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<ul>
<li>vector D = 4 cm North <br /></li>
<li>vector J = 4.5 cm West</li>
</ul>
<p>what is D+J?</p>
<p>In a more general sense, how can two 2D vectors that are perpendicular to each other be added?</p>
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<p>Hoping this is not a silly and stupid question let me ask for help in this problem.</p>
<p>I have a particle in an infinite square well (the box is from 0 to a), in the state described by the function </p>
<p>$\psi (x) = Ax(a-x) \qquad \mathrm{for }\qquad 0<x<a$, $\qquad 0 \qquad otherwise$.</p>
<p>I have to determine the most likely value of energy and the probability to obtain a value of $E = \frac{9\hbar^2 {\pi}^2}{2ma^2} $.</p>
<p>To solve the second question I thought that $E$ it's the classic solution for energy in a potential well with $n=3$. So I calculate $\langle3| \psi\rangle$ $-$ in which $3$ is the solution wave function with $n=3$ $-$ and that's it? Right?</p>
<p>But for the first question?
Do I have to calculate $\langle H \rangle$ and compare it with a solution of the potential well?</p>
<p>After I also have to determine the evolution of the wave function for $t>0$ when at $t=0$ we turn off the potential well, and I have no idea how to solve it.</p>
<p>Please forgive me for my English. Hope you'll give me some hints.</p>
| 5,191 |
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<p>A block of mass $m$ is pushed towards a moveable wedge of mass $\eta m$ and height $h$, with a velocity $u$. All surfaces are smooth. The minimum value of $u$ for which the block will reach the top of the wedge is:
<img src="http://i.stack.imgur.com/vaqFh.png" alt="enter image description here"></p>
<ol>
<li>$\sqrt{2gh}$</li>
<li>$\eta \sqrt{2gh}$</li>
<li>$\sqrt{2gh(1+\frac1\eta)}$</li>
<li>$\sqrt{2gh(1-\frac1\eta)}$</li>
</ol>
<p>What's confusing me is how to proceed knowing the wedge is moveable. I tried drawing the FBD of both the objects. But I am not getting any far with it.</p>
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<p>I have been really staring for a while in a MP-Beiser book and I totally disagree with a statement he does there. On a page 85 he states that photons act as they have a mass $m$. He derives this by stating that:</p>
<p>$$
\begin{split}
p &= m v\\
\frac{h\nu}{c} &= m c\\
m&= \frac{h \nu}{c^2}
\end{split}
$$ </p>
<p>But I totally disagree with this. We have learned and derived that momentum of a particle is:</p>
<p>$$
\begin{split}
p &= m v \gamma (v)\\
\frac{h \nu}{c} &= m c \gamma{(c)}\\
m &= \frac{h \nu}{c^2 \underbrace{\gamma(c)}_{=0}}\\
m &= 0
\end{split}
$$</p>
<p>Something here is totally wrong, but what? How can an author state what he does? I know that on Harvard they did an experiment resulting in different $\nu$ of a photons falling in a gravitational field, but they must have been wrong or something... Please someone explain.</p>
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<p>Imagine a large bandgap material which is irradiated by an intense laser beam.
If the photon energy is only high enough for 1/5 of the bandgap, is there a way to approximate the absorption by 5-photon excitation, i.e. the ratio of transmitted to initial Intensity?</p>
<p>All I found is related to 2 or 3 Photon absorption but not higher orders in the pertubation series ...</p>
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<p>While reading this question: <a href="http://physics.stackexchange.com/questions/32120/why-do-we-still-not-have-an-exact-definition-for-a-kilogram">Why do we still not have an exact definition for a kilogram?</a> , I had a crazy thought.</p>
<p>Using <a href="http://en.wikipedia.org/wiki/Polymerase_chain_reaction" rel="nofollow">PCR</a>, you make a known number of copies of a DNA strand where the length and composition is exactly known for all the copies. The number of copies is two to the power of the number of PCR cycles, and if you start with, say, all C-G DNA, you get a known number of duplicates which are the same molecular weight. If you only provide C and G nucleotides, you don't even have to worry too much about errors in copying, because whichever way they are fixed to make pair-matching DNA, you will get the same atomic weight. You can separate out the DNA and get a known huge macroscopic batch size of macro-molecules where we know the atomic composition exactly.</p>
<p>So we can easily make a batch of atoms where we know the number of atoms with potentially exact precision. Can you use this to make a mass standard? Just weighing the DNA won't work, because the water content will be uncertain. Drying the DNA won't work, because there will always be some completely random stuck protons or hard-to-evaporate water.</p>
<p>But I suspect this can be done, because here is a system with an exactly known macroscopic number of atoms. Perhaps if you centrifuge the DNA and compare the density of water to the density of DNA-water?</p>
<p>Question: Can you define the kg with PCR?</p>
| 5,195 |
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<p>So they say the remote observer will never see anything fallen to the black hole, because any object will slow down as it gets closer to the event horizon and eventually stop to stay there forever. Am I getting it right so far? With this said, it turns out nothing has fallen to the black hole up till now. That is for us, as remote observers, anything that ever tried to reach the black hole got stuck at its event horizon. So we can go and see it right away - it must be still there. At the same time they say that the black hole keeps sucking matter. So question is, to a remote observer, how does all this matter fit the event horizon provided it will never fall through it?</p>
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<p>I am reading this research paper authored by NS Manton on the <a href="http://www.sciencedirect.com/science/article/pii/0550321377902942" rel="nofollow">Force between 't Hooft-Polyakov monopoles</a>. I have a doubt in equation 3.6 and 3.7. We assume the gauge field for a slowly accelerating monopole to be $A_0 = \epsilon^2 a_i t A_1$, where $\epsilon^2$ is an infinitesimal. Also, we write $\partial_0 \phi = -\epsilon^2 a_i t \partial_i \phi$. Using this he writes $D_0\phi=-\epsilon^2 a_i t D_i \phi$, where $D_i\phi=\partial_i \phi + [A_i,\phi]$. Isnt the sign of the second term wrong?</p>
<p>Secondly, he says differentiation wrt t gives us, $D^0 D_0 \phi = \epsilon^2 a_i D_i \phi$. Shouldnt it be $\partial^0D_0 \phi$? Cause we are taking the actual derivative wrt t rather than the covariant derivative, WE should get some extra terms, do they cancel out? How does the minus sign disappear?</p>
<p>Does the covariant derivative behave like a normal derivative in any case?</p>
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] |
<p>From <a href="http://www.facebook.com/ConstellationEnergy" rel="nofollow">Constellation Energy</a></p>
<blockquote>
<p>Quick energy efficiency tip: To stay cool and manage your energy at
the same time, use ceiling fans to create a “wind chill” in rooms you
are using. The wind chill will help you feel cooler than the actual
temperature. <strong>Make sure the ceiling fan is set to turn in a
counter-clockwise rotation</strong>.</p>
</blockquote>
<p>I was wondering why the ceiling fan is set to turn in a counter-clockwise rotation, instead of the other way around?</p>
<p>Thanks!</p>
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<p>What is the influence of Hermitian condition ($\psi=\psi^{\dagger}$) of Majorana fermions operators in their statistical behavior?</p>
<p>A Majorana fermion gas must obey the Fermi-Dirac statistics, or their stastistical behavior may be anyonic?</p>
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<p>Wave-particle duality states that a particle has both wave properties and particle properties when one is <em>not</em> observing it.</p>
<p>1) What is an observer? Need it be anything living or can other particles also act as observers?</p>
<p>2) When doing the electron double slit experiment--shooting just one electron at a time, the electron goes through both slits at the same time (if one is not observing it). Does that say that the electron is on every single geographical point at the same time?</p>
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<p>Both the <a href="http://en.wikipedia.org/wiki/Black%E2%80%93Scholes" rel="nofollow">Black-Scholes PDE</a>{*} and the Mass/Material Balance PDE have a similar mathematical form of the PDE which is evident from the fact that on change of variables from Black-Scholes PDE we derive the heat equation (a specific form of Mass Balance PDE) in order to find analytical solution to the Black-Scholes PDE.</p>
<p>I feel there should be some physical similarity between the two phenomena which control these two analogous PDEs (i.e. Black-Scholes and Mass/Material Balance). My question is, can one relate these two phenomena physically through their respective PDEs? I hope my question is clear, if not please let me know. Thanks.</p>
<p><sup>*PDE=Partial Differential Equation</sup></p>
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<p>I am asking about an English translation of a Helmholtz paper: </p>
<blockquote>
<p>Ueber die physikalische Bedeutung des Princips der kleinsten Wirkung. <em>Journal für die reine und angewandte Mathematik (Crelle's Journal), Volume 100, Issue 2, 1887, Pages 137-166, and Volume 100, Issue 3, 1887, Pages 213-222.</em></p>
</blockquote>
<p><a href="http://www.degruyter.com/view/j/crll.1887.issue-100/crll.1887.100.137/crll.1887.100.137.xml?format=INT" rel="nofollow">http://www.degruyter.com/view/j/crll.1887.issue-100/crll.1887.100.137/crll.1887.100.137.xml?format=INT</a>. (Also see <a href="http://resolver.sub.uni-goettingen.de/purl/?GDZPPN002160013" rel="nofollow">link</a> and <a href="http://resolver.sub.uni-goettingen.de/purl/?GDZPPN002160072" rel="nofollow">link</a>.)</p>
<p>The title in English: <em>On the Physical Significance of the Principle of Least Action.</em></p>
<p>Has it ever been translated (to English)?</p>
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<p>I'm trying to understand Morin's example of a spring pendulum. What I don't get is his expression for $T$. I can understand the $\dot x^2$ term in the brackets. But I don't understand the $(l + x)^2\dot \theta^2$. </p>
<p>Also, it seems rather strange to break up Kinetic Energy into tangential and radial components when it is a scalar. </p>
<p><img src="http://i.stack.imgur.com/apteI.jpg" alt="enter image description here"></p>
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<p>Almost everybody knows that light is massless. But where this come from and how it can be proven (experimentally or theoretically)? I actually found <a href="http://open.salon.com/blog/kwatts59/2009/02/18/physics_what_is_the_mass_of_light" rel="nofollow">this article</a> which explains and calculates the mass of light at rest (which is not 0). So how do we know that light is massless?</p>
| 180 |
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] |
<p>If there are no orbitting electrons in a neutron star's makeup to interact with EM, what happens to light that strikes it? </p>
| 5,204 |
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<p>When we want to figure out the long-term evolution of a planet's atmosphere/orbit, when is perihelion more important than mean distance?</p>
<p>E.g. some processes (like Jeans Escape and escape of atmospheres) are disproportionately affected during perihelion (point of closest approach) rather than during aphelion.</p>
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] |
<p>Who has succeeded in demonstrating the <a href="http://en.wikipedia.org/wiki/Lense%E2%80%93Thirring_precession" rel="nofollow">Lense-Thirring effect</a>?</p>
<p>This effect is one that describes the rotational motion of the Earth from a space-time structure. This effect is the "drag" of the geometrical structure of space-time due to rotational motion of the object which causes the gravitational-inertial field.</p>
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<p>I've heard that differential forms are related to densities, however I'm still a little confused about that. I thought on the case of charge density and I came to that: let $U\subset\mathbb{R}^3$ be a region of $3$-space, and let $\rho : U \to \mathbb{R}$ give the charge density at every point of $U$. I can then create the $3$-form $\omega = \rho \ dx \wedge dy \wedge dz$, which in my understanding gives me the approximate amount of charge enclosed by a volume determined by $3$ vectors when they're given.</p>
<p>So, if I give the vectors $v, u, w$, the value of $dx \wedge dy \wedge dz(v,u,w)$ should be the volume enclosed by those vectors, and hence $\omega(u, v, w)$ should be an approximation of the charge enclosed. Is this correct?</p>
<p>My only problem is: in this point of view, the form isn't giving me the density, the density itself is being given by a scalar field, while the form gives me the charge instead of the density.</p>
<p>Is this correct? The form is always meant to give the charge instead of the density? The density should always be regarded as a scalar field?</p>
| 5,207 |
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<p>When reading the term <a href="http://en.wikipedia.org/wiki/Superselection#Superselection_Sectors">superselection sector</a>, I always wrongly thought this must have something to do with supersymmetry ... DON'T laugh at me ... ;-)</p>
<p>But now I have read in <a href="http://physics.stackexchange.com/a/56526/2751">this</a> answer, that for example for a free QFT highly excited states , that would need infinite occupation numbers to build them up, and that lie therefore outside the Fock space are said to lie in a (different?) superselection sector. If a state has finite or infinite energy depends on the Hamiltonian, and a finite energy and physically relevant Hilbert space can be obtainend from the inacessible infinite energy states of another Hamiltonian.</p>
<p>This makes me now want to really know what a superselection sector is. What are the key ideas behind the definition of a superselection sector? Are they an underlaying concept to derive quantum field theories with a physical hilbert space that has only finite energy states, or what is their common use in physics?</p>
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<blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="http://physics.stackexchange.com/questions/36469/are-the-protons-and-neutrons-in-the-nucleus-arranged-in-any-particular-way">Are the protons and neutrons in the nucleus arranged in any particular way?</a> </p>
</blockquote>
<p>Isotopes have different numbers of neutrons. Essentially my question is: how does the addition of neutrons affect this structure if there is one?
*The reason this question is different is since it attempts to discuss how isotopes affect the structure of the nucleus, the other question discussed similar isotopes as far as I could tell.</p>
| 181 |
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<p>In a tight binding model, we usually start from the atomic orbits and linearly combine them to get the wave function of the crystal energy band.</p>
<p>My questions are:</p>
<ol>
<li><p>Since this kind of tight binding is an approximate method due to using atomic orbits, is it exact to use the <a href="http://en.wikipedia.org/wiki/Wannier_function" rel="nofollow">Wannier function</a> formalism? If so, how do I get the Wannier function systematically?</p></li>
<li><p>What is the use of maximally localized Wannier functions?</p></li>
<li><p>Why can't we get the maximum localized Wannier function when the <a href="http://en.wikipedia.org/wiki/Geometric_phase" rel="nofollow">Berry phase</a> is not zero?</p></li>
<li><p>Also, in tight binding, formalism, taking atomic orbits or Wannier functions as the basis function, respectively, what does position operator (diagonal or not), velocity operator and angular momentum operator look like?</p></li>
</ol>
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<p>Experiments used to observe particle spin properties (such as <a href="http://en.wikipedia.org/wiki/Stern-Gerlach_Experiment" rel="nofollow">Stern-Gerlach</a>) rely on a varied magnetic field and a dipole-like reaction in the particle, deflecting it in one direction or another.</p>
<p>In the case of a point-particle such as an electron, how is it explained that a dipole with spatially separated poles exists in a single point?</p>
| 5,210 |
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<p>When we are on Earth, we look <strong>UP</strong> in the Sky to see the <strong>Moon</strong>. How do we have to look at Earth from Moon.. Is it the way ? If so, how are these bodies actually placed in the space?</p>
<p>Are all astronomical objects lined up on the same planar level? When we look at stars, are they actually above the Solar System planar level or below ?!</p>
<p>Because, in general, if we have to look up to see something on a building, then from the building we have to look down to see the same object. </p>
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<p>It is usually said that the idea of fields was introduced (electric and magnetic fields) in electricity and magnetism after Coulomb's law to cure the conceptual problems of action at a distance.</p>
<p>Could someone explain what are the conceptual and physical difficulties or contradictions that one might have with action at a distance?</p>
| 5,212 |
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<blockquote>
<p>An astronomer is trying to estimate the surface temperature of a star with a radius of $5 \times 10^8\ m$ by modeling it as an ideal blackbody. The astronomer has measured the intensity of radiation due to the star at a distance of $2.5 \times 10^{13}\ m$ and found it to be equal to $0.055\ W/m^2$. Given this information, what is the temperature of the surface of the star?</p>
</blockquote>
<p>How do I do this? The hint was to use $I=\sigma T^4$ (where $\sigma = 5.67 \times 10^{-8}$).<br>
Others: </p>
<blockquote>
<p>As the star radiates energy, the total power that flows through any
spherical surface concentric with the star remains constant. The total
power flowing through such a surface can be obtained by multiplying
the intensity (power per square meter) by the surface area of the
sphere. </p>
<p>Using the law of conservation of energy, the surface area of
the two spheres (one at the star's surface and one with the radius
equal to the observing distance), and the intensity measured by the
observer at the observing distance, you should be able to obtain the
intensity at the surface of the star.</p>
</blockquote>
<p>Don't really understand the 2nd paragraph too ... </p>
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<p>I remember when I was in primary school, the science teacher put me in charge of a mercury thermometer. I do not quite understand the mechanics behind except that mercury expands when it is hot and contracts when it is cold, and that this could be read off a temperature scale along the stem of the thermometer.</p>
<p>At the back of my mind, there is this doubt on how gravity affects the movement. Having seen most thermometers being hanged vertically on a wall, I did likewise, placing the thermometer on the floor, upright, leaning against the window sill beside my desk. This is fine until one day when a strong gust of wind blew and cause the thermometer to fall flat on the floor, smashing the glass casing and causing the mercury to leak out resulting in the evacuation of the whole class.</p>
<p>After that incident, I learnt my lesson and place the new thermometer lying flat instead of upright. But, the lesson that I didn't learn is how gravity affects the behaviour of such thermometer. Hope to learn something here.</p>
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<p>Basis is the ideal gas law: <a href="http://en.wikipedia.org/wiki/Ideal_gas_law" rel="nofollow">http://en.wikipedia.org/wiki/Ideal_gas_law</a></p>
<p>This law only states a mathematical relationship between many variables. It does not state what happens if one of the variables changes.</p>
<p>So we pin down the volume and increase moles of air by pumping in more gas. n obviously changes, so does P. But what happens to T? Does the temperature change at all?</p>
| 5,215 |
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<p>I have a physics question that I need some help with:</p>
<p>"Proxima Centauri is a star in the Alpha Centauri solar system, it’s the
nearest star to our sun (4.24light−years) <a href="http://en.wikipedia.org/wiki/Listofneareststars" rel="nofollow">http://en.wikipedia.org/wiki/Listofneareststars</a>.
How large of a solar sail would be needed to accelerate a solar sail of mass m
to a velocity which will get the ship to Proxima Centauri in two lifetimes?"</p>
<p>I'm assuming that the time is 160 years for 80 years each lifetime. Can anyone help?</p>
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<p>The question:</p>
<p>At what temperature is the RMS speed of Hydrogen molecules equal to the escape speed from the earth's surface? Values of radius of earth($r$) and gas constant $R$ has been supplied only.</p>
<p>We know that the escape velocity of a body on earth is given by $(2rg)^{1/2}$. Putting in the values of $r$ and $g$ we get $11.2\ \text{km/s}$ as the answer.</p>
<p>We also know that for a mole of ideal gas $PV=RT=\dfrac13M[c_{rms}]^2$,where $M$ is molecular weight of the gas. </p>
<p>So substituting for $T$ we get $T=\dfrac{M[c_{rms}]^2}{3R}$
I have put $11.2\ \text{km/s}$ in place of $c_{rms}$ and $M=2$ and put the value of $R$ and I have got $904.8\ \text{K}$ as my answer, which is equal to $631.8\ ^o\text{C}$.</p>
<p>For moon the answer came as $98\ ^o\text{C}$.</p>
<p>My question is that moon certainly has a temperature was lower than $98\ ^o\text{C}$ so then shouldn't it have an atmosphere of hydrogen? But the moon has no atmosphere!!</p>
<p>I cannot understand where i have gone wrong.</p>
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<p>The following Fierz relation does not seem so obvious to me :</p>
<p>\begin{equation}
\bar{\psi}_1 \gamma^\mu (1+\gamma_5)\psi_2 \bar{\psi}_3 \gamma_\mu (1-\gamma_5) \psi_4 = -2 \bar{\psi}_1 (1-\gamma_5) \psi_4 \bar{\psi}_3 (1+\gamma_5) \psi_2.
\end{equation}</p>
<p>As a first step I would have tried to do something like </p>
<p>\begin{equation}
\bar{\psi}_1 \gamma^\mu (1+\gamma_5)\psi_2 \bar{\psi}_3 \gamma_\mu (1-\gamma_5) \psi_4 = \bar{\psi}_1 (1-\gamma_5) \gamma^\mu \psi_2 \bar{\psi}_3 (1+\gamma_5)\gamma_\mu \psi_4
\end{equation}</p>
<p>but this does not help to get rid of the $\gamma_\mu$s. Am I going the wrong way ?</p>
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<p>I thought about posting this question in the Biology StackExchange site but really it is just the application of my question that applies to biology not the core of the question itself.</p>
<p>Can anyone help me find the math involved in modeling acoustic lubrication? My goal is to investigate whether it could be employed as a strategy to ease the passing of kidney stones which are extremely painful and can take days to pass. It would seem that if you could successfully achieve acoustic lubrication of a kidney stone that it might significantly reduce the time it takes for the kidney stone to travel down the ureter.</p>
<p>I found this which may help:</p>
<blockquote>
<p>The sound speeds were 4270 ± 80 m/s for calcium hydrogen diphosphate, 4320 ± 40 m/s for cystine, and 4330 ± 50 m/s for calcium oxalate monohydrate.
<a href="http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2913873/" rel="nofollow">http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2913873/</a></p>
</blockquote>
<p>Also the size range in question would be diameters ranging from 1mm to 5mm. Stones much larger than that are not considered passable.</p>
<p>I believe the challenge lies in the fact that the kidney stone may have a significant percentage of its surface in contact with the ureter walls so acoustic lubrication may not even be achievable, but I think it is worth investigating.</p>
| 5,219 |
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<p>A <a href="http://en.wikipedia.org/wiki/Supernova_remnant" rel="nofollow">supernova remnant</a> is the structure resulting from the explosion of a giant star. The supernova remnant is surrounded by an expanding shock wave that is formed from material ejected by the explosion and interstellar material swept away during the process.</p>
<p>When this material collides with the circumstellar or interstellar gas, it forms a shock wave that can heat the gas at high temperatures up to 10 million K, forming a plasma.</p>
<p>The first question is: exactly what is the material ejected by the supernova?</p>
<p>The other issue is: if we consider for a moment the idea that a "human" might be witnessing the exact moment of the explosion of a supernova, the "superhuman" would hear the explosion? That is, does the explosion produce some kind of sound?</p>
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<p>I am living up north in <a href="http://en.wikipedia.org/wiki/Norway" rel="nofollow">Norway</a>, 300 km above the <a href="http://en.wikipedia.org/wiki/Arctic_Circle" rel="nofollow">Arctic Circle</a>, which gives me six months per year of darkness and cold. I used to have a starter telescope when I was living in Spain, but I gave it away, and I want to build/buy a new system with the following requirements.</p>
<ul>
<li>operated remotely - by cables or by <a href="http://en.wikipedia.org/wiki/Wi-Fi" rel="nofollow">Wi-Fi</a>. I want to put it outside my house - maybe even 50 meters away and stay inside (warm) and do all the stuff from my computer: moving, viewing, zooming and saving information.</li>
<li>I would like the telescope to know how to track a specific star or nebula</li>
<li>I would like to be able to do long exposures with an attached camera for faint objects, probably hours, so I can get nice images for you :)</li>
</ul>
<p>What gear do I need and what things may I be missing?</p>
| 5,221 |
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<p>A ring placed along $y^{2}$ + $z^{2}$ = 4, x = 0 carries a uniform charge of 5 $\mu$C/m. Find D at P(3,0,0)</p>
<p>Should I be using Gauss's Law to solve this problem? I was considering using a spherical Gaussian Surface, and then using the formula D = $\epsilon_0 $E to find D, but I'm not sure how to set up my integral.</p>
| 5,222 |
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] |
<p>The <a href="http://en.wikipedia.org/wiki/Method_of_image_charges" rel="nofollow">method of image charges</a> is a well-known and very useful tool for solving problems in electrostatics.</p>
<p>Unfortunately, when I was taught this method, it was presented simply as an algorithm. No real physical justification was given for its usage, and there was a complete absence of rigorous mathematics. The method has always been a little hazy to me since.</p>
<p>If someone could give a rigorous first-principles derivation of this method from Maxwell's equations (perhaps simply Coulomb's law?), it would be most appreciated.</p>
| 5,223 |
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<p><img src="http://i.stack.imgur.com/hWGlJ.png" alt="alt text"></p>
<p>This problem is giving me a lot of problems. So $E=k*q/d^2$. We'd want to find the distance from q1 to P, which is .1 meters (not cm) using pythagorean thereom. So we know k, which is just $9x10^9$ times q1 which is $-2.4u$ where $u=10^{-6}$ divided by $r^2$ which is just $.1^2$. Then I get $-216,000,000$. To get x component, I take that number multiplied by $cos45$ to come up with a final answer of $-108,000,000$. And for point q2, I dont think there is an x component for the electric field since its right below the point P. </p>
<p>What have I done wrong? </p>
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<p>A week or so back I asked a <a href="http://physics.stackexchange.com/questions/21275/trying-to-understand-laplaces-equation">question</a> about the gravitational potential field
$$\phi=\frac{-Gm}{r}, \qquad r\neq 0, $$
and how to show the Laplacian of $\phi$ equals zero for $r\neq 0$? Eventually, (it took a while) I was able to understand that</p>
<p>$$\nabla\cdot\nabla\phi=Gm\left(\frac{2x^{2}-y^{2}-z^{2}+2y^{2}-x^{2}-z^{2}+2z^{2}-x^{2}-y^{2}}{\left(x^{2}+y^{2}+z^{2}\right)^{5/2}}\right)~=~0, \qquad r\neq 0,$$
which was a revelation. But now I'm wondering why Poisson's equation $$\nabla\cdot\nabla\phi=\nabla^{2}\phi=4\pi G\rho$$
doesn't always equal zero as well? Obviously it doesn't, so I'm assuming that inside a mass the gravitational potential field cannot be given by
$$\phi=\frac{-Gm}{r}, \qquad r\neq 0.$$
Is that correct? Also, is there a comparably easy formula for gravitational potential inside a mass or does it vary (horribly?) depending on the shape and density of the mass? </p>
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<p>Some lecture notes I was reading through claimed that a pressure pulse propagates through a liquid-filled tube (blood in a vein) with the speed</p>
<p>$$c=\sqrt{\frac{A}{\rho}\frac{dP}{dA}},$$</p>
<p>where $A$ is the cross-sectional area, $\rho$ the density and $P$ the pressure (which pressure?).</p>
<p>How can this result be derived? Why would the pressure vary with area? I suppose I'm missing some assumptions or misunderstand the expression.</p>
<p>FYI, the expression is also written, without explanation of the variables, as</p>
<p>$$c=\sqrt{\frac{Eh}{2\rho R}}$$</p>
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<p>Below is attached for reference, but the question is simply about whether vectors used in physics in a vector space can be represented by complex numbers and whether they can be divided.</p>
<hr>
<p>In abstract algebra, a <a href="http://en.wikipedia.org/wiki/Field_%28mathematics%29" rel="nofollow">field</a> is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms. The most commonly used fields are the field of real numbers, the field of complex numbers, and the field of rational numbers, but there are also finite fields, fields of functions, various algebraic number fields, p-adic fields, and so forth.</p>
<p><strong>Any field may be used as the scalars for a vector space</strong>, which is the standard general context for linear algebra. The theory of field extensions (including Galois theory) involves the roots of polynomials with coefficients in a field; among other results, this theory leads to impossibility proofs for the classical problems of angle trisection and squaring the circle with a compass and straightedge, as well as a proof of the Abel–Ruffini theorem on the algebraic insolubility of quintic equations. In modern mathematics, the theory of fields (or field theory) plays an essential role in number theory and algebraic geometry.</p>
<p>In mathematics and physics, a <a href="http://en.wikipedia.org/wiki/Scalar_field" rel="nofollow">scalar field</a> associates a scalar value to every point in a space. The scalar may either be a mathematical number, or a physical quantity. Scalar fields are required to be coordinate-independent, meaning that any two observers using the same units will agree on the value of the scalar field at the same point in space (or spacetime). Examples used in physics include the temperature distribution throughout space, the pressure distribution in a fluid, and spin-zero quantum fields, such as the Higgs field. These fields are the subject of scalar field theory.</p>
<p>In mathematics, an <a href="http://en.wikipedia.org/wiki/Algebra_over_a_field" rel="nofollow">algebra over a field</a> is a vector space equipped with a bilinear vector product. That is to say, it is an algebraic structure consisting of a vector space together with an operation, usually called multiplication, that combines any two vectors to form a third vector; to qualify as an algebra, this multiplication must satisfy certain compatibility axioms with the given vector space structure, such as distributivity. In other words, an algebra over a field is a set together with operations of multiplication, addition, and scalar multiplication by elements of the field.</p>
<p>A <a href="http://en.wikipedia.org/wiki/Vector_space" rel="nofollow">vector space</a> is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied ("scaled") by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex numbers, rational numbers, or even more general fields instead. The operations of vector addition and scalar multiplication have to satisfy certain requirements, called axioms,... An example of a vector space is that of Euclidean vectors which are often used to represent physical quantities such as forces: any two forces (of the same type) can be added to yield a third, and the multiplication of a force vector by a real factor is another force vector. In the same vein, but in more geometric parlance, vectors representing displacements in the plane or in three-dimensional space also form vector spaces.</p>
<p>In classical mechanics as in physics, the field is not real, but merely a model describing the effects of gravity. The field can be determined using Newton's law of universal gravitation. Determined in this way, the <a href="http://en.wikipedia.org/wiki/Gravitational_field#In_classical_mechanics" rel="nofollow">gravitational field around a single particle is a vector field</a> consisting at every point of a vector pointing directly towards the particle. The magnitude of the field at every point is calculated applying the universal law, and represents the force per unit mass on any object at that point in space. The field around multiple particles is merely the vector sum of the fields around each individual particle. An object in such a field will experience a force that equals the vector sum of the forces it would feel in these individual fields.</p>
<p>Because the force field is conservative, there is a scalar potential energy per unit mass at each point in space associated with the force fields, this is called gravitational potential.</p>
<p>Gauss' law for gravity is mathematically equivalent to Newton's law of universal gravitation, but is stated directly as vector calculus properties of the gravitational field.</p>
| 5,227 |
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<p>Suppose a car moves with a constant speed of $20 \text{m/s}$ a quarter of a circle, and completes the quarter in $5$ seconds. One way to calculate the circumference is simply $20 \cdot 5 \cdot 4 = 400 \text{m}$. However, I know that $a=\frac{v^2}{R}$ and the circumference is $2 \pi R$. The acceleration is defined as $\frac{\Delta v}{\Delta t}$ and thus from Pythagorean theorem we get $a = \frac{\sqrt{20^2+20^2}}{5}=4 \sqrt{2} ~~ \text{m/s}^2$. Now to find the circumference we plug our result into $2 \pi \frac{v^2}{a}$, thus the circumference: $2 \pi \frac{20^2}{4 \sqrt{2}} \approx 444.29 \text{m}$ which is clearly wrong. So why the second way gave me a wrong result?</p>
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<p>I am trying to understand the effect of effective pressure or effectiveness in a hydraulic system. I have seen different units for effective pressure or effectiveness - e.g. Nm/Bar Nm/L etc. and I am a bit confused how or why this is different.</p>
<p>Any help would be appreaciated.</p>
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<p>This is a question about evaporative cooling as used in residential evaporative cooling appliances. This type of cooling uses the heat in the ambient outside air to evaporate water and remove the heat from the air, then push the cooled air inside. The equation to predict the temperature of the resulting air after it's given up its heat to evaporate the water is as follows:</p>
<p>$$T_{output} = T_{dry} - (T_{dry} - T_{wet}) * \epsilon$$</p>
<p>where $T_{output}$ is the output air temperature, $T_{dry}$ is the air temperature of the dry bulb, $T_{wet}$ is the air temperature of the wet bulb, and $\epsilon$ is the cooling efficiency.</p>
<p>For example, on a very dry summer day (dry bulb 95 degrees, wet bulb 60 degrees) my evaporative cooler with 90% efficient media is capable of cooling the air to 63.5 degrees.</p>
<p>However, this equation does not seem to take into account the temperature of the water itself. Does it matter? Intuitively, it would seem to make sense to me that hotter water would be easier to evaporate, since it's closer to its boiling point. Or maybe colder water is better because it will absorb more heat from the air? Or maybe it's a wash because the same amount of heat is required, but with hotter water, more is needed because it will evaporate faster? Help me understand this.</p>
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<p>The FRW metric is given by:
$$ds^2=-dt^2+a^2(t)\ dr^2$$
where $ds$ is an interval of proper length, $dt$ is an interval of cosmic time, $dr$ is an interval of co-moving co-ordinate distance and $a(t)$ is the scale factor (also $c=1$).</p>
<p>If I take $dt=0$ then I find that an interval of proper distance $ds$ is given by:
$$ds = a(t)\ dr$$</p>
<p>Thus the proper distance between two nearby co-moving points is proportional to the scale factor - space expands.</p>
<p>If I take $ds=0$ then I obtain the null geodesic equation describing the path of a light ray:
$$dt = a(t)\ dr$$</p>
<p>Thus the light travel time between two nearby co-moving points is also proportional to the scale factor.</p>
<p>Does this imply that intervals of cosmic time expand along with intervals of space?</p>
<p>The natural clocks in co-moving co-ordinates are expanding light-clocks. </p>
<p>Maybe in order to derive the constant units of the "atomic" time that we experience, $d\tau$, from expanding cosmic time intervals $dt$, we need to use the conformal time equation:
$$d\tau = dt/a(t)$$</p>
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<p>I'm using a book from Griffiths, I got really stuck about how he arrived at the approximate solution, is it just by trying( trial solution method?), I really appreciate any help on this.</p>
<p>$$-\frac{\hbar^2}{2m}\frac{\mathrm{d}^2\psi}{\mathrm{d}x^2} + \frac{1}{2}m\omega^2 x^2\psi~=~E\psi.$$</p>
<p>Change variables for convenience:</p>
<p>$$\begin{align}
\xi &\equiv \sqrt{\frac{m\omega}{\hbar}}x \\
K &\equiv \frac{2E}{\hbar\omega} \\
\frac{\mathrm{d}^2\psi}{\mathrm{d}\xi^2} &= (\xi^2 - K)\psi.
\end{align}$$</p>
<p>For very large $\xi$:</p>
<p>$$\frac{\mathrm{d}^2\psi}{\mathrm{d}\xi^2} \approx \xi^2\psi.$$</p>
<p>Approximate solution:</p>
<p>$$\psi(\xi) \approx A e^{-\xi^2/2} + B e^{+\xi^2/2}.$$</p>
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<p>Almost in every textbook of condensed matter physics, the standard description of SSB could be formulated as follows: </p>
<p>Consider the lattice Heisenberg model in an external magnetic field $H=\sum_{ij}J_{ij}\mathbf{S}_i\cdot\mathbf{S}_j+hS_z$, where $h$ is the magnitude of magnetic field and $S_z=\sum_iS_i^z$. Now the average magnetization per site is a function of both magnetic field $h$ and number of lattice sites $N$, say $m\equiv \sum_i\left \langle S_i^z \right \rangle/N=m(N,h)$, where $\left \langle S_i^z \right \rangle\equiv tr(\hat{\rho }S_i^z)$ with $\hat{\rho }=e^{-\beta H}/tr(e^{-\beta H})$ the density operator. Then if $$\lim_{h\rightarrow 0}\lim_{N\rightarrow \infty }m(N,h)\neq 0$$, we say the system has SSB at temperature $T$. Now I get some questions:</p>
<p>(1)We know at finite $N$ and zero $h$, $m(N,h=0)=0$ due to spin-rotation symmetry. But <strong>there is no reason</strong> for that $$\lim_{h\rightarrow 0}m(N,h)=m(N,h=0)—[1]$$, right? Since the function $m(N,h)$ may <em>not be continuous</em> at $h=0$, from the math viewpoint.</p>
<p>(2)If Eq.[1] is correct, and hence $\lim_{h\rightarrow 0}m(N,h)=0$, then $\lim_{N\rightarrow \infty }\lim_{h\rightarrow 0}m(N,h)=0$, right?</p>
<p>(3)If Eq.[1] is wrong, say $\lim_{h\rightarrow 0}m(N,h)\neq m(N,h=0)$ and hence $\lim_{h\rightarrow 0}m(N,h)\neq0$, then what about $$\lim_{N\rightarrow \infty }\lim_{h\rightarrow 0}m(N,h)?$$ And why don't we use this identity to define SSB?</p>
<p>Thank you very much.</p>
| 5,233 |
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<p>I've been wondering about this question for a while. If you have alpha and beta particles released from a radioactive core, how do they ionise surrounding particles?</p>
| 5,234 |
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<p>In several papers (including a recent one by Banks and Seiberg) people mention a "folk-theorem" about the impossibility to have global symmetries in a consistent theory of quantum gravity. I remember having heard one particular argument that seemed quite reasonable (and almost obvious), but I can't remember it.</p>
<p>I have found other arguments in the literature, including (forgive my sloppiness):</p>
<ul>
<li><p>In string theory global symmetries on the world-sheet become gauge symmetries in the target space, so there is no (known) way to have global symmetries</p></li>
<li><p>in AdS/CFT global symmetries on the boundary correspond to gauge symmetries in the bulk so there again there is no way to have global symmetries in the bulk</p></li>
<li><p>The argument in the Banks-Seiberg paper about the formation of a black hole charged under the global symmetry</p></li>
</ul>
<p>I find none of these completely satisfactory. Does anybody know of better arguments?</p>
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<p>Let's say I have internal pipe flow with a constant wall temperature. If I evaluate the properties at either the mean inlet temperature of the average of the inlet and outlet, do I always have to account for property variation due to the wall temperature? For example, I would calculate the Nusselt number using a prescribed correlation, and then I would use </p>
<p>$$Nu_D=Nu_{D,m}(\frac{\mu_m}{\mu_s})^n$$</p>
<p>with $n$ changing whether the problem is heating the fluid or cooling. $\mu_m$ is the viscosity at the mean temp and $\mu_s$ is the viscosity at the surface temp. Would there ever be a time in turbulent pipe flow to not do this?</p>
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] |
<p>What are direction ratios and how are they useful in studying vectors?</p>
<p>How are these different from direction cosines?</p>
| 182 |
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<p>In the calculation of the Chern number within a 2D lattice model, let's take the <a href="http://prl.aps.org/abstract/PRL/v61/i18/p2015_1" rel="nofollow">Haldane model</a> for example, the Chern number$=\pm1$ has 2 contributions coming from 2 Dirac points described by
$$h_1(\mathbf{q})=q_y\sigma_x-q_x\sigma_y-\sigma_z$$ and
$$h_2(\mathbf{q})=-q_y\sigma_x-q_x\sigma_y+\sigma_z$$.</p>
<p>Both of the above 2 Hamiltonians contribute the same 1/2(or -1/2) Chern number with <strong>the same orientation (i.e., the sign of Chern number)</strong>.</p>
<p>My question is: How to judge whether two massive Dirac Hamiltonians(e.g. $h_1$ and $h_2$) have the same or opposite orientations <strong>simply from the form of the Hamiltonian</strong>?</p>
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<p>Let's suppose we have a particular light frequency emitter and relative sensor array, and that there is no external source of this light.</p>
<p>This emitter has a know angle respect the receiver, and emit a point of line. Because the angle is known, we can determine the distance just by looking at which point in the sensor we read the point. This is the triangulation method, used by <img src="http://i.stack.imgur.com/mesdR.png" alt="laser range finder">, <a href="http://www.posterus.sk/?p=11526" rel="nofollow">click here for an explanation</a></p>
<p>Now, I want to understand if it possible to use a line instead of a point, and a sensor matrix instead of an array, so we can get 2D reading.
This should work until the "wall" has a obliquity on the axis perpendicular to the light line, but then I can't find a way to "fix" this problem. I may use different pattern instead of a line, like point but then the problem rise if one or more of the point are out of range for the receiver. any thoughts?</p>
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<p>In the early days of quantum electrodynamics, the most popular gauge chosen was the <a href="http://en.wikipedia.org/wiki/Gupta%E2%80%93Bleuler_formalism" rel="nofollow">Gupta-Bleuler gauge</a> stating that for physical states,
$$\langle \chi | \partial^\mu A_\mu | \psi \rangle = 0.$$</p>
<p>However, this gauge is practically never used now. Why? Is there anything wrong or inappropriate with the Gupta-Bleuler gauge?</p>
<p>How is the Gupta-Bleuler gauge related to the $R_\xi$ gauge?</p>
| 5,239 |
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<p>I know that it must have something to do with (gauge?) theories with N=2 supersymmetry, BPS states and even black holes, but most papers on the subject are too technical for me. So what is wall crossing?</p>
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<p><img src="http://i.stack.imgur.com/3v6mf.png" alt="enter image description here"></p>
<p>Sorry for using this image, but I thought this was the most convenient way of asking this question. Please zoom in.</p>
<p>I do not understand from the line, "Now, in the body frame $T = (T_{x'}, T_{y'}, T_{z'})\ldots$"</p>
<p>How is that in the body frame? That should be in the inertial frame, since it is dL/dt.
In the body frame, it is dL'/dt. </p>
<p>How can you treat dl/dt as the torque in the body frame, and then derive the Euler's equation?</p>
| 814 |
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<p>The difference between pure and mixed states is the difference in their density matrix structure.</p>
<p>For density matrix $\rho$ of mixed state the trace of $\rho^{2}$ should be less than 1. For pure state corresponding trace $Tr(\rho^{2}) = 1$.</p>
<p>But when I tried to check the Bell two-qubit state, i got:
$$ \rho = \frac{1}{2}\begin{pmatrix} 1 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1\end{pmatrix}$$
$$ \rho^{2} = \frac{1}{4}\begin{pmatrix} 2 & 0 & 0 & 2 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 2 & 0 & 0 & 2 \end{pmatrix}$$
Trace of which is equal to 1.
As I understand, <a href="http://en.wikipedia.org/wiki/Quantum_entanglement#Reduced_density_matrices" rel="nofollow">reduced density matrix</a> is the right describing of bell states. But my matrix is not reduced.
Can you explain me how to find reduced matrix of bell state?</p>
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<p>Let us suppose I am running on a street. When my eyes are open, I can see many things moving backward, and thus it gives me an idea that I am moving wrt those things. Not even this, even if I close my eyes during the run, I can really <em>feel</em> that I am moving. </p>
<p>Now assume that I reach a world where there is absolutely nothing except my own body, and my eyes are closed. I am still feeling that I am running. The question is I am running wrt to what? </p>
<p>Even if I open my eyes I would feel that I am still running, may be very fast. But visually I would not able to perceive my movement. </p>
<p>Does it mean that my movement in space is an illusion?</p>
<p>If not, then I am moving w.r.t what? There is absolutely nothing around me! </p>
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<p>I can't find any law that states this (maybe the combined gas law does and I'm misinterpreting it?), but Feynman said that if you compress a gas, the temperature increases. This makes sense, for example, a diesel engine (or gas engine with insufficient octane or too high a compression ratio). Also, must thinking about a piston "hitting" particles as it is compressed makes sense that energy is imparted.</p>
<p>But he goes on to say that when the gas expands, there is a decrease in temperature. This used to make more sense to me, but the more I think about it, it doesn't at all. Why would the particles <em>lose</em> energy if the container expands?</p>
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<p>Neutrons have no charge so they would not, I think, interact with photons. Would a neutron star be transparent?</p>
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<p>I'm in doubt in the application of Gauss' Law to find electric fields when the charge distribution is symmetric. Well, first of all: I know how to find the magnitude of the field - we just enclose the charge distribution with a gaussian surface on which the electric field will not change it's magnitude, and then using Gauss's Law we can write it in terms of the total charge inside and the area of the gaussian surface.</p>
<p>My problem is: how do I find the direction of the field? I mean, in a spherical symmetric distribution it's easy, because we know what's the vector that points radially outwards (it's simply one of the unit vectors from spherical coordinates). But what about a cylindrical symmetric distribution ? Would I need to use the unit position vector of cylindric coordinates ?</p>
<p>In the general case I would need to switch to more appropriate coordinates to write the field ? Is there a general way of treating this ?</p>
<p>Sorry if this question is to silly or too basic. I'm just trying to understand how to use properly this law.</p>
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<p>So I know that magnetism can be obtained from the combination of electric fields and special relativity.
I am familiar with the way one can derive the magnetic field of a current carrying wire felt by a moving charge from length contraction.</p>
<p>Now: we all know that if a magnet falls through a coil, the magnetic flux through its cross sectional area will change with time and this gives rise to an induced current.</p>
<p>I would like to know if we can understand it solely from relativity, without having to assume that a change in magnetic flux implies an induced electric field.</p>
<p>Let's consider the situation in the <strong>frame of reference of the magnet</strong>:
the magnet sees the coil moving upwards; the coil is made of a conducting material so there are lots of delocalised (free to move) electrons.</p>
<p>There is a stationary magnetic field (produced by the magnet) and the delocalised electrons are moving upwards (with the coil) in it. The magnetic field lines are in the same direction as the upwards velocity of the electrons, so the <strong>v</strong> $\times$ <strong>B</strong> term in the Lorentz force is 0.</p>
<p>I tried to summarise it in this diagram (it's very simplistic and not very well drawn, sorry):
<img src="http://i.stack.imgur.com/wnoMw.png" alt="enter image description here"></p>
<p>How do we explain the induced current (i.e. the flow of electrons) in the coil without referring to Faraday's/Lenz's law?</p>
| 5,246 |
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<p>I know when we speak to the microphone, the pitch of our voice cause the vibration of magnet in the microphone, thus causing generation of different voltages of electrical signal.</p>
<p>But my question is: when we speak, our voice not only contains the pitchs, but also the content of our words. How are the words (such as word "hello" and "how are you") transferred into electrical signals?</p>
<p>I am trying to figure out how our voice data delivered as packets on the Internet when we are having voice chat through Internet. What does the packets contain specifically?</p>
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<p>SO I have a hamiltonian:
$$H=\alpha J_1\cdot J_2$$
And I am asked to find the eigenstates and eigenvalues of this Hamiltonian in terms of products of the eigenstates of the z components of the individual spins.</p>
<p>The wording of this question is kind of confusing me. I could solve it by rewriting it in terms of $(J_1+J_2)^2,J_1^2,J_2^2$ and then find the eigenvalues, but that is not in terms of the z components. How should I go about doing this? Thanks</p>
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<p>Even though a viscous fluid has a translation symmetry (invariance) for its Lagrangian , it still 'waste' Linear momentum. How come ?, isn't the rule that every symmetry yields a conservation law ? </p>
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<p>I was reading up on the history of $W/Z$ bosons today and I got a little puzzled. I always assumed that people measured $M_Z$ and $M_W$ and then derived the Weinberg angle. But it appears that they knew what the Weinberg angle was by the late 70's (but not with a lot of precision). How do you measure the Weinberg angle experimentally without using $W$ and $Z$ masses?</p>
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<p>As small as it may be, does every 'thing' have a gravitational pull? That is, something with mass or energy. No matter how obsolete or negligible it may be, is it there? If so, how is it calculated? What does 'it' affect?</p>
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<p>I'm musing about how to give students an intuitive feeling about density by letting them lift a same sized volume of different materials, e.g. 1 liter of water, a 10x10x10 cm cube of iron, lead etc. So far, the densest material accessible and affordable to a teacher would probably be mercury (I certainly remember my chemistry teacher letting us lift a small bottle, maybe 100ml, makes for an unforgettable impression).</p>
<p>Do I have any chance of getting anywhere above the 13g/cm$^3$ of mercury? I sorted the elements in the <a href="http://en.wikipedia.org/wiki/List_of_elements_by_density">Wikipedia Density article</a>. Gold and Platinum require to win the lottery first. A liter of Plutonium needs connections to evil people, plus a lot of safety measures, so is right out :-) Osmium? Rhenium?</p>
<p>Are there elements or compounds denser than mercury usable for a demonstration?
Are there heavy metal compounds that can be denser than the metal itself or is there a physical constraint prohibiting such a property? (I don't want to look at the far ends of a p/T phase diagram, standard temperature and pressure is okay).</p>
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<p>How do you actually define an <a href="http://en.wikipedia.org/wiki/Orbit" rel="nofollow">orbit</a>?</p>
<p>I believe, Newtonian Mechanics describes an orbit as one object in free fall around another where projectile paths become elliptical. I think, Einstein describes an orbit as an object taking the shortest distance through curved space. And in Quantum Mechanics, orbits are quantized orbitals or states. Is there a definition for orbits, where all these characteristics are true?</p>
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<p>I understand that it has to do with acceleration. Say a pilot does a quick maneuver and experiences a force of 5g. What exactly is happening here? </p>
<p>And what is this force relative to? </p>
<p>If someone can show an example with some calculations that would be really helpful. </p>
<p>Thank you </p>
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<p>We want to solve Legendre's equation:</p>
<p>$$
\frac{d}{dx}[(1-x^2)\frac{d}{dx}P(x)]+l(l+1)P(x)=0,\quad (1)
$$</p>
<p>Jackson writes $P(x)=x^\alpha\sum_{j=0}^\infty a_j x^j$, puts this in eqn. 1 and then comes up with:</p>
<p>$$
\sum_{j=0}^\infty\{(\alpha+j)(\alpha+j-1)a_j x^{\alpha+j-2}-[(\alpha+j)(\alpha+j+1)-l(l+1)]a_jx^{\alpha+j}\}=0,\quad (2)
$$</p>
<p>and from this, he equals each term of the series to zero, and finds the recurrence relation below for the coefficients of the series:
$$
\alpha_{j+2}=\frac{(\alpha+j)(\alpha+1+j)-l(l+1)}{(\alpha+j+1)(\alpha+j+2)}\cdot a_j,\quad (3)
$$</p>
<p>Now, I don't understand how he manages to obtain eqn. 2 from eqn. 1 and eqn. 3 from eqn. 2.</p>
<p>I rewrite eqn. 1 by doing the derivatives:
$$
(1-x^2)\frac{d^2P}{dx^2}-2x\frac{dP}{dx}+l(l+1)P=0,\quad (1\text{bis})
$$</p>
<p>and then I evaluate the first and second derivatives of $P(x)$:</p>
<p>$$
\frac{dP}{dx}=\frac{d}{dx}\sum_{j=0}a_jx^{\alpha+j}=\sum_{j=1}(j+\alpha)a_jx^{\alpha+j-1}=\sum_{j=0}(j+\alpha+1)a_{j+1}x^{\alpha+j}
$$
e
\begin{align}
\frac{d^2P}{dx^2}&=\frac{d}{dx}\sum_{j=0}(j+\alpha+1)a_{j+1}x^{\alpha+j}\\
&=\sum_{j=1}(\alpha+j+1)(\alpha+j)x^{\alpha+j-1}a_{j+1}\\
&=\sum_{j=0}(\alpha+j+2)(\alpha+j+1)a_{j+2}x^{\alpha+j}
\end{align}</p>
<p>and put the whole in eqn. $1\text{bis}$:</p>
<p>\begin{align}
\sum_{j=0}[(j+\alpha+2)(j+\alpha+1)x^{j+\alpha}a_{j+2}&-(j+\alpha+2)(j+\alpha+1)x^{j+\alpha+2}a_{j+2}\\
&-2(j+\alpha+1)x^{j+\alpha+1}a_{j+1}+l(l+1)a_jx^j]=0
\end{align}</p>
<p>what's next?</p>
<p>final edit: differentiation of the power series is wrong; see Marco81's answer for the correct development.</p>
| 5,255 |
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<p>I am to find an equation for the time it takes when one falls through a planet to the other side and returns to the starting point. I have seven different sets of values - mass of object falling, mass of planet, radius of the planet, and time. I'm not including friction in the calculations.</p>
<p>I think this qualifies as a harmonic oscillator, and thus I work with the formula</p>
<p>$$T = 2\pi \sqrt{\frac{m}{k}}$$</p>
<p>To find the spring constant $k$ I need force $F$, and this is where I get uncertain. Should I work with the gravitational force between the object and the planet when the fall begins? In other words</p>
<p>$$F = G\times\frac{m \times M}{R^2}$$</p>
<p>When I try this I find that</p>
<p>$$F = kx \iff k = \frac{F}{x}$$</p>
<p>$$\iff k = \frac{G\times\frac{m \times M}{R^2}}{2R} = \frac{G \times m \times M}{2R^3}$$</p>
<p>$$\Rightarrow T = 2\pi \sqrt{\frac{m}{\frac{G \times m \times M}{2R^3}}} \iff T = 2\pi \sqrt{\frac{2R^3}{G \times M}}$$</p>
<p>Using this equation for the values I have, however, I get the wrong results - $T = 7148$ instead of $T = 5055$. What am I doing wrong?</p>
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<p>Is the solar energy coming from the sun infinite and will continue to be radiated to our earth forever? (discarding any outer factors) what's the sun's fuel?</p>
| 5,257 |
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<p><img src="http://i.stack.imgur.com/lRJAa.jpg" alt="enter image description here"></p>
<blockquote>
<p>Two wires of diameter 0.25 cm, one made of steel and the other made of brass are loaded as shown in fig. The unloaded length of steel wire is 1.5 m and that of brass wire is 1.0 m. Compute the elongation of the steel and the brass wires.</p>
</blockquote>
<p>I'm having problem with brass that since it is not connected to a rigid support so then how to calculate the elongation in the brass wire? I think their will be less elongation in brass wire in this situation as compared to a situation where the brass wire is connected to a rigid support. Am I right? Why? </p>
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<p>At what angle the projectile should throw with initial velocity v in order to reach distance d? discard the air resistance, only gravitation acts. So far I got the equations for horizontal and vertical velocity. Can someone point me in right direction?</p>
| 5,259 |
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<p>I was wondering this: suppose you have two oxygen atoms. They will both have 8 protons and 8 neutrons in the nucleus (at least if they are the most common isotope). Now, will all those particles be arranged in the same way in both atoms? If they are, why would that be, and if not, does that affect the element's properties in any way?</p>
<p>But then I also thought that maybe the uncertainty principle doesn't let us even ask this question. Maybe you can't tell the particles' positions so accurately, so all you can say is that you have 8 protons and 8 neutrons all together in a small space.</p>
<p>So, which one is it? Can we even tell where all the particles are, and if we can, does it matter exactly how they are arranged?</p>
| 181 |
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<p>It is my understanding that when moving near the speed of light, time slows down relative to other things not moving so fast. Based on this principle, would it be theoretically possible to travel a thousand light years in a year, with the thousand years only having passed on the place you're moving relative to? </p>
<p>If something is wrong with my premises or question, that knowledge is welcomed, too. (I am still a bit shaky on the whole subject of time-dilation because I don't fully understand how it's possible without a privileged reference frame.) Understanding is the prime goal, here.</p>
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<p>It is fine to say that for an object flying past a massive object, the spacetime is curved by the massive object, and so the object flying past follows the curved path of the geodesic, so it "appears" to be experiencing gravitational acceleration. Do we also say along with it, that the object flying past in reality exeriences NO attraction force towards the massive object? Is it just following the spacetime geodesic curve while experiencing NO attractive force?</p>
<p>Now come to the other issue: Supposing two objects are at rest relative to each other, ie they are not following any spacetime geodesic. Then why will they experience gravitational attraction towards each other? E.g. why will an apple fall to earth? Why won't it sit there in its original position high above the earth? How does the curvature of spacetime cause it to experience an attraction force towards the earth, and why would we need to exert a force in reverse direction to prevent it from falling? How does the curvature of spacetime cause this?</p>
<p>When the apple was detatched from the branch of the tree, it was stationary, so it did not have to follow any geodesic curve. So we cannot just say that it fell to earth because its geodesic curve passed through the earth. Why did the spacetime curvature cause it to start moving in the first place?</p>
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<p>Looking at the decay chain, I saw it undergoes double beta decay. How is it feasible for something to undergo a simultaneous double decay?</p>
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<p>I'm looking for analytically solvable models in statistical mechanics (classical or quantum) or related areas such as solid state physics in which
the beta function for a certain renormalization procedure (preferably but not necessarily real space normalization) is exactly known. </p>
<p>I'm looking forward to your responses.</p>
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<p>How to systematically show that the resulting charges in <a href="http://en.wikipedia.org/wiki/Oil_drop_experiment" rel="nofollow">oil drop experiment</a> are integers multiplied by $e$ in other word how to extract $e$ from the data?</p>
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<p>Can someone explain how <a href="http://en.wikipedia.org/wiki/Isospin" rel="nofollow">isospin</a> and <a href="http://en.wikipedia.org/wiki/Hypercharge" rel="nofollow">hypercharge</a> can be used to label representations? What is the meaning of the term singlet, doublet etc in this context? In particular how can I use it to label representations of $SU(2)$ embedded in a larger gauge group. I had posted <a href="http://physics.stackexchange.com/q/48064/">this question</a>, but didn't get any response, so I thought asking it in a more general way would help.</p>
<p>Please feel free to close it down, if you feel it is the exact duplicate. </p>
<p>I have an idea of the isospin from 2 particle systems, and their <a href="http://en.wikipedia.org/wiki/Clebsch%E2%80%93Gordan_coefficients" rel="nofollow">Clebsch-Gordan coefficients</a>. In this case, isospin is the eigenvalue of the total angular momentum operator $J$, and the combination of tensor product states with this quantum number, forms a representation, called the doublet, singlet etc depending on its dimension. Is this correct?</p>
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<p>In a $N$ dimensional phase space if I have $M$ 1st class and $S$ 2nd class constraints, then I have $N-2M-S$ degrees of freedom in phase space. How can I calculate the degrees of freedom in configuration space?</p>
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<p>In Prof. Xiaogang Wen's theory, photons and electrons are described as quasi-particles appeared as a result of the existence of the string-net liquid, which is the topological order of the qubits that form the space. This is a little confusing to me. Consider when we talk about other quasi-particles, for example, phonons, we say they are excitons appear from the oscillation of the atoms that form the solid, which means that 'atoms' are 'real' while phonons are 'quasi-particles'. But when we make an analogy between photon and phonon, (topological order with lattice model, qubit with atoms), what are those qubit in essence? Are they some kind of ultimate thing that build up our world?</p>
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<p>Quantum optics all discuss the quantization of free electromagnetic radiation. The result is well established.</p>
<p>But what about an arbitrary electromagnetic field? For example, the simplest case, electric field generated by a point charge. How is that field quantized as an operator?</p>
| 5,268 |
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<p>I have a question that has been annoying me for a while. Going across many textbooks on quantum mechanics, looking at the postulates they list, we find that the number of postulates vary from one text to another. That means either some the postulates listed in some books are either not sufficient or are redundant.</p>
<p>That makes me wonder, what is really the minimum sufficient number of postulates of QM and what are they? isn't there supposed to be a consensus on that?</p>
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] |
<p>I got the impression that a regular iPhone charger can charge the iPhone and the iPhone won't become too hot while charging, and the charging time is standard, but if using the 10W iPad charger to charge the iPhone, then it is 2A and it can make the iPhone hotter while charging and the charging time will be less?</p>
<p>But <code>I = V / R</code>, so <code>V</code> is the same at 5V, and <code>R</code> is the same, so it seems like <code>I</code> should be the same, and it shouldn't affect charging time or making the iPhone become hotter?</p>
<p>Unless if the standard iPhone charger outputting 5W is below the required power, and so it is charging with a lower than needed power, so when the iPad charger is used, then now more current will in fact go through, and so up to a point, when a 50W, 100W, or 300W charger is used, it will all be the same?</p>
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<p>I know that heat flow from higher temperature to lower temperature, but theoretically, is it possible to build heat pump that can move energy from ocean which are at ~295 kelvin to a small boiler which could be at ~500 kelvin. </p>
<p>I am asking out of curiosity that If its at least theoretically possible or not ? i.e. if I put, say, 5 kw in to the heat pump, can energy of 15 kw be transfer from ocean to boiler ?</p>
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<p>Why is it when we look up into the night sky we can see stars. but when you see pictures taken from the ISS you don't see any stars. Why is this?</p>
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<p>I am not talking about any other attributes of particles, vacuum etc ruling out <em>Uncertainty Principle</em> thing. If talking about pure Space (which is continuous, not discrete, cf. e.g. <a href="http://physics.stackexchange.com/q/9720/2451">this</a> Phys.SE post), why is Planck Length lower measurement limit?</p>
<p><strong>Update:</strong><br>
I want to avoid <em>Uncertainty Principle</em> at all cost. After John's answer, I am clarifying my question with abstract Space notion. I have pure Space, but no Vacuum to rule out any Quantum Fluctuation to create energy based on <em>Uncertainty Principle</em>.</p>
<p>New related question: Is lower measurement limit due to Quantum Fluctuation of Vacuum? Or, there's more to it?</p>
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<p>I am not so much familiar with the computations tools of conformal field theory, and I just run into an exercise asking to demonstrate the following formula (related to the bosonic field case):</p>
<p>$$\cal{R}j(z_1)j(z_2)~=~\frac{1}{(z_1-z_2)^2}~+~:j(z_1)j(z_2):$$ </p>
<p>with $j$ defined as </p>
<p>$$j(z)~=~\sum_k \alpha_k z^{-k-1}.$$</p>
<p>My question is should I start the calculation form the Wick ordered term and make the two others appear, because starting from the left side, I don't see how could I develop some calculus?</p>
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<p>Suppose that I have a li-ion battery with voltage 10V (and some capacity, say 1000mAh).</p>
<p>Can I charge it completely using 5V voltage?</p>
<p>What will hapen if I charge it with 12V voltage?</p>
<p>Edit:
Found the <a href="http://batteryuniversity.com/learn/article/charging_lithium_ion_batteries" rel="nofollow">answer</a>.</p>
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<p>So, when biking, I noticed that when going up hills, it was less tiring if I went up them more quickly. This is not total Work done as is Force * Distance, as that should be the same.</p>
<p>But the longer one is going uphill, the longer gravity is pulling you backwards. And if you only are providing enough force to counteract the force of gravity (from a stop), you will not make it up the hill, yet you will feel quite tired afterwards. While if one pushes really hard, then one will hardly slow down at all.</p>
<p>I know that if you are coasting, then the conservation of energy applies, and $v_i^2 = v_f^2 + C$ where C is the gravitational potential energy at the top of the hill. But this doesn't explain why it is more taxing to go up a hill slowly than quickly. It's the same amount of energy transformed into gravitational potential anyways. </p>
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<p>We all know that predicting tsunami and earthquake is difficult, with too many variables involved.</p>
<p>But with the advent in data collection and computing power and better models, one should be able to predict tsunami better than in the past. How accurate is current tsunami prediction?</p>
| 5,277 |
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<p>Why do optical fibers usually have a cladding? Ok, if you put make a bundle of optical fibers this prevents that light leaks from one fiber to another fiber in contact. However, are there other reasons to use claddings? Are there applications of optical fibers without claddings?</p>
<p>(I am mainly interested in optical fibres which can be described by geometrical optics, i.e. multimode fibers.)</p>
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<p>I was surprised to read that we don't know how to analyze turbulent fluids. On page 3-9 of <em>The Feynman Lectures on Physics (Volume One)</em>, Feynman writes:</p>
<blockquote>
<p>Finally, there is a physical problem that is common to many fields, that is very old, and that has not been solved. [..] Nobody in physics has really been able to analyze it mathematically satisfactorily in spite of its importance to the sister sciences. It is the analysis of <strong>circulating or turbulent fluids</strong>.</p>
<p>If we watch the evolution of a star, there comes a point where we can deduce that it is going to start convection, and thereafter we can no longer deduce what should happen. A few million years later the star explodes, but we cannot figure out the reason.</p>
<p>We cannot analyze the weather.</p>
<p>We do not know the patterns of motions that there should be inside the earth [which cause earthquakes].</p>
<p>The simplest form on the problem is to take a pipe that is very long and push water through it at high speed. We ask: to push a given amount of water through that pipe, how much pressure is needed? No one can analyze it from first principles and the properties of water. If the water flows very slowly, or if we use a thick goo like honey, then we can do it nicely. you will find that in your textbook. What we really cannot do is deal with actual, wet water running through a pipe. That is the central problem which we ought to solve some day, and we have not.</p>
</blockquote>
<p>I'm no physicist, but I imagine he's saying that we have differential equations describing turbulent fluids, but no one has ever been able to explicitly solve them or sufficiently determine their properties.</p>
<p>However, Feynman's words were written over 50 years ago now. Has there been any progress in analyzing turbulent fluids since then?</p>
| 5,279 |
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<p>Is the entropy of every system zero at the absolute zero?</p>
<p>Or is it taken to be zero at the absolute zero?</p>
<p>Are there systems that doesn't reach zero entropy even till absolute zero?</p>
| 5,280 |
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<p>I'm confused about how to normalize the Hartle-Hawking state in 2D quantum gravity. We can compute the HH state for two circles of length $\ell_1$ and $\ell_2$ in the matrix model as $\langle W(\ell_1)W(\ell_2)\rangle$, using the notation of Ginsparg and Moore, <a href="http://arxiv.org/pdf/hep-th/9304011v1.pdf" rel="nofollow">http://arxiv.org/pdf/hep-th/9304011v1.pdf</a>. This is supposed to be $\Psi_{\text{HH}}(\ell_1,\ell_2)$. They do the computation and find (for pure gravity at the leading order in the genus expansion)</p>
<p>\begin{align*}
\Psi_{\text{HH}}(\ell_1,\ell_2)=\langle W(\ell_1)W(\ell_2)\rangle=2\sqrt{\ell_1\ell_2}\frac{e^{-2\sqrt{\mu}(\ell_1+\ell_2)}}{\ell_1+\ell_2}.
\end{align*}
This is (10.11) in their paper. The norm is then
\begin{align*}
\int \frac{d\ell_1}{\ell_1}\frac{d\ell_2}{\ell_2}\Psi_{\text{HH}}(\ell_1,\ell_2)^2.
\end{align*}
which clearly diverges logarithmically because of short circles. How should I interpret this?</p>
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<p>In quantum mechanics, is there an upper bound for the <a href="http://en.wikipedia.org/wiki/Uncertainty_principle" rel="nofollow">uncertainty principle</a>? I know that quantum harmonic oscillator (QHO) has the uncertainty relation $\sigma_x\sigma_p = \hbar(n+1/2)$, but I think the QHO becomes localized at two peaks spread out over a large distance?</p>
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<p>To be efficient, a phase-matching condition has to be fulfilled in many nonlinear optical processes. For instance, the phase-matching requirement for second-harmonic generation is</p>
<p>$k_{2\omega}=2k_{\omega}$ or $\Delta k = k_{2\omega}-2k_{\omega}=0$</p>
<p>It is often said that this is equivalent to momentum conservation. However, even if $\Delta k \neq 0$, the process still takes place - although with lower efficiency and a finite coherence length $L = \frac{\pi}{\Delta k}$.</p>
<p>How can the conversion process still occur while momentum is not conserved? Is there momentum transfer to the medium? I guess not, because in many nonlinear processes only virtual photons participate. Do the photons 'borrow' momentum to make the jump? In other words, how does this work?</p>
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<p>Im trying to show that the integral over a closed loop of a crossproduct stays the same if I choose a different origin with $\overrightarrow{r}=\overrightarrow{r}\prime+\overrightarrow{r_0}$ and $\oint{d\overrightarrow{F}}=0$</p>
<p>$$ \overrightarrow{N} = \oint{\overrightarrow{r} \times d\overrightarrow{F}} = \oint{(\overrightarrow{r}\prime+\overrightarrow{r_0} )\times d\overrightarrow{F}} = \oint{\overrightarrow{r}\prime \times d\overrightarrow{F}} + \oint{\overrightarrow{r_0} \times d\overrightarrow{F}}$$</p>
<p>How do I show $\oint{\overrightarrow{r_0} \times d\overrightarrow{F}}= 0 $, or why can I say $\oint{\overrightarrow{r_0} \times d\overrightarrow{F}} = \overrightarrow{r_0} \times\oint{ d\overrightarrow{F}}$?</p>
<p>To clarify, $\overrightarrow{dF}$ is the force on the current $I \overrightarrow{dl}$ in a loop in a uniform and constant magnetic field. So $\overrightarrow{dF} = I \overrightarrow{dl} \times \overrightarrow{B}$.</p>
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