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<p>I am new to Quantum Mechanics. I read the uncertainty principle - it says there are pairs of physical quantities which can't both be determined with certainty for a particle.</p>
<p>My question is does the same apply to a system of particles - the nucleus for example ? Can we determine both the position and momentum of a nucleus (containing more than one protons) with certainty?</p>
<p>Any help would be appreciated.</p>
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<p>The clumsy "almost-commutative graviton" is provocative. I use it on purpose, to ask two questions in one :</p>
<ul>
<li><p><strong>Is the observation of only one Higgs and no supersymmetric particle below 8 TeV (up to now) a sufficient evidence to substantiate the almost commutative spectral model?</strong> </p></li>
<li><p><strong>Can physicists consider now this kind of models pioneered by Connes and Chamseddine to be an <em>effective</em> (physical) and not only <em>formal</em> tentative unification of gravitation and Yang-Mill-Higgs interactions?</strong> </p></li>
</ul>
<p>Recent developments of the almost-commutative spectral model regarding the Higgs boson and its mass:</p>
<ul>
<li><a href="http://arxiv.org/abs/1304.0415" rel="nofollow">Grand Symmetry, Spectral Action, and the Higgs mass</a> /Devastato, Lizzi and Martinetti 2013;</li>
<li><a href="http://arxiv.org/abs/1208.5023" rel="nofollow">Asymptotic safety, hypergeometric functions, and the Higgs mass in spectral action models</a> /Estrada and Marcolli 2012;</li>
<li><a href="http://arxiv.org/abs/1305.3066" rel="nofollow">Noncommutative Geometry in the LHC-Era</a> /Stephan 2013.</li>
</ul>
<p>Last comments:</p>
<p>(motives for "graviton" as a metaphore and "almost commutative" as a pedagogical reminder)</p>
<ul>
<li>I know that <em>graviton</em> is a spin 2 gauge boson associated to the gravitational field in a tentative quantification of general relativity. <em>In the framework of Quantum Field Theory</em> it is thus an object independant a priori from the <em>Higgs</em> that is a scalar boson responsible for masses of elementary particles from the Standard Model. Nevertheless I remind that Higgs interaction can be considered as <em>derivated from gravitation in the noncommutative geometric setting</em> (following <a href="http://arxiv.org/pdf/hep-th/0111236.pdf" rel="nofollow">Thomas Schücker</a>). </li>
<li>The adjective <strong>almost-commutative</strong> has a precise technical meaning but I use it also in my question to underline the fact that in any theoretical framework non-commutativity is a necessary but not sufficient tool to describe quantum phenomena, therefore it is clear that gravitation has not been quantized yet!</li>
</ul>
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<p>I got this question in a assignment and haven't been able to figure out how to get to the correct result.</p>
<blockquote>
<p><em>A force of 6 newtons and a force of 10 newtons can be combine to form
a resultant of what magnitude?</em></p>
<p><em>Ans ( 8 newtons )</em></p>
</blockquote>
<p>I though this would simply be the sum of these forces, but it wasn't.</p>
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<p>This was a statement given in my class:</p>
<p>"<a href="http://en.wikipedia.org/wiki/Strangeness" rel="nofollow">Strangeness</a> is conserved in the <a href="http://en.wikipedia.org/wiki/Strong_interaction" rel="nofollow">strong</a> and electromagnetic interactions, but not in a <a href="http://en.wikipedia.org/wiki/Weak_interaction" rel="nofollow">weak</a> interaction "</p>
<p>But could someone please tell me how we differentiate between a weak and a strong interaction? An example would be helpful.</p>
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<p>Given the lack of financial efficacy in the medical industry, I was wondering if nuclear magnetic resonance imaging would be possible with a copper/silver coil if the use is short 10 second imaging bursts.
Or is the resistance simply too disruptive for any imaging to occur?</p>
<p>What I had in mind is a very small machine you could stick an arm or leg through for home scanning. Either completely uncooled or a very inexpensive cooling technique like simply running faucet water through it for the brief duration in a bathroom.</p>
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<p>I would like to know whether a ball bounces back normally if you throw it against an oily wall. Like a wall watered down with oil.
Basically I want to know if it will actually slip before bouncing back or not?</p>
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<p>Sometimes, when we have a certain line element and we are given a worldline parametrised by a path parameter that is not necessarily proper time I don't completely understand some of the standard manipulations done. A specific example:</p>
<p>Godel spacetime has the line element $ ds^2=a^2(dt^2-dr^2 + f(r)d\phi^2 + 2g(r)d\phi dt- dz^2)$. </p>
<p>I came across the following question: </p>
<blockquote>
<p>A spaceship travels n a closed timelike worldline given by
$$t(\lambda)=0,\, r(\lambda)=R, \, \phi(\lambda)=w\lambda, \,
> z(\lambda)=0,$$ where $w>0$ and $\lambda$ is a suitable path
parameter. Calculate the proper time to go around the loop once.</p>
</blockquote>
<p>The normal thing to do is to use $ds^2=c^2\ d\tau^2$. But then I don't understand if when we say that $d\tau=1/c\cdot \sqrt(f(R))d\phi$ we mean that we take $\lambda=\tau$ (are we allowed to do this?) or if we simply divide both sides by the infinitesimal $d\lambda$ then use the expressions for the worldline given and then "cancel out" the infinitesimal again. Both approaches seem a bit suspicious for me.</p>
| 10,563 |
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<p>I'm studying a diagrammatic approach to degenerate electron gas. Now I need to prove that the energy contribution, with an arbitrary potential, of polarization at first perturbative order given by the diagram:
<img src="http://i.imgur.com/pECT8IX.png" alt="Diagramma">
is null. I expanded it as an integral:
$$
\Pi_{(1)b}^*(q)=\lim_{p^o\to 0^+}\lambda\int d^4k\int d^4p\text{ }(G^{(0)}(k))^2e^{ip^0\eta}G^{(0)}(p)U_0(k-p)G^{(0)}(k-q)
$$
$$
E^{(1)}_b\propto\int d^4q\text{ }U_0(q)\Pi^*_{(1)b}(q)
$$</p>
<p>I've made several tries, but I can't prove it. Have you got any ideas? Thanks in advance. </p>
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<p>As a person who will be using the scattering measurement results (in Zemax), I was asked to prepare the specs for the <a href="http://en.wikipedia.org/wiki/Bidirectional_reflectance_distribution_function" rel="nofollow">BRDF</a> measurement. However I've never done it until now, so help would be greatly appreciated. I have a few questions:</p>
<ol>
<li>What is a realistic range of illumination / detection angles for BRDF measurement?</li>
<li>What angular step is typical for elevation and azimuthal angles?</li>
<li>What dynamic range should we expect?</li>
<li>How do we take polarization into account? Is there a standard procedure? I could find it neither in the <a href="http://kb-en.radiantzemax.com/Knowledgebase/BSDF-Data-Interchange-File-Format-Specification" rel="nofollow">Zemax explanation of BRDF</a>, nor anywhere online.</li>
</ol>
<p>The measurements will be executed at 10.6 um and possibly other IR wavelengths.</p>
<p><strong>UPDATE:</strong> For common interest, I will put here some information I already have.</p>
<p>Zemax cannot measure at 10 um (350...1050 nm only) and usually perform measurement at 5 incident angles in the range of 0...75°. They do not perform polarization-sensitive measurement. Information is <a href="http://www.zemax.com/services/measurement-services/scatter-and-appearance" rel="nofollow">available at their web-site</a>.</p>
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<p>what do they mean by "in a weak interaction, quark flavor can change"</p>
<p>I know that in a weak interaction strangeness is conserved... so is this what the statement is referring to.</p>
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<p>Suppose we have a qubit in state $| \Psi \rangle = \alpha | 0 \rangle + \beta | 1 \rangle$</p>
<p>Suppose we expose this to decoherence, which we will express as the state $| R \rangle$ such that
$$| 0 \rangle| R \rangle \rightarrow | 0 \rangle| R_0 \rangle$$
$$| 1 \rangle| R \rangle \rightarrow | 1 \rangle| R_1 \rangle$$</p>
<p>Where $| R \rangle, | R_0 \rangle$ and $| R_1 \rangle$ are all normalised states. </p>
<p>I'm trying to work how the density operator of the qubit changes when we apply $R$.
If we consider the density operator for $|\Psi \rangle$
$$\rho=|\Psi \rangle \langle\Psi |= \Big( \begin{matrix}
\alpha^2 & \alpha \beta \\
\alpha \beta & \beta^2
\end{matrix} \Big)$$</p>
<p>Assuming alpha and beta are real.</p>
<p>Next, we apply $|R\rangle$ to out qubit,
$$\rho=|\Psi \rangle \langle\Psi | \rightarrow |\Psi \rangle |R\rangle \langle\Psi | \langle R|$$
$$=\alpha^2 |0 \rangle |R_0\rangle \langle 0 | \langle R_0|+\alpha\beta(
|0 \rangle |R_0\rangle \langle 1 | \langle R_1|+
|1 \rangle |R_1\rangle \langle 0 | \langle R_0|
)+\beta^2
|1 \rangle |R_1\rangle \langle 1 | \langle R_1|$$</p>
<p>Next would we take the reduced density operator of $|\Psi \rangle$ to find it's density operator? This would produce a $2 \times 2$ density operator, which I'm hoping to express in terms of $\rho=|\Psi \rangle \langle\Psi |$</p>
<p>Is this the correct way to think about the relation between density operators and reduced density operators for entangled states? What about unentangled states? </p>
<p>Also would anyone be able to briefly explain what decoherence is and why we can describe it as such a operator $|R \rangle$? Any help would be greatly appreciated. </p>
| 10,567 |
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<p>The question asked me to find the potential at a distance $r$ from the center of a charged sphere, where $r>r_0$ of the sphere. </p>
<p>Actually, the question is answered, but what is confusing me is that the author took 2 points out of the sphere ($a$ & $b$) and calculated the potential difference between them, and getting the result that $$V= \frac{Q}{4\pi \epsilon_0 r} $$ at any point outside the sphere?</p>
<p>Why did he take these 2 points ?</p>
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<p>As far as I can tell within an Fourier Transform Spectrometer the spectral information is gained from changing the path length along one arm, this sounds very similar to a michelson interferometer but using two apertures instead of one.</p>
<p>So are there any underlying differences between the two? </p>
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<p>I am trying to find an approximate value of the electrical conductivity $\sigma$ of graphene in units of S/m or S/cm. <a href="http://en.wikipedia.org/wiki/Electrical_resistivity_and_conductivity#Resistivity_of_various_materials" rel="nofollow">This table on Wikipedia</a> gives $\sigma$ values for a variety of materials (including references), but I do not see graphene.</p>
<p>In the classic 2004 paper on graphene by Novoselov and Geim (Novoselov et al., <em>Science</em> <strong>2004</strong>, <em>306</em>, 666: <a href="http://www.sciencemag.org/content/306/5696/666.abstract?sid=1c214462-e89b-48d0-b4d4-02a60e4e0ee1" rel="nofollow">available here</a>), I see a plot of $\sigma \text{ (m} \Omega^{-1})$ versus $V_g \text{ (V)}$, where $V_g$ is the gate voltage:</p>
<p><img src="http://i.stack.imgur.com/0H2pT.gif" alt="Electrical conductivity"></p>
<p>Based on that plot, what can we say about the electrical conductivity $\sigma$?</p>
<p>One confusing thing about the above plot is that it is in $\text{m} \Omega^{-1}$. But $1 \text{ S} = 1 \text{ } \Omega^{-1}$, so in the plot is given in units proportional to S, not S/m. Why is this?</p>
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<p>in this video at arround 0:47 they deliver the 'dry pots' to the under water lab, bellow the water and they seem to swim up into an opening where it is completly dry.. how is this dry, surly the water would fill into that void seeing as its under water ?</p>
<p>Sorry if this is such a basic quesiton !</p>
<p><a href="http://www.youtube.com/watch?v=kuI7KFsKrBY&feature=player_embedded#" rel="nofollow">http://www.youtube.com/watch?v=kuI7KFsKrBY&feature=player_embedded#</a>!</p>
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] |
<p>Is there any current estimate of the electroweak vacuum lifetime based on the recent measurement of the Higgs boson and the top quark masses?</p>
| 10,572 |
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<p>Excitons can be observed when we excite electrons to the conduction band.</p>
<p>I don't know about excitons being observed when we excite the electrons to an electronic level that would eventually be in the middle of the bandgap.</p>
<p>Why? Is it related with the binding energy, density of states of the level, or is it a fundamental aspect that I am not aware of?</p>
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] |
<p>The BICEP2 paper reports a tensor/scalar ratio $r = 0.20_{-0.05}^{+0.07}$, but then says:</p>
<p>Subtracting the various dust models and re-deriving the r
constraint still results in high significance of detection. For
the model which is perhaps the most likely to be close to reality
(DDM2 cross) the maximum likelihood value shifts to $r = 0.16_{-0.05}^{+0.06}$</p>
<p>Which value is proper to compare with <a href="http://arxiv.org/pdf/1303.5082.pdf" rel="nofollow">Planck's</a> $r<0.11$?</p>
| 10,574 |
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] |
<p>How does the <a href="http://www.google.com/search?q=tester+screwdriver+work" rel="nofollow">tester screwdriver work</a>? If I put the tester screwdriver inside a "phase" hole of an electric socket, it lits up. If I press my finger against the metal cap on top of the screwdriver. This happens also if I stand on a surface of isolating material, such as wood. I read elsewhere that this happens due to stray capacitance formed by the "phase" and the ground. One has</p>
<p>$$z=r+1/jwc$$</p>
<p>for the impedance, so if $C$ is high enough, the impedance should be close to $r$, the "effective resistance" of the formed circuit. Here I get lost; why $r$ is small enough to cause a current in the range of $mA$ even if I stand on an isolating surface?</p>
<p>But even supposing that the majority of effective resistance is inside the screwdriver and I form a capacitor by pressing my finger on the metal cap (the capacitor "plates" being the metal cap and the ground, for whatever ground means), then why the capacity increases when I press my finger against the cap?</p>
<p>Even supposing the human body has a MUCH greater dielectric constant at 50Hz than air, aren't also the dielectric constants of the surface on which one stands etc involved here?</p>
| 10,575 |
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<p>I am tring to find the eigenvectors of a two spin system, with $j_1=3/2$ and $j_2=1/2$. To start,
$$m_1 =-3/2,-1/2,1/2,3/2$$
$$m_2=-1/2,1/2$$
For $j_1$, there are 4 possible states, and 2 possible states for $j_2$. $(2j+1)$. Now, I get confused. I am looking for
$$|S,m\rangle=\sum_{m_1+m_2=m}CG|s_1m_1\rangle|s_2m_2\rangle$$
I presume that these are the eigenvectors for total angular momentum and its z component. From here, I am not sure what to do. I write out all of the states, but then I am unsure of how to find the CG coefficients (we are not given a table). Any help would be appreciated. </p>
| 10,576 |
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<p>How much of a difference would it make if every roof, road and vehicle were painted white? This would certainly reduce the urban heat island effect but how much of a difference would it make to total scattered insolation? Apologies in advance for this the bizarre question.</p>
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] |
<p>It is well known that stable, traversable wormholes require exotic matter that violate the dominant energy conditions.</p>
<p>For the purpose of the question, let's assume enough exotic matter can be created for a brief period of time, where the wormhole mouths are stable and traversable. Additionally, suppose that during that time, a strong beam or wire is introduced across the throats. Now the question is:</p>
<blockquote>
<p><strong>Question:</strong> What happens if exotic matter is removed from the wormhole mouths? Are the tidal forces always enough to break any
material crossing the throats and disconnect the mouths?</p>
</blockquote>
| 10,578 |
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<p><strong>Problem statement</strong>: The Hamiltonian of a system is given by the formula:</p>
<p>\begin{equation*}
H = \frac{p_r^2}{2m} + \frac{p_\theta^2}{2mr^2} + V(r,\theta).
\end{equation*}</p>
<p>Under what condition is $f=p_\theta^2$ an integral of motion?</p>
<hr>
<p><strong>Attempted solution</strong>:</p>
<p>In order for $f$ to be an <a href="http://en.wikipedia.org/wiki/Constant_of_motion#Integral_of_motion" rel="nofollow">integral of motion</a>, according to Poisson theorem, is that:</p>
<p>\begin{equation*}
[f, H] =0,
\end{equation*}</p>
<p>where $[,]$ is the <a href="http://en.wikipedia.org/wiki/Poisson_bracket" rel="nofollow">Poisson bracket</a>.</p>
<p>Therefore:</p>
<p>\begin{align*}
\left[\frac{p_r^2}{2m} + \frac{p_\theta^2}{2mr^2} + V(r,\theta), p_\theta^2\right] &= 0 \Leftrightarrow \\
\left[\frac{p_r^2}{2m}, p_\theta^2\right] + \left[\frac{p_\theta^2}{2mr^2}, p_\theta^2\right] + \left[V(r,\theta), p_\theta^2\right] &= 0.
\end{align*}</p>
<p>But how do I proceed from here on?</p>
<p><strong>EDIT1</strong>: Regarding the first term:</p>
<p>\begin{align*}
\left[\frac{p_r^2}{2m},p_\theta^2\right]&=\frac{1}{2m}\left[p_r^2,p_\theta^2\right]=\frac{1}{2m}[p_r p_r, p_\theta^2]=\frac{1}{2m}\left(p_r[p_r,p_\theta^2]+p_r[p_r,p_\theta^2]\right)\\
&=\frac{1}{m}[p_r,p_\theta^2]=
\frac{1}{m}\left(p_\theta[p_r, p_\theta]+p_\theta[p_r,p_\theta]\right) = \frac{2p_\theta[p_r,p_\theta]}{m}
\end{align*}</p>
<p>So it boils down to what is $[p_r, p_\theta]$ equal to ? I'm not sure whether it can be answered directly without resorting to the definition of Poisson bracket.</p>
<p>The coordinates in our problem are $q_i = \{r, \theta\}, p_i = \{p_r, p_\theta\}$. Therefore:</p>
<p>\begin{align*}
[p_r, p_\theta] &= \sum_{i=1}^2 \left(\frac{\partial p_r}{\partial q_i}\frac{\partial p_\theta}{\partial p_i} - \frac{\partial p_r}{\partial p_i}\frac{\partial p_\theta}{\partial q_i}\right)\\ &=
\left(\frac{\partial p_r}{\partial r}\frac{\partial p_\theta}{\partial p_r} - \frac{\partial p_r}{\partial p_r}\frac{\partial p_\theta}{\partial r}\right) +
\left(\frac{\partial p_r}{\partial \theta}\frac{\partial p_\theta}{\partial p_\theta} - \frac{\partial p_r}{\partial p_\theta}\frac{\partial p_\theta}{\partial \theta}\right)\\
\end{align*}</p>
<p>And now I'm stuck again.</p>
<p>The way I understand it $p_r, p_\theta$ are two components of momentum along $r, \theta$ respectively. <strong>Does that imply that $p_r$ does not depend on $\theta$ and that it only depends on $r$</strong> ? If the answer is yes, then the result for the first term is zero.</p>
<p><strong>EDIT 2</strong>:
Based on the answers in Hamiltonian formalism all the canonical variables are taken independent to each other, therefore $[p_r, p_\theta]=0$. Similarly it may be shown that the 2nd term is also zero.</p>
| 10,579 |
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] |
<p>While mapping out the position, distance and movement of galaxies and <a href="http://en.wikipedia.org/wiki/Quasar" rel="nofollow">quasars</a> at the limits of the observable universe, have astronomers ever observed the motion of these object due to gravitational sources that may reside outside the limits of the observable universe?</p>
<p>If so, would this break the principle of information travelling faster than the speed of light?</p>
<p>If not, are these objects in fact the furthest objects there are?</p>
| 10,580 |
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] |
<p>I would like to learn how scientists go about measuring the large-scale curvature of the universe to determine if the universe is closed 'i.e. spherical', flat, or open 'i.e. saddle shaped'.</p>
<p>My simplistic thought is that you could measure the corner angles of a really large triangle and see if they add up to <180, 180, or >180 degrees.
However I can't imagine how you would do that in practice, (not without owning a Ningi anyway).</p>
<blockquote>
<p><em>Ref. Hitchhiker's Guide to the Galaxy: The 'Triganic Pu' is a
Monetary unit. Its exchange rate of eight Ningis to one Pu is simple
enough, but since a Ningi is a triangular rubber coin six thousand
eight hundred miles along each side, no one has ever collected enough
to own one Pu. Ningis are not negotiable currency, because the
Galactibanks refuse to deal in fiddling small change.</em></p>
</blockquote>
| 10,581 |
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<p>How does the optical zoom works..</p>
<ol>
<li>Does it brings the light closer to the object... Or</li>
<li>Does it just enlarges the picture of the object.</li>
</ol>
<p>In other words... Suppose I have telescope with 1 light-year zoom capacity, could I see stars 1 year ahead of its light reaching earth...</p>
<p>Ps : I'm not physics graduate, but interested in physics. I joined here, to learn more... Please excuse me if my question is trivial or stupid.</p>
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<p>I know that we have some solar system analogs. But is there a limit to it?</p>
| 10,583 |
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<p>Provided that I have the instantaneous positions and velocities of two particles that can separate in 3D with respect to time. I would like to characterize the change in length. What could be a good formulation for that? Thanks.</p>
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<p>How can I reduce the double image seen when looking through binoculars? I recently got a new pair, but have been having troubles using them because I am seeing double...</p>
| 10,585 |
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<p>It's known that single spherical lens cannot focus parallel beam of monochromatic light into single point.</p>
<p>Could you suggest how should aspherical lens look like to be able to focus in single point (again, light is monochromatic)? Do we need conic factor, or higher components?</p>
<p>Could you suggest any decent books on aspherical optics & it's manufacturing?</p>
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<p><strong>Background:</strong> I'm in a legal academic discussion about the status of electronic 'goods' and whether they qualify as 'goods' in the same way a chair and a pen do. In this context (and specifically at the exact circumstance under discussion), it matters if electricity is 'tangible'. So far, most authors have blindly assumed electricity to be a flow of electrons, making a literal analogy with water, making statements such as:</p>
<ul>
<li><p>Information is stored in capacitors in the form of electrons. When a capacitor is filled more than 50% with electrons, it's considered to be 'on' (a bit with value '1').</p></li>
<li><p>The information represented by a certain current (or rather, a series of on/off currents) has mass, because it consists of the electrons that flow through the wire.</p></li>
<li><p>A virtual object is tangible because it exists in memory in the form of electrons that are there (or not) in a certain pattern.</p></li>
</ul>
<p>Now I have a background in informatics but only a basic knowledge of electricity, and as good as none on the fundamental (physics) level. However I still feel this representation is wrong, and that you can't just say that information in a RAM chip has mass because it consists of electrons that are or are not in the capacitors on that chip. I have found hints in that direction on sites such as <a href="http://amasci.com/miscon/eleca.html#made" rel="nofollow">http://amasci.com/miscon/eleca.html#made</a>, but I can't quite make out what 'electricity' <em>is</em> and how it relates to <em>current</em> and <em>potential</em> and other words that are used interchangeably in these discussions but which are, I think, different things.</p>
<p>So my questions are (all just different angles of looking at the same underlying concept):</p>
<ul>
<li><p>What is 'electricity', really, on a fundamental level; but explained in terms a layman can understand? Is there an analogy with other things that is accurate, unlike the 'flowing water' analogy, which is sufficient for high school level but is a simplification? (at least, I think...)</p></li>
<li><p>Do 'electricity', an 'electrical current' and an 'electrical charge' have mass, apart from the object they are embodies in? Does the mass of a copper wire change when you put a current through it, because of the electrons coming in and going out?</p></li>
<li><p>How do electrons fit into this? Is electricity composed of a bunch of electrons that flow through a mass? I think not, reading the link I gave before, but I don't quite understand what their role is.</p></li>
<li><p>Most authors blindly assume that electricity is merely a flow of electrons through mass. In how far and under what assumptions is this correct?</p></li>
</ul>
| 10,587 |
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<p>To be more specific, what I mean is to measure it in a experiment.
And if the answer is no, I want to know if it is principally impossible, or just impossible due to the technic limitation of our time. </p>
<p>I am sorry I wan't clear enough at first.
For most of the times, in the progress of experiment, the system does not have enough time to get an equilibrium state. unless you are aiming to do a thermal dynamics experiment.
(in which the entropy of the system could be measured as $$\Delta S = \int_{T_1}^{T_2} \frac{C(T)}{T} \, \mathrm{d}T$$)
What I want to ask is, could we develop a normal way to measure the entropy in all kinds of experiments?</p>
<p>Thank you for your time:-P</p>
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<p>In the case of the B-mode detected in BICEP2, at first they assumed there was no circular polarization and then the Q and U stocks parameter of the CMB is given by E-mode and B mode. If we have circular polarization too, what is relation between the B-mode and circular polarization? Does this B- mode include the contribution of circular polarization or Not not?</p>
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<p>In all text books (and papers for that matter) about QFT and the classical limit of relativistic equations, one comes across the "minimal substitution" to introduce the magnetic potential into the equation (Schrödinger/Dirac/Klein-Gordon) through:
$$ \hat{p}^2 \rightarrow (\hat{p} - e \hat{A})^2$$</p>
<p>The wording suggests that this is an approximation for small? electromagnetic fields (or at least not strongly coupled). I understand this was chosen such that the "classical" Lorentz force is retrieved from the Hamiltonian, but not why exactly this form and not another that leads to the same result.</p>
<ol>
<li><p>To what extent is this "approximation" valid?</p></li>
<li><p>How can one improve this "minimal substitution"? Is there a more general expression, intuitively something like a series of the vector potential?</p></li>
</ol>
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<p>I know that integrating acceleration twice will give me position (acceleration-->velocity-->position) but how can I do all this when I all I have are a set of data points (ex: 1 second = some # m/s^2); how can I take these data points and integrate them?</p>
<p>Does this have something to do with <a href="http://en.wikipedia.org/wiki/Numerical_integration" rel="nofollow">numerical integration</a>?</p>
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<p>Why is straight gasoline (or whetever the mixture was before the introduction of ethanol) more efficient (ie, more miles/gallon) than E85? I've known since it's introduction that E85 was less efficient, but why is it?</p>
| 10,592 |
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<p>Our galaxies are speeding apart from each other and that is because of dark energy, but we don't know where it comes from right? If I'm not mistaken could dark matter have dark energy? (If they are the same thing please correct me.) Matter has gravity so could <a href="http://en.wikipedia.org/wiki/Dark_matter" rel="nofollow">dark matter</a> have <a href="http://en.wikipedia.org/wiki/Dark_energy" rel="nofollow">dark energy</a>, so sense there is more dark matter is that why our galaxies are being pushed apart?</p>
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<p>I'm looking for a pedagogic introduction to d-wave superconductivity.</p>
<p>Ideally, this would involve a derivation of d-wave superconductivity and the form of the gap parameter from some fundamental Hamiltonian (if such a derivation is even possible).</p>
<p>Books and papers are welcome.</p>
| 10,593 |
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<blockquote>
<p>If $55 034.175 \rm{kJ }$ of heat are transferred to $150 \rm{kg}$ of ice at a temperature of $-12.15 ^\circ \rm{C}$, calculate the temperature of the resulting water.</p>
</blockquote>
<p>Using $Q = mc(t_2-t_1)$ or $t_2 = \frac{Q}{mc}+t_1$,</p>
<p>$$t2 = \frac{55034.175}{150 \times 2.135} + (-12.15)$$</p>
<p>My answer is $178.6 ^\circ \rm{C}$</p>
<p>However, this does not seem possible to me because the boiling point of water is $100^\circ\rm{C}$ so therefore it would no longer be water. I feel my answer should be below $100^\circ\rm{C}$ if they are asking for the temperature of the resulting water.</p>
<p>Can someone please explain?</p>
| 10,594 |
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<p>A point-like particle $A$, coming from minus spatial infinity, heads at another one, $B$, with an <a href="http://en.wikipedia.org/wiki/Impact_parameter" rel="nofollow">impact parameter</a> of $b$.
Initial momenta are $p_A$ and $p_B=0$.</p>
<p>They repel each other via a <a href="http://en.wikipedia.org/wiki/Yukawa_potential" rel="nofollow">Yukawa potential</a></p>
<p>$$\ \ V(r_A,r_B)= +g^2\frac{e^{-\frac{|r_A-r_B|}{\lambda}}}{|r_A-r_B|} \ge 0.$$</p>
<blockquote>
<p>How does the angle of the incoming particle $A$ change, relative to how it was before?</p>
</blockquote>
| 10,595 |
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<p>I have never understood what's the meaning of the sentence "rolling without slipping". Let me explain.</p>
<p>I'll give an example. Yesterday my mechanics professor introduced some concepts of rotational dynamics. When he came to talk about spinning wheels he said something like: </p>
<blockquote>
<p>"If the wheel is rolling without slipping, what's the velocity of the point at the base of the wheel?? It is... zero! Convince yourself that the velocity must be zero. Since if it wasn't zero, the wheel wouldn't be rolling without slipping. So the wheel is rolling without slipping if and only if the point at the base has velocity zero, i.e. if and only if the tangential speed equals the speed of the center of mass."</p>
</blockquote>
<p>Well, what I really don't understand is this: is the "rolling without slipping" condition <em>defined as</em> "Point at the base has zero speed"? If not, what's the proper definition for that kind of motion?</p>
<p>Looking across the internet, I have found more or less the same ideas expressed in the quotation. Furthermore, if it was a definition, then it would be totally unnecessary to say "convince yourself" and improper to talk about necessary and sufficient conditions.</p>
<p>I'd like to point out that I'm not really confused about the mathematics behind this or with the meaning of the condition above. What puzzles me is why are those explanations always phrased as if the condition $v'=0$ (where $v'$ is the relative velocity beetween the point at base and the surface) is some necessary and sufficient condition to be "rolling without slipping". Seems to me that this <em>is</em> exactly the definition of "rolling without slipping" and not an "iff".</p>
<p>Any help is appreciated, thanks.</p>
| 10,596 |
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<p>Goldstein's mechanics book, on the chapter on relativistic mechanics says that "We cannot assume that all observers make the same division into time and space in the same way." What does it mean to "divide" spacetime?</p>
| 10,597 |
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<p>Im looking for some references here. I hope it is the right place to ask.</p>
<p>I need to find a table of (or a formula from which to extrapolate) the matrix elements of the radial functions of the hydrogen atom evaluated in powers of $r$ for both diagonal and off-diagonal elements. That is, I need this guys:</p>
<p>$<R_{nl}|r^k|R_{n'l'}>$</p>
<p>I found this article <a href="http://iopscience.iop.org/0953-4075/28/3/007" rel="nofollow">http://iopscience.iop.org/0953-4075/28/3/007</a> which is great, but Im looking for something more textbook-like, say suited for undergrad. I especially need positive powers until k=4.</p>
<p>Any help is very appreciated.</p>
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<p>I read somewhere what performing an orbifolding (i.e. imposing a discrete symmetry on what would otherwise be a compactification torus) is equivalent to "gauging the discrete symmetry". Can anybody clarify this statement in any way? In particular what does it mean to gauge a discrete symmetry if no gauge bosons and no covariant derivatives are introduced? </p>
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<p>I know that <a href="http://en.wikipedia.org/wiki/Classical_XY_model" rel="nofollow">XY statistical model</a> for $d=2$ doesn't show a regular phase transition , while the $3d$ has, I was wondering what is the behaviour for $2< d < 3$.</p>
<p>If it is simpler one could consider another model in its class of universality, like it's done for the $2+\epsilon$ expansion.</p>
<p>The simplest hypothesis is that exist a $d_{min}$ between 2 and 3 such that for values of dimension greater than $d_{min}$ there is phase transition.</p>
<p>Assuming this number has a meaning for every statistical model, I would like to ask if it is universal with respect to the simmetries of the order parameter.</p>
<p>Going further, what is the behaviour of the critical temperature around $d_{min}$, is that universal?</p>
| 10,600 |
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] |
<p>Newton's ring apparatus consists of a Plano-spherical glass which rests on its vertex on top if a horizontal surface. Reflection-interference occurs along the air wedge, and is seen as a series of concentric rings from above. You may assume the radius of convergence is much larger than the thickness of the wedge. </p>
<p>Now, to the actual problem.
When a liquid is introduced into the air between the glass and table, the radius of the tenth bright fring changes from $1.50$ cm to $1.31$ cm from the center of the pattern. Calculate the index of refraction of the liquid.</p>
| 10,601 |
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] |
<p>If we want to transform to another inertial frame of reference using <a href="http://en.wikipedia.org/wiki/Galilean_transformation" rel="nofollow">Galilean transformation</a> in 4-dimensional space-time, what is the Jacobian matrix of Galilean transformation?</p>
| 10,602 |
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<p>I was wondering if we have the chance to run on the Moon's surface, how would you expect it look like?
I expect our velocity will increase for the same work we do on Earth, but not sure if this will be multiples in term of gravity variations.</p>
<p>How do you think our maximum speed would reach?</p>
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] |
<p>Can we calculate the number of photons emitted per second by a radioactive source (gamma) ?
we can take 100g of barium-133 as an example</p>
| 10,604 |
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] |
<p>Gravitational lensing is an observed phenomenon. Can one have a gravitational mirror? </p>
<p>A slightly unrelated question: Can gravitational waves be reflected?</p>
| 10,605 |
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<p>So, I understand that for a NPN transistor to work the emitter-base junction needs to be forward biased and the collector-base junction needs to be reverse biased. I understand how current flows from the emitter to the base when forward biased, but I can't seem to wrap my head around how it flows from the emitter, through the base, and into the collector.</p>
<p>From what I understand about the depletion region of a PN junction diode, when forward biased, the electrons in the N-type material are repulsed by the negative charge of the supply (say a battery) and the holes in the P-type material are repulsed by the positive charge from the supply, thus narrowing the depletion region and allowing electrons to flow from the N region to the P region. I also understand that the holes and electrons are attracted to the source charges when reverse biased, widening the depletion region and blocking any electrons from flowing.</p>
<p>What I don't understand is what happens to the depletion region of the collector-base junction that allows electrons to flow from the emitter to the collector even though the collector-base junction is reverse biased.</p>
<p>Take a simple common emitter circuit for example. I apparently am not cool enough to post pictures yet so you'll have to use your imagination. So pretend we have the base connected to the positive terminal on a 1v battery (Vbb). The collector is connected to the positive terminal on a 10v battery (Vcc). The emitter and both negative terminals are connected to ground. Also pretend that there is some kind of current limiting. How does current get from the emitter through to the collector? I can see how they are at different potentials, but I don't understand how this overcomes the reverse biasing of the collector-base junction.</p>
<p>Can anyone tell me what actually happens at the junction to allow current flow and how changing Vbb will affect that current?</p>
| 10,606 |
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] |
<p>I'm working on a science project and need to know what makes a strong electromagnet - more wire turns or more voltage? I understand energy can be lost to heat with too many wire turns.</p>
| 10,607 |
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<p>Are there any packages that can simplify functional derivatives in path integrals? For instance the expression (integrate over, $x,y,z,u,v,r,s$): </p>
<p>$$\int\delta_{J_{A}^x}\delta_{J_{B}^x}\delta_{J_{B}^x}\int{J_{B}^y}\Delta^{yz}_B{J_{B}^z}\int{J_{B}^u}\Delta^{uv}_B{J_{B}^v}\int{J_{A}^r}\Delta^{rs}_A{J_{A}^s},$$
where $$\delta_{J_{A}^x}:= \frac{\delta}{\delta {J_{A}^x}}$$ and
$$\Delta^{rs}_A:=\Delta(r-s)_A.$$</p>
| 10,608 |
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<p>Suppose I have two neutrally-buoyant macromolecules diffusing in water. They have the same radius of gyration (i.e. same root-mean-square distance from their center of mass), but one of them is compact (its mass is roughly the cube of its size) and the other is extended (its mass is roughly the square of its size).</p>
<p>Since these molecules are the same size, do they have roughly the same diffusion coefficient? </p>
<p>Alternatively, their root-mean-square velocities should be different since they have different mass. Does this lead to substantially different diffusion coefficients?</p>
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<p><a href="http://physics.stackexchange.com/a/9910/22775">Kostya answered a question</a> that was asking what the diffraction pattern looks like for a hexagonal aperture in front of a lens. He lists an equation which was derived using a Heaviside function to describe the shape of the aperture.</p>
<p>I was wondering if anyone knows what units were used for the display of his Fourier Transform. He mentions he plotted that from -100 to 100 for (what I am assuming) is $\omega_x$ and $\omega_y$. The units seem to be degrees but I wanted to make sure, as sometimes these problems regarding FTs are treated as being unitless. </p>
<p>I have tried other approaches to predict what that diffraction pattern would look like, and so far Kostya's answer agrees by far the best with my own experimental results and with graphical Fourier Transforms of similarly shaped apertures.</p>
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<p>Given a Hamiltonian of the form</p>
<p>$H=\sum_k \begin{pmatrix}a_k^\dagger & b_k^\dagger \end{pmatrix}
\begin{pmatrix}\omega_0 & \Omega f_k \\ \Omega f_k^* & \omega_0\end{pmatrix} \begin{pmatrix}a_k \\ b_k\end{pmatrix}$</p>
<p>where $a_k$ and $b_k$ are bosonic annihilation operators, $\omega_0$ and $\Omega$ are real constants and $f_k$ is a complex constant.</p>
<p>How does one diagonalise this with a Bogoliubov transformation? I've seen an excellent answer to a similar question <a href="http://physics.stackexchange.com/questions/53158/validity-of-bogoliubov-transformation">here</a>, but I'm not quite sure how it translates to this example. Any hints or pointers much appreciated.</p>
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<p>I am nothing but a curious layman so don't go too technical on me.</p>
<p>First of all, I am well aware that a lot of people consider the question of determinism vs indeterminism to be unsolved and others claim it is solved. That is not what this post is about, so let's not digress into heated debates over the validity of the interpretation I am asking about here.</p>
<p>Now <em>if</em> something like the copenhagen interpretation or grw is true then there is inherent randomness in nature. Before measurement, there is no truth about what outcome will be. There is no mechanism deciding that the infamous Cat will die or survive. There is no fact until measurement is made. I.E. the future is 100% open.</p>
<p>But how can that be reconciled with the view we get from special relativity? That there is no difference between past-present-future?
If I have understood the consequence of relativity correctly, that means that the "now" we experience is basically just a slice within a huge block of time. So the past is as real as now and the future too. And if that is the case then the future has to be fixed, otherwise what sense would it make to say that it already exists?</p>
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<p>I would like to compute the heat distribution of a piece of metal with some surrounding material. The heat is assumed to propagate by diffusion, so inside the metal piece and also on the outside, the governing law would famously be
$$
\rho c_p \frac{\partial u}{\partial t} = \nabla(\kappa \nabla u).
$$
What would be the appropriate condition for the interface between the two materials?</p>
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<blockquote>
<p><em>Two point charges a distance $d$ apart in free space exert a force of $1.4\times10^{-4}N$. When the free space is replaced by a homogeneous dielectric medium, the force becomes $0.9\times10^{-4}N$. What is the dielectric constant $\epsilon_r$ of the medium?</em></p>
</blockquote>
<p>I have been staring at this problem for awhile. I can only think of using $E={F/Q}$. But the problem is this only applies to free space and I do not know $Q$. So I am not sure where to start to compare the free space and medium.</p>
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<p>When work is done on a positive test charge by an external force to move it from one location to another, does the potential energy increases or decreases, and does the electric potential increases or decreases?</p>
| 10,615 |
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<p>I have a given Lagrangian: </p>
<p>$$L= e^{st}\cdot\frac12\cdot(mv_y^2-ky^2)$$</p>
<p>And are asked to identify the equations of motions, the constants of motions and physical system. </p>
<p>Without the exp-time-term, this wold be a harmonic oscillator with the solution</p>
<p>$$\frac{d^2y}{dt^2} = -\left(\frac km\right)\cdot y$$ and $$y(t)=\Re\left(Ae^{i\omega t}+Be^{-i\omega t}\right)$$</p>
<p>I'm wondering:</p>
<ol>
<li><p>How will the $\exp(st)$-term will effect my solution?</p></li>
<li><p>What will happen to the Lagrangian if I do a transformation of variables $q=\exp(st)\cdot y$? </p></li>
</ol>
| 10,616 |
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<p>I have been studying scattering theory in Sakurai's quantum mechanics. The phase shift in scattering theory has been a major conceptual and computational stumbling block for me.</p>
<p>How (if at all) does the phase shift relate to the scattering amplitude?</p>
<p>What does it help you calculate?</p>
<p>Also, any literature or book references that might be more accessible than Sakurai would be greatly appreciated.</p>
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<p>Can someone please help me explaining how to convert units in problems like the one below? I've done many things, but I am very unsure of all my methods. Could you tell me what steps should I take?</p>
<p>$$5.66\ \frac{\mathrm{hm}}{\mathrm{min.}} \frac{\hspace{2cm}}{}\frac{\mathrm{km}}{\mathrm{hr.}}$$</p>
<p>Thank you.</p>
| 156 |
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<p>I have read that the fine structure <a href="http://www.physorg.com/news202921592.html">constant may well not be a constant</a>. Now, if this were to be true, what would be the effect of a higher or lower value? (and why?)</p>
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<p>Another Noether's theorem question, this time about electrical charge.</p>
<p>According to Noether's theorem, all conservation laws originate from invariance of a system to shifts in a certain space. For example conservation of energy stems from invariance to time translation.</p>
<p>What kind of symmetry creates the conservation of electrical charge?</p>
| 823 |
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<p>When it rains and water flows down an inclined street, ripples may form that are carried along with the current. Here's a picture with an example of what I'm talking about</p>
<p><img src="http://i.stack.imgur.com/b3rj5.jpg" alt="enter image description here"></p>
<p>I'd like to know what the fluid mechanical mechanism is for the formation of these waves. A quick google search reveals that they've been identified in some scientific literature as "rain waves" (if you have a jstor subscription you can see <a href="http://www.jstor.org/stable/4297089?seq=1" rel="nofollow">http://www.jstor.org/stable/4297089?seq=1</a>) but I've only been able to find phenomenological descriptions of their properties without any details of the physics giving rise to them. In particular, I'm curious if people think that the surface on which the water is flowing must have a particular texture in order to give rise to the waves.</p>
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<p>So in wired/wireless networking and radio, signals are sent in form of wave. Then the concept of bandwidth comes in, which is the difference between highest frequency and lowest frequency in a signal. But I do not get why bandwidth determines the maximum information per second that can be sent. If we are able to send signals of any frequency in the bandwidth, then as the number of signals that are of frequencies in an aggregated signal increases, information that can be sent increases without bound.</p>
<p>Is this not possible because when adding some frequency, information in some frequency is necessarily violated? And why does maximum information per second that can be sent depends only on bandwidth, not highest frequency in the aggregated signal?</p>
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<p>What formula defines a curve for a nose cone with the minimal possible drag?</p>
<p>The nose cone is attached to a cylinder (assume it goes on forever).
The volume of the nose cone is not relevant.
The nose cone isn't near the ground. And the object will be traveling at 20m/s max.</p>
<p>So far, I've been using the sears haack body but im not sure if it is actually the best possible curve.</p>
| 10,621 |
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<ol>
<li><p>Which transformations are <a href="http://en.wikipedia.org/wiki/Canonical_transformation" rel="nofollow">canonical</a>? </p></li>
<li><p>Why do canonical transformations <a href="http://en.wikipedia.org/wiki/Liouville%27s_theorem_%28Hamiltonian%29" rel="nofollow">preserve</a> the measure of integration in phase space? </p></li>
</ol>
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<p>I have the following setup:
<img src="http://i.stack.imgur.com/DTy61.png" alt="Tightening a string"></p>
<p>The string of negligible mass, causing little friction, is shown in dark blue and is fixed at the red circle. The taut string is shown in light blue (with excess string removed). I have shown an orange arrow which indicates the direction in which the string is being pulled. Importantly the force is always exerted at that location and any excess string is removed.</p>
<p>I would like to know how the string deforms over time as it is pulled from that point (orange). Does the end closest to the pulling force move first and straighten up, wrapping around the object 'A' early on, or does the whole string change shape continuously?</p>
<p>How does the direction of pulling force affect things if at all?</p>
<p>Thanks in advance.</p>
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<p>What's the difference between Feynman rules in momentum space for $\phi^3$ theory and for $\phi^4$ theory?</p>
<p>I know it's only a slight difference and perhaps found in the vertex factor? But for some reason I can't seem to find this difference anywhere.</p>
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<p><strong>Is there anyway to recreate this experiment with <em>detectors</em> at home?</strong></p>
<p>I want detectors because I want to get non interference pattern too.
I have a little know-how in electronics so If I need to buy a detector there wouldn't be a problem using it. The thing is I don't know which detector should I use.
<strong>Can I do this? Is It just too expensive for a DIY project?</strong></p>
| 10,625 |
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<p>If $S_F(x-y)$ is the Green's function for the Dirac operator $(i\gamma^\mu\partial_\mu-m)$, that is, I assume the following matrix equation holds: $$ (i\gamma^\mu\partial_\mu-m)S_F(x-y)=i\delta(x-y) $$</p>
<p>The adjoint dirac equation is:
$$ -i\partial_\mu\bar{\psi}\gamma^\mu -m\bar\psi=0 $$</p>
<p>I am looking for the Green's function of the above equation, in terms of $S_F(x-y)$, that is, if $F[S]$ is some function of $S$ (may include complex conjugation, transposing, etc), then I am looking for such an $F[S]$ such that this equation holds:
$$ -i\partial_\mu F[S]\gamma^\mu -mF[S]=i\delta(x-y) $$</p>
<p>What I have done so far:</p>
<ol>
<li>Dagger this equation: $(i\gamma^\mu\partial_\mu-m)S_F(x-y)=i\delta(x-y)$ and get $$(-i\partial_\mu S_F(x-y)^\dagger \gamma^{\mu\dagger} - S_F(x-y)^\dagger m)=-i\delta(x-y)$$</li>
<li>Multiply by $\gamma^0$ on the right to get: $$ (-i\partial_\mu [-\bar{S_F}(x-y)] \gamma^{\mu} - [-\bar{S_F}(x-y)] m)=i\delta(x-y)\gamma^0 $$</li>
<li>So $[-\bar{S_F}(x-y)]$ almost solves this equation, except you get a factor of $\gamma^0$ on the right which I am not sure how to handle.</li>
</ol>
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<p>Schwarschild found his solution to Einstein's field equations for vacuum ($T_{\mu \nu}=0$) by placing a point-mass in the center of origin.</p>
<p>Since the Ricci tensor $R_{\mu \nu}$ and the Einstein tensor $G_{\mu \nu}$ are trace-reversed, meaning they vanish identically:</p>
<p><strong>Which point of a universe with a mass in it would meet this equation: $R_{\mu \nu}=0$ (and therefore $G_{\mu \nu}=0$)?</strong></p>
<p>(to make sure, infinity is not a point, i.e. if gravity is infinite than you can't find a particular point, where it vanishes down to zero due to distance)</p>
| 10,627 |
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<p>I am studying diagonalization of a quadratic bosonic Hamiltonian of the type:</p>
<p>$$ H = \displaystyle\sum_{<i,j>} A_{ij} a_i^\dagger a_j + \frac{1}{2}\displaystyle\sum_{<i,j>} [B_{ij} a_i^\dagger a_j^\dagger + B_{ij}^* a_j a_i ]
$$</p>
<p>in second quantization.<br>
Also, $A = A^\dagger$ and $B = B^T$, so that the Hamitonian is Hermitian. </p>
<p>This can be written in matrix form as
$$ H = \frac{1}{2} \alpha^\dagger M \alpha - \frac{1}{2} tr(A)$$
where $ \alpha = \begin{pmatrix}
a \\
a^\dagger \\
\end{pmatrix} $
, $ \alpha^\dagger = \begin{pmatrix}
a & a^\dagger
\end{pmatrix} $
and M is $ M = \begin{pmatrix}
A & B\\
B^* & A^* \\
\end{pmatrix} $ </p>
<p>Question: Can I numerically diagonalize the matrix M to get eigenvalues and eigenvectors of the Hamiltonian?
If yes, then what would be the right way to write those eigenvectors in second quantization?</p>
<p>e.g. If for 2$\times$2 matrix M, one of the numerically calculated eigenvectors is $\begin{pmatrix}
p\\
q \\
\end{pmatrix} $ , then, should it be written as $$p \,a|0\rangle + q \, a^\dagger|0\rangle $$
(where the column $\alpha$ has been used as the basis)
or
$$p \,a^\dagger|0\rangle + q \, a|0\rangle $$
(where $\alpha^\dagger$ has been used as the basis)?</p>
<p>Note: Here $|0\rangle$ is the vacuum state for 'a' type (bosonic) particles.
End of Question.</p>
<p>Note : Consider a simpler Hamiltonian
$$ H = \displaystyle\sum_{<i,j>} A_{ij} a_i^\dagger a_j $$
and note that its eigenvectors are of the form
$$(a_1^\dagger a_2^\dagger ... ) |0\rangle $$</p>
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<p>The <a href="http://en.wikipedia.org/wiki/Renormalization_group">renormalization group</a> equation is given by:
\begin{equation}
\left[\mu \frac{\partial}{\partial \mu} + \beta \frac{\partial}{\partial g} + m \gamma_{m^2} \frac{\partial}{\partial m} - n \gamma_d \right] \Gamma^{(n)}(\{p_i\};g,m,\mu) = 0
\end{equation}
where:
\begin{align}
\beta & \equiv \mu \frac{\partial g}{\partial \mu} \\
\gamma_d & \equiv \frac{1}{2} \mu \frac{\partial \ln Z_\phi}{\partial \mu} \\
\gamma_{m^2} & \equiv \frac{1}{2} \mu \frac{\partial \ln m^2}{\partial \mu} = \frac{\mu}{m} \frac{\partial m}{\partial \mu}
\end{align}
Why do we refer to it as the renormalization <em>group</em> equation. Does it form a <a href="http://en.wikipedia.org/wiki/Group_%28mathematics%29">group</a>? And if so, what are the elements of the group, and how can one see that it satisfies the axioms of a group?</p>
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<p>I have always thought that the time is discrete (jumping), not continuous and the shortest time is <code>Planck Time</code>. One reason for that was the information on <code>Wikipedia</code> and another was the following note I read in a book:</p>
<blockquote>
<p>If you throw a rock to a tree, the rock will go half the way, then
half the way, then half the way and it will keep going the half of the
remaining way. Therefore, the rock will never hit the tree</p>
</blockquote>
<p>This could be easily applied on time. It will go half the time and half the remaining and so on. My point is:</p>
<ol>
<li><p>How come the self-appearing virtual particle duos live shorter than <code>Planck Time</code> if it is shortest possible time? The information on life time of virtual particles was in Lawrance Krauss' book titled as <code>Universe from Nothing</code></p></li>
<li><p>Is time continuous or discrete?</p></li>
</ol>
<p>I know there are many posts regarding my second question but my aim was to ask, if time is continuous (as mentioned in some posts), then how do you explain the rock-tree phenomena?</p>
| 10,630 |
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<p>This is what I understand so far: in a conductor, the ions have a weak pull on the valence electrons. So when an electric field is applied, the free electrons are able to easily move about. Makes sense.</p>
<p>In a neutral conductor with no electric field, the free electrons aren't bound to any ions. Why? I understand that the ions have a weak pull on the electrons, but what makes electrons leave the ion and stay free?</p>
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<p>I am a n00b to physics. What physics I know comes from high school and popular science books.</p>
<p>At school we were taught about atoms and what make up atoms, but we weren't actually told what the particles were!</p>
<p>So sure an atom has electrons orbiting around a nucleus made of protons and neutrons, but what exactly is an <a href="http://en.wikipedia.org/wiki/Electron" rel="nofollow">electron</a>?</p>
<p>And if we don't know, how can we fire them from guns, like they do in experiments ... etc</p>
<p>Any clarification on this would be great.</p>
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<p>This calculation is for a double slit experiment setup which is experiencing a far field radiation from an extended monochromatic thermal source. I assume the source is 1-D and it's length is $b$. Here is the schematics.</p>
<p><img src="http://i.stack.imgur.com/VRbiW.jpg" alt="enter image description here"></p>
<p>I think a single photon state emitted from such source is: $$|\Psi\rangle = \int^{\frac{b}{2}}_{-\frac{b}{2}} db \sum_n c_n \int d\omega f_n(\omega)\hat{a}^\dagger_n(\omega) e^{- i \omega F(z_n, b)} |0\rangle$$. But I am little lost what should be $F(z_n, b)$?</p>
<p>I think a coherent state representation of the same source is: $$|{\alpha}\rangle = \int^{\frac{b}{2}}_{-\frac{b}{2}} db \prod_{\bf k} |\alpha_{\bf k} e^{- i \omega F'(z_n, b, k)} \rangle$$. But I am little lost what should be $F'(z_n, b, k)$?</p>
<p>The value of n could be either 1 or 2 corresponding to the first or second slit respectively. Each $\omega$ is a mode coming from the point sub source. Are my expressions correct?</p>
| 10,633 |
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<p>I am looking for an expression for the velocity of a micron sized (1 - 10 micron diameter) sized particles under accelerating forces.</p>
<p>I have aerosols in mind.</p>
<p>This is what I have in mind
The resisting force acting on the particle is </p>
<p>$F_D = 3 \pi \mu Vd$ so its (mechanical) mobility is $\frac{V}{F_D} = \frac{1}{3 \pi \mu d}$</p>
<p>This expresses velocity of a particle per unit force. Does that mean that if the same force acts on a bunch of different sized particles, the smaller particles should move faster due to their smaller size and the dependance is $\frac{1}{d}$? Is this problem perhaps more complicated than this (full F=ma treatment)</p>
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<p>Suppose that a spherical metal sphere with mass $m=10^{-16}kgr$ radius $R=10μm$ charge $Q=10^{-9}C$ travels with $v=c/3$ and is trapped in the earth's magnetosphere at a distance around $r = 1000km$. <em>The exact numbers are not that important, I am providing them for order of magnitude considerations</em>. </p>
<p>Assuming a non-zero velocity component parallel to the field lines:</p>
<ul>
<li>Will magnetic mirroring and magnetic drift be the same as with plasma
(i.e. can one use the same equations to calculate the motion of the sphere)? Will the eddy currents in the sphere affect mirroring/drift and how? </li>
<li>If the sphere is already spinning, what effect will the spin have on its motion and vice-versa? </li>
<li>If instead of a solid sphere we had a spherical shell with the same characteristics, would it behave differently and how?</li>
</ul>
<p>I am looking for a qualitative answer, or a pointer to work that has been done along these lines. </p>
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<p>Is it true that every point in the universe is connected or could be so theoretically? If so how is this mediated?</p>
<p>Is it through the quantum nature of the fabric of space or is it through the interrelationships of the gravity fields throughout the universe. From the gravity fields of single solar systems affecting each other to galaxies and galactic clusters as if they gravitational energy of each field has a knock-on effect on the next field?</p>
| 10,636 |
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<p>I am performing a simulation involving growth of bacteria. This is an agent-based simulation where the solutes (glucose, oxygen etc.) are represented as a concentration field discretised over space, and the bacteria are discrete objects in a continuous space (spherical blobs with a size, mass and position).</p>
<p>The concentration fields diffuse throughout space, and the bacteria provide the <em>reaction</em> by taking up (or releasing) these solutes. The rate of uptake at any position in space is proportional to the amount of biomass at that position. Hence this becomes a <strong>reaction-diffusion</strong> problem.</p>
<p>The simulation is time discretised such that in one time step:</p>
<ul>
<li>concentration fields are solved to steady state by numerical integration of a reaction-diffusion equation for the current amount of biomass</li>
<li>bacteria grow (biomass in the system changes) with respect to the updated concentration fields</li>
</ul>
<p>I want to know, how valid is this approximation? I have always justified it that diffusion happens on a much smaller timescale than growth of the bacteria. Is it worth creating a transient solver where the reaction-diffusion equation is solved over a finite length of time and the bugs are grown for the same amount of time in a leapfrog-like scheme?</p>
| 10,637 |
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<p>If an external EM field (a laser, for example) act on a dielectric (a glass, for example) what will be the effect of this field on the dielectric constant and on the refractive index of the material? How, mathematically speaking, could I explain what is happen? </p>
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<p>I am looking for interesting ways to introduce the Doppler effect to students. I want some situations in nature or every day life, where a student is possibly surprised and may ask "how could it be"?</p>
<p>The common example for Doppler effect of sound waves is the siren of a moving emergency vehicle. Clearly the siren example is one way to introduce it starting from a phenomenon. However are there other interesting, perhaps astonishing phenomena in nature or in every day life which can explained by Doppler effect of sound waves?</p>
<p>In this question I am only interested in Doppler effect of sound waves, not that one of light.</p>
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<p>This question is in reference to the first ~15 minutes of this String Theory lecture by Prof.Shiraz Minwalla,</p>
<p><a href="http://theory.tifr.res.in/Videos/strings28_24sep08.mp4" rel="nofollow">http://theory.tifr.res.in/Videos/strings28_24sep08.mp4</a></p>
<p>Can one give a reference (paper/review/book) for this (to my mind very technical) issue being explained here? I have seen Shiraz mention this issue in the previous 2-3 lectures too and its one of those rare concepts for which I could not understand his explanation.</p>
<p>I can't relate as to what in Polchinski's book corresponds to this.
(though this course claims to follow that book). </p>
<hr>
<p>The final answer that Shiraz wants to get comes as two integrals (one for open and one for the closed case) on the top of the LHS of the board at 15:32 of the video. He claims something about being able to rewrite the moduli space integral in such a way that the b-c 0-mode ghost expectation value decouples from the "integrated vertex operators". I can't understand the point that he is making here about why since the ghost 0-mode is a constant one can evaluate it any any point and that somehow leads to some simplification etc. The initial few minutes of motivation part of the lecture is also not clear to me as to what is he trying to do - about the independence or not of the holomorphic and anti-holomorphic part of the b-ghosts and their boundary values on the worldsheet. </p>
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] |
<p>Why are these states called as such, and how do they differ? I vaguely understand that when $E > 0$ you obtain a scattering state, but when $E < 0$ you have a bound state.</p>
| 10,641 |
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<p>I need to process some images taken from a telescope to determine the intensity of an astroid.
This way I can determine the rotation period of this asteroid. </p>
<p>The pictures were taken the usual way (I guess, I'm not an astronomer, this is a school project), setting the telescope to track a nearby star. I would need to do dark frame and flat field correction on a lot of images (so batch processing would be great). Since the asteroid is moving in relation to the star, there would also have to be functionality to bring this into account.</p>
<p>Is there a piece of software (<strong>for mac</strong>) around that can do this? I know Iris exist for windows, but I don't have a PC with windows available, and learning a new piece of software on a computer that isn't mine (eg in the library) will only lead to frustration.</p>
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<p>As an aeroplane accelerates through the Earth's B-field, it experiences a changing flux $d\phi \over dt$ and a potential difference is induced along its wings. </p>
<p>Given the wings are made of metal and act as a continuous conductor, current flows along the wings in a direction perpendicular to the vertical component of the Earth's B-field. Does this mean that the aeroplane would experience a Lorentz force? </p>
<p>What other effects could the building up of charge have on an aeroplane, and what difference would it make to this process if the aeroplane's wings were made of an insulating material?</p>
<p>Also, in terms of bigger effects, what would happen to an aeroplane in a thunderstorm?</p>
<p>Update: I discovered that aeroplanes have something called "discharge wicks" that are designed to counter this problem (see pic). However. the build up of static charge seems more significant than the buildup of charge due to the effect I described as Gupta pointed out so these are probably used to dissipate static charges.</p>
<p><img src="http://i.stack.imgur.com/TF1vt.jpg" alt="enter image description here"></p>
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<p>I wonder if there is some simple theory of what determine <strong>efficiency ( speed, energy end force required )</strong> of cutting by edge ( <strong>blade , knife, sword</strong> )</p>
<hr>
<p>At least something phenomenological like in case of <strong>friction</strong> ( which I guess is very much related to cutting ) $F_F = \mu F_\perp$ (where $F_F$ is friction force, $\mu$ is <strong>friction coefficient</strong> which is tabulated for specific materials, and $F_\perp$ is normal force - the force you push the surfaces into each other )</p>
<p>Particular problems which should be solved by this theory:</p>
<ul>
<li><p>Given a material ( i.e. some coefficients) and total force $F_{tot}$ asserted on the knife, what should be the optimal <strong>tilt angle</strong> $\alpha$ relative to surface normal to achieve <strong>most efficient cutting</strong>. $\alpha$ determine decomposition of $F_{tot}$ into components perpendicular $F_\perp = \cos(\alpha) F_{tot}$ and parallel $F_\parallel = \sin(\alpha) F_{tot}$ to the surface.</p></li>
<li><p>How it depends on <a href="http://www.fooduniversity.com/foodu/food_c/basic_knife_skills/Knives/basic_knife_bevel_angle.htm" rel="nofollow">bevel angle</a> $\theta$ ( or possibly hardness of the tip and cut material )</p></li>
</ul>
<p>To have some hint what such theory could be based on - in <a href="https://www.youtube.com/watch?v=G5_4uC3-Xu8" rel="nofollow">this video (8:41)</a>
is said that the blade actually works like <strong>microscopic saw</strong>. So, it would be probably quite straightforward to develop such theory having in mind a saw model with considering saw-tooth with particular <strong>tooth tip angle</strong> $\omega$ or some statistical distribution of such teeth </p>
| 10,644 |
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<p>When calculating time period of simple pendulum (an approximation of SHM at small amplitudes) we take gravitational force greater than tension in string and resolve gravitational force in two orthogonal components and equate one of them with tension.While in case of conical pendulum we take tension greater than gravitational force and resolve it into two orthogonal components and equate one of them with gravitational force on metallic bob.Similarly in banked road problem we take normal reaction greater than gravitational force.</p>
<p>Why in one case tension is more and in other gravitation is more?</p>
| 10,645 |
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<p>I've been tasked to do a simple experiment on the elasticity of collisions. For this experiment I used two "pucks" (very light circular metal pieces of certain height but hollow) and a table that works like air hockey tables (decreases friction by blowing air from underneath). One puck was placed on this board and the other one was shot into it. Each puck had two reflective markers on them, one in the center and one on the edge. The positions of these markers were logged by two cameras shooting infrared light. I am now trying to understand this position data.</p>
<p>This is the data I have (I'm using Mathematica):</p>
<pre><code>m1Vel = Differences /@ {m11x, m11y};
m2Vel = Differences /@ {m21x, m21y};
m12Vel = Differences /@ {m12x, m12y};
m22Vel = Differences /@ {m22x, m22y};
m1DeltaX = m11x - m12x;
m1DeltaY = m11y - m12y;
m2DeltaX = m21x - m22x;
m2DeltaY = m21y - m22y;
angularVel[dy_, dx_] := Differences@ArcTan[dy/dx]
vectorNorm2[list_] := Sqrt[list[[1]]^2 + list[[2]]^2];
</code></pre>
<p>Using this to plot position data for the pucks M1 and M2:</p>
<pre><code>ListLinePlot[{m11x, m11y, m21x, m21y},
PlotLegend -> {"M1 X", "M1 Y", "M2 X", "M2 Y"}, LegendSize -> 0.5,
LegendPosition -> {1.1, 0}]
</code></pre>
<p><img src="http://i.stack.imgur.com/0a7e3.png" alt="Position data"></p>
<p>And then approximate the velocity for each puck, for the marker in the center:</p>
<pre><code>ListLinePlot[{vectorNorm2[m1Vel], vectorNorm2[m2Vel],
vectorNorm2[m1Vel] + vectorNorm2[m2Vel]}, PlotRange -> Full,
PlotLegend -> {"M1 v", "M2 v", "M1+M2 v"}, LegendSize -> 0.5,
LegendPosition -> {1.1, 0}]
</code></pre>
<p><img src="http://i.stack.imgur.com/r1jRV.png" alt="Velocity for center marker"></p>
<p>And for the marker on the edge:</p>
<p><img src="http://i.stack.imgur.com/FDS8P.png" alt="Velocity for edge marker"></p>
<p>And finally the rotation of each puck, using the approximation that the angle from a horizontal line is $\mathrm{arctan}(\frac{\Delta x}{\Delta y})$ and the angle velocity therefore the difference between the angle at one point and the angle at the next point, as seen in the function angularVel above.</p>
<pre><code>ListLinePlot[{MovingAverage[angularVel[m1DeltaY, m1DeltaX], 10]^2,
MovingAverage[angularVel[m2DeltaY, m2DeltaX], 10]^2},
PlotRange -> Full, PlotLegend -> {"M1 w", "M2 w"}, LegendSize -> 0.5,
LegendPosition -> {1.1, 0}]
</code></pre>
<p><img src="http://i.stack.imgur.com/bdVLN.png" alt="Rotation of each puck"></p>
<p><strong>Alright, so what's the matter?</strong></p>
<ol>
<li>I was expecting both of the velocity graphs to look like the first. Since the kinetic energy is proportional to the velocity squared, it is unacceptable to me that it goes up and down in the second velocity graph. It should decrease in a monoton manner. The first collision is with the other puck, but there are collisions after that with the walls.</li>
<li>The rotational energy is proportional to the angular velocity. I get that if in the collision with a wall some energy is transferred from translation to rotation and that the rotational energy therefore does not decrease in a monoton manner, but these really sharp peaks (even sharper without the moving average) I cannot understand.</li>
</ol>
<p>Since I'm studying the elasticity I really need only to understand what happens in the first collision. But I feel like a fraud if I write something up about that, neglecting the rest of the graph with all of its peculiarities. <strong>If you had to explain these things in a report, what would you write?</strong></p>
| 10,646 |
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<p>People always talk about finding good dark matter (DM) candidates of beyond Standard Model (SM) theories. To be a proper candidate the particles should not decay into the SM particles and could potentially be weakly interacting mass particles (WIMPs), but this doesn't need to be the case. If DM is not composed of WIMPs then can we just add DM through the simplest and most boring way,
\begin{equation}
\Delta {\cal L} = m_{DM}^2 \phi_{DM} ^\dagger \phi_{DM}
\end{equation}
to the SM? Is there something wrong with this naive addition (other than it not being predictive)?</p>
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<p>Consider a two-level energy gap with electronic energy states $E_1$ & $E_2$ and associated population densities $n_1$ and $n_2$ with $E_2>E_1$.</p>
<p>In the derivation of the Einstein coefficients for <em>Absorption</em>, <em>Spontaneous Emission</em> and <em>Stimulated Emission</em>, textbooks/notes quite often make statements like the transition from $E_2$ to $E_1$ <strong>only</strong> depends on population density $n_2$. As an example, for the case of spontaneous decay, this clearly is just not true as there is a limit to the number of electrons occupying $E_1$ .</p>
<p>My question then is: <em>how is this assumption justified?</em> and/or how is the maximum value of $n_1$ taken into account in general LASER theory?</p>
<p>All comments welcome.</p>
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<p>According to Newton, if no external net force acting on a system then the system will keep its initial condition whether at rest or moving uniformly in a straight line.</p>
<p>Let's consider a boy riding a bicycle. He exerts a force to the bicycle and the bicycle reacts with the same magnitude but opposite direction to the boy. From an observer outside the system of boy-bicycle, the net external force acting on the system is always zero so there will be no acceleration. However, in our daily life, we have seen that accelerating a bicycle is possible. It seems to me inconsistent, doesn't it?</p>
<p>What is wrong with my argumentation above?</p>
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<p>I've been looking for the relation between moment of a tennis racket and its maneuverability. If I define maneuverability as the easiness to get in position for quick reaction strokes like volleys and returns, does low moment indicate high maneuverability? I seem to get the answer from this site <a href="http://www.racquetresearch.com/sevencri.htm" rel="nofollow">http://www.racquetresearch.com/sevencri.htm</a> but can anyone explain further? In a quick reaction stroke, I still need to move the racket, so what is the relation between moment and the effort I need to move the racket?</p>
<p>By the way, if my racket has balance point at 32cm away from the racket butt, and I hold the racket at 7cm away from racket butt (which would be the axis of rotation). If I add some weight to the racket head, is there any way I can reduce the racket moment so I can increase swing weigh without sacrificing much maneuverability?</p>
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] |
<p>I would like to know the probability of return to the initial point in three dimensional Brownian motion. Does someone know an expression for the diffusion constant? (Suggestions of books on this would also be welcome)</p>
| 10,651 |
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<p>In <a href="http://motls.blogspot.com/2012/04/is-lhcs-125gev-higgs-boson-extra.html#more" rel="nofollow">this</a> article, Lumo introduces and explains the idea (presented by the original authors in <a href="http://prl.aps.org/abstract/PRL/v108/i14/e141602" rel="nofollow">this</a> paper) that the LHC signal at about 125 GeV could alternatively be interpreted as a higher dimensional <a href="http://en.wikipedia.org/wiki/Radion_%28physics%29" rel="nofollow">radion</a>. Such a higher dimensional radion would better fit to the branching ratios observed at the LHC so far (at the present state of data accumulation) than a SM Higgs.</p>
<p>While reading Lumo's article, I got very curious about the following:</p>
<p>a) Where would the Higgs hide, if this model is true and the 125 GeV signal is rather a radion than a Higgs?</p>
<p>b) Would a Higgs still be needed in this case for Electroweak Symmetry Breaking (EWSB)?</p>
<p>or </p>
<p>c) Could the radion itself play the role of the Higgs?</p>
<p><strong>"Acknowledgment"</strong>: Lumo has started to think about these questions at the end of the article too, so we are both very curious about the answers to these issues.</p>
| 10,652 |
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] |
<p>I don't know if this is the place to ask this kind of questions but I'm sure you guys can help me. I'm looking for a small (4 pages max.) technical explanation of nuclear energy, preferably of an academic source. It would be best if the physics and mathematics involved is understandable by last-year high school students. Can anybody help me out?</p>
<p>Thanks,</p>
<p>Arnoud</p>
| 10,653 |
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