question stringlengths 37 38.8k | group_id stringlengths 2 6 | sentence_embeddings listlengths 768 768 |
|---|---|---|
<p>My question was inspired by <a href="https://physics.stackexchange.com/questions/108526/does-the-mass-of-a-star-change-as-it-collapses-into-a-black-hole#108529">this question</a>, which got me thinking. According to <a href="http://en.wikipedia.org/wiki/Newton%27s_law_of_universal_gravitation" rel="nofollow">Newton's Law of Gravitation</a>,</p>
<p>$$F = G\frac{m_1m_2}{r^2},$$</p>
<p>the gravity of an object is inversely proportional to the square distance between the objects, meaning that the closer the objects, or, rather, their centers of mass, get, the higher the gravitational force between them. If this is the case, why are <a href="http://en.wikipedia.org/wiki/Black_holes" rel="nofollow">black holes</a> "special"? Seeing as a star is made of gas & plasma, would an object at what would become the event horizon after it becomes a black hole be "sucked in" and, assuming it isn't destroyed by heat or various pressures, not be able to get out of the gravitational pull?</p>
<p>If an object were extremely close to the center of gravity of a planet, whether it was solid or gas, would it be able to get out? The Schwartzchild radius for Earth, according to Wikipedia, is 8.87 millimeters. If someone were able to get that close to its center of gravity, would he/she be able to escape?</p>
<p>What about for smaller objects, which have a Schwarzchild radius measured in nanometers or smaller, which is the size of atoms & subatomic particles? I assume there is a limit where subatomic forces like the strong & weak forces take over, but what is that limit & why does it happen?</p> | g10194 | [
0.00465165451169014,
0.052343349903821945,
-0.005550842732191086,
0.01645893231034279,
-0.024101797491312027,
0.06326957046985626,
-0.00741785392165184,
0.028888586908578873,
-0.06395556032657623,
-0.002956710522994399,
-0.007026554550975561,
0.03301398828625679,
0.004067859146744013,
-0.0... |
<p>Is the following correct for the distance $d$ from the origin $(0,0)$ to point $(t,x)$ in the 2-dimensional
<a href="http://en.wikipedia.org/wiki/De_Sitter_space" rel="nofollow">de-Sitter</a> and <a href="http://en.wikipedia.org/wiki/Anti_de_Sitter_space" rel="nofollow">anti de-Sitter</a> spaces? Here, $t$ is time and the distance may be called the interval if you prefer.</p>
<p>1) de-Sitter:
$\sinh{d}=\frac{\sqrt{t^2-x^2}}{\sqrt{1-(t^2-x^2)}}$</p>
<p>2) anti de-Sitter:
$\sin{d}=\frac{\sqrt{t^2-x^2}}{\sqrt{1+(t^2-x^2)}}$</p>
<p>If the above is correct, it implies that the de-Sitter space is akin to the hyperbolic space in terms of the distance measure, whereas anti de-Sitter space is more like the elliptic space. So, then the de-Sitter space has negative curvature, whereas it is positive for the anti de-Sitter space.</p>
<p>EDIT:</p>
<p>I derived (1) and (2) from a formula for $\Psi_{pq}$ given on page 439 of "Perspectives on projective geometry" by Jurgen Richter-Gebert, taking p=(0,0,1),
q=(x,t,1) and A=B=diag(1,-1,+1) for the de-Sitter space and A=B=diag(1,-1,-1) for anti de-Sitter space. I've just checked my derivation and with the final amendments the formulas are correct as written. What I want to know is whether my identifications of the spaces is correct. Perhaps, my (1) should be for anti de-Sitter space and (2) for de-Sitter. However, I read somewhere that de-Sitter space has hyperbolic distance measure (in Beltrami-Klein model I'm using) in accord with my choice. Oh, and I haven't linked those wikipedia pages. I think it's confusing and unnecessary to view de-Sitter and anti de-Sitter spaces as sub-manifolds of a higher-dimensional Minkowski space.</p> | g10195 | [
0.016501931473612785,
-0.0019267331808805466,
-0.0272511038929224,
0.030922645702958107,
-0.011856681667268276,
-0.01375364325940609,
0.053716376423835754,
-0.004133044742047787,
-0.06403333693742752,
0.03266632929444313,
-0.02449239417910576,
0.03366948664188385,
0.03487909957766533,
0.01... |
<p>Breit Wigner Formula describes the cross section for interactions that proceed dominantly via a intermediate particle (O*) A+B → O* → C + D:</p>
<p>$$σ = \frac{2\Pi}{k^{2}}\frac{Γ_{i}Γ_{f}}{(E-E_{o})^{2} + (Γ/2)^{2}}$$</p>
<p>A short question: Does the formula apply to situations when the intermediate particle is actually virtual?</p>
<p>For example, in positron electron annihilation, they form a photon which might eventually decay into another two particles. Can we calculate the resonant cross section for this process with the Breit Wigner Formula as well? If it is possible, what should we put in for $E_0$, which is supposed to be the rest mass of the intermediate particle?</p> | g10196 | [
0.02737504057586193,
-0.031115073710680008,
0.004724761471152306,
0.017805321142077446,
0.04225127771496773,
0.00807587243616581,
-0.018150776624679565,
-0.015018398873507977,
-0.02430884540081024,
0.06215815618634224,
-0.005493056960403919,
0.036046352237463,
-0.03250082582235336,
-0.0260... |
<p>Is there a way in which one can use the BCH relation to find the equivalent angle and the axis for two rotations? I am aware that one can do it in a precise way using Euler Angles but I was wondering whether we can use just the algebra of the rotation group to perform the same computation?</p> | g10197 | [
0.04602557420730591,
-0.020551247522234917,
0.014620037749409676,
-0.06024396792054176,
0.02641960233449936,
-0.08388333767652512,
0.036327049136161804,
0.021936284378170967,
-0.021776597946882248,
0.040380630642175674,
-0.026370089501142502,
0.058820728212594986,
-0.013909390196204185,
0.... |
<p>In classical mechanics for two mass particles $a$,$b$ we assume the symmetric potential arising from $F_{ab}$ and $F_{ab}$ given by $$U_{ab}(r)=-\int^{r}_{r_{0}}F_{ab}(r')dr'$$ and $$U_{ba}(r)=-\int^{r}_{r_{0}}F_{ba}(r')dr'$$ </p>
<p>The book <strong>mechanics</strong> by <em>Florian Scheck</em> gives $$\frac{d}{dt}r_{a}\nabla_{a}U_{ab}+\frac{d}{dt}r_{b}\nabla_{b}U_{ba}=\frac{d}{dt}U_{ab}$$ because $$[\frac{d}{dt}r_{a}\nabla_{a}+\frac{d}{dt}r_{b}\nabla_{b}]U_{ab}=\frac{d}{dt}U_{ab}$$</p>
<p>I am confused how we get the summation form. My derivation goes as follows: Notice $F_{ab}=-\nabla_{b} U_{ab}$. Thus we should have $$\frac{d}{dt}U_{ab}=\frac{d}{dr}*\frac{dr}{dt}U_{ab}=\frac{dr}{dt}\frac{d}{dr}U_{ab}=\frac{dr}{dt}[-F_{ab}]=\frac{d}{dt}[r_{a}-r_{b}][-F_{ab}]=[\frac{d}{dt}r_{a}\nabla_{a}+\frac{d}{dt}r_{b}\nabla_{b}]U_{ab}$$</p>
<p>My simple question is just whether my derviation is correct, for I assume $r=r_{a}-r_{b}$ at here. </p> | g10198 | [
0.044036466628313065,
-0.01396024040877819,
0.004645662847906351,
-0.018916916102170944,
0.060927435755729675,
-0.0004888142575509846,
0.05040304735302925,
0.01122225821018219,
-0.02288195677101612,
-0.004854327999055386,
-0.02207333967089653,
0.02568109706044197,
0.044948216527700424,
-0.... |
<p>On the <a href="https://en.wikipedia.org/wiki/Bohr_radius" rel="nofollow">Wikipedia</a> I found that the Bohr radius is equal to: </p>
<p>\begin{align}
\boxed{r_b=\dfrac{4\pi\varepsilon_0\hbar^2}{m_e{e}^2}}
\end{align}</p>
<p>but while we have been learning Bohr's model we derived the equation for a radius of the electron orbit like they do it on the <a href="http://hyperphysics.phy-astr.gsu.edu/hbase/bohr.html" rel="nofollow">hyperphysics</a>: </p>
<p>\begin{split}
&2\pi r = N \lambda_b\\
&\phantom{2\pi}r = \frac{N \lambda_b}{2\pi}\longleftarrow \lambda_b = \tfrac{h}{p}~,~p=mv\gamma(v)\\
&\phantom{2\pi}r = \frac{N h}{2\pi m_e v\, \gamma(v)}\\
&\phantom{2\pi}r = \frac{Nh \sqrt{4 \pi \varepsilon_0 r m_e}}{2 \pi m_e Ze_0\, \gamma(v)}\longleftarrow v = \sqrt{\tfrac{Z{e_0}^2}{4\pi \varepsilon_0rm_e}}\\
&\phantom{\,~\,}r^2 = \frac{N^2h^2 4 \pi \varepsilon_0 r m_e}{4 \pi^2 {m_e}^2 Z{e_0}^2 {\gamma(v)}^2}\\
&\phantom{2\pi}r = \frac{N^2h^2 \varepsilon_0}{\pi m_e Z{e_0}^2 {\gamma(v)}^2}\\
\\
&\substack{\text{We assume that $v \ll c$ and therefore $\gamma(v) = 1$}}\\
\\
&~\,~\boxed{r = \dfrac{N^2h^2 \varepsilon_0}{\pi m_e Z{e_0}^2}}~\boxed{r=\dfrac{N^2}{Z\vphantom{{e_0}^2}}r_b}\xrightarrow{\text{this means that Bohr radius is} } \boxed{r_b = \dfrac{h^2\varepsilon_0}{\pi m_e {e_0}^2}}
\end{split}</p>
<p>Ok so I have two equations for Bohr's radius but I can't connect them, because if I insert $\hbar = h/(2\pi)$ into the one from Wikipedia I don't get the one from hyperphysics:</p>
<p>\begin{align}
r_b=\dfrac{4\pi\varepsilon_0\hbar^2}{m_e{e}^2} = \dfrac{4\pi\varepsilon_0h^2}{4^2\pi^2m_e{e}^2} = \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\underbrace{\dfrac{h^2\varepsilon_0}{4\pi m_e{e}^2}}_{\substack{\text{This is not the same as the}\\\text{equation on the Hyperphysics...}}}
\end{align}</p>
<p>Is Wikipedia wrong? Please someone fix it if it is...</p> | g10199 | [
-0.013555021025240421,
0.01139480248093605,
-0.01111556962132454,
-0.02788294479250908,
0.08381311595439911,
-0.007806119509041309,
0.03421750292181969,
0.036602701991796494,
-0.06007489934563637,
0.02392864227294922,
0.0019980049692094326,
0.01157806720584631,
-0.04991932213306427,
0.0035... |
<p>I am now working on a slab system, but encountered some problems on the classical Hartree potential. This slab system is infinity along x-y plane, and has finite size along $z$ axis $z\in[0,L]$. I found a paper [PHYSICAL REVIEW B 80, 235101 (2009)], in which the Hartree potential reads:</p>
<p>$$V_H=-2 \pi e^2\int_{-\infty}^\infty dz^\prime\ |z-z^\prime| n(z^\prime)$$</p>
<p>In this equation, $n(z)$ is the position-dependent induced electron density in the system (or, can be interpreted as the net charge density $n_e(z)-n_{iron}(z)$. In other words, usually for $n(z^\prime)$, it satisfies $\int dz^\prime n(z^\prime)=0$, this is the condition that I saw in most papers for such Hartree potential). Does anyone have any idea of the derivation of this equation? Or, some references are also very helpful.</p>
<p>Besides, in the above equation it doesn't account for the influence of permittivity. My system has the dielectric constant for surrounding medium $\epsilon_0$ and also for slab $\epsilon_{slab}$. In this case, what's the equation for the classical Hartree potential?</p>
<p>Thank you very much for any help.</p> | g10200 | [
-0.020533382892608643,
0.09313192218542099,
-0.01019082311540842,
-0.05162328481674194,
0.00008353313023690134,
-0.02600715681910515,
-0.0015333846677094698,
-0.011076869443058968,
-0.022556623443961143,
0.09025704860687256,
0.005613451357930899,
0.030875204131007195,
-0.016274062916636467,
... |
<p>I am confused between these two quantities: <strong>Isentropic pressure</strong> and <strong>Total pressure</strong>.
Are they the same?</p> | g10201 | [
0.0739804059267044,
-0.035747263580560684,
0.015397337265312672,
0.0007863689097575843,
0.06144236400723457,
-0.020554793998599052,
-0.017112096771597862,
0.021162865683436394,
0.0022766857873648405,
0.03270769119262695,
-0.015540418215095997,
0.0019624587148427963,
-0.031537000089883804,
... |
<blockquote>
<p><em>A particle of mass $m$, moves around a central force whose potential is $$V(r) = kmr^3 \; (k>0),$$
then<br>
$(c)$ if the particle is slightly disturbed from the circular motion, what is the period of small radial oscillation about $r=a$?</em></p>
</blockquote>
<p>I have the solution but I am not understanding it. It says that the equation of motion to the first order in $x$ will be
$$m \ddot x + \left( \frac{3V'(a)}{a} + V''(a)\right)x = 0$$
Where $a$ is the radius of stable orbit when the particle will be in circular motion, and $r$ being radial distance, $r = a + x(t)$. Taylor expansion of $$V(a+x) = V(a) + V'(a)x + \frac 1 2 V''(a) x^2 + \dots $$ has been used. I don't understand the logic behind it, I am not even sure if this is correct, kind of looks like hooks law. Please explain if it's correct and provide hints to correct answer if it's wrong.</p>
<p>The final answer is given as $\sqrt{15ka}$, which is angular frequency of the above equation.</p> | g10202 | [
0.05325906351208687,
0.0013715922832489014,
-0.0035005228128284216,
-0.012663890607655048,
0.03355534374713898,
0.019654355943202972,
0.05309218913316727,
-0.034569863229990005,
0.006952012423425913,
0.005836856085807085,
0.024176280945539474,
0.1041991338133812,
-0.018974781036376953,
-0.... |
<p>I'm very primitive with my thought. So please help if you can in layman terms with an answer to this question.
Here we go- by using a street power pole as the source responder, is it possible to read a speed of a traveling car by using two reflectors on the road which sonar back to the source? My thought is reflector A and B placed at a calculated distance apart on the road, and the source (positioned on the power pole) is measuring the continuous flow of sonar between both.
A car travels over reflector 'A' disturbing the sonar flow as it continues it then travels over reflector 'B'. The source reads the speed between both reflectors. </p> | g10203 | [
0.017669960856437683,
-0.007248282432556152,
-0.0007374865817837417,
-0.049571000039577484,
0.06459193676710129,
-0.045110560953617096,
0.02748238295316696,
0.04086683318018913,
-0.058929070830345154,
-0.018983988091349602,
0.018374985083937645,
-0.011690469458699226,
0.04677730053663254,
... |
<p>Let's say you have a short pulse of light which expands radially from a lightbulb, and it impinges upon a mirror and reflects towards a photodetector which you have places somewhere above the mirror.</p>
<p>If the light was monochromatic, then you could easily calculate the Fresnel zone and claim that there are interference effects from an ellipse with sharp boundary around the path of least time. However since the pulse is not monochromatic, then the traditional Fresnel zone is not well defined.</p>
<p>How can the Fresnel zone be defined when the source pulse is not monochromatic? How can the Fresnel zone effects be thought about productively?</p> | g10204 | [
0.06465025991201401,
0.035614412277936935,
0.0002159817813662812,
0.030581139028072357,
0.004079995211213827,
-0.009513823315501213,
0.04522692784667015,
0.022652406245470047,
0.04094161093235016,
-0.06456077098846436,
-0.015504931099712849,
0.07058534026145935,
0.0530853234231472,
0.01269... |
<p>I am deriving the results of renormalization for nonlinear sigma model using Polyakov approach. I am closely following chapter 2 of Polyakov's book--- ``Gauge fields and strings''. </p>
<p>Action for the nonlinear sigma model (NLSM) is
\begin{equation}
S= \frac{1}{2e_0^2}\int d^2x (\partial_\mu \mathbf{n})^2.
\end{equation}</p>
<p>Polyakov breaks the $N$-dimensional unit vector $\mathbf{n}$ into slow ($e_a,\mathbf{n}_0$) and fast ($\varphi_a$) variables as
\begin{equation}
\mathbf{n}(x) = \sqrt{1-\varphi^2}\mathbf{n}_0(x)+\sum_{a=1}^{N-1}\varphi_ae_a(x)
\end{equation}
where $\varphi^2 = \sum_{a=1}^{N-1}(\varphi_a)^2$. The vectors $e_a$ and $\hat{\mathbf{n}}$ are orthogonal unit vectors. He then introduce the gauge fields $A_\mu^{ab}$ and $B_\mu^a$ by
\begin{eqnarray}
\partial_\mu \mathbf{n}_0 &=& \sum_{a}^{}B_\mu^a e_a\\
\partial_\mu e_a &=& \sum_{b}^{} A_\mu^{ab} e_b - B_\mu^a\mathbf{n}_0
\end{eqnarray}
where $a,b=1,2,\dots,N-1$ denote the transverse directions; $\mathbf{n}_0 \cdot e_a=0$ and $e_a \cdot e_b=0$.</p>
<p>Using these parametrization, the action of NLSM becomes
\begin{equation}
S= \frac{1}{2e_0^2}\int \left\{ \left( \partial_\mu \sqrt{1-{\varphi}^2} -B_\mu^a\varphi^a\right)^2 + \left( \partial_\mu \varphi^a- A_\mu^{ab}\varphi^b +B_\mu^a\sqrt{1-{\varphi}^2}\right)^2\right\}d^2x
\end{equation}
Second order correction is given as
\begin{equation}
S^{(II)}= \frac{1}{2e_0^2}\int \left\{ \left( \partial_\mu \varphi^a -A_\mu^{ab}\varphi^b\right)^2 + B_\mu^a B_\mu^b\left(\varphi^a \varphi^b-{\varphi}^2 \delta^{ab}\right)\right\}d^2x+\frac{1}{2e_0^2}\int \left(B_\mu^a \right)^2 d^2x
\end{equation}
At this level, he clearly ignores terms like
\begin{equation}
B_\mu^a \partial_\mu \varphi^a\quad \text{and}\quad B_\mu^a A_\mu^{ab}\varphi^b.
\end{equation}
On which basis, these terms can be ignored? This is my first question. Secondly, how to perform integration over ${\varphi}$?</p>
<p>Moreover...</p>
<ul>
<li>How the terms in action $S^{(II)}$ change under the continuous rotation of transverse coordinate system? </li>
<li>How can we prove that the Lagrangian can only depend on derivatives of gauge fields? </li>
<li>What is importance of gauge fields? </li>
</ul>
<p>Hints of these questions are given Assa Auerbach book---``Interacting Electrons and Quantum Magnetism '' [Chapter 13; section 13.3 Poor Man's renormalization]. But it is not clear to me. I would appreciate very much if some one help me in understanding the Mathematics and Physics related to my questions.</p> | g10205 | [
0.007152761798352003,
-0.01607246696949005,
-0.003808669513091445,
-0.043579522520303726,
0.014694370329380035,
-0.015233860351145267,
0.08224716037511826,
0.056189000606536865,
-0.0026713141705840826,
0.04049456864595413,
-0.08917345106601715,
0.0480765625834465,
-0.02443779446184635,
0.0... |
<p>Related: <a href="http://physics.stackexchange.com/questions/59211/faradays-law-in-a-ring">Faraday's law in a ring</a></p>
<p>The ring launcher is a standard introductory physics demonstration that I assume almost everyone has seen (if not, YouTube it). The explanation of why the ring is launched is explained on many websites. For example, here is the explanation from the <a href="http://www.pasco.com/prodCatalog/EM/EM-8661_ring-launcher/" rel="nofollow">Pasco’s website</a>, which was exactly the same explanation that several of my professors have used:</p>
<p>The changing magnetic field from the AC powered coil causes a changing magnetic flux through the aluminum ring. The induced EMF in the ring sets up a current which produces a magnetic field. The induced magnetic field opposes the field of the coil, pushing the ring up.</p>
<p>Here is where my problem lies – I feel that this is an incomplete explanation and has a serious flaw. Since this is AC powered, I imagine that for one instant in time the current through the solenoid that is creating this changing magnetic flux induces a current in the ring, say, in the CCW direction and this results in pushing the ring up. However, a split second later, the current through the solenoid changes in the opposite direction and induces a current in the opposite direction (CW) in the ring. <strong>Doesn’t this imply that the ring would now be attracted to the solenoid instead of being pushed up?</strong> However, this doesn’t happen so why is the ring launched from the solenoid?</p>
<p>The key for me came after taking circuit analysis and learned how the voltage leads the current in an AC RL circuit (which the ring launcher is). If the induced voltage on the ring occurs before the “second induced current” (this would be the CW current above) comes in, then the ring has enough time to be pushed up before the attraction comes in from the second induced current. </p>
<p>I’ve never seen this explanation before so I am assuming that I am incorrect. Please correct me! In my opinion, there is no way that such a fundamental concept could have been missed. </p>
<p>Thank you in advance for any help on this question</p> | g10206 | [
0.02890542894601822,
0.0446525476872921,
-0.008896355517208576,
-0.035843826830387115,
0.05785408243536949,
0.054240696132183075,
0.04905715957283974,
-0.004955070558935404,
-0.029755424708127975,
-0.00561542296782136,
-0.05694713816046715,
0.030507244169712067,
-0.00004393733252072707,
0.... |
<p>If magnetic energy depends on the electron poles within two-fields within a permanent magnets void, how do invidiual atoms react within the attraction or repulsion of poles, and what incurrence does this live by in regards to molecular struction?</p> | g10207 | [
0.01927739754319191,
0.017196934670209885,
-0.0005830596201121807,
-0.055818360298871994,
0.004860807675868273,
0.0047983103431761265,
-0.025744982063770294,
0.05907607078552246,
-0.016417350620031357,
-0.03756840154528618,
-0.044410545378923416,
-0.005868900567293167,
-0.02526887319982052,
... |
<p>My book doesn't explain well how to build a doublet of antiparticles that transforms the same way the particle doublet $(p,n)^T$ (proton neutron) does. </p>
<p>They claim $$\tag 1 \vert I=1,I_3=1\rangle = -p\bar n$$ for a composite nucleon-antinucleon system. </p>
<p><strong>Why is $(1)$ true?</strong></p>
<p>Perhaps it's just bad notation in the book? I got confused because the <a href="http://en.wikipedia.org/wiki/Table_of_Clebsch%E2%80%93Gordan_coefficients#j1.3D1.2F2.2C_j2.3D1.2F2" rel="nofollow">Clebsch-Gordan for $(1)$ comes with a $+1$ and not a $-1$</a>, but perhaps one should insclude this negative sign into the CG coeff? That is $(1)$ should be </p>
<p>$$\tag 2 \vert I =1,I_3=1\rangle =\underbrace{\vert 1/2,1/2\rangle}_{p}~\Big(\underbrace{-\vert 1/2,1/2\rangle}_{-\bar n}\Big)?$$ </p> | g10208 | [
-0.019249873235821724,
0.02963325008749962,
-0.0026020302902907133,
-0.0068232291378080845,
0.12017468363046646,
0.020870473235845566,
0.021648243069648743,
0.040916383266448975,
-0.042192548513412476,
0.0022763963788747787,
-0.018387790769338608,
0.05891000106930733,
-0.0033715004101395607,... |
<p>In the following research paper (link provided) , I am not able to figure out how do we get the equation (7) . Please help me out. Also , please recommend the necessary pre requisite text to understand how to reach equation (7). I am able to figure out the equation (5) (inside the quantum dot - using Bessel Equation series solution method). Will the same method be applied to equation (7) ?</p>
<p>Link to research paper: <a href="http://www.ias.ac.in/pramana/v73/p573/abs.htm" rel="nofollow">abstract</a> <a href="http://www.ias.ac.in/pramana/v73/p573/fulltext.pdf" rel="nofollow">pdf</a></p>
<p>Relevant equations from this paper:</p>
<p><img src="http://i.stack.imgur.com/C5VNU.png" alt="enter image description here"></p> | g10209 | [
0.043893154710531235,
0.05094515532255173,
0.009322777390480042,
-0.10576285421848297,
0.05373484641313553,
0.014110386371612549,
0.04421252757310867,
-0.023924296721816063,
0.055206116288900375,
0.009315486066043377,
-0.05012562498450279,
0.06414806842803955,
0.03092114068567753,
-0.01412... |
<p>On page 410 of Griffiths QM 2nd Ed. book, he begins an analysis to evaluate the integral:
$$\frac{1}{2i}\int_{-\infty}^\infty \frac{s \sin{(sr)}}{(s-k)(s+k)}\mathrm{d}s.$$
To exploit Cauchy's formula, he dissolves the integral into:
$$\int_{-\infty}^\infty \frac{s e^{isr}}{(s-k)(s+k)}\mathrm{d}s-\int_{-\infty}^\infty \frac{s e^{-isr}}{(s-k)(s+k)}\mathrm{d}s.$$</p>
<p>Now he chooses a contour by which he overlooks the poles (at $\pm k$). The poles are (at least apparently) the most severe points. Is this a rigorous analysis? After all, for each of these two integrals I can choose a contour containing no poles at all, and hence each is zero. That is not the answer.
I looked for mathematical sources that discusses this particular point, and I found none. </p> | g10210 | [
0.008315366692841053,
-0.017844852060079575,
-0.011827711947262287,
-0.013706110417842865,
-0.004776965361088514,
-0.020398220047354698,
0.061469972133636475,
0.014928782358765602,
-0.022833680734038353,
0.002777289832010865,
-0.05366762727499008,
-0.007303144782781601,
0.04336534067988396,
... |
<p>As described in <a href="http://arxiv.org/abs/1104.4543" rel="nofollow">"A class of elementary particle models without any adjustable real parameters"</a>, <a href="http://arxiv.org/abs/1011.0061" rel="nofollow">"The Conformal Constraint in Canonical Quantum Gravity"</a>, and <a href="http://arxiv.org/abs/1009.0669" rel="nofollow">"Probing the small distance structure of canonical quantum gravity using the conformal group"</a>. (Also see <a href="http://arxiv.org/abs/0909.3426/" rel="nofollow">"Quantum gravity without space-time singularities or horizons"</a>.) </p>
<p>The motivating idea is that black hole complementarity is implemented by conformal transformations. This leads to an interest in theories of gravity+matter which possess an exact local scale invariance, spontaneously broken by a dilaton VEV. In order to avoid a particular divergence, it's suggested that the dilaton has nontrivial couplings to the matter fields, such that the beta function vanishes. Therefore, the coupling constants of such a theory are not free parameters; they must satisfy this "conformal constraint". </p>
<p>No concrete examples of such a theory are provided, but the papers do contain some reasoning about relations between the physical constants, such as the implications of a small nonzero cosmological constant. </p>
<p>I would appreciate any intelligent commentary on these papers, but especially from the perspective of string theory. For example, the whole process reminds me a little of what would be involved in constructing a worldvolume Lagrangian for a supermembrane (except that supersymmetry is playing no role here). </p>
<p>But I am also suspicious of theoretical approaches which say "such a theory wouldn't work unless all the natural constants cooperate to produce a miracle, therefore such a theory would predict all the natural constants, but I don't have a concrete example of such a theory". In the absence of quantitative evidence that nontrivial examples of the "miracle" exist, I am inclined to think that the required miracle is either <em>impossible</em> or <em>unnecessary</em>. After all, the opportunity for the miracle to appear and produce a predictive theory, only occurs after a series of guesses about how the theory should work. If one of these presupposed guesses is wrong, then we are dealing with a mirage, not a miracle. </p>
<p>However, it would be better to have an opinion that engaged with some of the technicalities in these papers, rather than just being based on high-level heuristics. So: Is this new landscape real? Is it relevant? Does it overlap with the string landscape? </p> | g10211 | [
0.025207998231053352,
-0.011529468931257725,
0.021761110052466393,
-0.030506961047649384,
-0.019726164638996124,
0.035352107137441635,
0.03532976284623146,
0.008121907711029053,
0.011573486030101776,
-0.01280154101550579,
-0.011277923360466957,
-0.0318971686065197,
0.03161395341157913,
0.0... |
<p>In the literature on topological insulators and superconductors the 'bulk-boundary correspondence' features quite heavily. One version of this conjecture says roughly: "At an interface between two materials belonging to the same symmetry class with bulk invariants n and m, precisely |n-m| gapless edge modes will appear". Are there any known counterexamples to this statement when the invariants are of the usual non-interacting Bloch band type? (specifically I have in mind the invariants appearing in the "periodic table" of T.I.s/T.S.Cs, see 0901.2696 and 0912.2157). As far as I know no comprehensive proof of the statement exists, although considerable supporting evidence has been found in a number of special cases. </p>
<p>EDIT: As some extra motivation, suppose that there are new bulk invariants waiting to be found protected by symmetries falling outside the usual classification schemes (e.g. the recently proposed topological crystalline insulators protected by point group symmetries). Is there good reason to be confident that the bulk-boundary correspondence will continue to hold in these cases?</p> | g10212 | [
0.02641306072473526,
-0.04093165695667267,
0.026905469596385956,
-0.0007743254536762834,
0.005815413314849138,
0.02581137791275978,
0.042146362364292145,
0.009202453307807446,
-0.002280517714098096,
-0.02856571413576603,
0.026511497795581818,
0.022657806053757668,
-0.039725713431835175,
-0... |
<p>Assume we know two different mixed states, p and q, and an efficient (quantum) algorithm for creating such two. Does it follow that there exists a computationally efficient method/measurement for optimally(that is, according to the trace norm distance) distinguishing between the two? </p> | g10213 | [
0.0021421650890260935,
-0.030733605846762657,
0.018988333642482758,
-0.008810736238956451,
0.019181519746780396,
-0.016917970031499863,
-0.008478255942463875,
0.026511263102293015,
0.038871798664331436,
-0.012433864176273346,
-0.004950298462063074,
0.012336291372776031,
0.037950266152620316,... |
<p><strong><em>Steady flow</em></strong> is the condition in which the flow velocity profile does not vary with time.</p>
<p>Mathematically this is translated to $\frac{\partial \mathbf{v}}{\partial t} = 0$, for example in the derivation of Bernoulli's principle for the Navier-Stokes equation.</p>
<p>Why is it not $\frac{d\mathbf{v}}{dt}$?</p>
<p>I know the mathematical differences between the two differente operators, but what is the physical difference?</p>
<p>if the flow profile is constant in time, shouldn't the <em>total</em> derivative of the vector field $\mathbf{v}$ be constant everywhere?</p> | g10214 | [
0.10285885632038116,
-0.00431683287024498,
-0.018020210787653923,
0.03448894992470741,
0.07067205011844635,
-0.007537378929555416,
0.06855054199695587,
0.03479595109820366,
-0.04849652945995331,
-0.04020904377102852,
-0.045036833733320236,
-0.03920672461390495,
-0.016675420105457306,
0.003... |
<p>If the Universe did start from a single point, then wouldn't all particles be fundamentally entangled? How then could there be a truly "pure" state?</p> | g10215 | [
0.016098856925964355,
-0.005907313898205757,
-0.0067574819549918175,
0.026463251560926437,
0.05324970930814743,
0.01093064621090889,
-0.010807770304381847,
0.07062076032161713,
0.05760077014565468,
-0.05248148739337921,
-0.03012896701693535,
-0.04466615244746208,
0.012832514010369778,
-0.0... |
<p><a href="http://en.wikipedia.org/wiki/Plasma_%28physics%29" rel="nofollow">Plasma</a> is the fourth state of matter. Wikipedia says, that all the gaseous atoms will be ionized into positive ions and free electrons at extremely high temperatures. But, what does this explain in a <a href="http://en.wikipedia.org/wiki/Plasma_globe" rel="nofollow">plasma-globe</a> and <a href="http://en.wikipedia.org/wiki/Plasma#In_physics" rel="nofollow">other applications</a>. Do all these artificial apps have a relation with the plasma in stars...? If so, how is this high temperature maintained in such a small area?</p>
<p>If this relation is perfect, then plasma could be used for conduction of electricity, as it has extra-ordinary number of free electrons. Am I correct in stating that plasma conducts electricity very well?</p> | g10216 | [
0.04560059681534767,
0.018056640401482582,
-0.02855302393436432,
-0.024406708776950836,
0.022702040150761604,
0.0088490666821599,
-0.029976746067404747,
0.042807891964912415,
-0.0229436457157135,
0.019374893978238106,
-0.04716554656624794,
0.026434743776917458,
0.026130961254239082,
-0.020... |
<p>To ask a more specific one for the rotation curves of elliptical galaxies, and hope from there to later understand the dynamics of spiral galaxies. </p>
<ol>
<li><p><strong>Treating the galaxy as an isothermal gravitational gas sphere, what is the equation of Density for an elliptical galaxy?</strong> </p></li>
<li><p><strong>Assuming the above density and modelling by the Virial Equation, what are the Velocities?</strong></p></li>
</ol>
<p><strong>Note:</strong> I’ve drawn on a paper by Fritz Zwicky 1937 <a href="http://ned.ipac.caltech.edu/level5/Sept01/Zwicky/paper.pdf" rel="nofollow">“On The Masses of Nebulae and of Clusters of Nebulae”</a> </p> | g10217 | [
0.003357240464538336,
0.01704704761505127,
-0.027716325595974922,
-0.07019665837287903,
0.056196656078100204,
-0.008750376291573048,
-0.004318591672927141,
-0.055846843868494034,
-0.03775554150342941,
0.017033422365784645,
0.012532833963632584,
-0.012015915475785732,
0.06520123779773712,
-... |
<p>Consider an object orbiting around a point with radius $r$ and angular velocity $\omega$. Here its linear velocity is $v=\omega r$. If we choose a large enough $r$ and reasonable $\omega$, $v$ might be greater than $c$.</p>
<p>If this fact is impossible, what prevents it from happening?</p> | g121 | [
0.03506476804614067,
0.054371774196624756,
-0.002148805418983102,
0.004666068125516176,
0.05789508298039436,
0.03198700025677681,
0.043987538665533066,
-0.012868588790297508,
-0.034017473459243774,
-0.0009163887589238584,
-0.027747217565774918,
0.0208498015999794,
0.012303483672440052,
-0.... |
<p>What are some molecules stable in outer space that are unstable under terrestrial conditions?</p>
<p>So there are many molecules that violently react on Earth because they're too charged, have unfilled valence electrons, or have extra electrons in their outer shells. But in space, they often can't really "find" another molecule to transfer/attract electrons - in a long period of time</p> | g10218 | [
-0.014554859139025211,
0.033996567130088806,
0.010409052483737469,
-0.015563266351819038,
0.03614562004804611,
0.04545842483639717,
-0.1075909435749054,
-0.004487267695367336,
0.018118515610694885,
0.00904586911201477,
-0.0009641433134675026,
0.03744940459728241,
0.021644672378897667,
0.00... |
<p>We can see the definition of quantum anomaly in terms of Lagrangian path integral formulation. What is the definition of quantum anomaly in terms of Hamiltonian operator approach or even more directly in terms of wave functions? What are the characteristics of anomalous Hamiltonians or wave functions?</p> | g10219 | [
0.0497874915599823,
-0.016994336619973183,
-0.0055912937968969345,
0.005430659279227257,
0.014217730611562729,
0.040327705442905426,
0.052140045911073685,
0.009486955590546131,
0.0022883564233779907,
0.0050564841367304325,
-0.01970822364091873,
0.05343290790915489,
-0.031365785747766495,
0... |
<p>The sun is made of fire but fire needs oxygen right? So.. </p>
<ol>
<li><p>Why can there be flames in space, while there's no oxygen? </p></li>
<li><p>Same idea as with the rocket engines of the spaceship, which also produce fire while there's no oxygen?</p></li>
</ol> | g10220 | [
0.021992860361933708,
0.017886551097035408,
0.005063948221504688,
0.021629784256219864,
0.006776789668947458,
0.058478668332099915,
-0.016124702990055084,
0.01955854706466198,
-0.02025497332215309,
-0.048767782747745514,
0.053576789796352386,
0.026156939566135406,
0.041895221918821335,
0.0... |
<p>I don't understand this sentence (emphasis added):</p>
<blockquote>
<p>A consideration of Bragg's law (nλ = 2dsinθ), i.e. the relationship between scattering angle (θ) and the interplanar spacing (d) shows that <strong>if the wavelength (λ) is increased the total diffracted intensity becomes less sensitive to the spacing or to changes in angle.</strong></p>
</blockquote>
<p>I'd say I'm fairly confident with Bragg's law, and it's a simple enough equation. I just don't see how this can be described as “sensitivity” of intensity (not even in the equation from what I understand?) for d or θ.</p>
<p>Can anyone help me fit this together? <a href="http://books.google.co.uk/books?id=qbHLkxbXY4YC&lpg=PA353&ots=c3NJijBi2z&dq=%22is%20increased%20the%20total%20diffracted%20intensity%20becomes%20less%20sensitive%20to%20the%20spacing%20or%20to%20changes%20in%20angle%22&pg=PA353#v=onepage&q&f=false" rel="nofollow">Source</a></p> | g10221 | [
0.04373467341065407,
-0.04549234360456467,
-0.002918519079685211,
-0.03310820460319519,
0.01782134920358658,
0.0010957306949421763,
0.03604469448328018,
0.04382626339793205,
-0.012367774732410908,
-0.011914225295186043,
-0.04231718182563782,
0.0337570421397686,
0.020510181784629822,
-0.016... |
<p>If the photons of a laser would produce a radiation pressure upon whatever it shown upon wouldn't it be accurate to say that the laser would be propelled in the opposite direction of its beam?</p> | g10222 | [
-0.0054696425795555115,
0.0038508703000843525,
0.014586861245334148,
0.0038720956072211266,
0.029868578538298607,
0.03588062897324562,
0.004318440333008766,
0.002545315306633711,
-0.021583925932645798,
0.0012758162338286638,
0.007471083197742701,
0.04004768654704094,
0.019032368436455727,
... |
<p>Of course aerodynamics factors into this question, and the faster you are moving the more air you have to push out of your way, the more energy you use. But would the difference be only a small percentage change or would it be a lot more than that. Essentially would a certain unit of energy be able to move a certain unit of mass a relatively fixed distance, or will that distance be reduced with increases in speed.</p> | g10223 | [
0.010525000281631947,
0.04635583236813545,
0.03002583235502243,
0.09646586328744888,
0.002003780333325267,
-0.01974969357252121,
0.0012294941116124392,
0.06157854199409485,
-0.04892262816429138,
-0.04084339737892151,
0.015115202404558659,
-0.022650349885225296,
-0.020822051912546158,
-0.00... |
<p>Today, we were being given a lesson on Thermodynamics when my brain encountered an error.</p>
<p>It is easy to derive that, $\Delta U$, the internal energy change for constant volume process is given by,
$$
\Delta U = nC_V \Delta T
$$
and that enthalpy change for a constant pressure process is given by
$$
\Delta H = nC_P \Delta T
$$</p>
<p>However, my teacher later mentioned that <strong>internal energy and enthalpy changes are given by the same equations for any thermodynamic process. They need not be constant volume/pressure processes.</strong></p>
<p>I can't figure out <strong>why should that be true</strong>. A hint shall be enough.</p> | g10224 | [
0.07705891877412796,
-0.03791225329041481,
0.019594307988882065,
0.03969388082623482,
-0.0020308333914726973,
-0.002827888820320368,
-0.03127481788396835,
0.07368029654026031,
-0.05860787257552147,
-0.003927609883248806,
-0.0025640390813350677,
0.03177020326256752,
-0.039842329919338226,
-... |
<p>(At the suggestion of the user <em>markovchain</em>, I have decided to take a very large edit/addition to the <a href="http://physics.stackexchange.com/questions/52211/is-the-quantization-of-gravity-necessary-for-a-quantum-theory-of-gravity#comment111930_52211">original question</a>, and ask it as a separate question altogether.)</p>
<p>Here it is:</p>
<p>I have since thought about this more, and I have come up with an extension to the original question. The answers already given have convinced me that we can't just leave the metric as it is in GR untouched, but at the same time, I'm not convinced we have to quantize the metric in the way that the other forces have been quantized. In some sense, gravity isn't a force like the other three are, and so to treat them all on the same footing seems a bit strange to me. For example, how do we know something like non-commutative geometry cannot be used to construct a quantum theory of gravity. Quantum field theory on curved non-commutative space-time? Is this also a dead end?</p> | g10225 | [
-0.011795170605182648,
0.010831199586391449,
-0.002973056398332119,
-0.02366737648844719,
0.06005343794822693,
-0.012499730102717876,
0.005695730447769165,
-0.04263980686664581,
-0.027920149266719818,
-0.032255422323942184,
0.02370326593518257,
-0.05737850442528725,
-0.023787304759025574,
... |
<p>1.In classical mechanics, using Newton's laws, the ellipticity of orbits is derived. It is also said that the center of mass is at one of the foci. </p>
<p>2.Each body will orbit the center of the mass of the system.</p>
<p>My question is : Are the assumptions in 1 and 2 correct? </p>
<p>Follow up question : Assuming the distance from the centre of the mass to each body remains the same, do we have two bodies orbiting the centre of the mass of the system in an elliptical or circular orbit? </p>
<p>Finally : With elliptical orbits, if the heavier mass is supposed to be in one of the foci, if there is any significance to second focus, what is it? Is it a Lagrange point by any chance or does it have some other mathematical property?</p> | g856 | [
0.012897830456495285,
0.009328486397862434,
0.017590533941984177,
-0.007982341572642326,
0.062050677835941315,
0.033978596329689026,
0.0201428085565567,
-0.02617698721587658,
-0.005879854783415794,
0.02956126444041729,
-0.019788725301623344,
0.005309389904141426,
0.056240055710077286,
-0.0... |
<p>I am aware that the field in General Relativity (the metric, $g_{\mu\nu}$) is not completely physical, as two metrics which are related by a diffeomorphism (~ a change in coordinates) are physically equivalent. This is similar to the fact that the vector potential in electromagnetism ($A_\mu$) is not physical. In electromagnetism, the equations can be written in terms of physical (i.e. gauge invariant) quantities -- the electric and magnetic fields. Why can't Einstein's equations similarly be written in terms of physical variables? Is it just that nobody has been able to do so, or is there some theorem/argument saying that it can't be done?</p>
<p>EDIT: Let me rephrase: Prove/argue that there is no explicit prescription that can be given which would uniquely fix coordinates for arbitrary physical spacetimes. I.e., show that there is no way to fix gauge in the full theory of general relativity (unlike in E&M or linearized GR where gauge can be fixed).</p> | g10226 | [
0.01901751011610031,
-0.025818079710006714,
-0.015887856483459473,
-0.019604774191975594,
0.049827203154563904,
0.03110387735068798,
0.049408506602048874,
0.04516417533159256,
-0.06989103555679321,
-0.0031600650399923325,
0.09333553165197372,
-0.052208997309207916,
0.017808815464377403,
-0... |
<p>Does a sound at 50dB at 1m have the same intensity of a sound of 51dB at 10m, and also the same intensity of a 52dB sound at 100m?</p> | g10227 | [
0.029763201251626015,
-0.006175902206450701,
0.019672902300953865,
-0.07179972529411316,
-0.0012218491174280643,
-0.03544940799474716,
0.02489461377263069,
-0.023611290380358696,
-0.011680015362799168,
-0.021977674216032028,
-0.06918126344680786,
0.031378500163555145,
-0.062190160155296326,
... |
<p>Is it possible for a liquid to exist in a high quality vacuume? For example, a few Torr.</p>
<p>If so what are the methods for doing this?</p> | g10228 | [
0.04644705355167389,
0.014721040613949299,
0.015039559453725815,
0.01576181873679161,
0.07200977206230164,
0.06188933923840523,
-0.0599982775747776,
0.05456935614347458,
-0.06828025728464127,
-0.0335211418569088,
0.01230932492762804,
0.005349969025701284,
0.013652466237545013,
0.0233686603... |
<p>What is <a href="http://www.google.com/search?as_q=thermal+undulation" rel="nofollow">thermal undulation</a> in the context of lipid bilayers? Is it another word for "thermal fluctuation"?</p> | g10229 | [
0.03965746983885765,
0.04793034866452217,
0.009337848983705044,
-0.000019649596652016044,
0.04888618364930153,
0.006852342281490564,
-0.0044306544587016106,
0.02173340693116188,
-0.05279922112822533,
-0.022757628932595253,
0.003298242576420307,
-0.008802393451333046,
0.11670994758605957,
-... |
<p>According <a href="http://gamedevelopment.tutsplus.com/tutorials/custom-2d-physics-engine-oriented-rigid-bodies--gamedev-8032" rel="nofollow">this tutorial</a>, formula number 5:</p>
<p>$$j = \frac{-(1 + e)((V^{A} - V^{B}) * t)}{\frac{1}{mass^{A}} + \frac{1}{mass^{B}}}$$</p>
<p>translates into formula number 6:</p>
<p>$$j = \frac{-(1 + e)((V^{A} - V^{B}) * t)}{\frac{1}{mass^{A}} + \frac{1}{mass^{B}} + \frac{(r^{A} \times t)^{2}}{I^{A}} + \frac{(r^{B} \times t)^{2}}{I^{B}}}$$</p>
<p>when dealing with oriented bodies. $j$ is a scalar by which you divide normal and friction impulses to simulate them.
This is only valid for the 2D case, though, where the inertia is a scalar. What if I have the inertia as a tensor (more specifically a 3x3 matrix)?</p> | g10230 | [
0.03528418019413948,
-0.00629679998382926,
-0.008339508436620235,
0.0007749394862912595,
0.03817829117178917,
-0.0012433902593329549,
0.0774453729391098,
-0.010514844208955765,
-0.01992267742753029,
-0.012023747898638248,
-0.019023293629288673,
-0.021309930831193924,
0.0016721971333026886,
... |
<p>How does <a href="http://en.wikipedia.org/wiki/Divergence_theorem" rel="nofollow">divergence theorem</a> holds good for electric field.
How does this hold true- </p>
<p>$$\iiint\limits_{\mathcal{V}} (\vec{\nabla}\cdot\vec{E})\ \mbox{d}V=\mathop{{\int\!\!\!\!\!\int}\mkern-21mu \bigcirc}\limits_{\mathcal{S}} {}\vec{E}\cdot\hat{n}\ \mbox{d}A$$ </p> | g326 | [
0.03850176930427551,
0.05480090528726578,
-0.01147905271500349,
-0.05506578087806702,
0.04405558109283447,
-0.020115802064538002,
0.0502353273332119,
-0.03148151561617851,
-0.013502155430614948,
-0.050416819751262665,
-0.09188298881053925,
0.045803070068359375,
-0.026429440826177597,
-0.00... |
<p>Can somebody explain the proof of Gauss's theorem / <a href="http://en.wikipedia.org/wiki/Divergence_theorem" rel="nofollow">divergence theorem</a> taking the vector as electric field</p>
<p>$$\iiint(\nabla\cdot\vec E)\mbox{ d} V=\iint \vec E \cdot\hat{n} \mbox{ d} A?$$</p> | g326 | [
0.056902095675468445,
0.02194378338754177,
0.0007288996130228043,
-0.047367412596940994,
0.03580906242132187,
0.02875431440770626,
0.045904066413640976,
-0.015080227516591549,
0.009658526629209518,
-0.020304519683122635,
-0.09846971929073334,
0.03201930969953537,
-0.004546488169580698,
-0.... |
<p>How would you prove Gauss' law for an asymmetrical closed surface? I can find it for symmetrical surface but couldn't for Asymmetrical surfaces.</p> | g327 | [
0.0207974910736084,
-0.018231123685836792,
0.027082238346338272,
0.022784944623708725,
0.04293600469827652,
0.06813688576221466,
0.021202873438596725,
0.0013023854698985815,
0.005246318411082029,
-0.02352064661681652,
-0.004781547002494335,
0.013532737269997597,
0.039216604083776474,
0.013... |
<p>I encountered the following question in a previous year paper of a graduation-based test.</p>
<blockquote>
<p>On a certain night the moon in its waning phase was a half-moon. At midnight the moon will be (choose one of the following)</p>
<ol>
<li>on the eastern horizon. </li>
<li>at 45 degrees angular height above the eastern horizon.</li>
<li>at the zenith. </li>
<li>on the western horizon. </li>
</ol>
</blockquote>
<p>I remember looking out of my window and finding the moon on left and sometimes on right. I never imagined that it would be having a fixed pattern. This question is making me think otherwise. But I am unable to decide what that pattern is. Is the position of the moon really determine-able?</p> | g10231 | [
-0.013643894344568253,
-0.02332192286849022,
-0.006940681487321854,
-0.007994874380528927,
0.025602173060178757,
-0.004184674471616745,
0.05651417374610901,
-0.007841894403100014,
-0.00816462654620409,
0.004824995994567871,
-0.0002097213437082246,
0.004564208444207907,
-0.020050104707479477,... |
<p>I'm dealing with angular momentum, or particularly spin, on my quantum mechanics course; I guess the Pauli matrices thing is a more general one, but I'd like to illustrate my doubt with them (maybe get a deeper answer). Say, for the electron ($s=1/2$), why does
$$S_z=\begin{pmatrix}\left\langle\frac{1}{2},\frac{1}{2}\middle|S_z\middle|\frac{1}{2},\frac{1}{2}\right\rangle&\left\langle\frac{1}{2},\frac{1}{2}\middle|S_z\middle|\frac{1}{2},-\frac{1}{2}\right\rangle\\[0.1in]\left\langle\frac{1}{2},-\frac{1}{2}\middle|S_z\middle|\frac{1}{2},\frac{1}{2}\right\rangle&\left\langle\frac{1}{2},-\frac{1}{2}\middle|S_z\middle|\frac{1}{2},-\frac{1}{2}\right\rangle\end{pmatrix}=\frac{\hbar}{2}\sigma_z$$
instead of
$$S_z=\begin{pmatrix}\left\langle\frac{1}{2},-\frac{1}{2}\middle|S_z\middle|\frac{1}{2},-\frac{1}{2}\right\rangle&\left\langle\frac{1}{2},-\frac{1}{2}\middle|S_z\middle|\frac{1}{2},\frac{1}{2}\right\rangle\\[0.1in]\left\langle\frac{1}{2},\frac{1}{2}\middle|S_z\middle|\frac{1}{2},-\frac{1}{2}\right\rangle&\left\langle\frac{1}{2},\frac{1}{2}\middle|S_z\middle|\frac{1}{2},\frac{1}{2}\right\rangle\end{pmatrix}=\frac{\hbar}{2}\begin{pmatrix}-1&0\\0&1\end{pmatrix}=-\frac{\hbar}{2}\sigma_z$$
What I mean to say is, <strong>why does the matrix elements go from $m_s=+s$ to $m_s=-s$ (left-upper corner to right-lower corner) instead of the other way, as usual</strong>?</p>
<p>We usually take rows and columns from smaller to bigger value, but why is this not the case? For instance, recently we've seen the matrix representation of the hamiltonian of a simple harmonic oscilator, with elements $H_{mn}=(n+1/2)\hbar\omega\,\delta_{mn}$ and it goes like $\hbar\omega\begin{pmatrix}1/2&0&\ldots\\0&3/2&\ldots\\\vdots&\vdots&\vdots\end{pmatrix}$, not as, say $\hbar\omega\begin{pmatrix}\vdots&\vdots&\vdots\\\ldots&3/2&0\\\ldots&0&1/2\\\end{pmatrix}$, for example. I'm just trying to make clearer my question. Here ($s=1/2$) is only a difference of a minus sign, but when I did it for $s=3/2$, the ladder (spin) operators would interchange ($S_-$ would have non-zero values 'above' the diagonal and $S_+$ 'below' the diagonal). I tried to find the reason but found none; is it a mere convention or definition? Or am I missing something important here? Thank you in advance.</p> | g10232 | [
0.023331686854362488,
-0.03698933124542236,
-0.020042315125465393,
-0.038153283298015594,
0.04284815862774849,
-0.04679358750581741,
0.0876847356557846,
0.011936342343688011,
-0.03417506814002991,
0.021375954151153564,
-0.02121977135539055,
0.053170833736658096,
0.023867614567279816,
-0.00... |
<p>I have used FEA for a couple of years now, but using it and using it correctly are two different things, safety factor is not the solution to everything. I have the feeling I won't be using it right unless I have a clear answer to that question:</p>
<p>I am aware elements must be close to their ideal shape (based on the Jacobian) in order to get accurate results.. But why? Since I understand it comes from a coordinate transform, unless two vectors of the element become colinear shouldn't the results be accurate no matter its shape?</p>
<p><strong>A step-by-step answer based on n illustrated example</strong> (arbitrary stress distribution) would be really appreciated, especially given that it is a relatively common question (but never well answered from what I have seen).</p> | g10233 | [
0.01872693933546543,
-0.012437104247510433,
0.003168815281242132,
-0.022060532122850418,
0.043807025998830795,
-0.0361977219581604,
0.03211748227477074,
0.02492423914372921,
-0.0396081879734993,
-0.019091226160526276,
-0.03180113062262535,
0.04020560905337334,
0.04670969396829605,
0.004506... |
<p>In a car which phenomenon, diffraction or the resonant frequency of the car, lends itself more to the ability of bass to go farther?</p>
<p>Related Answer: <a href="http://physics.stackexchange.com/questions/18090/why-do-bass-tones-travel-through-walls">Why do bass tones travel through walls?</a></p> | g10234 | [
0.043264102190732956,
0.06520850956439972,
0.01030677743256092,
-0.012180913239717484,
0.03883245214819908,
0.013473201543092728,
0.06335952132940292,
-0.015316656790673733,
-0.03541029989719391,
-0.04140545055270195,
-0.029455823823809624,
-0.0029643906746059656,
0.018556788563728333,
0.0... |
<p>''electric field always undergoes a discontinuity when you cross a surface charge $\sigma$'' GRIFFITHS</p>
<p>In the derivation; Suppose we draw a wafer-thin Gaussian Pillbox, extendind just barely over the edge in each direction. Gauss law states that:</p>
<p>$\int_{S} E \cdot A = Q_{enc}/ \epsilon $</p>
<p>and so $E_{above}^{perp} - E_{below}^{perp} = \sigma/ \epsilon $</p>
<p>My question is why not $2A$ ?
$\int_{S} E \cdot A = 2EA$
because the top area of pillbox and the bottom area of pillbox, just as because the 2 parts of the flux...</p>
<p>SO.. WHY NOT : $E_{above}^{perp} - E_{below}^{perp} = \sigma/ 2\epsilon $</p>
<p>And why there is tangencial component of electric field; not just perpendicular to the surface, which can be seen as flat just looking very close to the surface.</p> | g10235 | [
0.04637144133448601,
-0.016114840283989906,
-0.01564161665737629,
-0.043141450732946396,
0.050939563661813736,
0.059935443103313446,
0.01396818645298481,
-0.007377102971076965,
-0.027387654408812523,
-0.020516393706202507,
-0.027910141274333,
0.045979034155607224,
0.04887473210692406,
-0.0... |
<p>Concerning combustion of fuel droplets:</p>
<p>Why is the mass fraction of fuel on a fuel droplet surface slightly less than one?</p>
<p>It is known that the temperature is below the boiling point at the fuel droplet surface, so there is no evaporation at the surface and therefore my intuition tell me that the mass fraction of the fuel at the surface should be unity, but it isn't. So why is the mass fraction of fuel less than one, and what happened to that fraction of fuel mass?</p> | g10236 | [
0.045236244797706604,
-0.010791601613163948,
0.007565540727227926,
0.041912712156772614,
0.05220499262213707,
0.040884435176849365,
0.0033041660208255053,
0.05616839975118637,
-0.08649636059999466,
-0.051250480115413666,
0.022775771096348763,
-0.057451918721199036,
0.03849110007286072,
0.0... |
<p>I just started to read the book 'A Brief History of Time' by Stephen Hawking. Actually When he was talking of the idea of infinite density 'thing' before big bang suddenly the mathematical function came to my mind was delta Dirac impulse function I am familiar at engineering classes. But that was normally either in time domain or frequency domain and also it was one dimensional. The Fourier transform dual of delta function is flat constant right from minus infinity to plus infinity . How can I model a 2D and 3D delta Dirac function in space and find its corresponding Fourier transform dual. Also what will be dual of space in the Fourier domain?</p> | g10237 | [
0.021931426599621773,
-0.03590632602572441,
-0.00545524014160037,
-0.06285721808671951,
0.02686583437025547,
0.06931991875171661,
0.03290749341249466,
0.04771118983626366,
-0.07917042821645737,
-0.10944725573062897,
-0.021220054477453232,
-0.010219870135188103,
0.07347990572452545,
0.00822... |
<p>In connection with a <a href="http://scifi.stackexchange.com/questions/15825/in-back-to-the-future-why-the-energy-to-travel-in-time-was-1-21-gwatts">related question on Science Fiction & Fantasy</a> Stack Exchange page: does this number 1.21 GW make any sense? What is 1.21 GWatts? With what can you compare it? Can the <a href="http://en.wikipedia.org/wiki/Lightning" rel="nofollow">lightning</a> really produce this amount of <a href="http://en.wikipedia.org/wiki/Power_%28physics%29" rel="nofollow">power</a>?</p>
<p>Are there any idea of why this number was used in a physical sense?</p> | g10238 | [
-0.015199446119368076,
0.08978904038667679,
-0.04624342545866966,
-0.06620864570140839,
0.00436156103387475,
-0.02808324061334133,
0.018940387293696404,
0.023108819499611855,
-0.09419393539428711,
0.014562061987817287,
0.06661630421876907,
0.06508705765008926,
0.016075238585472107,
0.00036... |
<p>I'm interested in building a robot with a carbon shell, and was wondering where I could find some ressources about how to work with carbon. For example, where to find it, how to shape it, and how to work with it.</p>
<p>Thank you</p> | g10239 | [
0.07104416191577911,
0.038507577031850815,
0.032453957945108414,
0.0251200869679451,
0.05482908710837364,
0.0037388266064226627,
-0.0565815269947052,
0.00002472645246598404,
-0.00984167493879795,
-0.0645727887749672,
-0.02354772761464119,
-0.027301421388983727,
0.0751359611749649,
-0.01106... |
<p>Let's say we try to remove the event horizon of a Kerr black hole by throwing in matter with some large angular momentum. If it starts with GM > a, could we increase a at all? Would such a particle be able to enter the black hole?</p> | g10240 | [
0.008415729738771915,
0.005324354395270348,
0.031021688133478165,
-0.04840319603681564,
0.016679277643561363,
-0.0015736029017716646,
-0.005419767927378416,
0.01593657396733761,
-0.014180121012032032,
0.054394472390413284,
0.0009465470793657005,
0.011216994374990463,
0.0024750682059675455,
... |
<p>Reading some books there is something that i don't understand. Is the capacitors topics when they name a potential difference between conductors.
As i know, Potential difference is the difference in electric potential energy per unit charge between two points.</p>
<p>So, talking about capacitors, the potential difference between conductors is:</p>
<pre><code>a) The difference between the potential of each conductor.
b) The difference of potential between those 2 points: (conductor)· ·(conductor)
</code></pre> | g10241 | [
0.050556015223264694,
-0.0063310121186077595,
-0.001589388819411397,
-0.0013825236819684505,
0.026990629732608795,
0.04039305821061134,
0.03314458951354027,
0.011920491233468056,
-0.04665398970246315,
0.0029422915540635586,
-0.05102773755788803,
0.03785758092999458,
-0.002014875877648592,
... |
<p>We know that Quantum Theory should be considered as a framework in which all other theories/forces (Strong, Weak, EM and Gravity) exist.</p>
<p>For example, we have the Quantum Chromodynamics, Quantum Flavordynamics (Electroweak), Quantum Electrodynamics (but still no Quantum GR).</p>
<p>When I think about this, it then strikes me why gravity, and specifically special relativity, is part of the framework itself (because QFT is based on QM and SR). Why would a theory like SR (and maybe in future GR) be part of the framework? This looks like a circular logic.</p>
<p>I would appreciate it if someone can explain?</p>
<p>My understanding is that gravity is different because it exist everywhere, and are not a result of a charge (electric, color or flavor). That's why we cannot speak about a quantum theory of EM without considering gravity, but we can do it without considering strong or weak force. Is this correct?</p>
<p>EDIT: it would make more sense to me to see Gravity/GR considered only as a framework (a geometric one) rather than a force, or it can be considered a fictitious force, as with the centrifugal force. see first paragraph of first answer <a href="http://physics.stackexchange.com/questions/55213/why-is-gravity-so-hard-to-unify-with-the-other-3-fundamental-forces?rq=1">here</a>.</p> | g10242 | [
0.06491551548242569,
0.008430941961705685,
0.014186328276991844,
-0.006614504382014275,
0.03847931697964668,
0.03865305706858635,
0.011372881941497326,
-0.006788352038711309,
-0.016606882214546204,
-0.04387067258358002,
0.05266720801591873,
-0.008120118640363216,
-0.00012929923832416534,
0... |
<p>When an object is submerged in a fluid (e.g., water), there is a pressure on the object due to weight of water by p=mgh, but why is there an upward buoyant force?</p> | g10243 | [
0.03760341927409172,
-0.0017321701161563396,
0.004547758027911186,
-0.025221480056643486,
0.014088423922657967,
0.09071584045886993,
0.0034051884431391954,
0.0010217076633125544,
-0.05480312928557396,
-0.019049551337957382,
-0.010279946029186249,
0.006000847090035677,
0.013325233943760395,
... |
<p>If one would like to study <a href="http://en.wikipedia.org/wiki/CP_violation" rel="nofollow">CP violation</a>, what would be the prerequisites for it? </p>
<p>For example, until now I have not studied quantum field theory and have done very little classical field theory, but I have done some quantum mechanics—non-relativistic and relativistic. What else do I need?</p> | g10244 | [
-0.013200958259403706,
0.01767297461628914,
0.01622232422232628,
-0.016643397510051727,
0.03401411324739456,
0.005307668354362249,
0.02715875767171383,
0.027530023828148842,
0.0060569229535758495,
-0.05388905107975006,
-0.014090158976614475,
-0.004681444261223078,
-0.006868717260658741,
0.... |
<p>Already the early papers on supergravity were written using computer algebra software to do some calculations. What modern packages do people normally use for doing such calculations? Of course Mathematica and presumable a lot of other programs can be made to do such calculations, I am wondering if there are special packages or tutorials how to do such calculations in them.</p> | g10245 | [
0.06630825251340866,
0.03895585611462593,
-0.03182320296764374,
0.01815619319677353,
-0.043645117431879044,
0.011310291476547718,
0.04273531585931778,
0.026350855827331543,
-0.07231315970420837,
-0.010233888402581215,
-0.011205261573195457,
-0.01732901856303215,
0.048853058367967606,
0.050... |
<p>With regards to the equations: $P=VI$ ; $P=V^2/R$ and $P=I^2R$, if you are given $P$, $V$, $I$ and $R$ for a circuit, how do you know which equation to use?</p>
<p>Does the use of an equation have to do with whether the circuit is in series or in parallel?</p> | g10246 | [
0.04540633037686348,
-0.019755933433771133,
-0.013531404547393322,
0.008862496353685856,
0.03350342810153961,
-0.09826794266700745,
0.061396460980176926,
0.00984832365065813,
-0.0014937962405383587,
0.0352875292301178,
-0.06356004625558853,
0.0826309472322464,
-0.061076629906892776,
0.0370... |
<p>Say we are given the scattering cross section for neutrinos from $d$ and $\bar{u}$ quarks as $\frac{d\sigma^{d}}{dQ^2}=\frac{G_F^2}{\pi}$, $\frac{d\sigma^{\bar{u}}}{dQ^2}=\frac{G_F^2}{\pi} (1-y)^2$, and the $u,\bar{d}$ cross sections are zero, how is the <strong>average nucleon cross section</strong> determined?</p>
<p>The solution involves this expression:
$\frac{d\sigma^p}{dQ^2}=\displaystyle \sum_q \int _0^1 dx\left(q(x) \frac{d\sigma^{q}}{dQ^2} + \bar{q}(x) \frac{d \sigma^{\bar{q}}}{d Q^2} \right).$ </p>
<p>Why is this the cross section for the proton? What exactly is the meaning of those functions, $q(x)$? </p>
<blockquote>
<p>$q(x)dx$ is the expectation value of the number of $q$ quarks in the hadron whose momentum fraction is within $[x,x+dx]$.</p>
</blockquote>
<p>Why are we using $\frac{d\sigma}{dQ^2}$ and what does it represent? After this we can take $\frac{d}{dx}\frac{d\sigma^p}{dQ^2}$ do the same for the neutron and take averages. The result is:</p>
<p>$\frac{d^2 \sigma^N}{dQ^2 dx} = \frac{G_F^2}{2\pi}\left(u^p(x)+d^p(x)+(1-y)^2(\bar{d}^n(x) + \bar{u}^p(x))\right)$</p>
<p>But I don't understand the origin behind the different quantities used here and the logic behind the derivation very well.</p> | g10247 | [
-0.014697094447910786,
0.02269308641552925,
-0.01332875993102789,
-0.02034938335418701,
0.0828857496380806,
-0.005154292099177837,
0.027053596451878548,
0.04896625131368637,
-0.047243352979421616,
-0.05015776678919792,
0.007789271883666515,
0.017354410141706467,
0.04840360954403877,
-0.003... |
<p>Im trying to understand this line given by our prof. :<br/>
"Representing fluid parameters as a function of the spatial coordinates($x$, $y$, $z$) and time $t$.</p>
<p>For example:
$$\vec{V} = u(x,y,z,t) \vec{i} + v(x,y,z,t) \vec{j} + w(x,y,z,t)\vec{k}$$</p>
<p>NOTE: $\vec{i}$, $\vec{j}$ and $\vec{k}$ are the unit vectors.</p>
<p>So my question is how come the $x$ coordinate is represented as function of all spatial coordinates and time $u(x,y,z,t)$. (similarly for $y$ and $z$).</p> | g10248 | [
0.06766525655984879,
-0.04355442151427269,
-0.026810165494680405,
-0.02300565131008625,
0.05226294323801994,
0.010153496637940407,
0.0412699393928051,
0.008240556344389915,
-0.06084924563765526,
0.01836303621530533,
-0.010846962220966816,
0.020043691620230675,
0.04300918057560921,
0.017432... |
<p>After the data from the cosmic microwave background has been collected by WMAP or Planck, what types of analysis is needed to conduct in order to deduce the cold dark matter density and the distribution of matter in the universe? In other words - How the CMB anisotropy measurements giving us evidence for the existence of dark matter/energy? (Any further explanation on the analysis of data is also welcomed.)</p> | g10249 | [
0.038847532123327255,
-0.04471732676029205,
-0.005085548851639032,
-0.026550160720944405,
0.03783686086535454,
0.03889276832342148,
0.014577222056686878,
-0.017391759902238846,
-0.03138095885515213,
-0.046134065836668015,
0.03013725019991398,
-0.005442147608846426,
0.002985332626849413,
0.... |
<p>In classical mechanics, a center of <em>mechanical</em> momentum frame can always be found for a system of particles interacting with one another locally. For an electromagnetic system where the charges interact non-locally via the electromagnetic field, a center of <em>total</em> momentum can likewise always be found. I have two questions: </p>
<p>When can a center of <em>mechanical</em> momentum frame exist for an electromagnetic system?</p>
<p>When is this frame also inertial?</p> | g10250 | [
0.07377678900957108,
-0.013944800943136215,
0.0035842303186655045,
-0.017898716032505035,
0.11236940324306488,
-0.00759010249748826,
0.03828584775328636,
0.02669670060276985,
-0.030054843053221703,
0.04824744537472725,
-0.01975855976343155,
-0.038623351603746414,
-0.05582228675484657,
-0.0... |
<p>I've recently encountered a path integral of the form</p>
<p>$$\int \delta[a\phi+b\phi']\,L(\phi,\phi')\;\mathcal D\phi\mathcal D\phi'$$</p>
<p>(where $a$, $b$ are integers) and would like to eliminate one of the $\mathcal D\phi$ integrations.</p>
<p>Is this possible? A change of variables doesn't seem to work since e.g. for substitutions $u:=a\phi$ the path integral Jacobian isn't well-defined if $a\neq 1$.</p> | g10251 | [
0.013513924553990364,
0.03284229710698128,
0.028312252834439278,
-0.030200256034731865,
-0.015913210809230804,
-0.028396405279636383,
-0.0003000610158778727,
0.04135838523507118,
-0.013718817383050919,
0.050631605088710785,
-0.03376379236578941,
0.042217470705509186,
-0.061523035168647766,
... |
<p>Absence of magnetic charges is reflected in one of Maxwell's fundamental equations:
$$\operatorname{div} \vec B = 0 \text{ (1).}$$
This equation allows us to introducte concept of vector potential:
$$\vec B = \operatorname{rot} \vec A.$$
Using this concept, it is possible to express Maxwell's equations in a graceful symmetric form:
$$\nabla^2 \vec A - \frac{1}{c^2}\frac{\partial^2 \vec A}{\partial t^2} = = - \frac{\vec j}{\epsilon_0 c^2} \text{ (2)}$$
$$
\nabla^2 \phi -\frac{1}{c^2}\frac{\partial^2 \phi}{\partial t^2} = - \frac{\rho}{\epsilon_0} \text{ (3)}
$$</p>
<p>Noticing, that vector $\vec A$ and scalar $\phi$ potentials, as well as electric current density $\vec j$ and charge density $\rho$, form a 4-vector in Minkovsky space-time. Therefore, Maxwell's equations can be expressed in a covariant form, using dalambertian:
$$\nabla_{\mu}\nabla^{\mu} A_{\nu} = \frac{j_{\nu}}{\epsilon_0} \text{ (4)}$$.</p>
<p>If magnetic monopols exist, Maxwell's equation (1) will look as:
$$\operatorname{div} \vec B = \mu_0 c \rho_{magnet}$$</p>
<p>As the divergence of $\vec{B}$ isn't equal to zero, it impossible to introduct concept of vector potential. Thus, the equation in the form of (4) will not be possible to express. </p> | g10252 | [
0.034436292946338654,
0.010863518342375755,
-0.032836247235536575,
-0.018670853227376938,
0.07035183906555176,
0.024188494309782982,
0.10095646977424622,
0.0057623134925961494,
-0.05165911465883255,
0.018803704530000687,
-0.035545315593481064,
0.007031865883618593,
-0.005728540476411581,
0... |
<p>I am going over Coulomb's law and there is something that is a bit
confusing for me:</p>
<p>According to Coulomb's law, if I have a charge $q_{1}$ at a point
$\vec{r_{1}}$ and a charge $q_{2}$ at a point $\vec{r_{2}}$ then
the force that the first charge applies to the second charge is given
by
$$
F_{1,2}=\frac{Kq_{1}q_{2}}{|\vec{r_{2}}-\vec{r_{1}}|^{2}}(\vec{r_{2}}-\vec{r_{1}})=\frac{Kq_{1}q_{2}}{|\vec{r_{2}}-\vec{r_{1}}|^{3}}\hat{(\vec{r_{2}}-\vec{r_{1}})}
$$</p>
<p>I see that in the first expression is does look like the force is
proportional to $\frac{1}{r^{2}}$, but I don't understand why it
isn't the second expression that matters, giving us that the force
is proportional to $\frac{1}{r^{3}}$:</p>
<p>Say both charges are on the $x-$axis, then $\hat{(\vec{r_{2}}-\vec{r_{1}})}=\hat{x}$.
Increasing the distance between the charges to two times what is by
moving the second charge on the $x$- axis was, would decrease the
force $2^{3}=8$ times, since $\hat{(\vec{r_{2}}-\vec{r_{1}})}$ is
still $\hat{x}$.</p>
<p>What is the mistake in my reasoning ?</p> | g10253 | [
0.06176489219069481,
-0.004199828952550888,
0.0022905103396624327,
0.046534232795238495,
0.13421258330345154,
0.014440124854445457,
0.0316551998257637,
0.009108989499509335,
-0.06796229630708694,
0.03295457363128662,
-0.01440003328025341,
0.01806797832250595,
-0.007764054927974939,
-0.0164... |
<p>I'm analyzing the experimental data obtained during Frank-Hertz experiment (conducted with Hg atoms):
<img src="http://i.stack.imgur.com/D3K82.jpg" alt="enter image description here"></p>
<p>Accelerating voltage values were multiplied by 0.1 during measurement (i.e. the mean value of energies differences if not 0.508 eV, but 5.08 eV). The output voltage was measured on resistive load of the anode, therefore it is proportional to anode's current.</p>
<p>The first five minimas in output voltage (top graph) have constant voltage difference - this value is used to calculate the first excitation energy of Hg atom (bottom graph). </p>
<p>The sixth minima, however, can be seen to have larger increase in output voltage(top graph - the first five voltage peaks may be fitted by a linear graph, whereas the sixth is no longer fits to this graph) and a higher voltage offset from the previous minima (bottom graph).</p>
<p><strong>Question 1:</strong> Is the sixth minima corresponds to the higher order excitation of Hg, or ionization, or else?</p>
<p><strong>Question 2:</strong> The curve is clamped at $V_{out} = 5V$. Is there a physical effect which causes this, or it is just the limit of measurement equipment, or else?</p> | g10254 | [
0.025527184829115868,
0.03945491835474968,
-0.0001786493230611086,
0.04457143321633339,
0.052421245723962784,
-0.026087304577231407,
0.0038121745456010103,
0.024366773664951324,
-0.04871716350317001,
-0.030302105471491814,
-0.03708445280790329,
0.05269121378660202,
-0.029034214094281197,
-... |
<p>I have a pretty simple homework question, but I can't rap my head around it. </p>
<p>In the question a swimmer of $55 \mbox{ } \mathrm{kg}$, jumps off a stationary raft of $210\mbox{ }\mathrm{kg} $. The swimmer jumps off the raft with a speed of $4.6 \mbox{ } \mathrm{ms}^{-1} $. I need to work out the recoil velocity of the raft. </p>
<p>So because Momentum before = Momentum after, I went: </p>
<p>$p_i = 0$</p>
<p>Therefore $0 = p_f$ and $p_f = mv$, $m = 210\mbox{ }\mathrm{ } $, so the $v$ would have to equal $0$. Making the recoil velocity equal $0$. However that doesn't seem right. Could use some clarification or help, thanks. </p> | g10255 | [
0.060516826808452606,
-0.010466773062944412,
0.02181125618517399,
0.011134685017168522,
0.025633027777075768,
-0.00831780955195427,
0.02805262804031372,
0.0368378609418869,
-0.047045573592185974,
-0.05195242166519165,
-0.04524385184049606,
0.0352436788380146,
0.0029698244761675596,
-0.0195... |
<p>When I turn my microwave oven over the stove on, it will cause a motion sensor light in the hallway next to the kitchen to got off and on. This affect can be reproduced anytime. I did notice that we stand in front of the microwave instead of out of the room, the motion sensor light goes on then off then on repeatedly. </p>
<p>How is the microwave affecting the motion sensing light that is about 4 ft. away behind a plaster wall? Is it dangerous? </p> | g10256 | [
0.010926141403615475,
0.03667427971959114,
0.00938338227570057,
0.03802153468132019,
-0.0038944180123507977,
-0.035478997975587845,
0.03299868851900101,
0.020244421437382698,
0.023330358788371086,
0.0025385566987097263,
-0.05004572123289108,
0.039232101291418076,
-0.024623200297355652,
0.1... |
<p>When I was in the university (in the late 90s, circa 1995) I was told there had been research investigating the $2$ (the square of distance) in the Newton's law of universal gravitation.</p>
<p>$F=G\frac{m_1m_2}{r^2}$</p>
<p>Maybe a model like </p>
<p>$F=G\frac{m_1m_2}{r^a}$</p>
<p>with $a$ slightly different from $2$, let say $1.999$ or $2.001$, fits some experimental data better?</p>
<p>Is that really true? Or did I misunderstand something?</p> | g466 | [
0.015094236470758915,
0.00032791285775601864,
-0.01097174547612667,
-0.00029519377858377993,
0.020268768072128296,
0.05568980798125267,
0.01785055547952652,
0.017582565546035767,
-0.04684771969914436,
-0.032318245619535446,
0.05153460428118706,
-0.018994994461536407,
0.055254727602005005,
... |
<p>Suppose I have a spherically symmetric potential and I can find its cross section in configuration space (i.e position-space), $d\sigma / d\theta$. Now I need to find its distribution $d^2\sigma / p_rdp_rdp_\theta$ in momentum space. How can I do this? </p> | g10257 | [
0.03547104075551033,
0.021196134388446808,
-0.038796715438365936,
-0.05086679384112358,
0.06549843400716782,
-0.010998760350048542,
0.02908945642411709,
-0.00043837164412252605,
-0.027168240398168564,
0.019811702892184258,
0.015266559086740017,
-0.04434515908360481,
-0.013960578478872776,
... |
<p>I am trying to desgin a little penny balista toy (built with a 3d printer).</p>
<p>The short description is that there is a track with a mass ("hammer") that slides along it. There is a stack of pennies in a hopper at the front. The hammer goes from back to the front, slams into the penny at the end and launches it.</p>
<p>I had a debate with a friend about whether it would go farther if the penny started at the back and was accelerated along the full length of the track along with the hammer (a more difficult design).</p>
<p>My question is, all else being the same, which method would impart more energy to the penny? (my suspicion, based on those "newtons cradle" toys, is that it would be the same).</p> | g10258 | [
0.08721280843019485,
0.08568798750638962,
0.026503929868340492,
-0.03757689520716667,
0.012852353043854237,
0.04372261092066765,
0.02379477024078369,
-0.0312175415456295,
-0.08628123998641968,
-0.05524028465151787,
-0.0079659353941679,
-0.028545383363962173,
0.040229830890893936,
-0.013520... |
<p>I just read <a href="http://physics.stackexchange.com/questions/9584/stability-of-hypothetical-lunar-atmosphere">stability of hypothetical lunar atmosphere</a>. From the correct answer, i understand, the low escape velocity from Luna is part of the reason it is unable to retain an atmosphere.</p>
<p>Titan apparently has a comparable escape velocity</p>
<ul>
<li>Titan = 2.65 km/sec </li>
<li>Luna = 2.4 km/sec</li>
</ul>
<p>; yet Titan maintains an atmosphere. </p>
<p>Why? What have I missed? Does Luna's relative proximity to Sol make the difference?</p> | g10259 | [
0.07506325840950012,
-0.004070446826517582,
0.0003173445875290781,
0.0556047298014164,
0.07053139060735703,
0.038419656455516815,
0.022974768653512,
-0.01462637074291706,
-0.0488835833966732,
-0.0339677594602108,
0.03571326285600662,
0.024882396683096886,
0.0017841983353719115,
-0.00992688... |
<p>Are photovoltaic cell arrays on a satellite the same that are used within Earth, or is there some difference in their construction given the differing environment in which they are to operate? Does the efficiency/life of a photovoltaic cell change depending upon whether it is within/without Earth's atmosphere? </p> | g10260 | [
0.01889614574611187,
-0.001695253886282444,
0.03363288566470146,
0.02686978690326214,
-0.04316982626914978,
0.011492707766592503,
-0.016453441232442856,
-0.035650137811899185,
-0.008231204003095627,
0.0058984714560210705,
0.028889982029795647,
0.040659982711076736,
0.001903417520225048,
0.... |
<p>I am writing a physics engine to map the rotary and translatory movements of a uniformly dense solid cylinder within 3d space. If a vectored thrust is applied to one end of the cylinder at an arbitrary angle (say 30 degrees off center), I understand this will cause both a rotation and translation of the body.
ROTATION....... Torque= radius*force*sin(angle) . moment of inertia(of a uniform cylinder) = 1/12*mass*(3*radius^2+height^2). Angular acceleration= torque/moment of inertia. TRANSLATION...... linear acceleration= force/mass.</p>
<p>IF my thrust angle is 0 then sin0=0 so my torque=0. But my linear acceleration(in-line with thrust vector) is the same. Surely if some of the thrust is producing a rotation then I should have a lower translatory acceleration? I reason this as it takes energy to both rotate and translate, therefore I seem so be getting more energy with vectored thrust then when operating with a thrust angle of 0(through the cog). Maybe my intuition is wrong here.... thanks</p> | g10261 | [
0.05416492745280266,
-0.0042913067154586315,
0.0051042730920016766,
0.00344275776296854,
0.008637484163045883,
-0.014817730523645878,
0.029986735433340073,
0.03649303317070007,
-0.06821172684431076,
-0.008094600401818752,
0.0038947961293160915,
-0.01559742446988821,
-0.0022646349389106035,
... |
<p>An object has an initial velocity of <strong>v</strong> and should stop in <strong>d</strong> meters, in <strong>t</strong> seconds where <strong>v*t >= d</strong>. I can predict that this acceleration won't be constant but it is a function of time and this problem may not have one exact solution for given <strong>v</strong>, <strong>t</strong> and <strong>d</strong>. But I am not able to think of any solution, can any one help me to calculate this function?</p> | g10262 | [
0.0757218524813652,
0.01830942928791046,
0.020882586017251015,
-0.004667555447667837,
0.03524366766214371,
0.005440889857709408,
0.0464744009077549,
0.034591589123010635,
-0.07283473014831543,
0.014908717013895512,
-0.05725841224193573,
0.01867089234292507,
-0.02804146520793438,
-0.0006966... |
<p>About how fast can a small fish swim before experiencing turbulent flow around its body? An Engineering Problem! Please go through this question step by step. :D</p> | g10263 | [
0.046208225190639496,
0.023558547720313072,
0.012911912053823471,
0.007094451691955328,
0.04998622089624405,
0.007465600036084652,
0.09243148565292358,
0.021037312224507332,
-0.031733233481645584,
-0.016555294394493103,
-0.03362833335995674,
0.11410799622535706,
-0.003103723982349038,
0.00... |
<p>I'm a high school student. I still don't understand what <a href="http://en.wikipedia.org/wiki/Turbulence" rel="nofollow">turbulence</a> is. Please can you explain what it really is? This is what I think it is: rotating motion of water when a particle travels at a velocity of $V$. </p> | g10264 | [
0.023889876902103424,
0.053792666643857956,
-0.03516538813710213,
-0.007787687703967094,
0.06725487858057022,
-0.002801156137138605,
0.10625026375055313,
-0.029696524143218994,
0.004486754536628723,
-0.04836135730147362,
-0.010751094669103622,
0.050009023398160934,
0.02104620821774006,
-0.... |
<p>The following is the equation which, I want to know, if it is valid in relativistic domain. Consider two equal charges moving in same direction with velocity $v$ and charge $q$ at a separation of $d$. The magnetic force acting between them is $(1/4\pi\epsilon_0 c^2)v^2q^2/d^2$ and the electrostatic force is $(1/4\pi\epsilon_0 )q^2/d^2$.the ratio of the magnetic (attractive) force and the electrostatic (repelling) force is $v^2/c^2$ and therefore the conclusion provided was that two individual charges with same (or any) velocity can never attract. Is this equation valid relativistically. Moreover, since magnetic field is just the relativistic counterpart of electrostatic force, can we find a frame of reference wherein in the case stated, all the force is magnetic and must be attractive? I am comfortable only with basic special relativistic mechanics but an intuitive understanding would be better.</p> | g10265 | [
0.037852801382541656,
0.02482776902616024,
-0.0035201977007091045,
-0.011428020894527435,
0.0949086919426918,
0.03505440801382065,
0.05126899480819702,
-0.0016357491258531809,
-0.02877269871532917,
-0.0014579748967662454,
-0.03400420397520065,
0.01644846424460411,
-0.012805174104869366,
-0... |
<p>my physics teacher told me about the refraction and its applications one of them was<br>
<strong>2 minutes of early sunrise</strong> and after she explained this effect she concluded that days are<br>
<strong>2 minutes longer</strong> than one would naively presume. </p>
<p>However, I think that <strong>her conclusion is wrong</strong> because<br>
<strong>if sunrise is considered then sunset should also be considered</strong> and according to me <strong>sunsets should be 2 minutes late</strong> therefore <strong>the day time is increased by 4 minutes and not 2 minutes</strong> over the naive calculation. </p>
<p>According to me, the situation looks something like this:<br>
<img src="http://i.stack.imgur.com/O1YUG.jpg" alt="enter image description here"><br>
Is this idea of mine correct or not? And if we both are wrong, then what should be the right conclusion and why ?</p> | g10266 | [
0.025677749887108803,
-0.006497358437627554,
-0.0004729341308120638,
0.004087886307388544,
-0.016951540485024452,
0.025508590042591095,
0.055083900690078735,
-0.03820999711751938,
0.009287748485803604,
-0.02220194786787033,
0.03681746497750282,
-0.011701052077114582,
-0.00676747877150774,
... |
<p>I have an extremely ridiculous doubt that has been bothering me, since I started learning quantum mechanics. </p>
<p>If we consider the finite dimensional vector space for the spin$\frac{1}{2}$
particles, I guess it is nothing but $\mathbb{C}^2$. Each vector has two components (which is why it is two dimensional right ?), each of which can be any complex number.</p>
<p>Now coming to the case of position space (say one-dimension). I was taught this <a href="http://www.google.com/search?as_q=linear+vector+space">LVS</a> is <a href="http://www.google.com/search?as_q=infinite+dimensional+vector+space">infinite dimensional</a> (also continuously infinite, unlike the number operator basis). I am not able to understand this subtle thing that it is infinte dimensional (is it something like $\mathbb{R}^\infty$?). It is quite confusing every time I encounter this kind of space. Also in this <strong>each component</strong> (of the <strong>infinite no. of them</strong>) can <strong>take any real value (infinite number of them)</strong>? I learnt that the way to represent these can be in term of complex-valued functions, I would like have it elucidated. </p> | g10267 | [
0.013759064488112926,
-0.041741326451301575,
-0.010272779501974583,
-0.06280428916215897,
0.03640080615878105,
0.004525193478912115,
0.061957623809576035,
0.023987362161278725,
-0.0354720763862133,
-0.061442237347364426,
-0.0316225029528141,
-0.014563844539225101,
-0.0041891406290233135,
-... |
<p>I know that it is a very old question but still I don't find any satisfactory solution for <a href="http://en.wikipedia.org/wiki/Zeno%27s_paradoxes#Achilles_and_the_tortoise">Achilles Paradox</a>. Please explain me the fundamentals of Achilles paradox in terms of stage wise distance covered. Note that it is easily solvable in terms of time, but if you start analysing this event in terms of time, then there is not at all any paradox. So please explain in terms of stage wise distances only.</p> | g985 | [
0.022797776386141777,
0.0655762329697609,
-0.004369102884083986,
0.0020579667761921883,
0.04308299720287323,
0.0006351426709443331,
0.0419531986117363,
0.021488366648554802,
-0.007360497023910284,
0.030383992940187454,
-0.012051933445036411,
0.011463283561170101,
0.05944130942225456,
-0.04... |
<p>If I have a vector A=4i+3j and B=5i-2j, how can I find the vector product AxB? I know that given the angle, its C=AB sin theta, but how can I solve this without the angle?</p> | g10268 | [
0.06440727412700653,
0.0011579083511605859,
-0.010584894567728043,
0.019899150356650352,
-0.020111892372369766,
-0.054897740483284,
0.0533570796251297,
-0.03455430641770363,
-0.0031002925243228674,
0.025356914848089218,
-0.041042182594537735,
0.034838661551475525,
-0.025169145315885544,
-0... |
<p>Considering the typical situation of a <a href="http://www.youtube.com/watch?v=8H98BgRzpOM" rel="nofollow">rotating bicycle wheel held by one end of its axle by a rope tied to the ceiling</a>: gravity torque is the time derivative of the angular momentum, and in this case is perpendicular to the angular momentum vector which makes it revolve without changing its magnitude (it precesses). </p>
<p>My question is the following: the wheel stops turning eventually because of the friction in the string (it also slows down its own rotation because of friction in the bearing but let's ignore it), but could you clarify how, step by step? Is it because the gyroscope creates a torque on the string axis, but it's reduced by the friction torque, and as a result the friction torque translates back into a residual "gravity torque" (in other terms, the gyroscope "gives up" on a certain amount of gravity torque it can't counteract because of the string)?</p>
<p>To progress toward my actual problem in several dimensions, what if the friction torque is in fact torque given by another gyroscope on the same solid? </p> | g10269 | [
0.06695593893527985,
0.01998547837138176,
0.0272474717348814,
-0.021999845281243324,
0.06296229362487793,
0.023659178987145424,
0.12217499315738678,
-0.02527686394751072,
-0.05714535713195801,
-0.01892099529504776,
-0.06466250866651535,
-0.050007835030555725,
0.044323232024908066,
-0.03103... |
<p>Again, as usual Schwinger leaves me startled as he writes, the Hermitian displacement operator in 2D is
$$ G = p_1\delta x_1 +p_2 \delta x_2 $$
Now, we know clearly that this group is an Abelian group, therefore $ [p_1,p_2] = 0$. But the author suggests this is equivalent to saying that the commutator generates any unitary transformation without physical consequences, and goes on to write,
$$ \hbar ^{-1}[p_1,p_2] = \mathbb{1} \tag{1}$$</p>
<p>Now, effect of an infinitesimal change in an operator($X$) is given by $\delta X = -i[X,G] $ where G is the generator of an unitary transformation ($U$)</p>
<p>Hence I obtain,
$$ \delta p_1 = -i[p_1,G] = \delta x_2 $$
$$ \delta p_2 = -i[p_1,G] = -\delta x_1 $$</p>
<p>Which then means, $$ x_1 = -p_2 \equiv q \:\:\:\:and\:\:\:\: x_2 = p_1 = p $$ Now, we see the association to the single particle's phase space state $(q,p)$.</p>
<p>Further translations in this (phase-space) two-dimensional space are described by the three parameter group with the generator,</p>
<p>$$ G = p\delta q -q\delta p + \delta \phi \mathbb{1} $$</p>
<p>My questions are :</p>
<p>a) How is the choice of commutator in equation $(1)$?</p>
<p>b) What exactly is the author trying to do here, or what is he motivating from this exercise? Why does he connect the translation group in normal 2D space to phase space of a single particle?</p>
<p>c) What is this operator $\delta \phi \mathbb{1} $ in the last generator?</p> | g10270 | [
-0.01823311671614647,
0.012734818272292614,
-0.04586902633309364,
0.01386613305658102,
0.01899743638932705,
0.035560257732868195,
0.049987711012363434,
0.05696794018149376,
-0.04773237183690071,
0.010747738182544708,
-0.05428974702954292,
0.025795280933380127,
-0.015064042992889881,
-0.011... |
<p>Is there any way to annihilate matter without the use of anti-matter? And vice versa? I mean, for example is it possible to totally convert the mass of a proton into "pure energy" without use an anti-proton?</p> | g10271 | [
0.0354100838303566,
-0.013858931139111519,
0.042654816061258316,
-0.004346881061792374,
0.06477297097444534,
0.006324853748083115,
-0.07446297258138657,
0.05428449809551239,
-0.03254249691963196,
0.025212442502379417,
0.023543547838926315,
-0.028420548886060715,
-0.02242276817560196,
0.000... |
<p>What is <a href="http://en.wikipedia.org/wiki/Drift_velocity" rel="nofollow">drift velocity</a>? And why in some books it is expressed as <strong>drift speed</strong> and not <strong>drift velocity</strong>?Are these different ?</p>
<p>Does it mean that the electron will have extra velocity opposite to the direction of electric field i.e
<strong>thermal(random) velocity + drift velocity=Total velocity</strong> of electron,in the conductor with a specific direction (opposite to electric field)</p> | g10272 | [
0.02727874182164669,
0.05653415992856026,
-0.01791468821465969,
0.016904983669519424,
0.08986404538154602,
0.010585731826722622,
0.046648912131786346,
0.03851257637143135,
-0.02175763249397278,
-0.03423957899212837,
-0.03758769482374191,
0.04387601092457771,
0.01734722964465618,
0.02649911... |
<p>Does anyone know what Feynman was referring to in this interview which appears at the beginning of The Feynman Tips on Physics? Note that he is referring to something that did not appear in the Feynman lectures.</p>
<blockquote>
<p>I didn't like to do the second year, because I didn't think I had
great ideas about how to present the second year. I felt that I
didn't have a good idea on how to do lectures on electrodynamics.
But, you see, in these challenges that had existed before about
lectures, they had challenged me to explain relativity, challenged me
to explain quantum mechanics, challenged me to explain the relation of
mathematics to physics, the conservation of energy. I answered every
challenge. But there was one challenge which nobody asked, which I
had set myself, because I didn't know how to do it. I've never
succeeded yet. Now I think I know how to do it. I haven't done it,
but I'll do it someday. And that is this: How would you explain
Maxwell's equations? How would you explain the laws of electricity
and magnetism to a layman, almost a layman, a very intelligent person,
in an hour lecture? How do you do it? I've never solved it. Okay,
so give me two hours of lecture. But it should be done in an hour of
lecture, somehow -- or two hours.</p>
<p><strong>Anyhow I've now cooked up a much better way of presenting the electrodynamics, a much more original and much more powerful way than
is in the book.</strong> But at that time I had no new way, and I complained
that I had nothing extra to contribute for myself. But they said, "Do
it anyway," and they talked me into it, so I did.</p>
</blockquote>
<p>Did this approach to teaching electrodynamics appear in any of his later writing?</p> | g10273 | [
0.0005531674833036959,
-0.006608446594327688,
0.000585705682169646,
0.005332428030669689,
0.0958140417933464,
0.030157046392560005,
0.056919556111097336,
-0.02505195513367653,
-0.014598118141293526,
-0.010966409929096699,
0.007271503563970327,
0.02936323545873165,
-0.02854200080037117,
0.0... |
<p>Let us begin in a two-dimensional Euclidean plane. The vector is e.g.
$\vec{V}(x,y)$
It is often useful – but in this case, it's just a mathematical trick that doesn't make the complex numbers "fundamental" – to combine the components into a complex number,
$z=x+iy$.
The two-dimensional rotations in $SO(2)$ are fully determined by the angle $δ$. And the matrix acting on the vector $\vec{V}(x,y)$
$M=
\begin{bmatrix}+\cosδ & −\sinδ\\+\sinδ & +\cosδ\end{bmatrix}$
may be fully replaced by the complex coefficient $e^{iδ}$ that multiplies our complex coordinate $z$. Instead of $z$, however, we could have dealt with its power $z^p$. The rotation could be described by the complex transformation
$z^p→z^{′p},z^{′p}=e^{ipδ}z^p$.
In this context, the complex number $e^{ipδ}$ plays the role of the "transformation matrix" that rescales the one and only component of our tensor-spinor-whatever, $z^p$. In our overly trivial two-dimensional context, the coefficient or exponent $p$ may be anything you want. But the value $p=1/2$ may be identified with the spinors in two dimensions. The spinor in two dimensions may be represented as $\sqrt{z}$ where $z=x+iy$ encodes a vector; so the spinor $z^{1/2}$ is literally the square root of a vector (translated to a complex number) in this case.</p>
<p>So I got all this from a blog that tries to explain spinors which unfortunately I don't remember. I don't understand why the $p$ in $e^{ipδ}$ has to be there since any complex number can be rotated without the need for that coefficient. Is there some rule for rotating complex numbers that are raised to higher powers $p=2,...,n$ or am i missing something? </p> | g10274 | [
-0.006272299215197563,
-0.0273186806589365,
-0.011433751322329044,
-0.005303173325955868,
0.03449515253305435,
-0.035871975123882294,
0.0787384957075119,
0.04078259691596031,
-0.02147412858903408,
0.01744929701089859,
-0.02567131444811821,
0.0307602696120739,
0.07107600569725037,
-0.047457... |
<p>I'm working through some basic theory on periodic potentials, and I would appreciate help in understanding the crystal momentum. Suppose we have a Bravais lattice with lattice vectors $\textbf{R}$. There is an associated reciprocal lattice with lattice vectors $\textbf{K}$ such that $\textbf{K} \cdot \textbf{R} = 2\pi n$ for $n \in \mathbb{Z}$. The relationship between these two lattices ensures that plane waves of the form $e^{i \textbf{K} \cdot \textbf{r}}$ are periodic in the direct lattice. A consequence of <a href="http://en.wikipedia.org/wiki/Bloch_wave" rel="nofollow">Bloch's theorem</a> is that the states $
\langle x|\psi \rangle$ of a particle assume the form</p>
<p>$$
\psi_{n\textbf{k}}(\textbf{r}) = e^{i \textbf{k} \cdot \textbf{r}}u(\textbf{r}),
$$</p>
<p>where</p>
<p>$$
u (\textbf{r} + \textbf{R}) = u(\textbf{r}).
$$</p>
<p>For these wavefunctions, $\textbf{p} \equiv \hbar \textbf{k}$ is defined to be the crystal momentum. Canonical momentum is ill-defined for this problem since the crystal breaks translation symmetry. However, for any translation $T_{\textbf{R}}$ within a lattice vector, $[H, T_{\textbf{R}}] = 0$. My questions are:</p>
<ol>
<li><p>In the first equation, I currently believe that $\textbf{k}$ can be any vector, and is not necessarily in the set of reciprocal wave vectors (i.e., $\textbf{k} \notin \{\textbf{K}\}$ necessarily). Since this is true, what is $\psi_{n\textbf{k}+\textbf{K}}?$</p></li>
<li><p>Suppose a particle has crystal momentum $\textbf{p} = \hbar \textbf{k}$. How do we interpret $\textbf{p}' = \hbar (\textbf{k} + \textbf{K})$?</p></li>
<li><p>Although there is no <em>continuous</em> symmetry in the lattice, there is a discrete symmetry of the potential $U(\textbf{r} + \textbf{R}) = U(\textbf{r})$, and therefore of the Hamiltonian. If Noether's theorem does not apply here, what quantity is "conserved" in time, and how do we justify such a conservation in general?</p></li>
</ol> | g10275 | [
0.048611488193273544,
0.00967417936772108,
0.007271115202456713,
-0.02476261369884014,
0.01630324497818947,
-0.03020000085234642,
0.02822757512331009,
0.033273905515670776,
-0.044520385563373566,
0.04356565326452255,
-0.028964633122086525,
0.017320899292826653,
-0.09248469024896622,
0.0018... |
<p>If we live in more than three spatial dimensions, is it not right to conclude that all matter observable to us shares almost the same coordinates of extra dimensions. Or is it just that ordinary interactions (like electromagnetism) are simply independent of these extra dimensions?</p> | g10276 | [
0.10123015940189362,
0.02569466270506382,
0.027328774333000183,
-0.029864419251680374,
0.017974762246012688,
0.1172446459531784,
-0.020145010203123093,
-0.014219097793102264,
-0.014533051289618015,
0.015774503350257874,
0.034473028033971786,
-0.04063645750284195,
-0.0170725230127573,
0.023... |
<p>I currently reading <a href="http://www.damtp.cam.ac.uk/user/tong/string/string.pdf" rel="nofollow">Tong's text on String Theory</a>.</p>
<p>In Chapter 4.1.1 he alludes to a technique to derive conserved currents</p>
<blockquote>
<p>Recall that we can usually derive conserved
currents by promoting the constant parameter $\epsilon$ that appears in the symmetry to a
function of the spacetime coordinates.</p>
</blockquote>
<p>Since I don't know what he refers to, could someone provide me a proper reference?</p> | g10277 | [
0.05667456239461899,
0.01636400818824768,
0.007127632386982441,
-0.09262444078922272,
0.08766764402389526,
0.002428092062473297,
0.033371392637491226,
-0.015608719550073147,
-0.05297034978866577,
0.010958060622215271,
-0.04169298708438873,
0.05024388059973717,
0.021537818014621735,
0.04268... |
<p>In some supergravities you have the gravition, gravitino, graviphoton and graviscalar. Each is analogous to each other in only sharing gravitational properties and nothing else. They differ by spin in the order of 2, 3/2, 1 and 0. What is the spin 1/2 analogue named?</p> | g10278 | [
0.008573970757424831,
0.009513146243989468,
-0.008975955657660961,
-0.009681015275418758,
-0.011280715465545654,
0.025249727070331573,
0.02173202484846115,
-0.04880232736468315,
-0.005186238791793585,
-0.00667178351432085,
0.011247563175857067,
-0.01012436393648386,
0.06400900334119797,
-0... |
<p>I'm studying the behaviour of superconductors in different cases, and I can't understand this. </p>
<p>We have a superconducting cylinder with a coaxial hole inside (vacuum). When the cylinder is cooled below it's critical temperature, and later a magnetic field is applied (case a), the superconductor will exclude the field even from the hole (B=0 inside the hole). However, if we first apply the magnetic field and <em>then</em> we cool down the cylinder (case b), the magnetic field will be excluded from the superconductor but not from the empty hole (as seen in the image below). Thus, depending on the procedure, we have two possible final states.</p>
<p>My question is, why is there this difference? Why doesn't the superconductor allow a magnetic field inside the hole in case a? It would still be excluding it from all the superconducting volume.</p>
<p>(Sorry, I don't know the source of the image, I took it from my professor's notes)</p>
<p><img src="http://i.stack.imgur.com/LXXup.png" alt="Field exclusion, expulsion and retention for different geometry superconductors following different procedures."></p> | g10279 | [
0.0833669900894165,
0.02080179564654827,
0.02199212834239006,
-0.001223080325871706,
0.025125974789261818,
0.04925674945116043,
0.06292080879211426,
0.014962769113481045,
-0.007295839488506317,
-0.033116668462753296,
-0.03139839321374893,
0.023680824786424637,
-0.0028859612066298723,
0.000... |
<p>Its not about availability of oxygen for combustion of gases. I want to know since there is no air or friction providing things in vaccum, how the force applied by a jet engine get can make thrust to move the vechile</p> | g10280 | [
0.033671215176582336,
0.08644258975982666,
0.007487686350941658,
0.04251024127006531,
0.07686768472194672,
0.04707537963986397,
0.012286951765418053,
0.015229525044560432,
-0.04229653254151344,
-0.029481960460543633,
-0.031452324241399765,
-0.019541209563612938,
0.03682699054479599,
0.0311... |
<p>I have a copy of David Bohm's <a href="http://rads.stackoverflow.com/amzn/click/0415404258" rel="nofollow">Special Theory of Relativity</a> and also a copy of T.M. Helliwell's <a href="http://rads.stackoverflow.com/amzn/click/1891389610" rel="nofollow">Special Relativity</a>. I was wondering if anyone has used these texts and if they're sufficient in providing an in depth inquiry into special relativity. </p> | g328 | [
-0.0009851602371782064,
0.03844313696026802,
0.016218356788158417,
-0.006631023250520229,
-0.008778041228652,
-0.002183453179895878,
-0.007596330717206001,
0.0378553532063961,
0.0022882616613060236,
0.004074569791555405,
0.04383869841694832,
-0.05748576670885086,
0.038176193833351135,
-0.0... |
<p>Consider the experiment in <a href="http://www.kids-fun-science.com/air-pressure-experiments.html">this link</a>.</p>
<p>The experiment includes using a ruler as a lever, with an inflated balloon on one side and a balloon which is not inflated on the other.</p>
<p>The aim of the experiment is to show that air has mass.</p>
<p>I have seen many kids performing similar experiments.</p>
<p>But, if the air pressure inside the balloon is equal to that outside, then the buoyant force will cancel out the weight of the air inside the balloon, won't it?</p> | g10281 | [
0.06069063022732735,
0.01381833478808403,
0.04202994331717491,
0.012500690296292305,
0.0034383831080049276,
0.06476891040802002,
0.015202627517282963,
0.0020314964931458235,
-0.04816925898194313,
-0.050046201795339584,
0.005484323017299175,
-0.03937205672264099,
-0.017043327912688255,
-0.0... |
<p>Well, is the basic difference between the work that we learn in <a href="http://en.wikipedia.org/wiki/Work_%28physics%29" rel="nofollow">Mechanics</a> and that in <a href="http://en.wikipedia.org/wiki/Work_%28thermodynamics%29" rel="nofollow">Thermodynamics</a>? This is because in Mechanics, whenever work of magnitude $W$ is done on a system $S$, then the system $S$ also does a work of magnitude $W$ on its surroundings. If I take this consideration in Thermodynamics, then according to the sign conventions, the total work in any process would turn $0$ (once positive and once negative), which is not the case. I am really very confused regarding these. Please explain what exactly is work in Thermodynamics, and its differences with Mechanics.</p> | g10282 | [
0.05779173597693443,
-0.000969929969869554,
-0.02496318705379963,
-0.02014252543449402,
0.0034313099458813667,
-0.007559709716588259,
0.014589845202863216,
0.01864364743232727,
-0.03157602623105049,
-0.03624742850661278,
0.008626488037407398,
-0.002720317803323269,
0.02282283455133438,
-0.... |
<p>I am reading the following article:
<a href="http://www.kof.zcu.cz/st/dis/schwarzmeier/galaxy_models.html" rel="nofollow">http://www.kof.zcu.cz/st/dis/schwarzmeier/galaxy_models.html</a>
and am currently at section 5.6 (positions of bodies in a galaxy).</p>
<p>I am trying to redo the simulations myself in Python, but I have a questions regarding sampling of the distribution.</p>
<p>Given a distribution function (Hernquist distribution):</p>
<p>$$\rho(r)=\dfrac{M}{2\pi}\cdot \dfrac{a}{r(a+r)^3}$$</p>
<p>the article states that to simulate the distribution, one has to calculate the mass within a circle of radius $r$ like follows:</p>
<p>$$m(r) = \int_{0}^{r} 4 \pi r'^2 \rho(r') dr'$$</p>
<p>which is the cumulative mass distribution function.</p>
<p>The article states that this formula represents the PDF. However, looking at the shape this appears to me to be a CDF. Approaching infinity the function approaches 1.0 which is for me a clear indication of a CDF.</p>
<p>To sample this distribution the article cites the Von Neumann method where one has to generate an $r$ and a $m$ value, and scale them accordingly, and check whether or not they fall below the $m(r)$ graph. If they do they are accepted, else they are rejected.</p>
<p>Am I completely off by thinking this is wrong? If I do this I end up with the majority of stars ending up at the higher radii.</p>
<p>I have the feeling I am sampling an CDF here instead of a PDF. To get accurate results (e.g.: having the majority of the stars in the center) means that I have to perform the Von Neumann method with the $\rho(r)$ function.</p>
<p>I am unable to contact the author of the article, so that's why I am asking here.</p> | g10283 | [
0.006689879111945629,
-0.03845985606312752,
-0.007233663462102413,
-0.06783223152160645,
0.026968175545334816,
0.018349003046751022,
-0.03014828823506832,
0.007962938398122787,
-0.05897273123264313,
0.006925730966031551,
-0.004120860248804092,
0.013895015232264996,
0.03900989517569542,
0.0... |
<p>I have a problem understanding the connection between the accoustic peaks in the CMB spectrum and the baryon oscillation picture. On the one hand it is stated, that the odd accoustic peaks (1,3,5..) are compressions peaks, i.e. peaks which have just reached maximum compression as decoupling occured. On the other hand there is the BAO picture, nicely illustrated for example here: <a href="http://astro.berkeley.edu/~mwhite/bao/" rel="nofollow">http://astro.berkeley.edu/~mwhite/bao/</a> . An overdense region starts to expand, because of gas and photon pressure, right after inflation and stops at the moment recombination occurs. The postulated peak at the distance 150 Mpc (the sound horizon $c_{sound} t_{rec}$ at recombination) could be found in galaxy survey experiments. Furthermore it is stated this corresponds to the first accoustic peak in the CMB spectrum. Clearly this contradicts the statement made above: Here the first peak should be a maximum rarefaction peak.</p>
<p>Besides that i don't know how to understand higher accoustic peaks in this picture. For me there should be no reason why the next peak, a maximum rarefaction peak, should be at approximately half the wavelength of the first peak. I can't see that we have here aboundary condition that would force a certain wavelength. </p>
<p>Any ideas or references that enlightens this confusing connection a bit more would be much appreciated.</p> | g10284 | [
0.014635005965828896,
0.01510413084179163,
0.018756510689854622,
0.00531122786924243,
0.053260572254657745,
-0.0002314094454050064,
0.02850029617547989,
0.003407962154597044,
-0.012284593656659126,
0.003482801141217351,
-0.027228374034166336,
0.047186512500047684,
-0.027698248624801636,
0.... |
<p>The symmetry of the standard model between the fermions and each generation of them is strongly suggestive of components as proposed in the <a href="http://en.wikipedia.org/wiki/Rishon_model" rel="nofollow">Rishon model</a>. Having discovered the Higgs boson, we know the Higgs plays a strong role in the mass of the fermions. Having heard of vector-like fermions, is there something that could be called a scalar-like fermion? Is it possible for Rishon fermions to exist with scalar properties to act like a Higgs boson and provide a substructure to fermions that obeys the standard model?</p> | g10285 | [
0.012435457669198513,
-0.022683558985590935,
0.006134538911283016,
-0.016405316069722176,
0.06502896547317505,
0.005484515335410833,
-0.006604021415114403,
0.05093224719166756,
-0.03750190883874893,
-0.027119552716612816,
-0.045840971171855927,
-0.00579645112156868,
0.0010759943397715688,
... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.