tracks the development of a problem chronologically? Like pieces of a puzzle.
I would like it to be mathematical and trying to get into the heads of people trying to solve that problem.</p>
<p>Something like a case study.</p>
| 6,881 |
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<p>I understand that in nature wind would never get high enough, but I am just curious as to whether physics would allow this to occur or not.</p>
| 6,882 |
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<p>What are the conditions in order for the equations: <a href="http://en.wikipedia.org/wiki/Entropy" rel="nofollow">Entropy</a> $dQ = TdS$ and <a href="http://en.wikipedia.org/wiki/Work_%28thermodynamics%29" rel="nofollow">Work</a> $dW = -p dV$ to work?</p>
<p>I think for $dQ = T dS$, it must be a reversible process?</p>
<p>But for $dW = -p dV$, shouldn't it always hold?</p>
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<p>I am not sure in which SE site I have to put this question. But since I have learnt Shannon Entropy in the context of Statistical Physics, I am putting this question here.</p>
<p>In the case of Shannon Information theory, he defines information $I$ for an $i^{th}$ event as,</p>
<p>$$ I_i = -\ln P_i \qquad \qquad \forall i=1,...n. $$</p>
<p>Based on this definition we further define Shannon Entropy as average information,</p>
<p>$$ S_\text{Shannon} =\langle I\rangle = -\sum\limits_{i=1}^n P_i\log_2P_i .$$</p>
<p>My question is what is the motivation behind defining entropy as some function that is inversely related to probability? I was told by my professor that lesser the probability of an event more information it possesses, although am still not convinced about this fact. </p>
<p>Secondly, what is the reason in choosing the logarithmic function in this definition? Are there places where this definition of information is forfeited? </p>
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<p>I am looking for a site that publishing riddles and problems such as <a href="https://projecteuler.net/">Project Euler</a>
publishes problems in computer science. Is there any similar site that is hardcore in the area of physics? </p>
<p><a href="http://www.feynmanlectures.info/">This site</a> has some good problems set, but it seems like there isn't a community and sharing of solutions/ideas like in Project Euler.</p>
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] |
<p>Any entangled state represents a quantum possibility that is classically impossible.<br>
Is the converse true?<br>
That is, are all states that are quantum mechanically possible but classically impossible entangled in some way?<br>
If so, can you give a proof, or a reference to a proof?<br>
If not, can you give a counterexample?</p>
| 6,886 |
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<p>I'm looking at <a href="http://www.math.ias.edu/QFT/fall/faddeev4.ps" rel="nofollow">this reference</a> (sorry it's a postscript file, but I can't find a pdf version on the web. <a href="http://www.srv2.fis.puc.cl/~mbanados/Cursos/TopicosRelatividadAvanzada/FaddeevJackiw.pdf" rel="nofollow">This paper</a> describes a similar procedure).</p>
<p>The topic is the Faddeev-Jackiw treatment of Lagrangians which are singular (Hessian vanishes) - similar to what Dirac does, but without the need to differentiate between first and second class constraints. Just looking at classical stuff here, no quantization.</p>
<p>Starting with the Maxwell Lagrangian</p>
<p>$$\mathcal{L}=F_{\mu\nu}F^{\mu\nu}$$</p>
<p>where</p>
<p>$$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$$</p>
<p>we see that it's second order in time derivatives acting on A.</p>
<p>We choose to write it in first order form like this</p>
<p>$$\mathcal{L}=(\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu})F^{\mu\nu}-{1\over{2}}F_{\mu\nu}F^{\mu\nu}$$</p>
<p>where we're treating $F^{\mu\nu}$ now as an auxilliary, independent variable. Having defined this, Faddeev says</p>
<p>"we rewrite (the last equation) as:</p>
<p>$$\mathcal{L}=(\partial_{0}A_{k})F^{0k}+A_{0}(\partial_{k}F^{0k})-F^{ik}(\partial_{i}A_{k}-\partial_{k}A_{i})-{1\over{2}}(F^{0k})^2-{1\over{2}}(F^{ik})^2$$"</p>
<p>My question is how does he arrive at this from the previous equation ? I don't see how just expanding the indices into time and space values ever gets me to $A_{0}(\partial_{k}F^{0k})$</p>
<p>I can see how there's something special about $A_{0}$, since when I write out the EOM for the first order Lagrangian, $A_{0}$ drops out, which indeed it should do because we'll end up with it being a Lagrange multiplier. I just can't see how you end up with that term, with $A_{0}$ multiplying $\partial_{k}F^{0k}$. </p>
<p>It's clearly correct since $A_{0}divE$ is just the Gauss law constraint. </p>
| 6,887 |
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<p>With all the hype of the impending "2012 Mayan doomsday" I was thinking it might be interesting to see what principles of physics prevent the theories of doomsday from occurring. One overarching theory is that on December 21, 2012, the two magnetic poles will reverse as a result of a solar flare equivalent to 100 billion nuclear bombs, enough to reverse the geomagnetic poles.</p>
<p>Is there any way a solar flare could lead to reversal of the magnetic poles? If not, what prevents it from happening?</p>
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<p>Find the displacement and velocity of horizontal motion in a medium in which the retarding force is proportional to the velocity.</p>
<p>I kind of understand how to do this problem.</p>
<p>We know that the resistive force $F_r \propto v$. Since $F_r$ is the only force present in the x-direction, Newton's second law gives $$F_r=ma=m\frac{dv}{dt}.$$ My book then says that $F_r=-kmv$. So thus we have $$-kv=\frac{dv}{dt},$$ from which it is trivial to find expressions for $v(t)$ and $x(t)$ by using initial conditions and integration.</p>
<p>The only part about the problem I don't understand is why $F_r=-kmv$. Why does the retarding force depend on the mass $m$? Since $F_r \propto v$, shouldn't we just stick a proportionality constant $k$ in there and have $F_r=-kv$?</p>
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] |
<p>Are all wavelengths absorbed in the semiconductor regardless of material's absorption band? </p>
| 6,890 |
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<p>I am stumped by a mundane A Level Physics question (teacher of physics here obviously a bit short of practice!). My colleagues and I are stumped and were wondering if any one could help us.</p>
<p>It concerns the change in KE of a satellite in orbit, if a satellite drops it’s orbit to that of a lower radius, it’s change in GPE is given by the expression $GMm(1/r_1-1/r_2)$. Due to the conservation of energy, I would have thought that the gain in KE would therefore be equal. But I keep seeing in past exam papers that the answer is HALF of this value! However I cannot seem to find a satisfying reason why anywhere.</p>
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<p>I am doing an experiment to show the effect on the velocity of a falling mass due to change in mass. How can i justify my prediction that the velocity of the falling mass will increase as i increase the mass. </p>
<p>For my experiment i am dropping masses between 100-800g threw 1000ml of water and timing how long the mass takes to fall the 35cm height. From my results i can see that as the mass increases the time taken for the mass to reach the bottom of the 35cm decreases. Is this due to change in velocity and am i correct in using velocity and mass as my measurements? </p>
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<p>$\newcommand{\v}[1]{\vec #1}\newcommand{\i}{\hat i}\newcommand{\j}{\hat j}$
Problem statement <sup>(<a href="http://i.imgur.com/pM6NQNa.jpg" rel="nofollow">1</a>,<a href="http://i.imgur.com/TP36dQo.jpg" rel="nofollow">2</a>)</sup></p>
<blockquote>
<p>A shopper at the supermarket follows the path indicated by vectors $\v A, \v B, \v C, \v D$ in the figure. Given that the vectors have magnitudes $A=51\:\mathrm{ft}, B=45\:\mathrm{ft}, C=35\:\mathrm{ft}, D=13\:\mathrm{ft}$, find the total displacement of the shopper using (a) the graphical method and (b) the component method of vector addition. Give the direction of displacement relative to $\v A$</p>
<p><a href="http://i.stack.imgur.com/iJbOU.png" rel="nofollow"><img src="http://i.stack.imgur.com/iJbOUm.jpg" alt="enter image description here"></a></p>
</blockquote>
<p><a href="http://i.imgur.com/Nb4uODn.jpg" rel="nofollow">My work</a>:</p>
<p><a href="http://i.stack.imgur.com/RCyp8.png" rel="nofollow"><img src="http://i.stack.imgur.com/RCyp8m.jpg" alt="enter image description here"></a></p>
<p>$$\begin{align}A &=&0&\i + &51&\j \\B &=&45&\i + &51&\j\\C &= &10&\i + &-35 &\j\\ D &= &10&\i + &-13 &\j \\\text{Resultant} &=&54&\i+&65&\j\end{align}$$
Essentially I am adding all the components to get the resultant vector but it is not leading me to the right answer. What am I doing wrong here?</p>
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] |
<ol>
<li><p>What would be a good Internet link that would properly explain <a href="http://en.wikipedia.org/wiki/Fermi_energy" rel="nofollow">Fermi Energy</a>? </p></li>
<li><p>How does the Fermi Energy of a material vary with external influence, such as doping of the material, and applied electric and magnetic fields for instance? What other factors can effect Fermi Energy?</p></li>
<li><p>In single walled Carbon nanotubes, how does the Fermi Energy vary with the geometric shape, i.e. length and radius of the nanotube? How does chirality affect the Fermi Energy?</p></li>
</ol>
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<p>My quantum mechanics textbook says that when a particle (in the classical case) comes across a potential-step barrier of finite height, if it has sufficient energy to surmount the barrier, it will continue on with reduced kinetic energy. </p>
<p>I'm finding this hard to understand since force is given by $$F=-\frac{dV(x)}{dx}$$</p>
<p>For a step barrier, this should give an infinite force acting (in the opposite direction) on the particle when it comes in contact with the barrier. </p>
<p>The only other thing I can think of is to model the force on the particle by a dirac delta function, so we effectively see it getting an impulse in the opposite direction, which could lower its kinetic energy. Is this reasoning right? </p>
| 6,895 |
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<p>Suppose you have two stationary people, A and B, who are equally effective in terms of hearing ability, and A emits a sound that is heard by B at a certain intensity. If they remain stationary, and B makes the exact same sound, does A hear the sound at the same intensity as B?</p>
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<p>A lepton is an elementary particle. The best known of all <a href="http://en.wikipedia.org/wiki/Lepton" rel="nofollow">leptons</a> is the electron which governs nearly all of chemistry as it is found in atoms and is directly tied to all chemical properties.</p>
<p>The heavier <a href="http://en.wikipedia.org/wiki/Muon" rel="nofollow">muons</a> and taus will rapidly change into electrons through a process of particle decay, Does it happen at high energies (heavier leptons decay)?</p>
<p>So, It may be possible that electron decay in very low energies.</p>
| 6,897 |
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<p>Recall the normal ordering of bosonic operators in QFT is defined by a re-arrangement of operators to put creation operators to the left of annihilation operators in the product. This is designed to avoid accidentally annihilating $|0\rangle$ when looking at an expectation value in relation to the vacuum state. </p>
<p>$
: \hat{b}^\dagger\hat{b} : \: =\: \hat{b}^\dagger\hat{b} \\ : \hat{b}\hat{b}^\dagger: \: = \: \hat{b}^\dagger\hat{b}
$</p>
<p>In CFTs, I've seen defined the normal ordering of operators as the zeroth basis field of the Laurent expansion of the radial ordering product.</p>
<p>$\mathcal{R}(a(z)b(w)) = \sum_{n = -n_0}^\infty (z-w)^n P_n(w),$</p>
<p>and select</p>
<p>$P_0(w) = \: : a(w)b(w) : $</p>
<p>Is there an equivalence between these two definitions? What is the CFT analog of not annihilating the vaccuum/ how do we show this definition has that property?</p>
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<p>Say that we have an irreversible expansion process which extracts energy, like a turbine. Isentropic efficiency is commonly defined by the following relation, which applies in a similar fashion for pumps. Here, state 1 refers to the inlet conditions and state 2 refers to the outlet conditions, while the "s" indicates the isentropic pseudo-state. P, h, and s are pressure, enthalpy, and entropy.</p>
<p>$$ \eta = \frac{ h_1 - h_2 }{ h_1 - h_{2s} } $$</p>
<p>This has a monumentally huge, glaring problem, that has always bothered me - it can't be applied to sub-segments of a turbine. By that, I mean let's say we keep P1 and P2 the same, but instead of one turbine, we have two in series. Here is the basic situation on a hs diagram:</p>
<p><img src="http://i.stack.imgur.com/OkrAn.png" alt="Isentropic Process"></p>
<p>The general structure, isobaric lines, and 3 labeled states are cannon in the literature. The line should probably be curved slightly. But let's add another point <em>somewhere in the middle</em> of the 1->2 process. Let's label that midpoint "m". If you begin with the assumption that two irreversible expansion processes exist 1->m and m->2(by pressure) with the same isentropic efficiency as 1->2, then I'm saying that it follows that:</p>
<p>$$s_1 = s(P_1,h_1) \\
h_{2s} = h(P_2,s_1) \\
h_2 = h_1 - \eta ( h_1-h_{2s} ) \\
\text{Two-Step Process} \\
h_{ms} = h(P_m,s_1) \\
h_m = h_1 - \eta (h_1 - h_{ms} ) \\
h_{2s}' = h(P_2,s(P_m,h_m)) \\
h_2' = h_m - \eta (h_m - h_{2s}' ) \\
\text{The Problem:} \\
h_2' \ne h_2$$</p>
<p>In these equations, I intend for the 2-variable functions to be property lookups. I'm mostly using the steam tables.</p>
<p>A more correct way of saying this is that if you assigned values for $\eta_{1m}$ and $\eta_{m2}$ such that you wind up at the intended final point of 2, then both of these values will be different from the original $\eta$.</p>
<p>The topic has come up on the site before, <a href="http://physics.stackexchange.com/questions/22146/what-meaning-does-the-slope-of-the-efficiency-path-on-a-mollier-diagram-have-in">considering what the derivative of some expansion line might mean</a>. But that's already jumping the gun, because we don't even have any defined way to draw that line in the first place. I can write down $\frac{\Delta h}{\Delta s}$, but not $\frac{dh}{ds}$. Again, we could replace one turbine with two turbines, and assign them efficiencies <em>such that</em> point 2 has the same properties. But that leaves a free degree of freedom. You could make $\eta_{1m}$ and $\eta_{m2}$ the same, or you could have one a little higher and one a little lower, without point 2 being affected.</p>
<p>One simple way to phrase this question would be: how can I calculate P_m and h_m in a physically meaningful way?</p>
<p>My main interest is - in doing so, what metric defines the <em>degree of irreversibility</em>. Has anyone put forth a definition of a variable which will accomplish this in a truly <em>differential</em> sense along a non-ideal expansion line?</p>
| 6,899 |
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<p>Due to differences in air pressure, temperature, and other factors, the speed of sound varies with altitude on Earth. Does this affect the pitch of the sound in any meaningful way?</p>
<p>For example, if I had a tuning fork that vibrates at around 262 Hz, would I hear the same "<a href="https://en.wikipedia.org/wiki/Middle_c#Middle_C" rel="nofollow">Middle C</a>" from it while standing on the shores of the <a href="https://en.wikipedia.org/wiki/Dead_sea" rel="nofollow">Dead Sea</a> as I would while standing at the peak of Mount Everest - an altitude difference of 9,271 meters or 30,417 feet?</p>
<p>Would there be any difference at all, and would it be perceptible to the human ear at close range? Would the altitude difference affect how the sound is heard more at a distance than it would nearby? Or, am I not quite understanding properly how sound works?</p>
| 6,900 |
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<p>Does Kepler's law only apply to planets? If so why doesn't it apply to other objects undergoing circular motion?</p>
<p>By Kepler's law I'm referring to $T^2 \propto r^3$</p>
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<p><img src="http://img84.imageshack.us/img84/4082/collision.jpg" alt="Branes Collision - Big Bang">
Imagine universe occurred when two parallel branes collided, Momentum of Branes converted to big bang kinetic energy after Collision. Thus, high-energy quanta are high-Vibrating strings.</p>
<blockquote>
<p>what cosmological
issues can be particularly well
explained by such a brane
scenario?
what direct or indirect experimental evidence would support it? </p>
</blockquote>
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<p>If I were to try to find pi using a ruler and a compass, I would first try to find out how many rational line segments of the diameter I could fit around the interior circumference and then continue to refine until I began converging on some number. Before the foundations of Non-Euclidean geometry were made, people would not have even questioned that the euclidean pi had a unique nature. </p>
<p>Why not think of the standard model parameters in the same light? Just simply as ratios that might change with the geometry of space or some something similar?</p>
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<p>I have a number of measurements of the same quantity (in this case, the speed of sound in a material). Each of these measurements has their own uncertainty.</p>
<p>$$ v_{1} \pm \Delta v_{1} $$
$$ v_{2} \pm \Delta v_{2} $$
$$ v_{3} \pm \Delta v_{3} $$
$$ \vdots $$
$$ v_{N} \pm \Delta v_{N} $$</p>
<p>Since they're measurements of the same quantity, all the values of $v$ are roughly equal. I can, of course, calculate the mean:</p>
<p>$$ v = \frac{\sum_{i=1}^N v_{i}}{N}$$</p>
<p>What would the uncertainty in $v$ be? In the limit that all the $\Delta v_i$ are small, then $\Delta v$ should be the standard deviation of the $v_i$. If the $\Delta v_i$ are large, then $\Delta v$ should be something like $\sqrt{\frac{\sum_i \Delta v_i^2}{N}}$, right?</p>
<p>So what is the formula for combining these uncertainties? I don't think it's the one given in <a href="http://physics.stackexchange.com/a/23651/10696">this answer</a> (though I may be wrong) because it doesn't look like it behaves like I'd expect in the above limits (specifically, if the $\Delta v_i$ are zero then that formula gives $\Delta v = 0$, not the standard deviation of the $v_i$).</p>
| 892 |
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<p>By 'wrong reference state' I mean a state which cannot be transformed into desired ones via variational ansatz</p>
<p>$\left|\Psi\left[\mathbf{n}\right]\right\rangle =e^{i\hat{O}\left[\mathbf{n}\right]}\left|ref\right\rangle \;,\; H\left[\mathbf{n}\right]=\left\langle \Psi\left[\mathbf{n}\right]\right|\hat{H}\left|\Psi\left[\mathbf{n}\right]\right\rangle $</p>
<p>where $\mathbf{n}$ is the variational parameter (field).</p>
<p>To be concrete, if we started with the first excitation state in a gapped system, i.e. choosing (making a guess) this first excitation state to be the reference state, then it seems to me that we will never reach the true ground state, as well as the correct excitations. The manifolds are disconnected because of the gap.</p>
<p>Are there any concrete examples for my general question here? Maybe I have not properly described this question. Please let me know if you are interested but don't understand. I will try to clarify.</p>
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<p>I need to elaborate the equation ,and need to know what is the physical significance and how matrices will manipulate in the equation $$
\hat{H} = (\hat{\tau_3}+i\hat{\tau_2})\frac{\hat{p}^2}{2m_0}+ \hat{\tau_3}m_0 c^2 =
\left| \begin{array}{ccc}
1 & 1 \\
-1 & -1 \\
\end{array}\right| \frac{\hat{p}^2}{2m_0} + \left| \begin{array}{ccc}
1 & 0 \\
0 & -1 \\
\end{array}\right| m_0 c^2
$$</p>
<p>Where $$\tau_1 , \tau_2,\tau_3
$$ are Pauli matrices and Hamiltonian comes from "Schrodinger form of the free Klein_Gordon equation
And also why did we added Pauli matrices in the free Hamiltonian ?</p>
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<p>Definition clarification needed, please: I am hoping to get physical sense of an "inertial frame".</p>
<p>Do inertial reference frames all have zero curvature for their spacetime?</p>
<p>So is an inertial frame just a flat metric?</p>
<p>(Sorry that the question is not too profound. I have read the Wiki article for <a href="http://en.wikipedia.org/wiki/Inertial_frame_of_reference" rel="nofollow">inertial frame of reference</a>, but I'm just not entirely sure.)</p>
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<p>It is said that we can introduce local inertial coordinates for any timelike geodesic. But why only for timelike geodesics? What about null geodesics? Perhaps it has to do with invertibility or something?</p>
<p>Thanks.</p>
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] |
<p>What causes lightning to follow the path it does ?</p>
<p><img src="http://i.stack.imgur.com/U7fBT.jpg" alt="loopy lightning"></p>
<p>picture from BBC news: <a href="http://news.bbcimg.co.uk/media/images/62891000/jpg/_62891901_untitled-1copy.jpg" rel="nofollow">http://news.bbcimg.co.uk/media/images/62891000/jpg/_62891901_untitled-1copy.jpg</a> main page: <a href="http://www.bbc.co.uk/news/in-pictures-19597250" rel="nofollow">http://www.bbc.co.uk/news/in-pictures-19597250</a></p>
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<p>I've not come across any expression involving $\langle\Omega|f|\Omega\rangle$ in <a href="http://web.physics.ucsb.edu/~mark/qft.html" rel="nofollow">Srednicki's QFT book</a> (please correct me if these exist there). On the other hand, they are abound in Chapter 7 of Peskin&Schroeder, in relation to the LSZ reduction formula. Here $|\Omega\rangle$ is the ground state of the interacting theory and $f$ is anything you can imagine that makes sense. </p>
<p>Srednicki only uses $\langle0|f|0\rangle$ where $|0\rangle$ is the ground state of the free theory. </p>
<p>How come this is so? Can one always go from one expression to the other by rewriting in terms of creation and annihilation operators? </p>
<p>Srednicki also derives the LSZ but he only uses the above-mentioned $|0\rangle$. </p>
| 6,909 |
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<p>Since it's convex lens there in our eyes so image formed on our retina is inverted, so how come that we see upright images? </p>
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<p>The scenario that I'm having is such that a ball of radius $15mm$ is thrown from a location point $\vec{p}=(2, 5, 2)$ in a direction of $\vec{d}=(3, 0, 4)$. The initial velocity is $30m/s$. There were wind velocity at $15m/s$ in the direction of $\vec{v}=(0, 1, 0)$. The mass of the ball is $2kg$ and the gravity is $9.81m/s^2$.</p>
<p>I'm trying to calculate the time for the ball to hit the plane at $z=0$.</p>
<p>I started by making the throw direction as an unit vector:
$\hat{\vec{d}} = \frac{\begin{bmatrix}
3\\
0\\
4
\end{bmatrix}}{\left | \begin{bmatrix}
3\\
0\\
4
\end{bmatrix} \right |}=\begin{bmatrix}
\frac{3}{5}\\
0\\
\frac{4}{5}
\end{bmatrix}$</p>
<p>This is where the ball will reach at time $t$:
$\vec{r(t)} = \vec{p}+t \cdot \hat{\vec{d}}$</p>
<p>$r_z = 2+t \cdot \frac{4}{5}$</p>
<p>Since equation of plane is $z=0$, if $z<15$, ball hits the plane:</p>
<p>$r_z < 15 \Rightarrow 2+t \cdot \frac{4}{5} < 15 \Rightarrow t\geq16.25$</p>
<p>This would imply that when $t\geq16.25$ seconds, the ball hits the plane. But I realise this <em>doesn't</em> consider the mass, gravity and wind resistance! So this value is probably invalid from start.</p>
<p>I recalled that $F=ma$, so $F= 2kg \cdot 9.81m/s^2$. But $F$ is just a scalar in this case and I don't know how I can use it any where.</p>
<p>How can I calculate the time $t$ for when the ball hits the plane with the consideration of the mass, gravity and wind resistance?</p>
| 6,911 |
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] |
<p>In the context of classical physics,is there any renormalization method to avoid infinite energy of point charges?</p>
| 6,912 |
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<p>How does repulsion and attraction of a magnet work? I have a hypothesis. We all know that repulsion works when people throw balls at each other. This is used as an analogy of how virtual particles are used to explain how magnets repel each other in some cases. The other idea is that virtual particles have momentum and anti-momentum to explain attraction and repulsion. Is it possible to explain attraction by virtual particles leaving the south pole of one magnet at some angle, let's say less than 45 degrees and arriving at the second magnet's south pole at around 90 degrees? </p>
<p>As an after thought, the 45 degree angle would be obtained if the north pole particles were repelled by the south pole particles but now I seem t be creating a loop in my thinking.</p>
| 6,913 |
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<p>There's this "atomic" explanation of the freezing-point phenomena on Wikipedia that leaves me really intrigued.</p>
<blockquote>
<p>Consider the problem in which the solvent freezes to a very nearly pure
crystal, regardless of the presence of the solute. This
typically occurs simply because the solute molecules do not fit well
in the crystal, i.e. substituting a solute for a solvent molecule in
the crystal has high <strong>enthalpy</strong>. In this case, for low solute
concentrations, the freezing point depression depends solely on the
concentration of solute particles, not on their individual properties.
The freezing point depression thus is called a colligative property.</p>
<p>The explanation for the freezing point depression is then simply that
as solvent molecules leave the liquid and join the solid, they leave
behind a smaller volume of liquid in which the solute particles can
roam. The resulting reduced <strong>entropy</strong> of the solute particles thus is
independent of their properties. This approximation ceases to hold
when the concentration becomes large enough for solute-solute
interactions to become important. In that regime, the freezing point
depression depends on particular properties of the solute other than
its concentration.
(source: <a href="http://en.wikipedia.org/wiki/Freezing_point_depression" rel="nofollow">http://en.wikipedia.org/wiki/Freezing_point_depression</a>)</p>
</blockquote>
<p>I'd like to understand better how the enthalpy and entropy "behave" on this process and how they explain it.</p>
| 6,914 |
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<p>I believe the title is straightforward. Evaporating water leaves behind chalk. Assuming you evaporate the same amount of (the same type of) water, does the temperature at which this happens (and so also the time it will take) matter for the amount of chalk that is left behind? </p>
<p>Maybe a little side-question: Does the surface or container from which evaporation happens matter? </p>
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<p>I have a system with a number of measurables (in time). Some measurables are discrete some are continuous (within the measurement accuracy). How can I determine whether my system experiences criticality or not?</p>
<p>I am looking for many different ways to (dis)proof criticality.</p>
<p>See here for critical phenomena <a href="http://en.wikipedia.org/wiki/Critical_phenomena" rel="nofollow">http://en.wikipedia.org/wiki/Critical_phenomena</a>.</p>
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<p>Int he three-mass coupled oscillator problem, we often see it stated that you have three masses, (they can be equal or not, but we'll assume they are equal here) connected by two springs and then another set of springs connecting the masses to the walls on the end. We assume that the spring constants are all the same. (We're assuming one-dimensional motion here as well). </p>
<p>So the equations of motion are usually written thus: </p>
<p>$$m\ddot x_1+kx_1-k(x_2-x_1)=0$$
$$m\ddot x_2+k(x_2-x_1)+k(x_3-x_2)=0$$
$$m\ddot x_3+k(x_3-x_2)+k(-x_3)=0$$</p>
<p>(And lord knows tell me if this is wrong and I missed a concept)</p>
<p>Well, ok, but what if we assume that the three masses are not connected to anything else? That is, they are connected by two springs but there are no walls? Would we just have to pick a reference frame and go from there, declaring one mas "stationary"? For instance, if we connect the first mass to a wall but leave the other end unconnected, would the EoMs look like: </p>
<p>$$m\ddot x_1+kx_1-k(x_2-x_1)=0$$
$$m\ddot x_2+k(x_2-x_1)+k(x_3-x_2)=0$$
$$m\ddot x_3+k(x_1+x_2)+k(x_3)=0$$</p>
<p>Anyhow, I was just curious, because all the problems in this vein I see seem to assume the three masses are attached between two walls. </p>
<p>Thanks for your insights, in advance. </p>
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<p>How does one find the time dependent position expectation value for a wave function? I thought we could simple take the time dependent wave and apply the position operator like normal, but this gave me the wrong answer, as the time dependencies multiplied to one and left me with my solution for $t = 0$. </p>
<p>Added:
The given solution was to use Ehrenfest's Theorem to find the expectation of momentum at a given time, then to use Ehrenfest's again to find the expected position at a time. I still don't understand why I couldn't just apply the momentum operator to the time dependent wave function to get the time dependent momentum expectation value, and do the same for position.</p>
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<p>Hey i am by no means a scientist but i have a idea for a art work using smoke.</p>
<p>Basically what i want to know is whether it is possible to circulate smoke by means of a pump through a transparent box. If the box is completely sealed will the smoke disappear within time? When the smoke is pumped out and back in again, will it be have the same consistency as it had before it was circulated?</p>
<p>Thanks in advanced</p>
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<p>Is there some condition in the N=1 SUSY algebra telling that the spin of the superpartners of gauge bosons (either for colour or for electroweak) must be less than the spin of the gauge boson?
I am particularly puzzled because sometimes a supermultiplet is got from sugra that contains one spin 2 particle, four spin 3/2, and then some spin 1, 1/2 and 0. If this supermultiplet is to be broken to N=1 it seems clear that the graviton will pair with the gravitino and the rest of spin 3/2 should pair with spin 1, so in this case it seems that a superpair (3/2, 1) is feasible. Why not in gauge supermultiplets?</p>
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<p>A gluon string is a particular kind of open string terminated in two particles which are the sources for the field. Is it possible to have a similar arrangement with gluinos? At first glance, it seems to me that such object could not exist, or at least not as an extended object: you need bosons to mediate between the two sources, and you need bosons to build a classical field extended in space.
On other hand, the fact that in the world-sheet such structure is just an object with fermionic coordinates --and without bosonic coordinates-- could be telling that you can build it, but it is not extended in space, just a point. But I have never read of such description, so surely intuition is failing here. </p>
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<p>Is it possible to use the data that is readily available (e.g. of the <a href="http://www2.jpl.nasa.gov/srtm/" rel="nofollow">Shuttle Radar Topography Mission</a>) to predict the flooding of a region?</p>
<p>Would it be necessary to use the refined geoid of Gravity Probe B?</p>
<p>Is there free software that can do this?</p>
<p>I always wondered about this and the current events near Fukushima
convinced me to ask this question:
<a href="http://search.japantimes.co.jp/cgi-bin/nn20110730a5.html" rel="nofollow">http://search.japantimes.co.jp/cgi-bin/nn20110730a5.html</a></p>
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<p>I am trying to teach myself some particle physics. </p>
<p>There are too many particles and its too much for me. </p>
<p>I hated biology just because of this sort of stuff. Too many names and it was all Greek to me. </p>
<p>Is there a good cheat sheet/ reference sheet of elementary particles? </p>
<p>It will also be very helpful if you share how you people manage to remember these things. </p>
<p>This is the first time I have ever hated studying physics. :(</p>
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<p>Someone can explain me what's the rule behind the correct expression of a quantity $K$ with its error $\Delta K$ as $K \pm \Delta K$?
They must have the same number of significant figures? Or the error should have in general 1-2 significant figures? For example, if I have:</p>
<p>$$K = 8510.33 \pm 56.97~.$$
This expression is uncorrect? Maybe should be expressed as: $$K= 8510 \pm 57~? $$</p>
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<p>Say there is a circuit with two 1.5V cells, and a 100 ohm resistor.</p>
<p>If you connect two cells in series, then the total emf is 3V. And the current will be 3/100 = 0.03 A. (Using V = IR):<br>
<img src="http://i.stack.imgur.com/qQmhp.png" alt="enter image description here"></p>
<p>If you have the cells in parallel, then the total emf is 1.5V, as the terminals of the cells are electrically the same point. So the current will only be 0.015 A:<br>
<img src="http://i.stack.imgur.com/aHMqv.png" alt="enter image description here"></p>
<p>But if you just had one 1.5V cell, so the total emf is again 1.5V. The current will still be 0.015 A.<br>
<img src="http://i.stack.imgur.com/4TowK.png" alt="enter image description here"></p>
<p>So what benefit does adding a second cell have? The emf and the current is the same no matter if you use 2 cells in parallel or just one cell.</p>
<p>I'm assuming that all the cells are identical, and internal resistance is negligible.</p>
| 6,925 |
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<p>I am trying to understand <a href="http://en.wikipedia.org/wiki/R%C3%B8mer%27s_determination_of_the_speed_of_light" rel="nofollow">Rømer's determination of the speed of light</a> ($c$). The geometry of the situation is shown in the image below. The determination involves measuring apparent fluctuations in the orbital period of Io. (Jupiter's moon)</p>
<p><img src="http://i.stack.imgur.com/J5dyj.png" alt="Geometry of the problem"></p>
<p>The Earth starts from point A. $r(t)$ is the distance between the Earth and Jupiter. $r_e$ is the radius of the (assumed) circular orbit of the Earth around the Sun, while $r_0$ is the same for Jupiter. $T$ is the period of the Earth's orbit.</p>
<p>Under the assumption that the Jupiter-Io system is stationary, $r(t)$ can be expressed as</p>
<p>$$r(t) = \sqrt{r_E^2 + r_0^2 -2r_0 r_E \cos \left(\frac{2\pi t}{T}\right)}$$</p>
<p>If we further assume that the period of Io's orbit around Jupiter, $\Delta t$ is much smaller that $T$, then it can be shown that the distance the Earth moves, $\Delta r$ when Io completes one orbit is:</p>
<p>$$\Delta r = \frac{2\pi r_E \Delta t}{T} \sin\left( \frac{2\pi t}{T} \right)$$</p>
<p>The point I am stuck is about why is there an apparent fluctuation in Io's orbit as observed on the Earth? And how can we derive the observed delay using these expressions?</p>
| 6,926 |
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<p>Consider fields $\rho \left( \vec{r} \right)$, $\vec{J} \left( \vec{r} \right)$, $\vec{E} \left( \vec{r} \right)$ and $\vec{B} \left( \vec{r} \right)$ in $\mathbb{R}^3$, with their usual meaning as per Electrodynamics. </p>
<p>Take any finite volume $V_s$ outside of which $\vec{J}\left(\vec{r}\right)$ and $\rho\left(\vec{r}\right)$ are $0$. Then, we know that $\vec{E}\left(\vec{r}\right) $ and $\vec{B}\left(\vec{r}\right) $ for any $\vec{r}$ are as follows:</p>
<p>$$
\vec{E} \left( \vec{r} \right) =
\frac{1}{4\pi \epsilon_0}
\iiint_{V_s}
\frac{\rho\left(\vec{r}_s\right)}{\left|\vec{r}-\vec{r}_s\right|^3}
\left(\vec{r}-\vec{r}_s\right)
\space dV\left(\vec{r}_s\right)
$$
$$
\vec{B} \left( \vec{r} \right) =
\frac{\mu_0}{4\pi}
\iiint_{V_s}
\frac{\vec{J}\left(\vec{r}_s\right)}{\left|\vec{r}-\vec{r}_s\right|^3}
\times \left(\vec{r}-\vec{r}_s\right)
\space dV\left(\vec{r}_s\right)
$$
Now my question is, is it possible to <em>mathematically</em> prove that the following surface integral will always evaluate to zero? If so, what is the proof? (To clarify, $\partial V_s$ is the bounding surface of the volume $V_s$ mentioned earlier)</p>
<p>$$
\frac{1}{\mu_0} \oint_{\partial V_s} \left(\vec{E}\left(\vec{r}\right) \times \vec{B}\left(\vec{r}\right)\right)\cdot d\vec{S}\left(\vec{r}\right)
$$
Motivation: I'm looking for proof that time invariant sources (static charges and constant currents confined to a volume) cannot radiate any energy, and I'm trying to do that without invoking the Hertzian dipole and Fourier analysis.</p>
<p>Thanks...</p>
<p><strong>Update</strong></p>
<p>As pointed out below, since $\vec{E}$ and $\vec{B}$ are time invariant, applying Poynting's theorem this boils down to proving that the following volume integral is zero:
$$
- \iiint_{V_s}
\left(
\vec{J}\left(\vec{r}\right)
\cdot
\vec{E}\left(\vec{r}\right)
\right)
\space dV\left(\vec{r}\right)
$$
So can <em>this</em> be proved, given that $\vec{J}$, $\vec{E}$, $\vec{B}$ and $\rho$ are all time invariant, beyond the trivial cases of $\vec{J} = 0$, $\vec{E} = 0$ or $\vec{E} \perp \vec{J}$ for all $\vec{r} \in V_s$?</p>
<p>Or, if this can't be proved, is there a counterexample?</p>
<p>Thanks...</p>
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<p>I am watching this <a href="http://upload.wikimedia.org/wikipedia/commons/c/ca/Collision_carts_inelastic.gif" rel="nofollow">example</a> from Wikipedia: </p>
<p><img src="http://upload.wikimedia.org/wikipedia/commons/c/ca/Collision_carts_inelastic.gif" alt=""></p>
<p>I am wondering what factors would indicate that the collision of 2 objects will be <a href="http://en.wikipedia.org/wiki/Inelastic_collision" rel="nofollow">inelastic</a> (I know macroscopic scale impacts are never perfectly <a href="http://en.wikipedia.org/wiki/Elastic_collision" rel="nofollow">elastic</a>). If these are 2 wagons of the same mass and they have both hard surface, there is no friction and collision takes place (mostly) in 1 dimension shouldn't it be more close to elastic. </p>
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] |
<p>Is there any tools except helmholtz coil to cancel out earth's magnetic field to calibrate magnetometers in practice.</p>
| 6,929 |
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] |
<p>Can I measure the value of g using only a metre stick and a ball? I am not supposed to use a stopwatch and that has been the problem.
NOTE: I do not know if a solution exists or not.</p>
| 6,930 |
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<p>I don't just mean reactions that require heat to proceed, storing surplus energy in chemical bonds. I wonder about strongly endothermic reactions that suck heat out of environment.</p>
<p>You take some substance A (e.g. Ammonium Nitrate), and some substance B (e.g. water), both at room temperature. You mix them together in a beaker which is room temperature, all performed in room temperature air. As the reaction begins, the two substances binding into substance C, the beaker cools down quite a bit below room temperature.</p>
<p>This is what we observe on macroscopic scale.</p>
<p>I wonder what happens on the microscopic scale - chemical particles, atoms, elementary particles, their energy. Normally entropy would suggest everything would remain in equilibrium but suddenly we have a higher energy concentration within the substance at cost of energy of the environment. It is not easily reversed. What happens that makes particles "want" to bind so much that they specifically "steal energy" from the environment just so that they can react?</p>
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<p>I didn't know whether to pose this question on <em>Physics.stackexchange</em> or <em>Math.stackexchange</em>. But since this is the last step of a development involving the eigenfunctions of an Harmonic oscillator and a <em>shift operator matrix</em>, I thought it'd be better to post it here.</p>
<p>I have to calculate the integral</p>
<p>$$\frac{1}{2^nn!\sqrt{\pi}}\int_{-\infty}^{+\infty}H_n(x)e^{-x^2+kx}H_l(x)\;\mathrm{d}x$$</p>
<p>where $H_n(x)$ is the $n^{th}$ Hermite polynomial and prove that it equals</p>
<p>$$\sqrt{\frac{m_<!}{m_>!}}\left(\frac{k}{\sqrt{2}}\right)^{|n-l|}L_{m_<}^{|n-l|}\left(-\frac{k^2}{2}\right)\exp\left(\frac{k^2}{4}\right)$$</p>
<p>where $m_<$ and $m_>$ denote the smaller and the larger respectively of the two indices $n$ and $l$ and where $L_n^m$ are the associated Laguerre polynomials.</p>
<p>The last term is $\exp(k^2/4)$, hence I suppose that I begin with </p>
<p>$$\frac{1}{2^nn!\sqrt{\pi}}\int_{-\infty}^{+\infty}H_n(x)e^{-x^2+kx-\frac{k^2}{4}}e^{\frac{k^2}{4}}H_l(x)\;\mathrm{d}x$$
$$\frac{1}{2^nn!\sqrt{\pi}}e^{\frac{k^2}{4}}\int_{-\infty}^{+\infty}H_n(x)e^{-(x-\frac{k}{2})^2}H_l(x)\;\mathrm{d}x$$</p>
<p>but here I'm stuck... No matter what or how I can't go further.</p>
<p>Thanks for your help!</p>
| 6,932 |
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<p>I know it is not safe when viewing a solar eclipse to look directly at the sun. I know you can purchase solar eclipse glasses online but how do you make your own solar eclipse glasses that are safe to use for solar eclipse viewing (and let us throw in the transit of Venus)? </p>
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<p>Consider spherical symmetric$^1$ masses of radii $R_1$ and $R_2$, with spherical symmetric density distributions $\rho_1(r_1)$ and $\rho_2(r_2)$, and with a distance between the centers of the spheres $d$. What is the exact force between them? I know point masses are a good approximation, but I'm looking for an exact formula. This would be useful for a gravity-simulation toy software.</p>
<p>--</p>
<p>$^1$ Assume for simplicity the idealization where <a href="http://en.wikipedia.org/wiki/Tidal_force">tidal</a> or centrifugal forces do not deform the spherical symmetric, i.e., the various mass parts are held in place by infinitely strong and rigid bonds.</p>
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<p>If the Hubble constant is extremely large, what will happen with quark confinement?</p>
<p>I guess that quarks will remain confined because of asymptotic freedom. But can gravity or dark energy have any effect on asymptotic freedom? Can gravitational collapse, or a huge cosmological constant, rip a proton apart?</p>
| 6,935 |
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<p>It was very cold outside, this morning, when I took the car that slept in the snow, with a simple cloth on the windshield. I entered the vehicle, drove a kilometer or so. The air inside was so cold I could see my breath (or maybe I forgot to brush my teeth). Warm air was blowing on the windshield from the inside and suddenly…</p>
<p>The glass in front of my became opaque, starting from the bottom (the place where the hot air was blowing) and freezing up, up, up until the whole screen was filled in about 5 seconds. It reminded my a little bit of how some baterias spread.</p>
<p>When I tried to remove the mist, I realised it was ice <strong>in</strong> the car. My windshield instant froze.</p>
<p>I know the theory behind supercooling: a very cold and still liquid can freeze when moved. I'm not sure what happened here, but there were some pretty big turns in the road before it froze. It really looked like the hot air froze the place and not the movement.</p>
<p>Any idea about what happened ?</p>
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<p>Physicists proposed the idea of multiverses to explain extraordinary fine tuning of the cosmic constant and physical laws which is essential for rise of life on earth.</p>
<blockquote>
<p>Is there any experiment that can test the validity of multiverse hypothesis?</p>
</blockquote>
| 6,937 |
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<p>$\int \frac{Q_{rev}}{T} = \Delta(k_B\ln\Omega)=\Delta S$ <BR>
Could anyone give some definite proof for this?</p>
<p>I was able to prove that the two definitions of change in entropy are equivalent for an isothermal process carried out on a gas (by quantizing space and then limiting the quantization to infinity), but my proof makes the absolute entropy of the gas infinite. If the process is not isothermal, the particle's velocities come into the picture and I don't know how to deal with that. I tried making various assumptions (quantizing time, etc), but it didn't work. I know that once I prove it for another process, it will be proven for any process carried out on ideal gases(as I can write any process as the combination of isothermal and another process).</p>
<p>Could someone please nudge me in the right direction/give a proof?</p>
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