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<p>Every country is trading with other countries around the world, some more than others. I was wondering if there would be any change to the Earth's rotation because of the imbalance of trade between countries. </p>
| 52 |
<p>How does <a href="http://en.wikipedia.org/wiki/Mass%E2%80%93energy_equivalence">$E=mc^2$</a> put a upper limit to velocity of a body? I have read some articles on speed of light and they just tell me that it is the maximum velocity that can be acquired by any particle. How is it so? What is violated if $v>c$ ?</p>
| 2,532 |
<p>First post to this site, and I've got at most a high school background in physics - I really appreciate any answer, but I may not be able to follow you if you're too advanced.</p>
<p>I suppose this goes for regular planes too, but I'm especially interested in supersonic planes.</p>
<p>I read some reports in the news about various people working on commercial supersonic travel, but there were a lot of comments attached to these news posts listing essentially what were physics constraints that would make such travel severely cost ineffective:</p>
<ol>
<li>"skin friction," causing high heat and stress, leading to different metaled (thus more expensively researched/manufactured) and heavier (thus less efficient) airplanes.</li>
<li>increased drag, requiring more fuel to overcome.</li>
<li>sonic booms.</li>
</ol>
<p>I'll leave alone sonic booms - I understand as well as I can why this could be hard to engineer around, and why various countries have made generating them over land illegal.</p>
<p>The other two I don't get. After spending some time on wikipedia this evening, if I've got this right, it seems that, holding the shape of the airplane constant, skin friction, lift, and drag are each equal to a scalar times density times velocity squared.</p>
<p>Density drops as the altitude raises, which seems to mean to me that you could keep drag, lift, and skin friction constant when increasing speed by merely increasing altitude.</p>
<p>I assume this is right, so I guessed that the "gas mileage" issue had to do with needing to burn too much more gas to achieve thrust required for the higher velocity. And yet, the wikipedia article on jet engines states that the concorde was actually more fuel efficient than some conventional subsonic turbofan engines used in 747's.</p>
<p>Given all this, what did I get wrong? Why can't supersonic planes just fly higher to be as cost effective, or more, than conventional subsonic commercial jetliners, using the same construction materials? Relatedly, why do current jets have service ceilings and max speeds (assuming it's not just about the high stress of breaking through the sound barrier)?</p>
<p>Thanks!</p>
| 2,533 |
<p>I was told that the total integral of the stress over the surface of a swimmer (i.e. the total force exerted by the swimmer on the fluid) always vanishes, because there are no external forces applied on it. That seems fair by the 3rd Newton Law.</p>
<p>But, how does it take into account the effects of the 2nd Newton's Law? If, for example the swimmer starts from a stationary state, and at the time $t_1$ reaches a velocity $v_1$, where does the force generating the acceleration $v_1/t_1$ come from?</p>
<p>I was told that this can happen just because in the case of a swimmer with density different than that of the fluid, so gravitational forces are present. But it seems bulls**t to me.
Could you help me to clarify all this? </p>
| 2,534 |
<p>I have read that heat pumps are more efficient than generating heat directly for heating homes, until the temperature gets down to a critical value.</p>
<p>I just asked how to calculate temperature of gasses: <a href="http://physics.stackexchange.com/questions/81661/behavior-of-gasses-ideal-and-otherwise">Behavior of gasses, ideal and otherwise</a></p>
<p>and have a little octave code</p>
<pre><code>function T2 = compressedgas(cp, cv, P1, V1, T1, P2)
gamma = cp/cv; % approx 7/5 for dry air
R = 8.3;
V2 = V1 * ( P1/P2)^(1/gamma)
n = P1 * V1 / (R * T1);
T2 = P2 * V2 / (n * R);
end
</code></pre>
<p>For air at STP, T = 298, P = 102kPa, compressed to P2 = 714kPa, I get a temperature of 1005 K given gamma = 1.4</p>
<p>Assuming a pressurized tank allowed to radiate heat until it cooled down, this seems like a very efficient heat source indeed.</p>
<p>When I change the assumption to very cold air T = 248 the temperature is still 839K.</p>
<p>This seems hot enough that I don't understand why heat pumps should ever be less efficient than directly heating the house, so can anyone identify the engineering details that make this difficult?</p>
| 2,535 |
<p>I know that electromagnetic waves induce electric currents in conductors and that's the basis for radio, wi-fi etc.</p>
<p>I also know that light is also an electromagnetic wave. So, can light induce a current in a conductor (like a metal wire? or a coil?). And, if the answer is yes, is the same visible for other high-frequency waves (X-rays, gamma)?</p>
<p>I heard about the photoelectric effect, but it seems related to the particle theory of light (photons transferring their energy to electrons).</p>
<p>So, do high-frequency electromagnetic waves generate electric currents? Is it possible to measure them? Is the skin effect relevant here?</p>
| 2,536 |
<p>My text claims that Gauss's Law has been proven to work for moving charges experimentally, is there a non-experimental way to verify this?</p>
| 2,537 |
<p>Suppose we have an ideal gas performing an irreversible cycle composed by:</p>
<ol>
<li>an isothermal transformation at $T_{1}$;</li>
<li>an isobaric transformation at $P_{A}$;</li>
<li>an isothermal transformation at $T_{2}$;</li>
<li>an isobaric transformation at $P_{B}$.</li>
</ol>
<p>with $\frac{T_{1}}{T_{2}}=\frac{3}{4}$ and $\frac{P_{B}}{P_{A}}=\frac{4}{5}$. </p>
<p>We want to calculate $\Delta S_{universe}$.</p>
<p>I know that it must be </p>
<p>$$
\Delta S_{universe}>0
$$
and that </p>
<p>$$
\Delta S_{system}=0
$$
because the system performs a cycle, thus </p>
<p>$$
\Delta S_{universe}=\Delta S_{ambient} + \Delta S_{system}=\Delta S_{ambient}
$$
In the isothermal transformations:</p>
<p>$$
Q^{1}_{ambient}=-Q^{1}_{system}=-nRT_{1}\ln\left(\frac{P_{B}}{P_{A}}\right)=nRT_{1}\ln\left(\frac{P_{A}}{P_{B}}\right)
$$
$$
Q^{3}_{ambient}=-Q^{3}_{system}=-nRT_{2}\ln\left(\frac{P_{A}}{P_{B}}\right)=nRT_{2}\ln\left(\frac{P_{B}}{P_{A}}\right)
$$
and thus:</p>
<p>$$
\Delta S^{1}_{ambient}=\frac{Q^{1}_{ambient}}{T_{1}}=nR\ln\left(\frac{P_{A}}{P_{B}}\right)
$$
$$
\Delta S^{3}_{ambient}=\frac{Q^{3}_{ambient}}{T_{2}}=nR\ln\left(\frac{P_{B}}{P_{A}}\right)
$$
but then</p>
<p>$$
\Delta S^{1}_{ambient} + \Delta S^{3}_{ambient}=0
$$</p>
<p><strong>Edit (according to the "on-hold" valutation, and according to the answers of user31748 and user139981)</strong></p>
<p>In the isobaric processes we have</p>
<p>$$
Q^{2}_{system}=\Delta U^{2}_{system} + W^{2}_{system}=nc_{V}\Delta T + P_{A}\Delta V=\frac{5}{8}nRT_{2}
$$</p>
<p>and</p>
<p>$$
Q^{4}_{system}=\Delta U^{4}_{system} + W^{4}_{system}=nc_{V}\Delta T + P_{B}\Delta V=-\frac{5}{8}nRT_{2}
$$</p>
<p>In order to calculate the entropy change of the ambient in $2$ and $4$ we assume that during $2$ the ambient is at $T_{1}$ and during $4$ it is at $T_{2}$, thus</p>
<p>$$
\Delta S^{2}_{ambient} + \Delta S^{4}_{ambient}=-\frac{Q^{2}_{system}}{T_{1}} - \frac{Q^{4}_{system}}{T_{2}}=-\frac{5}{24}nR
$$</p>
<p>Now two issues arise.</p>
<p>The first is whether or not the assumption of constant temperature of the ambient is justified, moreover how is it possible to consider the ambient being at $T_{1}$ during $1$ and $2$ and then suddenly change to $T_{2}$ during $3$ and $4$?</p>
<p>The second, assuming the validity of the previous hypotesis, regards the fact that:</p>
<p>$$
\Delta S_{universe}=\Delta S_{ambient} + \Delta S_{system}= \Delta S_{ambient}>0
$$ </p>
<p>Indeed we have:</p>
<p>$$
\Delta S_{ambient}= \Delta S^{1}_{ambient} + \Delta S^{2}_{ambient} + \Delta S^{3}_{ambient} + \Delta S^{4}_{ambient}= \Delta S^{1}_{ambient} + \Delta S^{3}_{ambient} -\frac{5}{24}nR
$$
and thus it can not be </p>
<p>$$
\Delta S^{1}_{ambient} + \Delta S^{3}_{ambient}=0
$$
as stated above.</p>
<p>This means that the irreversibility of the cycle plays a crucial role in determining the entropy change of the ambient during $1$ and $3$. However during $1$ and $3$ the system (ideal gas) does not alter its internal energy, thus:</p>
<p>$$
Q^{i}_{system}=W^{i}_{system}
$$
Now during $1$ the system increases its pressure, thus, being at constant temperature, its volume decreases and $W^{1}_{system}<0$, this means that the system receaves heat. Clearly the opposite happens during $3$. With simple calculations we are back at </p>
<p>$$
\Delta S^{1}_{ambient} + \Delta S^{3}_{ambient}=0
$$
So where is the error?</p>
<p><strong>II edit (according to a comment due to user31748</strong></p>
<p>The fact that the cycle is irreversible does not prevent us from considering it to be quasistatic, moreover I think this is the only way one has to proceed in order to solve the problem without using non-equilibrium thermodynamic (as it is implicitely assumed in the exercise). </p>
<p>In the case of a gas performing a quasistatic irreversible isobaric process (without friction), the system evolves through different equilibrium states close one to another, thus the expression</p>
<p>$$
W=\int pdV=p\Delta V
$$
is valid with the understanding of $p$ being the actual pressure of the gas (the one we can calculate with the equation of state $pV=nRT$).</p>
<p>However in the case of a gas performing a non-quasistatic irreversible isobaric process (without friction), the system does not evolves through different equilibrium states, nevertheless the external pressure is constant (because the process is isobaric) and the expression</p>
<p>$$
W=\int p_{ext}dV=p_{ext}\Delta V
$$
is valid with the understanding of $p_{ext}$ being the external pressure.</p>
<p>This is why I think that my derivation of the value of the work in $2$ and $4$ is correct.</p>
<p>Obviously I do not want to sound arrogant, I know I could be completely wrong, and thus I hope will answer (moreover I think the question can no longer be seen as a mere homework because of the sublety of the reasoning involved and thus could be made "off-hold").</p>
| 2,538 |
<p>While waiting for a 3D movie to start, I was playing with the glasses they give you. I understand each lens has different polarized filters, so the left and right superimposed images on the screen go to the correct eyes.</p>
<p>The first thing that tripped me up was that rotating the glasses didn't affect the light that passes through it. After searching I bit I discovered about <em>circular</em> polarization, which ignores the angle of the filter and seems to be the standard for cinema glasses.</p>
<p>The second thing that tripped me up were the wall lamps. When looking through one lens, the light seemed to have a bluish color. From the other, a reddish/orange color. It was subtle, but other people confirmed seeing it.</p>
<p>I figured the lamp's white light had blue and red components, which are of different wavelengths (almost opposite in the visible spectrum, if I remember correctly), but what does the wavelength have to do with circularly polarized filters? And if this line of thought is correct, why does it seems to divide the visible spectrum?</p>
| 2,539 |
<p>If we understand spacetime as a $4$-dimensional manifold $M$, from the point of view of physics what are the consquences of a subset of it being compact? My point here is simple: in math we usually think of compactness as some analogue of finiteness because it shares many properties with finite sets, but what are the consequences of this when we deal with physics? </p>
<p>Of course, we need not to go into relativity, we can even think about the usual three space $\mathbb{R}^3$. What are again the consequences of a set $A \subset \mathbb{R}^3$ being compact? Are there any cool things we can get out from this, or we simply use compactness in physics to grant the mathematical properties desired without having any direct impact in the way we understand and interpret those sets?</p>
<p>Thanks very much in advance.</p>
| 2,540 |
<blockquote>
<p>It was found centuries ago that these materials: wool cloth and paraffin wax, glass rod and silk cloth when rubbed against each other attracted one another. While two glass rods when rubbed against their respective silk cloths repelled each other and the same befell two pieces of paraffin wax. These forces of attraction and repulsion also manifested in the pieces of cloths that were used for rubbing. Hence, it was found that any pair of materials that demonstrated properties of attraction and repulsion after begin rubbed together could be classified into one of two categories: 1) Attracted to wax and repelled by glass 2) Repelled by glass and attracted to wax. Early experimenters suggested that there might have been fluids that transferred from one material to another that caused the attractive and repulsive forces. Charles Dufay suggested that there are exactly two types of changes that befell the pair of certain materials when rubbed together. This suggestion was based on the fact that there exist two types of forces of attraction and repulsion. The hypothetical fluid transfer became known as <em>charge</em>. On the contrary, Benjamin Franklin suggested that there was only one fluid exchanged between the two objects when rubbed together and that the two <em>charges</em> were nothing more than the excess or deficiency of this one fluid. He performed an experiment on wax and wool, and suggested that the wool takes some of the fluid from wax and this imbalance causes the attractive force because the fluid tends to be re-balanced. <em>Postulating the existence of a single fluid that was either gained or lost through rubbing <strong>accounted best</em></strong> <em>for the observed behavior: that all these materials fell neatly into one of two categories when rubbed, and most importantly, that the two active materials rubbed against each other always feel into opposing categories as evidenced by their invariable attraction to one another. In other words there was never a time where two materials rubbed against each other both became either positive or negative</em>.</p>
</blockquote>
<p>I know these are the old concepts, but ignoring the modern concepts My question is that, why was Franklin's Benjamin suggestion the best explanation to the observed behavior that all the materials fell into one of the two categories when rubbed? The suggestion that there were two types of fluids that were transferred from one material to another could also explain the same thing. </p>
| 2,541 |
<p>What is the current status or acceptance of <a href="http://en.wikipedia.org/wiki/Eternalism_%28philosophy_of_time%29" rel="nofollow">block time</a> as it relates to Einstein's theory of relativity? Has quantum mechanics ruled it out or is it still the favored view of the world? Perhaps there can't be any consensus as we cannot link quantum mechanics with relativity?</p>
| 2,542 |
<p>Let the tight-binding Hamiltonian be $\sum\limits_{ij} {{t_{ij}}\left| i \right\rangle \left\langle j \right|}$. Where ${\left| i \right\rangle }$ is the atomic orbit at lattice site $i$.</p>
<p>My question is, is it correct to express the usual operators in this representation like this:</p>
<p>Position operator:
$\mathop {\bf{r}}\limits^ \wedge = \sum\limits_i {{{\bf{r}}_i}\left| i \right\rangle \left\langle i \right|} $, where ${{\bf{r}}_i} = \left\langle i \right|\mathop {\bf{r}}\limits^ \wedge \left| i \right\rangle$ is just the lattice position of ${\left| i \right\rangle }$ if the atomic orbit is symmetric.</p>
<p>Velocity operator:</p>
<p>$\mathop {\bf{v}}\limits^ \wedge = \sum\limits_{ij} {{{\bf{v}}_{ij}}\left| i \right\rangle \left\langle j \right|} $</p>
<p>${{\bf{v}}_{ij}} = \left\langle i \right|\mathop {\bf{v}}\limits^ \wedge \left| j \right\rangle = \frac{{ - i}}{\hbar }\left\langle i \right|\left[ {\mathop {\bf{r}}\limits^ \wedge ,H} \right]\left| j \right\rangle = \frac{{ - i}}{\hbar }\left( {{{\bf{r}}_i} - {{\bf{r}}_j}} \right){H_{ij}}$</p>
<p>If the wavefunction of the Hamiltonian is </p>
<p>$\begin{array}{l}
\left| \psi \right\rangle = \sum\limits_i {{\psi _i}\left| i \right\rangle } \\
{\psi _i} = \left\langle i \right|\left. \psi \right\rangle \\
\end{array}$</p>
<p>Is it right to write:</p>
<p>$\begin{array}{l}
\left\langle \psi \right|\mathop {\bf{r}}\limits^ \wedge \times \mathop {\bf{v}}\limits^ \wedge \left| \psi \right\rangle = \sum\limits_{ij} {\left\langle i \right|\psi _i^*\left( {\mathop {\bf{r}}\limits^ \wedge \times \mathop {\bf{v}}\limits^ \wedge } \right){\psi _j}\left| j \right\rangle } \\
= \sum\limits_{ij} {\psi _i^*{\psi _j}\left\langle i \right|\left( {\mathop {\bf{r}}\limits^ \wedge \times \mathop {\bf{v}}\limits^ \wedge } \right)\left| j \right\rangle } = \sum\limits_{ij} {\psi _i^*{\psi _j}{{\bf{r}}_i} \times {{\bf{v}}_{ij}}} \\
\end{array}$, </p>
<p>where ${\psi _i^*{\psi _j}}$ is not operated by ${\left( {\mathop {\bf{r}}\limits^ \wedge \times \mathop {\bf{v}}\limits^ \wedge } \right)}$?</p>
<p>I ask this because if this is done in continuum real space, the wavefunction ${\psi _i^*{\psi _j}}$ should also be operated by ${\left( {\mathop {\bf{r}}\limits^ \wedge \times \mathop {\bf{v}}\limits^ \wedge } \right)}$ since it's position dependent.</p>
<p>Comment 1:</p>
<p>To calculate $\widehat{\bf{r}} \times \widehat{\bf{v}}$ at lattice site $i$, according to my formalism, in the tight-binding formalism it should be $\left\langle i \right|\widehat{\bf{r}} \times \widehat{\bf{v}}\left| i \right\rangle = \sum\limits_k {\left\langle i \right|\widehat{\bf{r}}\left| k \right\rangle \times \left\langle k \right|\widehat{\bf{v}}\left| i \right\rangle = } \frac{{ - i}}{\hbar }\sum\limits_{k,l} {{{\bf{r}}_i} \times \left( {{{\bf{r}}_i}{h_{ii}} - {h_{ii}}{{\bf{r}}_{i}}} \right)} = 0$, while it should be proportional to angular momentum at site $i$, for $p_z$ orbit, it should be nonzero, why I got this Contradictory result?</p>
| 2,543 |
<p>I'm currently searching for a good and reliable source where precision measurements on the wavelength of the so called sodium doublet ($D_1$ and $D_2$ lines) at approximately 589.0 nm and 589.6 nm are made.</p>
| 2,544 |
<p>The event horizon of a black hole is where gravity is such that not even light can escape. This is also the point I understand that according to Einstein time dilation will be infinite for a far-away-observer.</p>
<p>If this is the case how can anything ever fall into a black hole. In my thought experiment I am in a spaceship with a powerful telescope that can detect light at a wide range of wavelengths. I have it focused on the black hole and watch as a large rock approaches the event horizon.</p>
<p>Am I correct in saying that from my far-away-position the rock would freeze outside the event horizon and would never pass it? If this is the case how can a black hole ever consume any material, let alone grow to millions of solar masses. If I was able to train the telescope onto the black hole for millions of years would I still see the rock at the edge of the event horizon?</p>
<p>I am getting ready for the response of the object would slowly fade. Why would it slowly fade and if it would how long would this fading take? If it is going to red shift at some point would the red shifting not slow down to a standstill? This question has been bugging me for years!</p>
<p>OK - just an edit based on responses so far. Again, please keep thinking from an observers point of view. If observers see objects slowly fade and slowly disappear as they approach the event horizon would that mean that over time the event horizon would be "lumpy" with objects invisible, but not passed through? We should be able to detect the "lumpiness" should we not through?</p>
| 447 |
<p>I'll have to perform some simple experiments, like measuring the period and damping of a pendulum, focal length of a lens, …</p>
<p>As a result I will end up with some X, Y data points and need to calculate derivation, means, kovariance and provide a fit for a liniarized function.</p>
<p>It explicitly says that we can use software or self written programs.</p>
<p>I could implement all the statistics formulas on the help sheet in say Python and run it, but that seems like a huge duplication of effort.</p>
<p>Is there software that would assist me with those kind of things?</p>
<p>I currently have:</p>
<ul>
<li>Mathematica</li>
<li>Octave</li>
<li>xmgrace</li>
<li>(ROOT)</li>
</ul>
| 2,545 |
<p>In calculating transition amplitude for Klein-Gordon real-scalar field, I encountered the integral,</p>
<p>$$ \frac{-i}{2(2\pi)^2\Delta x} \int^{\infty}_{-\infty} \,dk \frac{ke^{ik\Delta x}}{\sqrt{k^2+m^2}} $$</p>
<p>I can see here the integrand has branch cuts at $ k= \pm im $
However, later they do a change of variables $ z= -ik $ and then the integral becomes, </p>
<p>$$ \frac{1}{2(2\pi)^2\Delta x} \int^{\infty}_{m} \,dz \frac{ze^{-z\Delta x}}{\sqrt{z^2-m^2}} $$</p>
<p>And it is said that they can wrap the contour around the upper branch cut for $ \Delta x > 0 $</p>
<p><img src="http://i.stack.imgur.com/9qmWP.jpg" alt="enter image description here"></p>
<p>I am not able to see how this transformation happens and how the contour can be wrapped around the upper branch. Thanks for your inputs.</p>
| 2,546 |
<p>Either through doping or gating. What are some good terms to search for if I'm looking for some experimentally obtained values for particular materials? I'm particularly interested in what the limit is for graphene, if anyone knows.</p>
<p>For example, the DOS for regular graphene has states between -3t and +3t, but what is the maximum fermi energy we can examine experimentally? Is there a breakdown of the sample at some limit of doping or voltage?</p>
| 2,547 |
<p><img src="http://i.stack.imgur.com/7xqyI.png" alt="http://thecuriousastronomer.wordpress.com/"></p>
<p>My doubt is in the equation (1) and (2). Aren't x,y and z also the radiuses?</p>
<p><strong>EDIT</strong>
Thank you guys for trying to give a wonderful explanation but I figured out the answer myself and it was just my silly interpretation. I thought the the value of ct on x axis would be equal to ct and forgot that the value is given by a perpendicular and not an arc (of the sphere). I was thinking that where the sphere touches x axis, that is the value of ct but it isn't. The value of ct for x is given by a point on the axis that lies perpendicular to the axis.</p>
| 2,548 |
<p>Why does light have the speed it does? why is it not considerably faster or slower than it is? I can't imagine science, being what it is, not pursuing a rational scientific explanation for the speed of light. Just saying "it is what it is" or being satisfied saying it is 1 ($c=1$), does not sound like science.</p>
| 283 |
<hr>
<p>What would a transistor level design of an XOR gate look like?</p>
<hr>
| 2,549 |
<p>The Thin Shell Formalism (<a href="http://en.wikipedia.org/wiki/Gravitation_%28book%29" rel="nofollow">MTW</a> 1973 p.551ff) is used to properly paste together different vacuum solutions to the Einstein equations. At the junction of the two solutions is a hypersurface of matter – the so-called thin shell. The thin shell formalism not only permits timelike and lightlike thin shells, it permits spacelike ones. A spacelike shell implies that when a stationary observer’s timelike world line encounters the shell (by simply moving forward in time), the observer would experience the momentary existence of a surrounding volume of matter – like momentarily finding yourself underwater. I have two questions about this:</p>
<p>1) How is it that the momentary appearance of a spacelike thin shell, apparently permitted by the formalism, is not a violation of energy conservation?</p>
<p>2) Imagine the maximally extended Schwarzschild solution with $r = R$ in the black hole sector of the solution being identified with (pasted onto) $r = R$ in the white hole sector. If $R$ is less than the radius of the event horizon, $r = R$ describes a spacelike hypersurface, which the thin shell formalism seems to have no problem with. Does this mean that this static wormhole construction (a black hole with an aperture beneath its event horizon that connects to a white hole) is perfectly valid?</p>
<p><strong>Update:</strong></p>
<p>This is my attempt to answer my own question.
Energy conservation in GR actually means that the divergence of the stress-energy tensor is zero. This in turn means that any change in energy within any 4-volume is due to flows of energy through its bounding 3-surface. This permits the instantaneous appearance/disappearance of a thin shell of matter. The matter could have entered a given 4-volume through its boundaries and left the same way. For it to be spacelike shell existing at a single instant, its speed in doing so, however, would have to have been infinite. The only argument I have for infinite speed is that it isn’t forbidden by Special Relativity per se. Rather, it’s the transition from subluminal to superluminal speed that’s forbidden.</p>
<p>I found evidence that physicists are perfectly happy to consider these spacelike thin shells. Here is an example of a summary of someone’s talk at <a href="http://gravitation.web.ua.pt/sites/gravitation.web.ua.pt/files/aveiroJPSLbhwregularbhs.pdf" rel="nofollow">a conference on regular black holes</a> in December 2011:</p>
<p>“<em>Can construct regular black holes by filling the inner space with matter up to a certain surface and make a smooth junction, through a boundary surface, to
the Schwarzschild solution as was done in (Mars CQG 1996, Magli RMP
1999, Elizalde and Hildebrandt PRD 2002, Conboy and Lake PRD 2005).
The junction to Schwarzschild is made through a spacelike surface, rather
than an usual timelike surface. This means the junction exists at a single
instant of time.</em>”</p>
| 2,550 |
<p>The action for $N=1$ supergravity in an $4$ spacetime dimenions is</p>
<p>$$
S= \int e\left( R + \overline{\psi}_a \gamma^{abc} D_b \psi_c \right)
$$
Here $R$ is the scalar curvature, $e=\det(e_{a\mu})$, and $e_{a\mu}$ is the frame field. $\psi = \psi_{\mu} dx^{\mu}$ is a spinor valued one-form. The indices $a,b\ldots = 0\ldots 3 $ are internal indices that transform under the Lorentz group. The frame field $e_{a\mu}$ can be used to 'convert' spacetime indices to internal indices, and vice-versa. The gamma matrices obey $\gamma^a \gamma^b +\gamma^b \gamma^a = \eta^{ab} $, with $\eta^{ab}$ the 'internal metric'. $\gamma^{ab\ldots z} = \gamma^{[a} \gamma^b \ldots \gamma^{z]} $ denotes an antisymmetrised product of gamma matrices. The covariant derivative is</p>
<p>$$
D \psi = d \psi + \frac{1}{2} \omega_{ab} \gamma^{ab}
$$</p>
<p>My question is the following: The RS field $\psi_{\mu}$ has a spacetime index and a spinor index, yet in the above action there is no affine connection part in the covariant derivative. The contribution from the torsion-free part of the affine connection vanishes because it is symmetric in two indices which get contracted with the antisymmetrised product of gamma matrices. But that still leaves the contorsion part. In the first order formalism, the spin connection and frame field are taken to be independent variables, so in general the spin connection may have torsion. In the second order formalism, the spin connection is not torsion free due to the presence of fermions. So in either case the contorsion tensor is non-zero. This leads me to believe that not having it in the above action will mean that the action is not invariant under diffeomorphisms. </p>
<p>Secondly, because the torsion is not in general zero, it seems to me that the RS action should actually be split into two pieces</p>
<p>$$
e\left( \overline{\psi}_a \gamma^{abc} D_b \psi_c - (D_b \overline{\psi}_a) \gamma^{abc} \psi_c \right)
$$</p>
<p>This is because a complex conjugation should send each of those terms to each other, so that the action is real. However, if you only have one of those terms and the torsion is non-zero, when you complex conjugate you have to use integration by parts to 'move the covariant deriative to the other side', whereupon you pick up torsion tensor contributions from the covariant derivative acting on $e$, and the action is not real.</p>
<p>I'd be grateful if anyone could shed some light on either of these issues.</p>
| 2,551 |
<p>I understand that <a href="http://en.wikipedia.org/wiki/Rectenna" rel="nofollow">rectifying antenna</a> (rectenna) is supposed to convert electromagnetic energy to electric current however I do not understand how it's really working.</p>
<p>I do get that it's kind of like how transformer works, but beyond that I am clueless. Could someone explain this to me?</p>
| 2,552 |
<ol>
<li><p>What causes <a href="http://en.wikipedia.org/wiki/Precession" rel="nofollow">precession</a> in a spinning object?</p></li>
<li><p>What causes <a href="http://en.wikipedia.org/wiki/Nutation" rel="nofollow">nutation</a> in a spinning object? </p></li>
<li><p>What causes a top, gyroscope, and the earth to wobble? </p></li>
</ol>
<p>Just because it's a simple question, I'm not expecting a simple answer, but please do summarize whatever you say in laymen terms, thanks.</p>
| 2,553 |
<p>This question appeared quite a time ago and was inspired, of course, by all the fuss around "LHC will destroy the Earth". </p>
<p>Consider a small black hole, that is somehow got inside the Earth. Under "small" I mean small enough to not to destroy Earth instantaneously, but large enough to not to evaporate due to the <a href="http://en.wikipedia.org/wiki/Hawking_radiation">Hawking radiation</a>. I need this because I want the black hole to "consume" the Earth. I think reasonable values for the mass would be $10^{15} - 10^{20}$ kilograms. </p>
<p>Also let us suppose that the black hole is at rest relative to the Earth.</p>
<p>The question is:<br>
How can one estimate the speed at which the matter would be consumed by the black hole in this circumstances? </p>
| 2,554 |
<p>I am sending a couple of questions which seem a bit more specific than others on this site, partially to probe if there is a point in doing so. Not sure what is the range of expertise here, and no way to find out without trying, so here goes:</p>
<p>I am wondering what is known about QCD, or other field theories, in the regime of large density and low temperatures, specifically studied in the large N limit. I know of the qualitative picture at finite N, but lots of the instabilities (e.g. the superconducting ones) are suppressed at large N and replaced by other interesting phenomena. I am only aware at the moment of the DGR instability to form chiral density waves, and I am wondering what else exists in the vast and possibly quite old literature. Any pointers or entry points to that literature will be appreciated</p>
| 2,555 |
<p>It's been a couple years since I've taken a physics class, and I have been wondering about this basically since I bought my car:</p>
<p>I drive a 2010 Honda Fit, which has a street weight of <a href="http://consumerguideauto.howstuffworks.com/2010-honda-fit-13.htm" rel="nofollow">approximately 2489 pounds</a> (1129 kg). One feature of the Fit is that the back seats can sit upright like a normal car, or they can completely fold down (<a href="http://autocarmanual.com/automotive/fold-the-honda-fit-rear-magic-seat-down-to-reveal-more-space.jpg" rel="nofollow">this image</a> from <a href="http://autocarmanual.com/gallery-honda/2010-honda-fit-user-manual-overview/" rel="nofollow">this page</a> shows a folded seat next to an upright seat).</p>
<p>Since I rarely have more than one passenger in my car, I can leave the rear seats folded down almost all the time. Having the seats folded down will result in a lower center of gravity which should increase my fuel efficiency (I think!).</p>
<p>I haven't been able to find an estimate on the weight of the rear seats, other than "pretty substantial" according to <a href="http://www.fitfreak.net/forums/fit-interior-modifications/46083-back-seat-delete.html#post709667" rel="nofollow">this forum post</a>. But I'm not sure how much of a difference it will make overall, anyway.</p>
<p>The first part of my question is, assuming that raising or lowering the center of gravity affects acceleration and deceleration, how does one quantify that?</p>
<p>Then, given that I may be able to move, say, 100 pounds of seat from the "dead center" of my vehicle to, say, 1-2 feet closer to the wheels, how can I estimate how that will affect my fuel economy? Perhaps a more specific question would be, how can I estimate how that will affect my acceleration? (which is of course when I get the worst fuel economy, besides being stopped)</p>
| 2,556 |
<p>I am sending a couple of questions which seem a bit more specific than others on this site, partially to probe if there is a point in doing so. Not sure what is the range of expertise here, and no way to find out without trying. This one is also not terribly focused, but nonetheless here goes:</p>
<p>I am wondering if there are some well-known and well-studied examples of large N matrix models (in which the fields are adjoint rather than vectors) which are of use in describing some condensed matter phenomena. </p>
<p>There are lots of applications of matrix models in anything between nuclear physics to number theory, and there are well-known vector models which are useful in CM physics, but off the top of my head I cannot think about matrix models which are used to solve some condensed matter problems. Quite possibly I am missing something obvious...References or brief descriptions will be appreciated.</p>
| 2,557 |
<p><a href="http://en.wikipedia.org/wiki/Cooper_pair" rel="nofollow">Cooper pairs</a> are one of the models how <a href="http://en.wikipedia.org/wiki/Superconductivity" rel="nofollow">superconductivity</a> is explained.</p>
<p>What still baffles me is how a vibration of the crystal lattice (the so-called <a href="http://en.wikipedia.org/wiki/Phonon" rel="nofollow">phonon</a>) can interact with the electron (an actual particle), in such a way that it then creates a coupled pair with an other electron...</p>
<p>What is the explanation for this behaviour? What is the maths behind it?</p>
| 2,558 |
<p>I was revising the harmonic oscillator for my intro to quantum course and realised I'd sort of accepted a change of variable result without actually being able to get to it. It says:</p>
<p>The stationary state Schrodinger equation of energy $E$ is</p>
<p>$$-\frac{\hbar^{2}}{2m}\frac{d^{2}\psi}{dx^{2}}+\frac{1}{2}m\omega^{2}x^{2}\psi=E\psi\tag{5.2}$$</p>
<p>The first thing to do is to redefine variables so as to remove the various physical constants:</p>
<p>$$\epsilon=\frac{2E}{\hbar\omega}, \xi=\sqrt{\frac{m\omega}{\hbar}}x$$</p>
<p>so that (5.2) becomes</p>
<p>$$-\frac{d^{2}\chi}{d\xi^{2}}+\xi^{2}\chi=\epsilon\chi$$ where $$\psi(x)=\chi(\xi)=\chi(\sqrt{\frac{m\omega}{\hbar}}x).$$</p>
<p>So, I've tried working with the algebra but can't seem to get to this. I'm probably missing something really obvious, but it's getting quite frustrating! Can anyone help?</p>
| 2,559 |
<p>I would like to see what type of noise I would get if I used just the frequencies in my voice. I created a matlab array using <code>fft</code> to get a <code>[frequency,amplitude,phase]</code> array of my voice. I then to reproduced my vocal signal using $A*\cos(ft+\phi)$</p>
<p>I would like to take this file/data and use it to create pink noise $1/f$. When I use $1/f$ for the frequency the numbers become very small. Does anyone have any ideas how to use my own vocal frequencies that I get from doing a fft in matlab to create $1/f$ pink noise?</p>
| 2,560 |
<p>I know that the direction of torque is along the axis of rotation, but would it be acceptable to say, for example considering a vertical thin rod in the x-y plane with a force acting on the bottom end towards the left, 'it is a clockwise torque about the centre of mass' (even though strictly speaking the torque's direction is into the page?)</p>
<p>It just seems to me easier to describe what's happening that way.</p>
| 2,561 |
<p>Does anyone know of websites or texts that have an abundance of <a href="http://www.google.com/search?as_q=Wick%27s+theorem+examples" rel="nofollow">examples</a> of computing time-ordered products of fields using <a href="http://en.wikipedia.org/wiki/Wick%27s_theorem" rel="nofollow">Wick's Theorem</a> for both bosons and fermions?</p>
<p>I'm not just talking about the simple product of 4 fields, something such as </p>
<p>$T(\phi^4(x) \phi^4(y))$ </p>
<p>$T(:\phi^4(x): :\phi^4(y):)$</p>
<p>$T(\bar{\psi}(x)\psi(x)\bar{\psi}(y)\psi(y))$.</p>
<p>These are actually the only examples I have and they're found in "Problem Book in Quantum Field Theory" by Voja Radovanovic.</p>
| 2,562 |
<p>I have been thinking is it possible that using a rather strong magnet on the human body's blood stream we could enact a small case of artificial gravity on the human body via the iron in the hemoglobin in the blood stream. </p>
<p>There are 4 Problems currently with this theory:</p>
<ol>
<li>The could be residual effect on the human body </li>
<li>No one has ever tried to affect the blood stream directly using a powerful magnet</li>
<li>Can we create an electromagnet or a non normal magnet to cause slight magnetization but only of the feet and legs to ensure no abnormal strain is placed on the heart</li>
<li>If the iron in the blood is magnetized so will every other magnetic metal item on board such as tools or weapons</li>
</ol>
| 2,563 |
<p>I have specific questions about Lorentz transformations specifically about length contraction.</p>
<ol>
<li><p>Why does length contraction only occur in the direction of travel, (not in all directions) when approaching the speed of light?</p></li>
<li><p>From an outside observer's perspective watching an object traveling close to the speed of light, why does the obsever notice the object contract?</p></li>
</ol>
<p>These Ideas are accepted by physicists, but for 1 I don't understand how they know that length contraction is in only one direction without they themselves being in the reference frame of someone or something traveling close to the speed of light.</p>
<p>For 2 Let's say that I'm driving in a car 5 meters long and I'm driving on a road that has white markers on it. Where the markers are in one line and each marker is 5 meters apart from the next one and the line of markers is parallel to my direction of travel. At normal speeds I would observe that at a given moment one marker would line up with back of the car and the next marker would line up with the front of the car (the car occupies 1 marker space). So now if I were to travel at speed such that my Lorentz factor is 2. I see one of two cases. In the first case I don't observe any change in myself or my car and I observe a length contraction of 2. Meaning that I would see a change in my environment not in myself or my car. So I would now observe that at a given moment the car would take up the length of 2 marker spaces. Therefore an outside observer would then see the car <strong>expanded</strong> by a factor of 2 not contracted. In the second case I do notice a change in myself. I contract with the space so things look very wierd in my perspective, Everything has contracted by a factor of 2 in the direction of travel. So my arms, my legs, the carseat, the hood of the car, etc. has all shrunk by a factor of 2. However because the road has shrunk by the same amount as the car, I take up only one marker space. Therefore an outside observer would not see any change in the length of me or the car. So in both cases the car doesn't contract from an outside observer's perspective.</p>
| 2,564 |
<p>When you have a drink with an ice cube and twirl the glass, the liquid itself seems to twirl but the ice cube stays roughly in the same place. Why is this?</p>
| 2,565 |
<p>What happens with very small spherical objects ($d=1\mu m$, e.g. a bacterium) in air? Do they fall? How quickly? Does it depend on their mass?</p>
<p>We often see objects of little mass e.g. leaves falling from trees, but these object ususally have a very large surface, so their behaviour is very different to spherical objects.</p>
<p><em>Disclaimer: I'm a biologist.</em></p>
| 2,566 |
<p>I have a question with conformal field theory in Polchinski's string theory vol 1 p. 51.</p>
<p>For $bc$ conformal field theory
$$ S=\frac{1}{2\pi} \int d^2 z b \bar{\partial} c $$
$$ T(z)= :(\partial b) c: - \lambda \partial (: bc:) $$
with central charge $c=-3 (2 \lambda-1)^2 +1 =1$. Introducing $\psi$ and $\bar{\psi}$ to replace the anticommuting fields $b$ and $c$ as following
$$b \rightarrow \psi =2^{-1/2} (\psi_1 + i \psi_2 )$$ and $$ c \rightarrow \bar{\psi} =2^{-1/2} (\psi_1 - i \psi_2)$$
It is claimed that
$$S=\frac{1}{4\pi} \int d^2 z \psi_1 \bar{\partial} \psi_1 + \psi_2\bar{\partial} \psi_2 (2.5.18b)$$
$$ T=- \frac{1}{2} \psi_1 \partial \psi_1 -\frac{1}{2} \psi_2 \partial \psi_2 (2.5.18c)$$</p>
<p>I cannot obtain the above expressions of $S$ and $T$. Here is my derivations. First I try to recover the anti-commuting characters of fields $b$ and $c$ by $\psi_1$ and $\psi_2$.
For $$bc+cb=0$$ I have $$\psi_1 \psi_1 + \psi_2 \psi_2 =0$$
Then for the action
$$S=\frac{1}{4\pi} \int d^2 z \left( \psi_1 \bar{\partial} \psi_1 + i \psi_2 \bar{\partial} \psi_1 -i \psi_1 \bar{\partial} \psi_2 + \psi_2 \bar{\partial} \psi_2 \right)$$
(1) [Solved] Why the term $i \psi_2 \bar{\partial} \psi_1 -i \psi_1 \bar{\partial} \psi_2$ does not contribute to the action? </p>
<p>(2) How to derive (2.5.18c)? </p>
| 2,567 |
<p><img src="http://i.stack.imgur.com/BB7m9.jpg" alt=""></p>
<p>Hello everyone.</p>
<p>Imagine an object moving around a certain point on a circular orbit. Magnitude of the velocity is constant during the motion ($|v|$). The orbit radius is $r$. (I'd better notice that we're just talking about <strong>kinematic</strong> view of this motion.)</p>
<p>According to the image I've uploaded, we'll have:</p>
<p>$\large v_x(\theta)=|v|\cdot \cos\theta$</p>
<p>$\large v_y(\theta)=|v|\cdot \sin\theta$</p>
<p>Since perimeter of the circular path is $2\pi r$, and magnitude of the velocity is constant, we'll have:</p>
<p>$\large\theta (t)=\frac{|v|\cdot t}{2\pi r} \times 2\pi =\frac{|v|\cdot t}{ r}$</p>
<p>Now we can combine these equations:</p>
<p>$\large v_x(\theta)=|v|\cdot \cos(\frac{|v|\cdot t}{ r})$</p>
<p>$\large v_y(\theta)=|v|\cdot \sin(\frac{|v|\cdot t}{ r})$</p>
<p>By this point, everything is okay. But the problem occurs here, where we try to get derivative of $v_x(t)$ and $v_y(t)$ in order to find $a_x(t)$ and $a_y(t)$. As we know by differentiation we have:</p>
<p>$\cos^{\prime}(x)=-\sin(x)$</p>
<p>$\sin^{\prime}(x)=\cos(x)$</p>
<p>And we know that acceleration(time) function is derivative of velocity(time). So:</p>
<p>$\large a_x(t)=(v_x(\theta))'=|v|\cdot -\sin(\frac{|v|\cdot t}{ r})$</p>
<p>$\large a_y(t)=(v_y(\theta))'=|v|\cdot \cos(\frac{|v|\cdot t}{ r})$</p>
<p>Well, now something is wrong: These two equations result a $m/s$ unit (or something like that) for acceleration, but that's wrong. Acceleration unit must be $m/s^2$, (or something like that).</p>
<p>The question is that: <strong>Where does this problem come from?</strong> I couldn't figure it out at all. I don't know, maybe some kind of misunderstanding about derivative concepts cause that. So please try to answer simple, clean as much as possible.</p>
| 2,568 |
<p>Long before statistical mechanics, entropy was introduced as:</p>
<p>$dS = \frac{dQ}{T}$</p>
<p>At the time when entropy was introduced in this manner, was it known that entropy represents how "disordered" a system is? If so, how can one tell that entropy represents the disorder of a system from the above definition?</p>
<p>If however it wasn't known that $S$ was related to the disorder of a system, what was the interpretation of $S$ at the time of formulation of the above formula?</p>
| 2,569 |
<p>This is based off question 4.30 from Griffith's <em>Introduction to Quanum Mechanics</em>. It asks for the matrix $\textbf{S}_r$ representing the component of spin angular momentum about an axis defined by: $$r = \sin{\theta}\cos{\phi}\hat{i}+\sin\theta\sin\phi\hat{j}+\cos\theta\hat{k}$$</p>
<p>for a spin = $1/2$ particle.The problem is, I can't visualize how the spin vectors relate to spatial coordinates. $$\chi_+ = \begin{pmatrix} 1 \\ 0\end{pmatrix}\ \text{ and } \chi_- = \begin{pmatrix} 0 \\ 1 \end{pmatrix}. $$</p>
<p>So these form a basis of some kind of space, but this is a space I don't understand, for example, why does:
$$\chi^{(x)}_+ = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\1 \end{pmatrix} ?$$</p>
<p>I was going to try to use a rotation matrix with angles corresponding to my axis of rotation to try and "rotate" the spin vector onto it, but then I realized that not only do the dimensions not match up, but the angle is π/4, which doesn't make much sense to me. So I guess my question is, how does the 'geometry' of spin work and how can I transform the spins corresponding to a transform I do in space?</p>
<p>Thanks.</p>
| 2,570 |
<p>The 2D vector field (x,-y) does not transform like a vector under rotation(Arfken Vol. 1)! Does this mean we cannot have such a vector field physically?</p>
| 2,571 |
<p>The EEP is used to justify that if an observer on the ground shoots a beam of light towards a tower, then when the light reaches the tower, it will be red shifted. This is because of what happens in an accelerating spaceship.</p>
<p>The books seem to say this implies time dilation, but I don't completely see why. Could it not just be like any other doppler effect? If I send a sound wave towards you and you are moving towards me, the frequency you observe will be greater than what I send out, but that doesn't mean your clock ticks slower (assume his speed is non-relativistic). Why does it necessarily imply time dilation?</p>
<p>In the derivation used to derive this redshift, from the light beam on a spaceship, special relativity doesn't even seem to come into play, just newtonian physics is used. Anyway if the physics on the ground is indistinguishable from the physics in the spaceship, would that not mean that at the back of the spaceship a clock ticks slower, rather than just appearing to click slower, relative to someone at the front? Otherwise it would seem to me that the EEP would just say that we can't do any experiment to distinguish the physics between the rocket and the gravitational field, though the physics wouldn't really seem to be the same.</p>
| 2,572 |
<p>When two wavefunctions are orthogonal we can write that</p>
<p>$$\langle\Psi_n|\Psi_m\rangle=\delta_{mn}$$</p>
<p>This means that </p>
<p>$$\langle\Psi_1|\Psi_2\rangle=\langle\Psi_2|\Psi_1\rangle=0$$</p>
<p>But if the two wavefunctions aren't orthogonal but $\langle\Psi_1|\Psi_2\rangle=d \in\mathbb{R}$ is real, can we then conclude that $\langle\Psi_2|\Psi_1\rangle=\langle\Psi_1|\Psi_2\rangle$?</p>
| 2,573 |
<p>Why do some conductors follow <a href="http://en.wikipedia.org/wiki/Ohm%27s_law" rel="nofollow">Ohm's law</a> and some do not? Isn't there any universal law that can explain the flow of current?</p>
| 2,574 |
<p>Wikipedia defines the <a href="http://en.wikipedia.org/wiki/Willmore_energy" rel="nofollow">Willmore energy</a> as: </p>
<p>$$e[{\mathcal{M}}]=\frac{1}{2} \int_{\mathcal{M}} H^2\, \mathrm{d}A,$$</p>
<p>where $H$ stands for the mean curvature of the manifold $\mathcal{M}$.</p>
<p>What is the Willmore energy of the Earth, or the geoid?</p>
| 2,575 |
<blockquote>
<p>A ball moving with velocity $1 \hat i \ ms^{-1}$ and collides with a friction less wall, afetr collision the velocity of ball becomes $1/2 \hat j \ ms^{-1}$. Find the coefficient of restitution between wall and ball.</p>
</blockquote>
<p>I approached it like: </p>
<p><img src="http://i.stack.imgur.com/ANFIk.png" alt="enter preformatted text here"></p>
<p>Now, $$e=cot^2\theta$$ but $\theta$ is not known. So,We equate velocity $$\sqrt{e^2 sin^2\theta+cos^2 \theta}=1/2$$
But this is hard to solve as $\cot^4 \theta$ get's involved. Is there any other method to do this or any easy method to solve these?</p>
| 2,576 |
<p>Calculate the potential and the intensity of the gravitational field at a distance $x> 0$ in the axis of thin homogeneous circular plate of radius $a$ and mass $M$.</p>
<p><img src="http://i.stack.imgur.com/nSXTf.png" alt="enter image description here"></p>
<p>Could anybody describe how to calculate this? Slowly and in detail. I'm helpless.</p>
<p>Answer is: potencial $\phi = - \frac {2 \kappa M}{a^2}(\sqrt{a^2+x^2}-x)$
and intesity $K = \frac {2 \kappa M}{a^2} \left( \frac{x}{\sqrt{a^2+x^2}}-1 \right)$</p>
| 313 |
<p>In the derivation of the Poisson-Boltzmann equation, my <a href="http://rads.stackoverflow.com/amzn/click/0716787598" rel="nofollow">textbook</a> arrives at the expression</p>
<p>$$
\rho_i = c_+ z_+ F + c_-z_-F = c_+^oz_+^oFe^{-z_+e\phi_i/kT} + c_-^oz_-Fe^{-z_-e\phi_i/kT}
$$</p>
<p>where $c_\pm$ is the concentration (of positive or negative ions), $z_\pm$ is the valence of the ions, $F$ is the Faraday constant ($=eN_A$) and $c_\pm^o$ is the concentration in the bulk. The rest is standard notation.</p>
<p>The book now writes that</p>
<blockquote>
<p>Because the average electrostatic interaction energy is small compared with $kT$, we may write</p>
</blockquote>
<p>$$
\rho_i = (c_+^o z_+ + c_-^oz_-)F - (c_+^oz_+^2 + c_-^oz_-^2)\frac{F^2\phi_i}{kN_AT} + \cdots
$$</p>
<p><strong>How do they know that $ze\phi \ll kT$?</strong></p>
<p>To make an estimate, I find the electric potential of a proton (or an $\text{Na}^+$ ion) in vacuum at 1 nm away.</p>
<p>$$
V^{vac} = \frac{1}{4\pi\epsilon_0}\frac{+1e}{1nm} \approx 1.43 \text{V}.
$$</p>
<p>In water ($\epsilon = \epsilon_0\epsilon_r = \epsilon_078$),</p>
<p>$$
V^{sol} = \frac{1}{4\pi78}\frac{+1e}{1nm} \approx 0.02 \text{V}.
$$</p>
<p>So in solution a $\text{Cl}^-$ ion would have the energy</p>
<p>$$
w=(-1e)V^{sol} \approx -3.2 \cdot 10^{-21}J
$$</p>
<p>while at room temperature $kT \approx 3.9 \cdot 10^{-21}J$.</p>
<p>Is this roughly the idea behind the above assumption?</p>
| 2,577 |
<p>I have came across this equation for <a href="http://en.wikipedia.org/wiki/Quantum_harmonic_oscillator" rel="nofollow">quantum harmonic oscillator</a></p>
<p>$$
W \psi = - \frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} + \frac{1}{2} m \omega^2 x^2 \psi
$$</p>
<p>which is often remodelled by defining a new variable $\varepsilon = \sqrt{m\omega/\hbar}\,x$. If i plug this in the equation above I know how to derive this equation (it is easy all i needed was a definition of differential): </p>
<p>$$
\frac{d^2 \psi}{d x^2} + \left(\frac{W}{\left( \tfrac{\hbar \omega}{2} \right)} - \varepsilon^2 \right)\, \psi = 0
$$</p>
<p>From this equation most of authors derive the energy equation:</p>
<p>$$W = \left(\tfrac{\hbar \omega}{2}\right)\cdot (\underbrace{2n+1}_{odd???})$$</p>
<p><strong>QUESTION 1:</strong> I don't quite understand how this is done. Could anyone please explain this? I dont know why energy has to be odd function nor how this comes into play.</p>
<p><strong>QUESTION 2:</strong> I am only guessing that if I would plug this in an equation above I would get I an equation below from which I could calculate possible $\psi$ functions for harmonic oscilator (please confirm).: </p>
<p>$$
\frac{d^2 \psi}{d x^2} + \left( (2n +1)- \varepsilon^2 \right)\, \psi = 0
$$</p>
<p><strong>QUESTION 3:</strong> Where do Hermitean polinoms (which are often mentioned) come into play?</p>
| 2,578 |
<p>Ok, let me make myself clear. I saw all the other questions related to the question, but none of them actually asks the question the way I would put it and therefore no one answers it the way I want it answered, so here it is, I'll try to formulate it.</p>
<p>I perfectly understand that every single point in the Universe may be taken as the "location of the Big Bang", as it happened, because everything was in the very same place - the singularity (zero volume and infinite mass), in terms of 3-dimensional space.</p>
<p>As far as I understand, once all this energy was released and the Big Bang occurred, it created and started expanding what we now perceive as the 3-dimensional space of our Universe.
And as far as I understand it was expanding (and still is) equally in all directions. In <strong><a href="http://en.wikipedia.org/wiki/Metric_expansion_of_space" rel="nofollow">Wikipedia</a></strong> the model given is that of a bread muffin, so that we see that all reference points expand equally, given a relative center, no matter where you are in the Universe. However, this is not true if you are close to the outer surface of the muffin, and the muffin has finite size and has an edge, where the Universe ends. Given the example with the muffin, with finite Universe, there is a single point that never formed a vector of movement, ever since the Big Bang took place. And this is the center of the muffin. <strong>That would be the center of the Universe, given the possibility that it is finite.</strong></p>
<p>However, if the Universe has no edge and no boundaries and is infinite in terms of 3-dimensional space, there would be no center in that space and I would be perfectly happy :)
But if it is finite, as I described above, it would have a center. Or am I wrong?</p>
<p>Please do not mark my question as duplicate, since no one puts it that way in the other questions, I've made my research :)</p>
| 2,579 |
<p>If two black holes collide and then evaporate, do they leave behind two naked sigularities ore? If there are two, can we know how they interact?</p>
| 2,580 |
<p>How to calculate number of exchanged virtual pohotons per unit of time between two electromagnetically interacting objects?</p>
| 2,581 |
<p>This <a href="http://arxiv.org/abs/1302.6165" rel="nofollow">paper</a> proposes a microscopic mechanism for generating the values of $c, \epsilon_0, \mu_0$. They state that their vacuum is assumed to contain ephemeral (meaning existing within the limits of the HUP) fermion/antifermion pairs. This affects the mechanism of photon propagation as follows:</p>
<blockquote>
<p>When a real photon propagates in vacuum, it interacts
with and is temporarily captured by an ephemeral pair.
As soon as the pair disappears, it releases the photon to
its initial energy and momentum state. The photon continues
to propagate with an infinite bare velocity. Then the
photon interacts again with another ephemeral pair and
so on. The delay on the photon propagation produced by
these successive interactions implies a renormalisation of
this bare velocity to a finite value</p>
</blockquote>
<p>Now this description of what happens to the photon sounds awfully like a heuristic description you might give to the photon self energy contribution in standard QED perturbation theory, where, although it's used in the renormalization procedure, it most certainly doesn't make any changes to the velocity of light.</p>
<p>Can someone explain how what's being proposed here relates to standard QED. I don't see how they fit together.</p>
| 2,582 |
<p>VEVs of Wilson loops in Chern-Simons theory with compact gauge groups give us colored Jones, HOMFLY and Kauffman polynomials. I have not seen the computation for Wilson loops in Chern-Simons theory with non-compact gauge groups. I think that what keep us from computing them is due to infinite dimensional representations of non-compact gauge groups. </p>
<p>Are there any references which describe this problem explicitly? What are the issues in computing Wilson loops in non-compact gauge groups? Is there any proposal to calculate them? Especially I am interested in the simplest cases of $SL(2,R)$ and $SL(2,C)$.</p>
| 2,583 |
<p>How do I derive the General Linear Wave Equation $$d^2y/dx^2=(1/v^2)d^2y/dt^2?$$ </p>
<p>My teacher differentiated the general wave function $f(x + vt)+g(x - vt)$ twice with respect to both variables to get $d^2y/dx^2$ and $v^2d^2y/dt^2$, and then somehow combined them to get $$d^2y/dx^2=(1/v^2)d^2y/dt^2,$$ but I don't remember how he did it.</p>
| 2,584 |
<p>The RPP note on quarks masses has traditionally carried, and it is still there, the comment that</p>
<blockquote>
<p>It is particularly important to determine the quark mass ratio mu/md,
since there is no strong CP problem if $m_u$ =
0.</p>
</blockquote>
<p>But in <a href="http://pdg.lbl.gov/2011/reviews/rpp2011-rev-quark-masses.pdf">recent versions</a> they have also added some remarks about how the calculations from the MILC and RBC collaborations show that this mass is no zero.</p>
<p>So, does it still survive some argument for $m_u=0$? And, by the way, does the strong CP problem request exactly 0, or is it enough if it is approximately zero?</p>
| 2,585 |
<p>I guess I never had a proper physical intuition on, for example, the "<a href="http://en.wikipedia.org/wiki/KMS_state" rel="nofollow">KMS condition</a>". I have an undergraduate student who studies calculation of Hawking temperature using the Euclidean path integral technique, and shamefully his teacher is not able to give him a simple, intuitive argument for it. What would it be?</p>
<p>Added on October 21:</p>
<p>First of all, thanks Moshe and S Huntman for answers. My question was, however, looking for more "intuitive" answer. As Moshe pointed out it may not be possible, since after all time is "imaginary" in this case. So, let me be more specific, risking my reputation.</p>
<p>I should first say I understand there are formal relation between QFT and statistical mechanics as in well-known review like "<a href="http://www.amsta.leeds.ac.uk/~siru/papers/p26.pdf" rel="nofollow">Fulling & Ruijsenaars</a>". But, when you try to explain this to students with less formal knowledge, it sometimes helps if we have an explicit examples. My motivation originally comes from "<a href="http://arxiv.org/abs/gr-qc/9812028" rel="nofollow">Srinivasan & Padmanabhan</a>". In there, they says tunneling probability calculation using complex path (which is essentially a calculation of semi-classical kernel of propagator) can give a temperature interpretation because "In a system with a temperature $\beta^{-1}$ the absorption and emission probabilities are related by</p>
<p>P[emission] = $\exp(-\beta E)$P[absorption]. (2.22) "</p>
<p>So, I was wondering whether there is a nice simple example that shows semi-classical Kernel really represents temperature. I think I probably envisioned something like two state atom in photon field in thermal equilibrium, then somehow calculate in-in Kernel from |1> to |1> with real time and somehow tie this into distribution of photon that depends on temperature. I guess what I look for was a simple example to a question by Feynman $ Hibbs (p 296) pondering about the possibility of deriving partition function of statistical mechanics from real time path integral formalism. </p>
| 2,586 |
<p>Suppose we try to apply supersymmetry in quantum mechanics to a particular potential. If you come up with two partner potentials, and two partner Hamiltonians, and then look at the energy of the ground state, is it true that at most one of them can be zero?</p>
<p>If they are both zero, does that mean that supersymmetry is just not preserved by the vacuum states of the partner Hamiltonians? Can we still apply supersymmetry to this system if the energy of both ground states are zero?</p>
| 2,587 |
<p>It seems most books about QM only talk about position and momentum operators. But isn't it also possible to define a acceleration operator? </p>
<p>I thought about doing it in the following way, starting from the definition of the momentum operator:</p>
<p>$\hat{p} = -i\hbar \frac{\partial }{\partial x}$</p>
<p>Then we define a velocity operator in analogy to classical mechanics by dividing momentum by the mass $m$</p>
<p>$\hat{v} = \frac{-i\hbar}{m} \frac{\partial }{\partial x}$</p>
<p>In classical mechanics acceleration is defined as the time derivative of the velocity, so my guess for an acceleration operator in QM would be</p>
<p>$\hat{a} = \frac{-i\hbar}{m} \frac{\partial }{\partial t} \frac{\partial }{\partial x}$</p>
<p>Is that the general correct definition of the acceleration operator in QM? How about relativistic quantum mechanics?</p>
| 2,588 |
<blockquote>
<p>A particle travels with speed $50 m/s$ from the point $(3,-7)$ in the
direction $7i-24j$ . Find its positional vector after 3 seconds.</p>
</blockquote>
<p><strong>My approach</strong>:
It has travelled a distance of 150m in the direction given by the unit vector $\frac{7\hat i - 24 \hat j}{25} $
.</p>
<p>So, its position is now $210\hat i -720 \hat j$. But it started from $(3, - 7)$, so I have to subtract that to get $197\hat i - 713\hat j$. But I think this answer is awkward, and I may have gone wrong. So, please tell me if I am right, and if yes, can you suggest any shorter way of doing it?</p>
| 2,589 |
<p>When placing ice cubes in a fizzy drink such as Prosecco, ice makes a cracking sound, after which the fizzy bubbles more than usual. What is the physics of this phenomenon?</p>
<p><img src="http://i.stack.imgur.com/hKlxy.jpg" alt="enter image description here"></p>
| 2,590 |
<p>Consider 3 particles. All 3 particles travel along the x-axis.</p>
<ul>
<li>The 1st particle possesses some mass, m, and its initial position is somewhere on the negative x-axis. It has some (positive) velocity v.</li>
<li>The 2nd particle possesses some mass, A*m, and its initial position is at the origin of the x-axis (0). It has no velocity (at rest).</li>
<li>The 3rd particle possesses some mass, B*m, and is initially situated somewhere on the positive x-axis. Like particle 2 it has no velocity (at rest).</li>
</ul>
<p>Find the relation between A and B, under which the 1st and 2nd particles will collide with each other more than once.</p>
<p>note: it is assumed the collisions are elastic as well as there being no external forces considered. also the collisions are such that all particles remain on the x-axis</p>
| 2,591 |
<p>So I gather the way you (and Vera Rubin) calculate a galaxy's mass is by measuring a star's orbital velocity $v$ and its distance $R$ from the galactic center, and then plugging them into this equation derived from Newton's second law:</p>
<p>$$M_{gal}=Rv^2/G$$</p>
<p>($G=6.67\times10^{-11}$. Units of $v$ and $R$ are km/sec. and km., respectively)</p>
<p>The value obtained for $M_{gal}$ this way famously disagrees with the value you would obtain by measuring the brightness of the galaxy, leading to the dark matter theory.</p>
<p>I was playing around with some <a href="http://hyperphysics.phy-astr.gsu.edu/hbase/solar/soldata.html#c1">solar system data</a> and found that if I calculate the mass of the sun by plugging this data into the equation above the result is too low by 9 orders of magnitude. $\sim 1.98 \times 10^{21}$ kgs. instead of the actual solar mass (according to wikipedia) of $1.98 \times 10^{30}$ kgs.</p>
<p>Is there some "dark vacuum" in the solar system? Or where have I gone wrong in the calculation?</p>
<p>Are there any datasets of $v$ and $R$ for stars in a particular galaxy out there that can be downloaded? I've heard that thousands of galaxies have now been observed to have stars "going too fast." Has any of that data been made available?</p>
| 2,592 |
<p>Noether's theorem says that if the following transformation is a symmetry of the Lagrangian</p>
<p>$t \to t + \epsilon T$</p>
<p>$q \to q + \epsilon Q$</p>
<p>Then the following quantity is conserved</p>
<p>$\left( \frac{\partial L}{\partial \dot{q}}\dot{q} - L \right) T - \frac{\partial L}{\partial \dot{q}} Q$.</p>
<p>Suppose our Lagrangian is given by</p>
<p>$ L = \frac{1}{2}m \dot{q}^2 - \ln t$</p>
<p>Then is not the Lagrangian invariant under the transformation given by </p>
<p>$T = t$</p>
<p>$Q = 0$?</p>
<p>Making this transformation contributes only an additive constant to the Lagrangian, which will not affect the dynamics, and so we should conclude that such a transformation is indeed a symmetry of the Lagrangian. However, the quantity</p>
<p>$ \left( \frac{\partial L}{\partial \dot{q}}\dot{q} - L \right) t = \left(\frac{1}{2}m\dot{q}^2 + \ln t \right)t$</p>
<p>is clearly not conserved. The E-L equations imply that the kinetic energy is constant, and so this function is clearly an increasing function of time.</p>
<p>Where is my error?</p>
<p>Thank you.</p>
| 2,593 |
<p>I am trying to obtain the polchinski's equation 4.3.16
which is following</p>
<p>$ Q_B^2 = \frac{1}{2}\{ Q_B, Q_B \} = -\frac{1}{2}g^{K}_{IJ}g^{M}_{KL}c^Ic^Jc^L b_M =0$</p>
<p>Where
$ Q_B = C^I(G_I^m +\frac{1}{2}G_I^g)$
and $C^I$, $b^J$ are anticommuting(ghosts)</p>
<p>and
$[G_I, G_J]=ig^K_{IJ} G_K$,<br>
$G_I^g = -ig^K_{IJ} C^Jb_K $ are ghost parts and $G_I^m$ are matter part and they satisfy above commutation relations</p>
<p>What I have done are</p>
<p>$\{ Q_B, Q_B \} = \{ C^I(G_I^m +\frac{1}{2}G_I^g), C^J(G_J^m +\frac{1}{2}G_J^g)\}
=\{C^IG_I^m, C^J G_J^m\} +\frac{1}{2} \{C^IG_I^m, C^JG_J^g\}
+\frac{1}{2} \{C^I G_I^g, C^JG_J^m\} +\frac{1}{4} \{C^IG_I^g, C^JG_J^g \}
= C^IC^J [G_I^m, G_J^m ] +\frac{1}{2} C^IC^J [G_I^m, G_J^g] +\frac{1}{2}C^IC^J[G_I^g,G_J^m]+\frac{1}{4}C^IC^J[G_I^g, G_J^g] =C^IC^J [G_I^m, G_J^m ]+\frac{1}{4}C^IC^J[G_I^g, G_J^g] = C^IC^J ig^K_{IJ}G_K^m+\frac{1}{4}C^IC^J ig^{K}_{IJ}G_K^g
=C^IC^J ig^K_{IJ}G_K^m+\frac{1}{4}C^IC^J g_{IJ}^K g^M_{KL}C^Lb_M
$</p>
<p>compare with the textbook
$\{ Q_B, Q_B \} = -g^{K}_{IJ}g^{M}_{KL}c^Ic^Jc^L b_M$</p>
<p>My calculation is something wrong. How can I fix it?</p>
| 2,594 |
<p>I have a series of number theoretic phenomena (mathematics) that I can describe exactly by the superpositions or linear combination of the below function (I know it is an inverse Fourier type). Does anybody knows from parallels in physics (quantum, particle, waves) where exactly this function is applied? Are there perhaps other notations in physics of this function. The question is very important for me trying to investigate this equation.</p>
<p>$$G(x)=\sum_{k=-j}^j \frac{2 \pi}{j}\,e^{(i\,2\pi\,(k-1)\,x / j)}$$</p>
<p>while $i=\sqrt{-1}$ and angular frequency:</p>
<p>$$\omega_j=\frac{2 \pi}{j}$$</p>
<p>PS: interestingly if $j\rightarrow \infty$ we get a ${\rm sinc}$ function that if squared then corresponding to pair-correlation function for the eigenvalues of a random Hermitian matrix a model of the energy levels in heavy nuclei.</p>
<p>Note: of course taking the first and second derivative of the function $\frac{2 \pi}{j}\,e^{(i\,2\pi\,(k-1)\,x / j)}$ we can construct easily the differential equation of a harmonic oscillator.</p>
| 2,595 |
<p>My friend and I had a little discussion about <a href="http://en.wikipedia.org/wiki/Added_mass" rel="nofollow">added mass forces</a>. </p>
<p>I always interpreted $F=ma$ as a cause-effect relationship, so I find rather uneasy to accept that the cause can instantaneously depend on the effect.</p>
<p>Is it fine to have a force which depends on an acceleration, in classical mechanics?</p>
<p>I came up with some possible solutions to this:</p>
<ol>
<li>It's perfectly fine for <strong>F</strong>=m <strong>a</strong> to be an implicit equation with respect to <strong>a</strong>.</li>
<li>The time derivative of the velocity appears as the result of an approximation of a time-delay.</li>
<li>It arises due to assumptions made on the nature of the fluid (i.e. incompressible).</li>
<li>None of them</li>
</ol>
| 2,596 |
<p>I'm confused about the Kronecker delta. In the context of four-dimensional spacetime, multiplying the metric tensor by its inverse, I've seen (where the upstairs and downstairs indices are the same):</p>
<p>$$g^{\mu\nu}g_{\mu\nu}=\delta_{\nu}^{\nu}=\delta_{0}^{0}+\delta_{1}^{1}+\delta_{2}^{2}+\delta_{3}^{3}=1+1+1+1=4.$$
But I've also seen (where the upstairs and downstairs indices are different):</p>
<p>$$g^{\mu\nu}g_{\nu\lambda}=\delta_{\lambda}^{\mu}=\left(\begin{array}{cccc}
1 & 0 & 0 & 0\\
0 & 1 & 0 & 0\\
0 & 0 & 1 & 0\\
0 & 0 & 0 & 1
\end{array}\right).$$</p>
<p>How can there be two different answers to (what appears to me to be) the same operation, ie multiplying the metric tensor by its inverse? Apologies if I've got this completely wrong.</p>
| 2,597 |
<p>Triangle defined by points <strong>OA</strong>, <strong>OB</strong> and <strong>OC</strong> : (-<strong>i</strong>,3 <strong>j</strong>,-4 <strong>k</strong>), (<strong>i</strong>,2 <strong>j</strong>,2 <strong>k</strong>) and (3 <strong>i</strong>,7 <strong>j</strong>,- <strong>k</strong>) where <strong>i</strong>, <strong>j</strong>, <strong>k</strong> are unit vectors along x,y,z axis. Point mass is placed at <strong>OA</strong>. Triangle rotates with angular velocity <strong>ω</strong> about <strong>BC</strong> axis. Calculate linear velocity of mass as it passes <strong>OA</strong>.</p>
<p>My Work:
$$v=\omega \times r $$
Does <strong>r</strong> have to be the vector perpendicular to <strong>BC</strong> to the point <strong>OA</strong> or can it be any vector from the line <strong>BC</strong> to <strong>OA</strong>??? If so can you explain? I know the answer as my tutor told me I got it right but I didn't need to find the vector perpendicular to <strong>BC</strong> and I forgot to ask why...</p>
| 2,598 |
<p>While looking through the questions, a came across a section about black holes. I immediately though; what would happen if an atom is orbiting a black hole and emitted a photon perpendicular to the event horizon, going away from the black hole. </p>
<p>How would light going away from a black hole react to the gravity?</p>
<p><a href="http://physics.stackexchange.com/questions/82984/photon-stuck-on-the-event-horizon-of-a-black-hole">Photon "stuck" on the event horizon of a black hole</a> actually talks about a photon stopping at the event horizon, but my question is about a photon just outside of the event horizon and if the photon is slowed down.</p>
| 2,599 |
<p>Consider a cube of ice in a flat based container(the base is very broad).The temperature of the system is at first fixed at a minus Celsius temperature, but then the system is left on a table with the top open to atmosphere.<br>
The ice starts melting, and finally there is only water, spread over the container's base(water doesn't touch walls of the container,so it looks like water spilled on the floor)<br>
Here, the centre of mass of the $H_2O$-container system moves downward because of the lowering of height of water molecules.<br>
But,this system experiences no net external force on it,all the time.Hence, the centre of mass of this system shouldn't accelerate at all!<br>
How is this contradiction sorted out?I feel that since melting is very slow, the centre of mass might move only very slowly, but still that doesn't explain things.For example if the room was very hot, melting wouldn't have been slow, right?</p>
<p>Edit:Many people are probably getting confused regarding what perspective I'm taking.Let the container and water(solid/liquid) be a single system.We can think of them together as a point mass.This point mass is at equilibrium,and at rest,situated at the position of the centre of mass of ice+container system(let's say,at a height 'h' above the table.<br>
There may be internal forces happening inside the point mass,but the net external force(resultant) is zero,all the time.Hence, according to Newton's first law,the point mass must remain at equilibrium and hence,at rest.But,when ice melts,the position of center of mass of ice+container system has moved down!<br>
Hence the point mass has to move down, in the absence of any resultant external force.The gravitational force on ice+container is cancelled by normal reaction of table on container.</p>
| 2,600 |
<p>I am reading some lecture notes which demonstrate how various models in <strong>SUSY QM</strong> can be used to obtain topological invariants such as the Euler characteristic from the Witten Index. </p>
<p>The following lagrangian has been used directly, said to be the supersymmetric generalization of the bosonic $\sigma$ model. What is the motivation to consider this Lagrangian? How do I obtain the lagrangian? What is the sigma model being considered and how does it generalise?</p>
<p>Please provide a direct answer or references as deemed necessary.
(I googled to learn more, but most of the reviews start from lagrangians in TQFT which I have no knowledge about. I would like a more elementary explanation for the lagrangian. )</p>
<p>$\phi^i(t)$ are maps from $R$ or $S^1$ to a Riemannian manifold $M$ with metric $g_{ij}$.</p>
<p>$$L =
\frac{1}{2}g_{ij}(\phi)\dot{\phi}^i \dot{\phi}^j+\frac{i}{2}g_{ij}\bar{\Psi}^i \gamma^0 \frac{D\Psi^j}{dt} + \frac{1}{12}R_{ijkl}\bar{\Psi}^1 \Psi^j \bar{\Psi}^k\bar{\Psi^l}
$$</p>
<p>$\frac{D}{dt}$ is the covariant derivative with $\dot{\phi}$ as a connection, and $\bar{\Psi}^i_{\alpha}=\bar{\Psi}^i_{\beta}\gamma^0_{\beta \alpha}$</p>
<p>How can I understand this from the physics of quantum mechanics? and what are the maps $\phi(t)$?</p>
| 2,601 |
<p>I've observed many times that if you drop a lot of a 'granular' substance in one place and keep the nozzle out of which the substance flows, that the shape of the pile created very much resembles a bell curve. The situation is a bit hard to explain so for example image somebody holding a small pipe vertically above the ground and dropping a large amount of sand through that small pipe. The shape created by the sand on the ground largely resembles a bell curve wrapped around the vertical axis. I come from a math background and am really unsure of how to even start proving or disproving the proposition that the shape formed by the grains is a bell curve. Any help and input would be appreciated. </p>
| 2,602 |
<p>Calculate the probability current density vector $\vec{j}$ for the wave function : $$\psi = Ae^{-i(wt-kx)}.$$</p>
<p>From my very poor and beginner's understanding of probability density current it is :</p>
<p>$$\frac{d(\psi \psi^{*})}{dt}=\frac{i\hbar}{2m}[\frac{d\psi}{dx}\psi^{*}-\frac{d\psi^{*}}{dx}\psi]$$</p>
<p>By applying the RHS of the above equation :</p>
<p>$$\frac{i\hbar}{2m}[-A^{2}ikxe^{-i(ωt-kx)}e^{i(ωt-kx)}-A^{2}ikxe^{i(ωt-kx)}e^{-i(ωt-kx)}]$$</p>
<p>This gives :</p>
<p>$$\frac{-2iA^{2}ik\hbar}{2m}=\frac{k \hbar A^{2}}{m}$$</p>
<p>This is not the correct answer. :( What have I done wrong ?</p>
<p>In the model workings instead of A in the complex conjugate of the wave function they have written $A^{*}$. Why is this necessary since $A$ is likely to be a real number anyways ?</p>
| 2,603 |
<p>Consider a single isolated rotating mass (for example planet), is it possible to extract energy out of its rotation? If yes, how could that theoretically be achieved?</p>
| 2,604 |
<p>Consider you have a quantum field theory that undergoes spontaneous symmetry breaking at some critical temperature. It doesn't necessarily have to be a continuous symmetry that's broken, I don't think that matters for my question. For simplicity, think of the field being a scalar field.</p>
<p>I (think I) do understand how symmetry restoration works in a background of some temperature. One looks at the mean field approximation, and the mean field then has statistical properties (temperature, pressure) and contributes to the Lagrangian. That adds to the symmetry breaking potential so that at the critical temperature the nature of the minima changes. That works fine eg in the early universe.</p>
<p>But now consider you look at a highly energetic scattering experiment, high energy meaning a momentum transfer much much larger than the critical temperature (arbitrarily large if you want), with ingoing and outgoing asymptotic states and so on. The mean field approximation doesn't make much physical sense in this case because the scattering event itself is a large fluctuation. But the mean field value itself must still exist, or does it? I am thus wondering what's the mean field value of the field whose symmetry is broken in that scattering event. I do not mean the vacuum expectation value - if there is something that's scattering, it's not vacuum. And again, I know that if you have enough energy for a large perturbation, the mean field approximation isn't useful. </p>
<p>I think I am missing something here. I am not even sure how the mean field would be defined for the scattering event. </p>
| 2,605 |
<p>This may look like a philosophical question, but I'm looking for physical explanations (if there's any), that's is why I'm asking it here.</p>
<p>What is the ability of thinking? We are all creatures consist of flesh and bones. Our brains are also nothing but flesh and water? Why are we thinking? What happens to this thinking power when we die? How and why does it disappear? Does it really disappear, or go to somewhere else in a sense we don't know and understand yet?</p>
<p>We observe that, different creatures have different capabilities of thinking. The creature which has the most advanced thinking ability is human. Besides human cats can think up to some degree; they know how to hunt, they decide where to hide and where to find food from. Ants think too, even if it is weaker then cats can do; they find food and quickly run away when you try to pick one of them from floor with your hand, because they sense danger and decide to escape from there. Cells can also think in some sense, but it is much less capable that we don't even call it "thinking". Most living creature sense themselves as separate being.</p>
<p>What happens if we stop the time, we compile and exact verbatim copy of a normal human being, then start the flow of time once again? Would the second copy think too as the original one? Or would the second copy start his life in vegetative state? Or would it just be a stack of dead flesh?</p>
<p>How does physics explain this? Is there any particle that causes us to think? One century ago, we didn't even dream that this many sub atomic particles existed. Can the ability of thinking be a cause of some unknown physical particle of a flow of vector field (like electromagnetic field) which hasn't been discovered yet? Do you expect that in the future some scientist(s) would discover this mystery? Is there any research going on this? What do we know about "thinking" today?</p>
<p>(Note: There wasn't appropriate tags for my question. I would appreciate if another user with higher reputation could add some tags for this question and remove this note.)</p>
| 2,606 |
<p>Are there any known potentially useful, nontrivial, irreducible representations of the Lorentz Group $O(3,1)$ of dimension more than $4$? Examples? A $5$-dimensional representation? EDIT: Is there some deep reason why higher-dimensional representations (other than infinite-dimensional representations) are less useful?</p>
| 2,607 |
<p>I don't understand how the zeroth order intensity maximum works. The intensity of a transmission diffraction grating is:</p>
<p>$$I=I_0\sin^2(β/2)/(β/2)^2\sin^2(Nγ/2)/\sin^2(γ/2)$$</p>
<p>Where $γ =kd\sin(θ)$, $β=kb\sin(θ)$, $d$ is the grating constant and $b$ is slit width. For the zeroth order intensity maximum, $θ =0$, so $γ =β=0$. </p>
<p>I understand how $\sin^2(β/2)/(β/2)^2$ goes to $1$ as $θ$ tends to zero. But what happens with $\sin^2(Nγ/2)/\sin^2(γ/2)$?</p>
<p>Answer:</p>
<p>Taking L'Hôpital's rule twice ($lim γ→0$) I get:</p>
<p>$$ [0.5N^2(dγ/dt)^2(cos^2(Nγ/2)-sin^2(Nγ/2))]/[0.5(dγ/dt)^2(cos^2(γ/2)-sin^2(γ/2))] =
[N^2(1-2sin^2(γ/2))]/(1-2sin^2(γ/2)$$</p>
<p>Which approaches $N^2$ as $γ$ goes to zero.</p>
| 2,608 |
<p>So based on the topic, this is the question.</p>
<p>"A particle with mass $m$ is moving along x-axis with $v_0$ at $t=0$ and $x=0$. The particle is acted by an opposing force with magnitude proportional to the square of velocity. Find out the a) velocity b) position and c) acceleration of particle at any time ($t>0$)"</p>
<p>I have the answers and solution for this but hardly understand them (I dont understand at all actually. Plus where does β come from?)</p>
<p>Basically I just need a clarification on question a) as b) and c) are related to the first question. </p>
<p>This is a self study for final exam. Answer provided by my friend but she also copied it from a source. So without understand it we do it correctly. But it seems to be useless. So I hope you guys can help me.</p>
<p>This is the answer.</p>
<p>a) velocity
<img src="http://i.stack.imgur.com/nU9kp.jpg" alt="Part a"></p>
<p>b) position
<img src="http://i.stack.imgur.com/PknJE.jpg" alt="Part b"></p>
<p>c) acceleration (t>0)
<img src="http://i.stack.imgur.com/cY0hx.jpg" alt="Part c"></p>
| 2,609 |
<p>What kinds of inconsistencies would one get if one starts with Lorentz noninvariant Lagrangian of QFT? The question is motivated by <a href="http://arxiv.org/abs/1203.0609" rel="nofollow">this preprint arXiv:1203.0609</a> by Murayama and Watanabe. </p>
<p>Also, what is the basic difference between </p>
<ul>
<li><p>a theory that is non-relativistic approximation to relativistic theory, and </p></li>
<li><p>a fundamentally Lorentz non-invariant theory?</p></li>
</ul>
| 2,610 |
<p>While searching, i found this page: <a href="http://timesofindia.indiatimes.com/home/stoi/Why-does-a-magnet-attract-iron/articleshow/4298171.cms" rel="nofollow">http://timesofindia.indiatimes.com/home/stoi/Why-does-a-magnet-attract-iron/articleshow/4298171.cms</a> but it does not have full explanation. So Please tell full explanation of why this happens?</p>
| 2,611 |
<p>Is there any formulated lagrangian (density) for M-theory? If not, why is there no lagrangian?</p>
<p>If not, is this related to many vacua existing?</p>
<p>Thnx.</p>
| 2,612 |
<p>Why are ceramic containers used for hot beverages? For example, is it dangerous to use plastic? Are there any other reasons other than to protect hands from heat?</p>
| 2,613 |
<p>I'm reading the book "Fundamental Physics 2: Electromagnetism" by Alonso and Finn.
I understand everything up to the point where everything is "unified".</p>
<p>The following example is given in the book:</p>
<blockquote>
<p><em>Take 2 observers $O$ and $O'$ that are moving with a constant velocity $v$ relative to each other and 2 charges $q$ and $Q$ that are at rest relative to $O'$. For observer $O'$ there's only an electrical interaction between $Q$ and $q$; he measures the force $\vec{F'}=q\vec{E'}$ where $\vec{E'}$ is the electrical field intensity caused by $Q$ in $q$, as measured by $O'$. Because $O$ sees the charge $Q$ moving, he observes an electrical field $\vec{E}$ and a magnetic field $\vec{B}$ caused by $Q$ and because he sees $q$ move with a velocity $\vec{v}$, the force exerted on $q$ by $Q$ according to $O$ equal to $\vec{F} = q(\vec{E}+\vec{v}\times\vec{B})$.</em></p>
</blockquote>
<p>Now a bit further it says that we choose the x and x' axis so that $\vec{v}=\vec{e_x}v$.<br>
Because of this: $\vec{v}\times\vec{B} = -\vec{e_y}v\vec{B_z}+\vec{e_z}v\vec{B_y}$.</p>
<p>I don't understand how this last conclusion is made.</p>
| 2,614 |
<p>I have an object in free space (no gravity) with angular momentum $ = \omega_i $, and some velocity vector $=\vec{V_i}$. To simplify we will say it has a mass-less rigid rod length $ = \ell $, connecting two small masses both of mass $ = M $. The masses are small in the sense of a radius equal to the rod radius both much smaller than $ \ell $. Of course for simplicity keep this in the 2D plane. I know precession can play a role, but accounting for it can make the problem easier, I've already done that.</p>
<p>Now, we want to change the velocity of the object by $ \Delta V $. We must do this by taking a bit of the matter of mass $ = m $ off the object and get it moving away at a velocity $ = \vec{V_m} $. While keeping $M>>m$ What will be the most energy efficient mechanism to do this? What will be the most momentum efficient mechanism to do this? </p>
<p>The final answer, like with the precession related solution, gives some $ \vec{V_m} $ value parallel or perpendicular to $\vec{V_i}$, and some mass "$ m $" proportional to "$ M $". </p>
<p>The goal will be to obtain maximum $ \Delta V $ with minimum energy or momentum. While keeping $M>2m$</p>
<p>Right now I am getting different solutions based on minimizing energy vs. momentum, is this logical? Why?
<a href="http://physics.stackexchange.com/questions/18692/converting-angular-velocity-to-linear-velocity-through-friction">This has helped me get to my solution thus far.</a> Thanks <a href="http://physics.stackexchange.com/users/6299/jcooper">JCooper</a> and <a href="http://physics.stackexchange.com/users/6671/maksim-zholudev">Maksim Zholudev</a>.</p>
<p><strong>EDIT:</strong> The answer should be at least partially derived using formulas. Some relation between $ \Delta V $ and the input energy or momentum must be shown.</p>
| 2,615 |
<p>I came across a method for writing the constraint equations known as "The Virtual Work Method".I am quoting the exact language of the text(well,not exactly the exact)-</p>
<blockquote>
<p>Consider the atwood machine shown.When one holds both the masses such that the string is slack,there is no tension.Once released,the string will become taut and it will oppose the motion of one block where as it will support the motion of other one.</p>
<p><img src="http://i.stack.imgur.com/UT1Oh.png" alt="fig"></p>
<p>Thus it is transferring the effort of one block to the other one.so it will not do work on its own.Since sum of the work done by string on each block is 0</p>
<p>$$\vec T\cdot\vec x+\vec T\cdot\vec y=0 $$
$$\Rightarrow \vec x+\vec y=0$$
<em>Differentiating twice</em></p>
<p>$$\vec a_x+\vec a_y=0$$</p>
</blockquote>
<p>Well here i do not not understand why sum of the work will be 0 and another thing is why is the work called virtual?</p>
| 2,616 |
<p>In high-school level books (for example the german standard text: "Dorn-Bader") I have often seen an explanation of the Lorentz force as on the following picture:</p>
<p><img src="http://i.stack.imgur.com/eHiQk.gif" alt="enter image description here"></p>
<p>The textbooks consider the superposition of the circular field of the wire and the homogenous field of the magnet (sure the homogenity doesn't matter here). Then the net field as you can see on the picture above on the right is larger on one side of the wire (here on the right) and smalle on the other one. So far so good. </p>
<p>However why does this <strong>explain</strong> the occurence and direction of the Lorentz force. Do do so, one would need another principle for example that the wire always wants to go to the weaker field regions or something like this. And this principle should be somehow more evident than the Lorentz-force itself (which you can "see" experimentally).</p>
<p>But how is this needed principle exacly forumlated? Why it is correct? Is there any good reason that it is more evident than just taking the Lorentz-force as an experimental fact?</p>
<p>Would be great if someone could clarify the logic of this, evaluate the soundness of the argument and embedd it conceptually and mathematically in the big picture of electromagnetic theory.</p>
<p>Additionally I want to know if the above cited "explanation" has any common name and if there are university level textbooks which proceed in a similar way. I feel that this argument goes back to Michael Faraday (just by the style of reasoning) - so if someone has a reference to the orgin of the argument I would be interested in it, too.</p>
<p>By the way: The magnetic field in the above cited book ("Dorn Bader") is introduced by the interaction of permament magnets...</p>
| 2,617 |
<p>It's interesting because we don't normally consider the "vaccuum of space" as a fluid, but it's becoming more apparent that it's an ocean of subatomic stuff.</p>
<p>Here's a link to a book: <a href="http://rads.stackoverflow.com/amzn/click/1463441541" rel="nofollow">Unified Fluid Dynamic Theory Of Physics</a></p>
<p>Do you think this is crack-pot science or a worth-while book to study?</p>
<p>Thanks.</p>
| 2,618 |
<ol>
<li><p>Can anyone outline the theory of plane wave <a href="http://en.wikipedia.org/wiki/Born_approximation" rel="nofollow">Born approximation</a> for direct nuclear reactions in detail?</p></li>
<li><p>Also What are the modification introduced in the distorted wave Born approximation?</p></li>
</ol>
<p>I was unable to find the right material/books/web resources on this. </p>
| 2,619 |
<p>I personally cringe when people talk about scientific theories in the same way we talk about everyday theories. </p>
<p>I was under the impression a scientific theory is similar to a mathematical proof; however a friend of mine disagreed. </p>
<p>He said that you can never be absolutely certain and a scientific <a href="http://en.wikipedia.org/wiki/Theory" rel="nofollow">theory</a> is still a theory. Just a very well substantiated one. After disagreeing and then looking into it, I think he's right. Even the Wikipedia definition says it's just very accurate but that there is no certainty. Just a closeness to potential certainty. </p>
<p>I then got thinking. Does this mean no matter how advanced we become, we will never become certain of the natural universe and the physics that drives it? Because there will always be something we don't know for certain? </p>
| 2,620 |
<p>Why are all observable <a href="http://en.wikipedia.org/wiki/Gauge_theory" rel="nofollow">gauge theories</a> not vector-like? </p>
<p>Will this imply that the electron and/or fermions do not have mass? </p>
<p>How is this issue resolved? </p>
<p>Background: </p>
<p><a href="http://en.wikipedia.org/wiki/The_Standard_Model" rel="nofollow">The Standard Model</a> is a <a href="http://en.wikipedia.org/wiki/Nonabelian_group" rel="nofollow">non-abelian</a> gauge theory with the symmetry group U(1)×SU(2)×SU(3) and has a total of twelve gauge bosons: the photon, three weak bosons and eight gluons.</p>
<p>Massless fermions can either have their spin pointing along their direction of travel or opposed to it, these two types of massless fermions are called right-handed and left-handed. A fermion mass can turn a left-handed fermion into a right-handed fermion (this is technically called a Dirac fermion mass). The ordinary electron for example has both left-handed and right-handed pieces. For this to be allowed in a gauge theory the left-handed and right-handed fermions must have the same charge. If all the fermions in the gauge theory can be paired up in this fashion so that they are all allowed to have masses, then the gauge theory is called vector-like. </p>
| 2,621 |
<p>I'm studying the oscillations of systems with more than one degree of freedom from Landau & Lifshitz's <em>Mechanics</em> Third Edition (for those who have the book, my question corresponds roughly to explaining the last paragraph before equation 23.9).</p>
<p>First I give a brief background to my problem to show what I understand. So far I understand that given the Lagrangian, the equation of motion is:</p>
<p>$${m_{ik}\ddot{x}_k + K_{ik}x_k = 0}\tag{23.5}$$</p>
<p>Using the usual <em>ansatz</em> $x_k = A_k e^{i \omega t}$ then, we get the set of equations:</p>
<p>$$(-\omega^2m_{ij} + k_{ik})A_k = 0\tag{23.7}$$</p>
<p>where it can be easily seen that the section in brackets must have determinant 0, leading us to the <em>characteristic equation</em>:</p>
<p>$$|k_{ik} - \omega^2m_{ik}| = 0\tag{23.8}$$</p>
<p>where furthermore it can be easily shown that for each solution to the characteristic equation $\omega^2_\alpha$, $\alpha = 1, \ldots, S$ where $S$ is the degrees of freedom of the system, $\omega_\alpha$ must be positive and real. </p>
<p>I now quote in full the paragraph I'm having difficulty with:</p>
<blockquote>
<p>The frequencies $\omega_\alpha$ having been found, we substitute each of them in equations (23.7) and find the corresponding coefficients $A_k$. If all the roots $w_\alpha$ of the characteristic equation are different, the coefficients $A_k$ are proportional to the minors of the determinant (23.8) with $\omega = \omega_\alpha$. Let these minors be $\Delta_{k\alpha}$. A particular solution of the differential equations (23.5) is therefore $x_k = \Delta_{k\alpha}C_{\alpha}\exp{(i\omega_\alpha t)}$ where $C_\alpha$ is an arbitrary complex constant.</p>
</blockquote>
<p>I think my main problem here is my poor knowledge of linear algebra. I don't understand why the roots all being different implies that the coefficients are proportional to the minors of the determinant. Furthermore I'm slightly confused by the wording "minors of the determinant" since to me, a determinant is a scalar value, it seems like they should be saying "minors of the matrix $(k_{ik} - \omega^2m_{ik})$" Can someone explain the linear algebra behind all this to me? I've taken a Linear Algebra course but I was quite poor at it.</p>
| 2,622 |
<p>This is a question about the negative energy solutions to the free particle Dirac Equation in the first quantized picture. We need both the positive and negative energy solutions to have a complete set of states.</p>
<p>For example, the bound states of the hydrogen atom consists of superpositions of positive and negative energy free particle solutions. So then, what is the physical interpretation of a superposition of positive and negative energy states?</p>
<p>Suppose we had a hydrogen atom, and we immediately removed the nucleus. If we measured the energy of the resulting free electron, what would we see?</p>
| 2,623 |
<p>A DIY fractal antenna project is described at <a href="http://www.htpc-diy.com/2012/04/diy-flexible-fractal-window-hdtv.html" rel="nofollow">http://www.htpc-diy.com/2012/04/diy-flexible-fractal-window-hdtv.html</a></p>
<p>Mainly idle curiosity, but I was wondering what the principles behind these might be, and if there was an optimal shape for them.</p>
<p>I'm guessing the idea is that self-similarity, up to a point, optimizes sensitivity over a wider frequency range.</p>
| 2,624 |
<p>A question is given as </p>
<blockquote>
<p>Consider a fluid of density $ \rho(x, y, z, t) $ which moves with velocity $v(x, y, z, t) $ without sources or sink. Show that $ \nabla \cdot \vec J + \frac{\partial \rho }{\partial t} = 0 ;$ where $ \vec J = \rho \vec v \hspace{0.5 cm}$ ( $\vec v$ being velocity of fluid and $ \rho $ density). </p>
</blockquote>
<p>In the solution it assumes, $ - \nabla \cdot \vec J $ is the change in $ \vec J$ within the volume element which should be equal to $ \frac{\partial \rho}{\partial t}$ <em>(why equal?)</em>. I don't understand this part and I doubt if this question is correct. I think the question should be
$$ \rho (\nabla \cdot \vec v) + \frac{\partial \rho}{\partial t} = 0$$
Is the question correct? If it's correct help me to understand the solution (if it's right) or please provide me correct answer. Thank you!!</p>
| 2,625 |
<p>I'm familiar with the equations for friction for a static object and an object moving at steady speed over a surface from high school physics. But we never learned how an object moving only due to momentum experiences friction. This is something I've modeled several times while building simple 2D games, but I have no idea if what I have made matches reality in any way. </p>
<p>Is the force of friction dependent on the speed of the object across the surface? Or is the force constant as the object decelerates? Is there a simple equation giving the acceleration (deceleration) of the object depending on it's mass and velocity?</p>
| 2,626 |
<p>Does the <a href="http://en.wikipedia.org/wiki/Casimir_effect" rel="nofollow">Casimir Effect</a> violate the <a href="http://en.wikipedia.org/wiki/Quantum_inequalities" rel="nofollow">Quantum Inequalities</a>? From what I understand, the Casimir Effect is able to produce negative energy densities for an indefinite amount of time, or is that wrong?</p>
| 2,627 |
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