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<p>I have read on several sites that this is a theory that cannot be tested. Why is this a theory that can not be tested? It seems to me that if the mass of particles is continually increasing, then the wavelength of a spectral line emitted by an atom would also be continually changing. This means that over a length of time (possibly not human time scale), if we could detect a shift in the emitted wavelength from an atom (stationary compared the observer), then we could determine if this interpretation is correct. We can obviously tell the difference in wavelength between the light emitted 10 billion years ago and that emitted today, so it should just be a matter of how fine our resolution is. The more narrowly we can resolve the wavelength, the shorter time span needed to make the measurement. If red shift is caused by movement relative to the observer, no red shift should be found. If red shift is caused by the mass change over time of the particles, red shift should be observable.</p>
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<p>It is often stated, particularly in popular physics articles and videos, that if one measures a particle A that is entangled with some other particle B, then this measurement will immediately affect the state of the entangled partner.</p>
<p>For example, if Alice and Bob share an entangled pair of electrons and Alice measures her spin in the $x$ direction, then Bob's spin will also end up spinning in that direction, and similarly if she measures in the $z$ direction. Moreover, the effect will be instantaneous, regardless of the spatial distance between the two particles, which seems at odds with special relativity.</p>
<p><strong>Can I use a scheme like this to communicate faster than light?</strong></p>
| 61 |
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<p>When you attach a bolt to something using a nut, it is clear what the roles of the nut and bold are. </p>
<p><img src="http://i.stack.imgur.com/JC94D.jpg" alt="enter image description here"></p>
<p>The more you tighten the bolt the more secure your fastening. However, you are often also told to use a washer as well.</p>
<p><img src="http://i.stack.imgur.com/JEZK3.jpg" alt="enter image description here"></p>
<p>I know this somehow prevents the bolt from loosening but from a physics/mathematics point of view, what is the role of the washer?</p>
| 9,260 |
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<p>Let there be two objects that have zero relative velocity with respect to each other in an inertial frame. If they both undergo identical accelerations, but one starts the acceleration at t = T1 and the other starts the acceleration at t = T2, then per special relativity the maximum distance that rest frame can measure between the two objects is (T1-T2) * c
What acceleration equations show that the separation between these two objects as t approaches infinity approaches c * (T1-T2)</p>
| 9,261 |
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<p>$xp$ is not a hermitian operator and hence doesn't represent an observable. Then, how can we interpret the expression</p>
<p>$$
\langle x p \rangle \text{,}
$$</p>
<p>the expectation value of position times momentum?</p>
<p>How to interpret the expectation value of a hermitian operator, say $p$, is clear: make several measurements on identically prepared systems and get the expectation value from the measured quantities.</p>
<p>But now we are talking about the expectation value of an operator that doesn't represent an observable. The discussion above hence isn't valid. However, computing $\langle x p \rangle$ knowing $\Psi$ according to the mathematical definition can be done, so $\langle x p \rangle$ can clearly be deduced if we know the wave function - but how can we interpret the quantity we get?</p>
<p>(The question arises from a line in my textbook - "in a stationary state, $\frac{d}{dt} \langle x p \rangle = 0$, ..." without further explanation - why is this obvious?)</p>
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<p>I was hoping someone could help me with understanding why a row of polarizes reflects a light wave when the whole row is the same length as the wavelength of the light.</p>
<p><img src="http://i.stack.imgur.com/kdcDG.gif" alt="Liquid crystals"></p>
<p>I pretty much get the physics behind the rest, just don't understand this little part. Why only the light of that wavelength and does this also apply for light coming in at a different phase than the crystals?</p>
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<p>Which is the smallest particle that has been actually seen by the scientists? When I say "actually seen", (may be using some ultra advanced microscope or any other man made eye, using any wavelength or phenomena) I really mean it; just like we have seen the <a href="http://www.wellcome.ac.uk/stellent/groups/corporatesite/@msh_publishing_group/documents/image/wtdv030817.jpg" rel="nofollow">red blood cells</a>. </p>
<p><a href="http://physics.stackexchange.com/questions/104523/which-is-the-smallest-known-particle-that-scientists-have-actually-seen-with-th/104526#104526">Davidmh's answer</a> is pretty much in line with what I am asking</p>
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<p>If there was a standing sound wave tube and a flammable gas was introduced then ignited, would the combustion be more forceful and more efficient since its following a standing wave, than just a gas ignited within a tube?</p>
| 9,265 |
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<p>This question is refered to those who are familiar with the concept of weak measurement.</p>
<p><strong>In short:</strong></p>
<p>How can the polarization of a photon be coupled to the position of a pointer state? What is the pointer state? The position of some particle? How to realise that in a lab?</p>
<p><strong>In detail:</strong></p>
<p>Let's say we want to weakly measure the polarization of a photon. We write the observable $A=|H\rangle\langle H| - |V\rangle\langle V|$ and the initial state shall be $|\psi\rangle = \alpha |H\rangle + \beta |V\rangle$.
Now we weakly couple it to a pointer by sending it through a birefringent element, i.e. depending on the polarization the photon needs more time to pass the birefringent element.</p>
<p>Formally:</p>
<p>The Hamiltonian whichs represents this coupling is given by $H=\lambda A \otimes P$ where $\lambda$ is the coupling strenght and $P$ the momentum operator of the pointer. Now $\exp{(-\text i H t)}$ will act as follows:</p>
<p>$|\psi\rangle \otimes |g\rangle = (\alpha |H\rangle + \beta |V\rangle) \otimes |g\rangle \;\longrightarrow\; ( \alpha |H\rangle \otimes|g_+\rangle + \beta |V\rangle\otimes|g_-\rangle)$</p>
<p>where the first degree of freedome is clearly the polarization of the photon and the second is the pointer state (a gaussian say). Both $g_+$ and $g_-$ refere to small shifts due to the birefringent element.</p>
<p>Then there is this well known fact that a post-selection whichs leads to an imaginary weak value $A_w$ causes a shift in the momentum-space of the pointer beeing proportional to $\text{Im} A_w$.</p>
<p>The weak value is defined as $A_w = \frac{\langle \phi|A|\psi\rangle }{\langle \phi|\psi\rangle}$ where $|\phi\rangle$ is the post-selected polarization state of the photon.</p>
<p>(reference: <a href="http://arxiv.org/abs/0706.4207" rel="nofollow">http://arxiv.org/abs/0706.4207</a> or <a href="http://arxiv.org/abs/0911.5139" rel="nofollow">http://arxiv.org/abs/0911.5139</a>)</p>
<p>Mathematically this is easy to understand, but I hace a very conceptual problem with all that: How do I have to understand the pointer state? Is it a position of a particle? If yes, how can its position be coupled (in a physically understandable way) to a photon? And how can a post-selection affect its momentum?</p>
| 9,266 |
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<p>I have been study about Veneziano amplitude and Virasoro Shapiro amplitude.
I want to summarize this two amplitude in the following way, please check that i am understand them properly.</p>
<p>Veneziano amplitude : they are open string amplitude for disk.
Usually we called four-open string tachyon on disk as Veneziano amplitude</p>
<p>Virasoro-Shapiro amplitude : they are closed string amplitude for sphere.
Usually we called four-closed string tachyons on shpere as Virasoro-Shapiro amplitude. </p>
<p>The similarity between Veneziano and Viraosoro-Shaprio amplitude are they both have Regge and hard scattering limit.</p>
<hr>
<p>Is there are more difference and similarity for Veneziano and Virasoro-Shaprio amplitude?</p>
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<p>Let $V_1$, $I_1$, $R_1$ and $V_2$, $I_2$, $R_2$ be voltages, currents, resistance in first, second circuit.</p>
<p>And we assume $R_1=R_2$, $I_1\neq I_2$.</p>
<p>The conservation of electrical power $P=V_1I_1=V_2I_2$ holds in the transformer circuits.</p>
<p>Hence $I_1^2R_1=I_2^2R_2$, $R_2=(I_1^2/I_2^2)R_1$ which leads to a contradiction.</p>
<p>What am I missing?</p>
| 9,268 |
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<p>Is entanglement necessary for quantum computation?
If there was no error in the computation,superposition of states would be sufficient for quantum computation to be carried out.Is this right?</p>
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<p>I live in Seattle and am thinking of purchasing a telescope. Is fall/winter a decent time of year for viewing (aside from summer)? Are there any major viewings/events during that this of year?</p>
<p>I know I live in a city of rain, but there are some nice nights. </p>
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<p>Is this photo "real"? Are the stars not super-imposed in the image?</p>
<p> <img src="http://i.stack.imgur.com/BvJZ0.jpg" alt="galaxy"></p>
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<p>Quarks and leptons are considered elementary particles, while phonons, holes, and solitons are quasiparticles. </p>
<p>In light of emergent phenomena, such as fractionally charged particles in fractional quantum Hall effect and spinon and chargon in spin-charge separation, are elementary particles actually more <em>elementary</em> than quasiparticles? </p>
<p>Does the answer simply depend on whether one is adopting a reductionist or emergent point of view? </p>
| 9,272 |
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<p>Consider this penny on my desc. It is a particular piece of metal,
well described by statistical mechanics, which assigns to it a state,
namely the density matrix $\rho_0=\frac{1}{Z}e^{-\beta H}$ (in the
simplest model). This is an operator in a space of functions depending
on the coordinates of a huge number $N$ of particles. </p>
<p>The <B>ignorance interpretation of statistical mechanics</B>, the orthodoxy to
which all introductions to statistical mechanics pay lipservice, claims
that the density matrix is a description of ignorance, and that the
true description should be one in terms of a wave function; any pure
state consistent with the density matrix should produce the same
macroscopic result.</p>
<p>Howewer, it would be very surprising if Nature would change its
behavior depending on how much we ignore. Thus the talk about
ignorance must have an objective formalizable basis independent of
anyones particular ignorant behavior.</p>
<p>On the other hand, statistical mechanics <em>always</em> works <em>exclusively</em> with the
density matrix (except in the very beginning where it is motivated).
Nowhere (except there) one makes any use of the assumption that the
density matrix expresses ignorance. Thus it seems to me that the whole
concept of ignorance is spurious, a relic of the early days of
statistical mechanics.</p>
<p>Thus I'd like to invite the defenders of orthodoxy to answer the
following questions:</p>
<p>(i) Can the claim be checked experimentally that the density matrix
(a canonical ensemble, say, which correctly describes a macroscopic
system in equilibrium) describes ignorance?
- If yes, how, and whose ignorance?
- If not, why is this ignorance interpretation assumed though nothing at all depends on it?</p>
<p>(ii) In a though experiment, suppose Alice and Bob have different
amounts of ignorance about a system. Thus Alice's knowledge amounts to
a density matrix $\rho_A$, whereas Bob's knowledge amounts to
a density matrix $\rho_B$. Given $\rho_A$ and $\rho_B$, how can one
check in principle whether Bob's description is consistent
with that of Alice? </p>
<p>(iii) How does one decide whether a pure state $\psi$ is adequately
represented by a statistical mechanics state $\rho_0$?
In terms of (ii), assume that Alice knows the true state of the
system (according to the ignorance interpretation of statistical
mechanics a pure state $\psi$, corresponding to $\rho_A=\psi\psi^*$),
whereas Bob only knows the statistical mechanics description,
$\rho_B=\rho_0$.</p>
<p>Presumably, there should be a kind of quantitative measure
$M(\rho_A,\rho_B)\ge 0$ that vanishes when $\rho_A=\rho_B)$ and tells
how compatible the two descriptions are. Otherwise, what can it mean
that two descriptions are consistent? However, the mathematically
natural candidate, the relative entropy (= Kullback-Leibler divergence) $M(\rho_A,\rho_B)$, the trace of $\rho_A\log\frac{\rho_A}{\rho_B}$,
[edit: I corrected a sign mistake pointed out in the discussion below]
does not work. Indeed, in the situation (iii), $M(\rho_A,\rho_B)$
equals the expectation of $\beta H+\log Z$ in the
pure state; this is minimal in the ground state of the Hamiltonian.
But this would say that the ground state would be most consistent with
the density matrix of any temperature, an unacceptable condition.</p>
<p>Edit: After reading the paper
<a href="http://bayes.wustl.edu/etj/articles/gibbs.paradox.pdf">http://bayes.wustl.edu/etj/articles/gibbs.paradox.pdf</a>
by E.T. Jaynes pointed to in the discussion below, I can make more precise the query in (iii): In the terminology of p.5 there, the density matrix $\rho_0$ represents a macrostate, while each wave function $\psi$ represents a microstate. The question is then: When may (or may not) a microstate $\psi$ be regarded as a macrostate $\rho_0$ without affecting the predictability of the macroscopic observations? In the above case, how do I compute the temperature of the macrostate corresponding to a particular microstate $\psi$ so that the macroscopic behavior is the same - if it is, and which criterion allows me to decide whether (given $\psi$) this approximation is reasonable? </p>
<p>An example where it is not reasonable to regard $\psi$ as a canonical ensemble is if $\psi$ represents a composite system made of two pieces of the penny at different temperature. Clearly no canonical ensemble can describe this situation macroscopically correct. Thus the criterion sought must be able to decide between a state representing such a composite system and the state of a penny of uniform temperature, and in the latter case, must give a recipe how to assign a temperature to $\psi$, namely the temperature that nature allows me
to measure. </p>
<p>The temperature of my penny is determined by Nature, hence must be determined by a microstate that claims to be a complete description of the penny.</p>
<p>I have never seen a discussion of such an identification criterion, although they are essential if one want to give the idea - underlying the ignorance interpretation - that a completely specified quantum state must be a pure state.</p>
<p>Part of the discussion on this is now at:
<a href="http://chat.stackexchange.com/rooms/2712/discussion-between-arnold-neumaier-and-nathaniel">http://chat.stackexchange.com/rooms/2712/discussion-between-arnold-neumaier-and-nathaniel</a></p>
<p>Edit (March 11, 2012): I accepted Nathaniel's answer as satisfying under the given circumstances, though he forgot to mention a fouth possibility that I prefer; namely that the complete knowledge about a quantum system is in fact described by a density matrix, so that microstates are arbitrary density matrces and a macrostate is simply a density matrix of a special form by which an arbitrary microstate (density matrix) can be well approximated when only macroscopic consequences are of interest. These special density matrices have the form $\rho=e^{-S/k_B}$ with a simple operator $S$ - in the equilibrium case a linear combination of 1, $H$ (and vaiious number operators $N_j$ if conserved), defining the canonical or grand canonical ensemble. This is consistent with all of statistical mechanics, and has the advantage of simplicity and completeness, compared to the ignorance interpretation, which needs the additional qualitative concept of ignorance and with it all sorts of questions that are too imprecise or too difficult to answer.</p>
| 9,273 |
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<p>It's obvious that as you increase the payload of a rocket, you need to also increase the amount of rocket fuel. Yet, as you increase the amount of rocket fuel, you are also increasing the overall mass of the rocket and, therefore, require even more fuel to get the rocket off the ground.</p>
<p>From this observation, one could potentially conclude that as you double the payload of a rocket, you must roughly quadruple its fuel (I am making an assumption here, so correct me if I'm wrong on that one). With that said, is there a theoretical/practical upper limit to the size and payload of a chemical rocket?</p>
| 305 |
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<p>In both string theory and soliton theory, moduli spaces are frequently used. </p>
<p>As far as I known, for soliton theory, moduli spaces are something like collective coordinates for solitons, and for string theory, moduli spaces is the spaces of all metrices divided by all conformal rescalings and diffeomorphisms. It seems like these two definitions(?) of moduli spaces are quite different, but the same terminology is used in both cases. I also learned that the name 'moduli spaces' comes from abstract geometry, but I don't know if that's any help here. </p>
<p>My question is the following: Could anyone provide an intuitive connection between the two uses of moduli spaces, or highlight the differences?</p>
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<p>A conductor that passes a magnetic field, decreases its velocity due to the opposing magnetic forces created by the Eddy Currents.</p>
<p>This "magnetic breaking" effect can be reduced if we change the shape of the conductor from a bar, to a this conductor like this:</p>
<p><img src="http://i.stack.imgur.com/0NFr5.jpg" alt="enter image description here"></p>
<p>What I understand is that conductor above has a higher resistance and reduces the eddy currents. However, what if the air gaps are conductors? Here is where I find laminated conductors to be useful like so:</p>
<p><img src="http://i.stack.imgur.com/cAvF7.gif" alt="enter image description here"></p>
<p>However, what is the difference? I believe the laminated conductor can carry more current(i/e higher current capacity) than the one above. The whole point is reducing eddy currents(the magnetic breaking)in electric motors, or generators... and not affecting its electric properties by increasing its resistance and reducing its current capacity. </p>
<p>Am I correct here or is my understanding incorrect?</p>
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<p>It is one of the axioms of special relativity that the <a href="http://physics.stackexchange.com/q/16018/2451">photon has no rest frame</a>; light travels at speed c when measured in any inertial frame of reference. As a corollary, it is often said that if one <em>were</em> in a photon's rest frame, infinite time-dilation and length-contraction would make the universe appear (from a photon's perspective) to be unchanging, with zero length in its direction of motion. In other words a measurement would not be possible in the reference frame of a photon, because there would be no time or space in which one could conduct it.</p>
<p>My question is: how does this accord with the fact that parallel-moving photons can interact with each other via loop diagrams[*]? Photons can therefore be used to perform measurements in the photon's rest frame. I suppose an answer may be "the act of measurement changes the photons momentum vector and therefore its co-moving frame is non-inertial" however this can be said of <em>any</em> measurement.</p>
<p>[*] Or can they? I can't find an explicit amplitude to check this (for, say, the photon-photon "box" diagram), however as far as I can tell from the Euler-Heisenberg Lagrangian nonlinear effects should be present for parallel photons. </p>
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<p>Is there a word or phrase for when someone is testing for a certain result thereby skewing his findings? </p>
| 9,277 |
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<p>What substance will have the largest specific heat capacity integrated from T=0 to, say, room temperature? In other words, given a finite amount of mass, what object or collection of objects has the largest number of degrees of freedom that can be excited as it absorbs energy starting from T=0? Would it be a complicated molecular polymer that can be tangled in all sorts of ways, or some kind of gas of low-mass particles, or maybe a spin lattice of some sort? Is there some kind of fundamental limit in the universe of the number of quantum degrees of freedom per mass or perhaps per volume that is allowed? </p>
| 9,278 |
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<p>I've read some papers recently that talk about gapped Hamiltonians or gapless systems, but what does it mean?</p>
<p><strong>Edit:</strong> Is an XX spin chain in a magnetic field gapped? Why or why not?</p>
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] |
<p>Does Wick rotation work for quantum gravity? The Euclidean Einstein-Hilbert action isn't bounded from below.</p>
| 9,280 |
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<p>Does a relativistic version of quantum thermodynamics exist? I.e. in a non-inertial frame of reference, can I, an external observer, calculate quantities like magnetisation within the non-inertial frame? </p>
<p>I'd be interested to know if there's a difference between how to treat thermodynamics in a uniformly accelerated reference frame and in a non-uniformly accelerated reference frame. </p>
<p>Thanks!</p>
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<p>The Higgs mechanism allows massless fields to acquire mass through their coupling to a scalar field. But if the masses cannot be predicted because the couplings have to be fixed, what really is the utility of the Higgs mechanism? Instead of saying "Here are <em>a priori</em> couplings; the Higgs mechanism generates mass.", I could just as well say "Here are <em>a priori</em> masses. Period.".</p>
<p>I understand that the Higgs mechanism is crucial to electroweak unification, but I have the same question there. Why does electromagnetism and the weak force have to be unified? Even if the couplings of the photon, Z and W bosons became related on unification, this is still at the cost of introducing new parameters - so it's not really clear to me that something has been explained or tidied up.</p>
<p>Do either the Higgs mechanism or electroweak unification tell us something new? Do either make any predictions that don't come at the cost of extra parameters? (I'm not really challenging anything here; I'm sure the answer to both questions is 'yes' - I just want to fill the gaps in my understanding as I study the Standard Model)</p>
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] |
<p>Classical electromagnetism says an accelerated charge would radiate. Is this understood in QED? </p>
<p>Is there QED derivation of Larmor's formula?</p>
| 9,283 |
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<p>I have applied for a summer project in <a href="http://en.wikipedia.org/wiki/Nuclear_magnetic_resonance_quantum_computer" rel="nofollow">NMR quantum computing</a> as I want to learn this field, but my professor wants me to tell him the project title. I have no idea about the field and what projects are good. Please can you suggest me some good 2 month projects. </p>
<p>I have done advanced course in quantum mechanics and also I am familiar with NMR machine.</p>
| 9,284 |
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<p>I had this problem for a candidacy exam, but wasn't able to get the complete answer. Their spring constants and masses are not the same, find the equilibrium position and frequencies of the system. Apparently once you find their equilibrium position, i.e. when gravity and the spring forces cancel out, you can transform to about that position and ignore gravity. Why do you get to ignore gravity? This is a normal modes problem, but I couldn't justify why gravity just disappears, but it does. Anyone have a simple reason why?</p>
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] |
<p>I'd guess it has to do with the structure of helium-3 allowing for greater Coulomb repulsion between the protons, but I'm unsure.</p>
| 9,286 |
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<p>When you stick ice in a drink, AFAICT (the last physics I took was in high school) two things cool the drink:<ul><li>The ice, being cooler than the drink, gets heat transferred to it from the drink (Newton's law of cooling). (This continues after it melts, too.)<li>The ice melts, and the cool liquid mixes with the drink, which makes the mixture feel cooler.</ul>My first question is, to what extent does each of these cool the drink: which has a greater effect, and how much greater?<br><br>Secondly, how much of the cooling by the first method (heat transfer) is without melting the ice? That is, is there any significant amount of heat transfer to each speck of ice before it's melted, and how much does that cumulatively affect the drink's temperature?<br><br>I suppose all this will depend on a bunch of variables, like the shape and number of ice cubes and various temperatures and volumes. But any light that can be shed, let's say for "typical" situations, would be appreciated.</p>
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-0.005350753664970398
] |
<p>Consider a closed system (say a box) of $n$ particles. There is a well-known idiom/meme/law in classical mechanics that says that the position and momentum of those $n$ particles is all that is needed to determine the future and past configuration of the system. (A minor question, does this idiom have a name?)</p>
<p>Why do we not consider two other pieces of information to be as important as the position and momentum of the $n$ particles?</p>
<ul>
<li><p>The force field (force as a function of space, and preferably of time too).</p></li>
<li><p>The particle interaction rules (When two particles touch each other, do they whiz by or do they scatter? if they scatter what are the rules for determining angles?)</p></li>
</ul>
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<p>Analogy: an infinite number of lines can pass through a point.
Is there a limit on the number of lasers that can pass through a point?
Obviously, with lasers the “point” would be a sphere with the diameter of the beams. </p>
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<p>I know that for close by stars (<50 LY) we can use the parallax effect. And for distant galaxies we use red-shift (& hubble's constant). So how do we measure how far is a star lets say 50,000 LY from the earth?</p>
<p>I know I am missing something, I just don't know what. </p>
<p>How can we assume there is a relationship between red-shift and distance when the stars </p>
<ul>
<li>act like a (turbulent) fluid within the galaxy and </li>
<li>can be moving in any sort of different manner?</li>
</ul>
<p><strong>Edit:</strong>
It is a physics question, because I really want to know what is the star velocity model within the galaxy in order to use red-shift. My instinct tell me that stars move like a swarm within the galaxy with an overall rotation about the galaxy center of gravity (In agreement to the edited horowitz answer).</p>
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<p>Let us consider a rod having a young's modulus $Y$. Let it be of length $l$, and suppose it is suspended from a point P. Let us pull the rod with a force $F$ at a point Q which is at a distance $2/3l$ from the point P. Next let us apply the same force at the other end point of the rod. Is the length of the rod in both the cases same?</p>
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<p>Let us consider an ideal gas. Let it be present initially in a state $(p_1,v_1,t_1)$. Now let it be driven to another state $(p_2,v_2,t_2)$. Why is it so that during this process the maximum work can be obtained if the ideal gas is transformed from one state to another by a reversible process ?</p>
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<p>I'm having trouble understanding the motivation for the definition of the expectation of a self adjoint operator $A$:</p>
<p>$$\langle A \rangle _\psi=\int_{\mathbb{R}}\psi^*A\hspace{0.2cm} \psi \hspace{0.2cm} dx$$</p>
<p>where $\psi(x,t)$ is a normalised state.</p>
<p>I can understand the expectation of the position operator in terms of basic probability: one of the assumptions of QM being that $|\psi|^2$ is the probability to find the particle at $x$. The expected value is just the sum of the positions multiplied by their probabilities:</p>
<p>$$\langle x \rangle _\psi=\int_{\mathbb{R}}x \hspace{0.1cm} |\psi(x,t)|^2 dx=\int_{\mathbb{R}}\psi^* \hspace{0.2cm} x \hspace{0.1cm} \psi \hspace{0.1cm}dx$$ by commutativity.</p>
<p>I don't understand how this might work with the momentum operator, for example. The lecture notes for my course say that the expectation of self adjoint operators are defined in analogy to this, so that</p>
<p>$$\langle p \rangle _\psi=\int_{\mathbb{R}}\psi^*\hspace{0.1cm}p \hspace{0.1cm}\psi\hspace{0.1cm} dx$$</p>
<p>I can't satisfy myself with this explanation; $p$ is a differential operator and so can't be moved about within the integral like $x$ can. For example</p>
<p>$$\int_{\mathbb{R}}\psi^*\hspace{0.1cm}\psi\hspace{0.1cm} p\hspace{0.1cm} dx$$ makes no sense to me, as the operator hasn't been applied to anything. In any case, $\int_{\mathbb{R}}|\psi(x,t)|^2 p dx$ doesn't mean anything to me probability-wise.</p>
<p>So basically I'm looking for an explanation as to why the expectation of self-adjoint operators are so defined. Thanks for any replies!</p>
| 9,293 |
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<p>Suppose an ideal gas in a piston cylinder has some initial pressure $p_1$, volume $V_1$, and temperature $T_1$. A pin, which holds the piston in place, is suddenly removed, and the gas quickly expands. Once equilibrium is reached, the final pressure of the gas is $p_2$.</p>
<p>I am looking at a textbook solution which claims that the work done during this process is $$W = \int p\,dV = p_2\left(V_2-V_1\right),$$ which implies that the pressure is constant ($p=p_2$) throughout the whole process, as soon as the pin is released. Is this really the case, and are there not irreversibilities in this process that must be considered?</p>
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<p>Does back emf occur in transformers and if so does it reduce the efficiency of the transformer and can it be reduced?</p>
| 9,295 |
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<p>A large thick copper wire(or bar if you like,) passes a magnetic field. What formula is best to calculate the $EMF$ here?</p>
<p>$$\epsilon = \frac{\Delta \Phi \
}{\Delta t} $$</p>
<p>or,</p>
<p>$$\epsilon = -vBL $$</p>
<p>Example:</p>
<p><img src="http://i.stack.imgur.com/n8r6i.png" alt="enter image description here"></p>
<p>This image is used to demonstrate Lorentz force, however, it can be used to illustrate my question, lets imagine the conductor in the middle is moving and its passing the magnetic field, when it reaches that point in the diagram it is connected to a circuit to allow current to flow. </p>
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] |
<blockquote>
<p>Two thermally iso- lated cylinders, A and B, of equal volume, both equipped with pistons, are connected by a valve. Initially A has its piston fully withdrawn and contains a monatomic gas at temperature $T_0$, while B has its piston fully inserted and the valve is closed. Calculate the final gas temperature in each cylinder after the following operations, each of which starts with the same initial configuration. Assume the thermal capacity of the cylinders is negligible.</p>
<p>(a) Piston B is fully withdrawn and the valve opened slightly. The gas is then driven as far as it will go by pushing in piston A at such a rate that the pressure in A remains constant; the cylinders are in thermal contact.</p>
<p>(b) As in (a), but the cylinders are thermally isolated from each other.</p>
<p>(c) Piston B is fully withdrawn and the valve is then opened; the cylinders are in thermal contact.</p>
<p>(d) As in (c), but the cylinders are thermally isolated from each other.</p>
</blockquote>
<p>I think I can do (a) by equating the work done (i.e. due to the piston moving in) to the change in internal energy - but it is (b) when they are thermal isolated that I am stuck on how to approach (b) .</p>
| 9,297 |
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<p>I have a homogeneous thin rod $AB$. $A$ can slide along $z$-axis, $B$ along $xy$ plane. There is no friction. We have any initial conditions. Now the question is to write down the equation of motion of the rod.</p>
<p>It is assumed that I don't know Lagrangian mechanics. I believe I should write some conservation theorem but I'm not quite sure how to do it. </p>
<p>I guess I could finish that problem if I would get some direction.</p>
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<p>I am building a truck from trash as materials. I have one small motor and a a few small gears, but no other engineered materials are allowed. The truck must carry a load for a distance of 3m. The winning truck is the truck with the greatest ratio of load carried:time.</p>
<p>What do you think would be the best approach to this project in terms of the gears and the radius of the driven wheels?</p>
<p>I don't know what would be the best approach in terms of torque and angular velocity.</p>
<p>Any other tips or inputs would be greatly appreciated! </p>
<p>Thanks!</p>
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] |
<p>Electroweak symmetry breaking is manifested by the nonzero masses of the weak W and Z bosons, and requires the existence of a Higgs boson or some other yet unseen mechanism. To solve the problem of flavor.</p>
<ul>
<li><p>why there are six quarks and six leptons forming three families?</p></li>
<li><p>why these particles exhibit such a peculiar pattern of masses and mixing?</p></li>
</ul>
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<p>We start with the metric tensor
\begin{equation}
g_{\mu\nu}(x) = \eta_{\mu\nu} + h_{\mu\nu}(x)
\end{equation}
in the linearised theory, or
\begin{equation}
g_{\mu\nu}(x) = \bar{g}_{\mu\nu}(x) + h_{\mu\nu}(x)
\end{equation}
in the more general case with a curved background. However, in both cases, we apparently need
\begin{equation}
\vert h_{\mu\nu}\vert \ll 1.
\end{equation}
This condition demands that the components of the perturbation are much less than 1, in the coordinate system where the components of the background metric tensor are of order 1. Why do we need this condition?</p>
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] |
<p>I have been under the impression that chirality is at least in part a way to sneak in a Lorentz invariant version of helicity for particles that mass. <a href="http://www.quantumdiaries.org/2011/06/19/helicity-chirality-mass-and-the-higgs/">Flip Tanedo</a> seems to espouse this view in <a href="http://www.lepp.cornell.edu/~pt267/">a somewhat cutesy fashion</a>, for example, though he seems to be using the math rightly. (<em>Mea culpa</em> BTW, with cute little bee-like particles $e$lly $p$eter $p$aul and $\mu$ary in fact.) Tanedo makes this observation:</p>
<blockquote>
<p>... is there some property which is equivalent to helicity in the massless limit
[and] all observers in valid reference frames would measure to be the same for a given particle[?] The good news is that such a property exists; it is called chirality.</p>
</blockquote>
<p>So, having convinced myself that chirality must somehow manage to remain experimentally independent of helicity -- since helicity <em>is</em> of course frame dependent -- imagine my surprise when I noticed <a href="http://physics.stackexchange.com/q/57285/7670#comment116188_57285">Michael Brown's comment</a> about helicity suppressing weak decays of the $\pi^{\pm}$:</p>
<blockquote>
<p>Indeed, you see "helicity" suppressed weak decays of the $\pi^{\pm}$. You could just as well call it chirality suppression since what is being measured is the rotation between chirality and helicity eigenstates which is the result of the mass term.</p>
</blockquote>
<p>Eh?</p>
<p>Someone help me with this: Even if the various constituent components of such decays are <em>by themselves</em> Lorentz independent, what good does that do if their relative ratios are dependent on helicity, which would then as best I can understand it would again make the decay rate of $\pi^{\pm}$ decay frame dependent?</p>
<p>I'm sure I'm missing something here, but I honestly don't know what. I've been looking for the papers on the original experiments, but I was hoping someone could point out where my understanding took an, um, left-hand turn...</p>
| 9,302 |
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<p>I'm developing a 2D game including collisions between many disks. I would like to know how I can get the angle corresponding to the new direction of each disk.
For every disk I have this information : <strong>direction</strong> (integer), <strong>speed</strong> (float), and <strong>position</strong> ($x$;$y$, which are both integers).</p>
<p>Of course I know that the new direction will be the opposite of where the collision happens (disk hit on the bottom-left will go to the top-right), but I don't know how to calculate exactly the new angle.</p>
| 136 |
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<p>Many introductory quantum mechanics textbooks include simple exercises on computing the de Broglie wavelength of macroscopic objects, often contrasting the results with that of a proton, etc.</p>
<p>For instance, this example, taken from a textbook:</p>
<blockquote>
<p>Calculate the de Broglie wavelength for</p>
<p>(a) a proton of kinetic energy 70 MeV</p>
<p>(b) a 100 g bullet moving at 900 m/s</p>
</blockquote>
<p>The pedagogical motivation behind these questions is obvious: when the de Broglie wavelength is small compared to the size of an object, the wave behaviour of the object is undetectable.</p>
<p>But what is the validity of the actual approach of applying the formula naively? Of course, a 100g bullet is not a fundamental object of 100g but a lattice of some $10^{23}$ atoms bound by the electromagnetic force. But is the naive answer close to the actual one (i.e. within an order of magnitude or two)? How does one even calculate the de Broglie wavelength for a many body system accurately?</p>
| 379 |
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<p>Consider the concept of these videos <a href="http://www.youtube.com/watch?v=Zip9ft1PgV0&sns=em" rel="nofollow">http://www.youtube.com/watch?v=Zip9ft1PgV0&sns=em</a> and <a href="http://www.youtube.com/watch?v=cUhgKFV5Ri4&sns=em" rel="nofollow">http://www.youtube.com/watch?v=cUhgKFV5Ri4&sns=em</a> but set in a zero gravity environment and in a fully encapsulated and high pressure container.
My questions are as follows:
What shapes will the water conform to at increasing speeds?
Will the water conform to a parabola and then ultimately a hollow cylinder as it does on earth? </p>
<p>Is it possible that without an opening to spit and by continuously adding more water to the centrifuge, that the water could ultimately fold in on itself and conform to various states of torus, by creating vortices within itself?</p>
<p>[Edit] Also, taking into account 0G, will the water even conform to its bounding container in a manner in which it receives traction and can be spun?</p>
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<p>How do I calaculate the amount of heat energy radiated from the sun in one minute?? Well, i tried some stefan method, but the answer seems far off from the correct one....well, i would thus strongly want to have some help in this regard..</p>
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<p>From what I have read, the problem with modern semiconductors/electronics seems to be quantum tunnelling and heat. The root of these problems is the size of the devices. The electrons are leaking out, and currents are causing active materials to melt.</p>
<p>How far have we become in this regard? Can we make our devices even smaller? What is being done to maintain advancements in computing power? What is the main research, particularly in quantum mechanics and in solid state physics, being done to compute faster using less energy and space?</p>
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<p>Say we have a single, solid circular loop of wire rotating at a constant angular velocity in a uniform magnetic field. There is an induced EMF creating a current in the wire. <em>At the absolute minimum</em>, we should be able to say that the potential V is 2pi periodic (it should have a single value at a single position on the loop). In my understanding of the problem, the wire ought to be equipotential, even. While the wire is radiating energy due to its resistance, that energy is getting put back into the system by whatever is spinning it at constant angular velocity.</p>
<p>Now, break the loop and attach a voltmeter to the two ends of the wire. It will read a potential drop. I am struggling to rectify these two ideas. Help?</p>
<p>(I know there are some similar questions to this, but I have not seen an answer that specifically addresses this issue.)</p>
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] |
<p>The current is maximum through those segments of a circuit that offer the least resistance. But how do electrons know beforehand that which path will resist their drift the least?</p>
| 510 |
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<p>I've read that free neutrons decay into a proton, electron and neutrino with an average lifespan of about 15 minutes. Is there anything physically different about a neutron that has existed for 14 minutes and one that has only existed for one minute? Does a random outside event trigger the decay or is something internally in the neutron slowly falling apart?</p>
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<p>Okay. I have two ways of working out the height of the atmosphere from pressure, and they give different answers. Could someone please explain which one is wrong and why? (assuming the density is constant throughout the atmosphere)</p>
<p>1) $P=h \rho g$, $\frac{P}{\rho g} = h = \frac{1.01\times 10^5}{1.2\times9.81} = 8600m$</p>
<p>2) Pressure acts over SA of Earth. Let r be the radius of the Earth. Area of the Earth is $4 \pi r^2$</p>
<p>Volume of the atmosphere is the volume of a sphere with radius $(h+r)$ minus the volume of a sphere with radius $r$.
$\frac{4}{3}\pi (h+r)^3 - \frac{4}{3}\pi r^3$
Pressure exerted by the mass of the atmosphere is:</p>
<p>$P=\frac{F}{A}$</p>
<p>$PA=mg$</p>
<p>$4\pi r^2 P = \rho V g$</p>
<p>$4\pi r^2 P = \rho g (\frac{4}{3}\pi (h+r)^3 - \frac{4}{3}\pi r^3)$</p>
<p>$\frac{4\pi r^2 P}{\rho g} = \frac{4}{3}\pi (h+r)^3 - \frac{4}{3}\pi r^3$</p>
<p>$3 \times \frac{r^2 P}{\rho g} = (h+r)^3 - r^3$</p>
<p>$3 \times \frac{r^2 P}{\rho g} + r^3 = (h+r)^3$</p>
<p>$(3 \times \frac{r^2 P}{\rho g} + r^3)^{\frac{1}{3}} - r = h$</p>
<p>$(3 \times \frac{(6400\times10^3)^2 \times 1.01 \times 10^5}{1.23 \times 9.81} + (6400\times10^3)^3)^{\frac{1}{3}} - (6400\times10^3) = h = 8570m$</p>
<p>I know that from Occams razor the first is the right one, but surely since $h\rho g$ comes from considering the weight on the fluid above say a 1m^2 square, considering the weight of the atmosphere above a sphere should give the same answer?</p>
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<p>If both electrons are in ground state, characterized by $(1s)^2$, this means that n =1, l = 0. The two electrons occupy the same energy and space (why doesn't pauli exclusion principle prevent this?), so the spatial part is symmetric. </p>
<p>So in order for the state to obey fermi-dirac statistics, the overall wavefunction (spatial*spin) must be anti-symmetric.</p>
<p>Does this mean that the spin part is necessarily anti-symmetric? Thus it must be a tripplet state.</p>
| 9,309 |
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<p>It happened to me quite often (most recently with my wireless keyboard) that a battery stopped to work and then, if I unplug it and then plug it back in it works again, not just for a couple of minutes but even for a day or two, then you can repeat the process but this time it will last less and so on until it finally dies. The same trick works for my tv control. I can't really understand why, can somebody help? </p>
| 9,310 |
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<p>I recently learned about breeder reactors, which are able to create more fissile material than they consume. They seem to be superior to the types of nuclear reactors currently in use. What are the reasons why breeder reactors are not more prevalent.</p>
| 9,311 |
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<p>I had this Homework Problem with a capacitor (parallel plate) that has a group of 3 dielectrics between it like so : </p>
<p><img src="http://i.stack.imgur.com/WbQx2.jpg" alt="Descriptive Image"></p>
<p>Now We were asked to find the equivalent capacitance and the distance of separation between the plates was $d$ and its area was $A$.</p>
<p>Now i assumed that the 3 individual dielectrics would act as individual capacitors and K2 would be parallel to K3 and their resultant series with K1</p>
<p>This was also the method I found in many other books.</p>
<p>$$C_{k1}~=~ \frac{2K_1\epsilon_oA}{d} $$
$$C_{k2}~=~ \frac{K_2\epsilon_oA}{d}$$
$$C_{k3}~=~ \frac{K_3\epsilon_oA}{d}$$</p>
<p>Finally $$C_{2,3}= \frac{K_2\epsilon_oA}{d} + \frac{K_3\epsilon_oA}{d} \implies \frac{(K_2+K_3)\epsilon_oA}{d}$$</p>
<p>And $$C_{(2,3),1} = \frac{\frac{(K_2+K_3)\epsilon_oA}{d}.\frac{2K_1\epsilon_oA}{d}}{\frac{(K_2+K_3)\epsilon_oA}{d}+\frac{2K_1\epsilon_oA}{d}}$$</p>
<p>Our teach told us that the answer was incorrect , When I told him about the books he told us that the books had it all wrong. So I asked him for the solution .</p>
<p>He divided the First Dielectric Into two parts along the line joining the bisector of area :</p>
<p><img src="http://i.stack.imgur.com/eWBHL.jpg" alt="enter image description here"></p>
<p>And Did the shown Charge distribution.</p>
<p>Later he equated the potentials and stuff like so </p>
<p>$$V_{Q'_1,Q'_2}=\frac{Q'd}{K_1\epsilon_oA}+\frac{Q'd}{K_2\epsilon_oA}~~~~....1$$
$$V_{Q''_1,Q''_3}=\frac{Q''d}{K_1\epsilon_oA}+\frac{Q''d}{K_3\epsilon_oA}~~~~....2$$ </p>
<p>then said that $V_{Q'_1,Q'_2}=V_{Q''_1,Q''_3}$ he then put the values of $Q',Q''$ in the equation $$C=\frac{Q'+Q''}{V}$$</p>
<p>Basically getting a result that was quite different from the result given in books . </p>
<ul>
<li>I asked him how a single plate , the one in contact with the K1 dielectric could have Two different charges on the same surface.</li>
<li>He told me that it was because of the accompanying dielectrics </li>
<li>Was my question valid the first point ?</li>
<li>Whose solution is correct?</li>
</ul>
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<p>From the perspective of a far-away observer, matter falling into a black hole never crosses the boundary. Why doesn't a basic symmetry argument prove that Hawking radiation is therefore also frozen on the boundary, and therefore not observable? Wouldn't the hawking radiation have to have started its journey before the formation of the black hole? Furthermore, wouldn't the radiation be infinitely red-shifted?</p>
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<p>To my knowledge if you calculate the coordinate time (time experienced by an observer at spatial infinity) it takes an infinite amount of time for an object to fall past the horizon of a Schwarzschild black hole. Doesn't this imply that it takes an infinite amount of coordinate time for a Schwarzschild Black Hole to form since the last bit of in-falling matter won't ever ross the horizon as observed by someone at spatial infinity? If so, is it possible for other types of black holes (Kerr etc.) to form in finite coordinate time? </p>
| 9,314 |
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<p>Is there an easy (aka intuitive) way to understand that the conserved quantity from time translation symmetry is just what we call energy?</p>
<p>In other words, we use two definitions of energy. One is with Noethers theorem, and I've been told this is the fundamental one. The other is what is you learn in school and is mentioned in the examples below. The question is how to connect this two definitions.</p>
<p>Examples</p>
<ul>
<li>I can lift weights, so they get more energy.</li>
<li>I can boil water for tee, so it gets more energy.</li>
<li>I can burn $CaO$ with carbon, so I get carbide, that has more energy.</li>
<li>..</li>
</ul>
<p>If I define energy as conserved quantity, how do I arrive at my examples..?</p>
<p>(Well, "easy/intuitive" is in the eye of the beholder. Thank you nevertheless.)</p>
<p>Side question: We have energy conservation in thermodynamics. I have never seen a Lagrangian formulation of thermodynamics. Can I only hope for an answer of my main question in theories that have such a formulation?</p>
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<p>Can a Feynman diagram end with more matter than it began with? The answer should be no because it would break the law of conservation of mass, right?</p>
<p>Example:
I learned that two up-quarks could interact to create a Higgs boson (via what I learned as "vector fusion"). If this new Higgs were to decay into an up-quark pair, wouldn't the new up-quark be able to interact with the old one to produce another Higgs, creating a potentially infinite amount of particles?</p>
<p>I'm new to this site (and physics), so if this is unclear, please comment.</p>
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<p>The equation of motion of a general anharmonic oscillator includes a position-dependent force that can be expanded in a Taylor series as</p>
<p>$$m\ddot{x}+2\mu\dot{x}+k_0+k_1x+k_2x^2+k_3x^3\ldots=F.$$</p>
<p>I am considering a case with force $F$ higher than the critical amplitude, which means I'm in the <a href="http://www.scholarpedia.org/article/Duffing_oscillator" rel="nofollow">Duffing</a> regime.</p>
<p>The <a href="http://en.wikipedia.org/wiki/Duffing_equation" rel="nofollow">Duffing equation</a> is</p>
<p>$$m\ddot{x}+2\mu\dot{x}+k_1x+k_3x^3=F,$$</p>
<p>where the third order term is the Duffing perturbation to the simple harmonic oscillator. This destroys the symmetry around the resonance of, say, amplitude plotted against frequency.</p>
<p>I get that we drop the $k_0$ term with a translation to the equilibrium point. However, I don't get why the second order term disappears. The books I've read (in particular Cleland's <a href="http://link.springer.com/book/10.1007%2F978-3-662-05287-7" rel="nofollow"><em>Foundations of Nanomechanics</em></a> and Radons' <a href="http://books.google.co.uk/books?id=L7f_BnmrYvUC&lpg=PR5&ots=Q85wXT7-OX&dq=Nonlinear%20Dynamics%20of%20Nanosystems&lr&pg=PP1#v=onepage&q=Nonlinear%20Dynamics%20of%20Nanosystems&f=false" rel="nofollow"><em>Nonlinear Dynamics of Nanosystems</em></a>) say it's because the harmonic potential is quadratic, hence symmetrical, and hence the sign of $x$ doesn't matter, but that isn't very clear to me; it's not making click. It's probably something very basic but maybe someone can explain it to me simply and briefly? </p>
<p>I've also read that even-ordered nonlinearities only mix signals to DC and twice the drive frequency, as opposed to odd-ordered ones which mix back to the operating frequency band, but I don't quite understand the use of the word "mix" in that context.</p>
| 9,317 |
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<p>In this video, a pool of liquid nitrogen in a metal bowl can cool it enough so that oxygen from the air condenses at the bottom:</p>
<p><a href="http://www.youtube.com/watch?v=Ojl10zMCYkY&list=PLmyvBUHJ4pVC3FrSXsT_5S7dFMKPLG_pS&index=1" rel="nofollow">http://www.youtube.com/watch?v=Ojl10zMCYkY&list=PLmyvBUHJ4pVC3FrSXsT_5S7dFMKPLG_pS&index=1</a></p>
<p>Thermodynamically, it seems that the bowl can't get cold enough. The bowl can't cool off below the temperature at which the nitrogen pressure in the air (80kPa, I got all the vapor pressure information form Air Liquide) equals the vapor pressure of the liquid nitrogen; at that point the rate of evaporation equals the rate of condensation from the pool. This happens at ~76K. This is also the threshold where oxygen can condense (vapor pressure of 20kPa). However, 76K is a lower bound. The actual temperature will be higher when you include condensation of water into/onto the bowl, convective/radiative heat transfer, and the fact that there is additional nitrogen/oxygen above the surface of the pool beyond the 80kPa used in the calculations. In the face of these calculations, why does it seem to work? </p>
<p>My reasoning is that it still gets cold enough for a <em>mixture</em> of nitrogen and oxygen to condense which has enough oxygen to burn the tissue paper, is this correct?</p>
| 9,318 |
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<p>Imagine we have two thermodynamic systems, one a mass of ice and the other an equal mass of liquid water, with both at 273.16K. Each system is isolated, except that they can interact with each other across a boundary that permits the exchange of heat but not matter or work.</p>
<p>What will the two systems look like at equilibrium? Somehow I want to automatically imagine that each system will be identical, a combination of liquid water, ice, and water vapor at 273.16K. But if this is true then the two systems were initially at the same temperature but not in thermodynamic equilibrium, an apparent violation of the zeroth law of thermodynamics. </p>
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<p>There is a somewhat widely accepted argument that the second law of thermodynamics exists because the universe began in a low-entropy state. I'm writing a paper that mentions this (and must be finished today), but I've realised that I don't know who originated this idea. So does anyone know the original reference for this?</p>
<p><strong>Progress</strong></p>
<p>My paper is now submitted, but I'm still curious about the origins of this idea. @Sundak below gives the hint that Boltzmann might have been its originator. Based on this and a bit of searching, I was able to turn up his <a href="http://books.google.de/books/about/Lectures_on_Gas_Theory.html?hl=de&id=-I7QzCXnstEC" rel="nofollow">Lectures on Gas Theory</a>, §90, in which he says "That in nature the transition from a probable to an improbable state does not take place as often as the converse, can be assumed by assuming a very improbable intial state of the entire universe surrounding us...", which sounds very much like the low-entropy big bang idea. However, what he's actually suggesting is that our local region of the Universe had fluctuated into a low-entropy state at some point in the past. He proposes this because he believes the Universe to be infinite in both time and space, implying that the universe as a whole must be in thermal equilibrium. He then goes on to say that at "distances $10^{10^{10}}$ times the distance of Sirius" there might be other civilisations for whom the arrow of time points in the opposite direction. Boltzmann is a fun guy to read. But the idea he's proposing here is not the same as the modern notion that the Universe simply started out in a low entropy state.</p>
<p>Skipping forward many decades, in the abstract for his 1979 paper <a href="http://www.springerlink.com/content/x3m1618084153548/" rel="nofollow">Singularities and Time's Arrow</a>, Roger Penrose describes the attribution of the second law to the universe's boundary conditions as being part of "the normal point of view". I suspect that the references in Penrose's paper might be a good place to look, but it seems that only the abstract is available online.</p>
<p>So that narrows it down to some time between 1896 and 1979, but probably towards the beginning of that period. It could well be that Boltzmann himself proposed the idea in a different work. If anyone has any further insight it'd be much appreciated.</p>
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<p>So on a homework assignment, we are give the width of a well, $a$, and the mass of the particle $m$ and we want to find the average velocity of the particle at the n=1 state. So here is my attempt at the solution:</p>
<p>Using, $\psi (x) = \sqrt{\frac{2}{a}} sin(\frac{\pi x}{a})$, I was able to integrate
$$\langle p^2\rangle=\int_0^adx\psi^*(x) (-i)^2\hbar ^2\frac{d^2}{dx^2} \psi (x)$$
which yielded
$$\langle p^2 \rangle = \frac{\hbar ^2\pi ^2}{a^2}$$</p>
<p>I'm stuck on the remaining parts. So the problem gives us a hint, to use the relation $p^2 = (mv)^2$. I'm not sure how I would go about finding average velocity from $\langle p^2\rangle$, since $\langle v^2\rangle$ and $\langle v\rangle^2$ are not equivalent.</p>
<p>And according to the problem, the speed is supposed to come from the kinetic energy, which I don't understand (I derived average KE by using $\langle p^2 \rangle$, so I don't understand why finding KE is necessary?). </p>
<p>This last part has been giving me trouble, and I would really appreciate any help.</p>
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<p>I'm reading Quantum Field Theory and Critical Phenomena, 4th ed., by Zinn-Justin and on page 154 I came across the statement that the functional integral of a functional derivative is zero, i.e.
$$\int [d\phi ]\frac{\delta F[\phi]}{\delta\phi^{\alpha}(x)} = 0$$ for any functional
$F[\phi ]$. </p>
<p>I would be most thankful if you could provide a mathematical proof for this identity.</p>
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<p>Herbert Dingle and H. Bondi debated the twin paradox on BBC radio before 1971. Does anyone have a link to the audio of this debate? thanks</p>
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<p>If I have a grounded conducting material, then I know that $\phi=0$ inside this material, no matter what the electric configuration in the surrounding will be. </p>
<p>Now I have a conducting material that is not grounded, then there will be (as long as I am dealing with static problems) no electric field inside this material. Therefore the potential will be constant inside this material, right?</p>
<p>Question 1:Therefore, is there any difference in the boundary conditions if I am dealing with a grounded conducting material and an insulator around or a non-charged insulated conducting material and an insulator around? </p>
<p>Question 2:Is it possible to get a non-zero potential inside an uncharged insulated conducting material? Especially, would you get a non-zero potential inside a conducting insulated material due to image charges?</p>
<p>Question 3:Of course, I read a few pages in Jackson's book about this and saw that he substituted the problem of a charged insulated conducting sphere in an external field with the one of having a grounded conducting sphere in the external field that has a charge sitting in the center of the sphere. Then, the magnitude of the extra charge was given by the difference of the initial charge of the sphere minus the induced image charge on the grounded conducting sphere. </p>
<p>Is it possible to make a general substitution like this: Thereby I mean, that we substitute a charged insulated conducting material carrying a charge by a grounded conducting material that has an additional charge(magnitude given by the difference of total charge-image charge) sitting on its surface? So, I would solve the grounded problem and would add the difference of the total charge-image charge to the surface of the material and add this field to the field calculated for the problem of the grounded material. </p>
| 9,324 |
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<p>I have been doing some research on all kinds of sound-related topics lately and have been a bit confused by the different uses of the term "frequency". Of course, the most general meaning of frequency is just how many times something happens during a certain period of time (a second, usually). However, I've also come across many texts referring to a sound containing multiple frequencies, which seemed weird to me at first. Now, after reading up on stuff like Fourier analysis I've come to understand that "frequencies" in this case refers to the frequencies of the sine waves that make up the sound. So I guess my question comes down to this:</p>
<ol>
<li>Is the above correct, or am I still not understanding it correctly in some way?</li>
<li>Assuming it is, is this not pretty ambiguous terminology? You could for example say that a square wave with a frequency of 440 Hz (meaning the waveform repeats 440 times per second) consists of many different frequencies (meaning it consists of many sine waves with different frequencies). It's not hard to see how this could be confusing.</li>
<li>And in that case, why can I not easily find a clear explanation of this anywhere? It took me quite some time to piece this together myself and seems like it really should be pretty basic stuff. For example, I first really started thinking about this when I ran into the <a href="http://en.wikipedia.org/wiki/Nyquist_frequency" rel="nofollow">Nyquist frequency</a>. It seemed logical to me that you needed at least two samples per period to represent any kind of change, so the maximum frequency of the signal could only be half of the sampling rate, but when I started involving different kind of waveforms I realized there would be no way to distinguish between them. It took me a lot of time to figure out that the Nyquist frequency implicitly refers to the maximum frequency that any sine wave present in the signal could have, making it impossible to sample any other waveform at the Nyquist frequency.</li>
</ol>
<p>Of course, if anyone could explain or point me in the direction of some material explaining the importance of sine waves in a more general fashion, and why these things seem to be supposed to be so obvious, that would be appreciated, but the three questions above are what I'm really wondering about.</p>
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<p>As far as i know Zeppelins were abandoned because they were not safe, back in time. Is it still the case in 2011 though? In my opinion it is more economical to operate a Zeppelin than a plane and they are more stable as there is less acceleration so the interior has tables and chairs etc.</p>
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<p>Black hole complementarity states that two observers, one falling into a black hole, and one observing outside, experience two different histories but since they can not communicate there is no inherent contradiction..imagine an observer far from a black hole watching some matter fall into a black hole. The observer sees the matter freeze at the horizon, but the matter itself does fall in. Now, if you measure the mass of the black hole before and after the matter falls in, would you not measure a higher mass? And wouldn't this violate the principal of complementarity? You don't see the matter fall in, yet you see the black holes mass increase. Help?</p>
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<p>My question might be based on a false premise, so here's why I asked. If you look up the meaning of the moment magnitude scale for measuring earthquake size, the articles usually say that each increase in magnitude by 1 point represents a tenfold increase in the amplitude of the wave, but a 31.6 increase in energy. Which suggests the energy is going up as the three halves power. With other waves the energy goes up with the square of the amplitude, so what is going on here?</p>
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<p>I was watching an outtake of Prof. Brian Cox talking to a tv producer about "gravity waves". Their discussion got a bit side-tracked, because the non-scientist didn't seem to understand what a wave "would look like", and also because there seemed to be some difficulties with the phrase "moving through", with Prof Cox saying that a wave doesn't move through space-time and then saying the ripples do.</p>
<p>I'm now not sure what moves when a speaker plays a sound; the air vibrates, but does "ripple outward"?</p>
<p>So, what is a wave?</p>
<p>Obviously, I'm really after a much simplified, probably broken, metaphor that allows me to visualise what a wave is. I understand that it will have flaws.</p>
<p>I forgot to include the youtube link, so here it is:
<a href="http://www.youtube.com/watch?v=6EJ8r9Rhfw8" rel="nofollow">http://www.youtube.com/watch?v=6EJ8r9Rhfw8</a></p>
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<p>What is the exact dependence of the refractive index of air and the temperature? Is there a theoretical derivation of it?</p>
| 9,330 |
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] |
<p>When reporting barometric pressure, weather stations normalize the
result to sea level using a well-known formula (approximately, add 1
inch to the measured pressure for every 1000 feet above sea level, but
I believe the formula they use is more precise). </p>
<p>Does a similar concept exist for temperature? As altitude increases,
temperature normally decreases. Is there a way to compensate for this
effect? </p>
<p>Reason: I often notice "unusually cold" spots on temperature maps,
which are usually the result of high-altitude stations. It'd be nice
to see what the temperature would look like if all stations were at
sea level. </p>
| 9,331 |
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<p>If the universe is made up of ~95% dark matter, and it interacts only gravitationally then why didn't Newton and Kepler discover it before ? Why does it show itself only in the radial velocity profile of stars in galaxies and not in that of planets around the Sun ?</p>
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<p>The mass of the car is 1500 kg. The shape of the body is such that its aerodynamic drag coefficient is $C_D=0.330$ and the frontal area is $2.50 m^2$. Assuming that the drag force is proportional to $v^2$ and neglecting other sources of friction, calculate the power required to maintain a speed of $100 km/h$ as the car climbs a long hill sloping is $3.20^\circ$.<br>
Use the formula:
$$
F_D=\frac{1}{2}C_D\rho_{air}Av^2
$$
where $\rho_{air}=1.2kg/m^3$</p>
<p>I don't know how to incorporate power into the formula. I get that $F_d$ is proportional to $v^2$ so am I able to cancel them out?</p>
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<p>I'm solving the heat equation with time dependent boundary conditions numerically in a 2D system using the <a href="http://en.wikipedia.org/wiki/Alternating_direction_implicit_method" rel="nofollow">ADI scheme</a>. For the purpose of this question, let's assume a constant heat conductivity and assume a 1D system, so
$$
\rho c_p \frac{\partial T}{\partial t} = \lambda \frac{\partial^2 T}{\partial x^2}.
$$
This works very well, but now I'm trying to introduce a second material. This one differs slightly in heat capacity and density but has a very different heat conductivity and is connected to the other material by a sharp interface, i.e. a stepwise change in $\lambda$.</p>
<p>How should this be treated numerically in the <a href="http://en.wikipedia.org/wiki/Alternating_direction_implicit_method" rel="nofollow">ADI scheme</a>? I can think of different approaches:</p>
<ol>
<li>Treat the two materials as independent domains and connect them by a boundary condition calculating the heat flow in and out of the interface in terms of temperature on the other side of the interface in the last time step. Use a simple forward difference for that on both sides of the interface.</li>
<li>Treat it as one domain and use a very fine discretization close the interface as compared to the homogeneous material. Use a scheme like
$$
\lambda_{left} \frac{T_i - T_{i-1}}{\Delta x_i} = \lambda_{right} \frac{T_j - T_{j+1}}{\Delta x_j},
$$
where $i$ and $j$ are the points left and right of the interface, instead of the the standard ADI for those points.</li>
<li>Drop the assumption of constant heat conductivity and use
$$
\rho c_p \frac{\partial T}{\partial t} = \frac{\partial \lambda}{\partial x}\frac{\partial T}{\partial x} + \lambda \frac{\partial^2 T}{\partial x^2}.
$$
But in order do so one needs to approximate derivative of lambda at the step position, i.e. introduce an unknown characteristic width $s$ of the <strong>sharp</strong> interface. I assume, that the (more or less arbitrary) choice of this width will significantly influence the system's behaviour.</li>
</ol>
<p>Any advice?</p>
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<p>Suppose a particle p of three momentum $\vec p$ decays into two particles of 3-momentum $\vec p_1$ and $\vec p_2$. I know the question might sound stupid but right now my brain is full stop:
Is the following "ok" $$\langle \vec p_1, \vec p_2 \vert \vec p\rangle = (2\pi)^3 \delta^3(\vec p-\vec p_1-\vec p_2)?$$ If one wants a Lorentz invariant normalization one should use (see eg Srednicki 2011, eq. 9.3) now four vectors!</p>
<p>$$\langle p_1 p_2 \vert p\rangle = 2p_0 (2\pi)^3 \delta^3(\vec p-\vec p_1-\vec p_2)$$ but my question then is how this translates to nonrelativistic physics/normalizations? Or perhaps there can't be a nonrelativistic limit of this because in ordinary quantum mechanics particle number is fixed?</p>
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<p>Is something special for a quantum system with infinite degeneracy like free particle levels?
$E=\frac{\hbar^2 \vec{k}.\vec{k}}{2m}$</p>
<p>Edit: I mean what is physical (or mathematical) significance of infinite degeneracy? </p>
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<p>From Sakurai eq. 6.4.21a we have that
$$\langle {\bf k} \vert E,l,m \rangle=\frac{\hbar}{\sqrt{M k}}\delta\left(E-\frac{\hbar^2 k^2 }{2M}\right) Y_l^m({\bf\hat k}),$$ where $M$ is the mass of the particle in question. Now I wonder what the generalization of this formula is. What if we have a two particle system where the particles have 3-momenta ${\bf p_1}$ and ${\bf p_2}$ respectively. What is then
$$\langle {\bf p_1}, {\bf p_2} \vert E,l,m \rangle=~???$$</p>
<p>The simplest thing is to copy:
$$\langle {\bf p_1}, {\bf p_2} \vert E,l,m \rangle=\frac{\hbar}{\sqrt{M_1 k_1}}\delta\left(E_1-\frac{\hbar^2 k_1^2 }{2M_1}\right) Y_l^m({\bf\hat k_1}) \frac{\hbar}{\sqrt{M_2 k_2}}\delta\left(E_2-\frac{\hbar^2 k_2^2 }{2M_2}\right) Y_l^m({\bf\hat k_2})$$
but I really don't know if this makes any sense? </p>
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<p><strong>Quantum entanglement</strong> is one of the most fascinating and mysterious phenomena in nature. It needs no interactions, or any sort of exchange for it to take place. It is possible, not against any rules of physics as far as we know, that all matter that was created in the early universe was in an entangled state. </p>
<p><strong>The question is:</strong></p>
<p>Is it possible to explain the uniformity and isotropy of matter in the universe, by means of quantum entanglement in the early 'days' of universe? If that could be possible, would it mean that there would be no need for the inflationary model any more?</p>
<p>If this problem has been researched in detail, any references posted will be appreciated.</p>
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<p>I’ve a got a question; and I am hopeful that you can provide any information or direct me to a better resource. I'm not a physicist; so please correct me if I'm wrong.</p>
<p>Scanning Probe Microscopy (SPM) can be used to probe and map the surface of electrical potential distribution. For example, let’s say that a Scanning Kelvin probe microscopy (SKPM) was used in some experiment to provide quantitative values of surface potential by determining the potential difference between the conductive probe tip and the sample surface (say NW assemblies). </p>
<p>I was wondering whether there is any physical formula to model/describe the behavior of the electrical potential distribution at nano/atomic scale.</p>
<p>Also, do you think Maxwell's equations holds?</p>
<p>I appreciate any input.</p>
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<p>According to the paper <a href="http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.47.6573&rep=rep1&type=pdf" rel="nofollow">Quantum Mechanics Beyond Hilbert Space</a> by J.P. Antoine, several mathematical structures have been devised to make mathematical sense of Dirac's formulation of quantum mechanics (specifically, QM with operators having continuous spectra). I was wondering if there was a general set of axioms that could be specified that we would want any such structure to satisfy for it to be a sufficient mathematical image of quantum mechanics. </p>
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<p>I am looking for a book explaining physics from, say, Galileo and Newton till now;
the book should be written using some math, similarly to my physics books when
I was at the university (Halliday Resnik). I am searching a book that take care of explaining the true "evolution" of reasoning, discoveries, experiment to prove or disprove something.</p>
<p>I do not like a book that gives definitions without explaining what had happened.
I also like to find, on that book, explanations on how actual measurements are taken
(how can we measure the speed of light, the mass of celestial bodies, the attributes of electron, and how were the measurements taken in the past - how was possible to measure the speed of light in 1700-1800?).</p>
<p>I am not very interested in recent theories like strings, branes.... so it is OK if these are not discussed.</p>
<p>I like to find a lot of discussions on concept that are related, like for instance the implications on space dimensionalities, ether non-existence, strange facts, missing explanations.</p>
| 9,341 |
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<p>Which will go further if a batter manages to hit it with a baseball bat: a baseball thrown to the batter at 90 miles per hour or one thrown at 60 miles per hour?</p>
| 9,342 |
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<p>I am in the middle of a discussion with a friend about the meaning of the term "clockwise".</p>
<p><a href="http://en.wikipedia.org/wiki/Clockwise">Wikipedia</a> indicates that a clockwise rotation goes as <strong>top-right-down-left</strong>.</p>
<p>However, my friend argues that "the clock is facing opposite to you. So if you rotate top-right-down-left facing the side the clock is facing then it is anti-clockwise."</p>
<p>Our question, therefore, is</p>
<blockquote>
<p>does "clockwise" mean <strong>top-right-down-left</strong> or <strong>left-down-right-top</strong>?</p>
</blockquote>
| 9,343 |
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<p>This might be a little naive question, but I am having difficulty grasping the concept of irreducible tensors. Particularly, why do we decompose tensors into symmetric and anti-symmetric parts? I have not found a justification for this in my readings and would be happy to gain some intuition here. </p>
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<p>I once had an old microscope, that included a projection screen that could be mounted instead the eyepiece. It showed a quite decent palm-sized image. </p>
<p>Now I have a new microscope, and removed the eyepiece to mount a single lens reflex camera. However, the image projected on the ground glass, and also on the imaging sensor , shows a brightened fluff around the image center. By moving the microscope in respect to the camera, the spot can be moved a little, but would always stay inside the field of view. What is the reason of this effect ? </p>
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<p>I was going through Griffiths chapter on potentials and fields just to brush up on a few old things. He gets to <a href="http://en.wikipedia.org/wiki/Jefimenko%27s_equations" rel="nofollow">Jefimenko's equations</a> by this general path: </p>
<ol>
<li><p>Maxwell's equations</p></li>
<li><p>Introduce scalar and vector potentials</p></li>
<li><p>Reduce MW's equations to the two equations with only potentials (no fields)</p></li>
<li><p>Choose the Lorentz gauge, which frames the two equations as 4-D Poisson equations: $$\Box^2V = -\rho/\epsilon, \quad \Box^2 \vec A = -\mu\vec J$$</p></li>
<li><p>We already know the solutions to the static case ($\dot{\rho},\dot{\vec J} = 0$ means $\Box^2$ becomes $\nabla^2$)</p></li>
<li><p>He says and then briefly proves that to get the non-static answers, we just use the retarded time $t_r = t - |r - r'|/c$ in the integrals for the static case</p></li>
<li><p>So now we have the potentials for a non-static source, and we just plug them into $\vec E = -\nabla V - \frac{\partial \vec A}{\partial t}$ and $\vec B = \nabla \times \vec A$ which apparently give us Jefimenko's equations (say $R = |r - r'|$, $\hat R$ is the unit vector in the direction of $R$, and $J$ and all $r$'s are vectors):</p></li>
</ol>
<p>$$E(r, t) = (1/4\pi\epsilon)\int[\frac{\rho(r',t_r)}{R^2}\hat{R} + \frac{\dot{\rho}(r',t_r)}{cR}\hat{R} - \frac{\dot{J} (r',t_r)}{c^2R}]d\tau '$$</p>
<p>$$B(r, t) = (\mu/4\pi)\int[\frac{J(r',t_r)}{R^2} + \frac{\dot{J}(r',t_r)}{cR}] \times \hat{R} \ d\tau '$$</p>
<p>So it all makes sense to me, the derivation anyway. But is there a simple intuitive reason for why the $E$ equation has a $\dot J$ term but the $B$ term has no $\dot \rho$ term? I'd suspect it has something to do with the "asymmetry" in Maxwell's equations with respect to $E$ and $B$ (which I also don't really understand), but that doesn't answer much.</p>
<p>Also, what are some concrete examples of the difference between $\dot \rho$ and $\dot J$? What I mean is, it seems like they must usually be very linked, because if you have a changing amount of charge $\dot \rho$ at some point, that implies that there is a changing amount of current $\dot J$ at the point also, because the charge at that spot has to go somewhere (so $\dot J \neq 0$) for $\rho$ to change. I guess I could think of some mythical (to me anyway, because I don't know if this is possible) chemical reaction where charge at a point "disappears" and "reappears" without having to actually move spatially.</p>
<p>I have one more small point of confusion about Jefimenko's equations: Griffiths says "In practice Jefimenko's equations are of limited utility, since it's typically easier to calculate the retarded potentials and differentiate them, rather than going directly to the fields." Isn't that exactly what he did in the process I described?</p>
| 9,346 |
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<p>Of course, I mean that the edge of the disk is traveling at the speed of light. This is a question that popped into my head a few years ago when I was learning about some basic relativity in high school.</p>
<p>According to what I learned, if an object were to move at near-light-speed, it would be "compressed" along the direction of movement (for a stationary observer), but I'm asking myself what would happen if a disk were to spin along its axis. <strong>Would its circumference also get compressed</strong>?</p>
<p>Even though there is no such material that would resist the extreme centrifugal forces of this spinning disk, I think that a stationary observer would see this disk <strong>shrink</strong> as it approaches the speed of light.</p>
<p>But if the disk shrinks, a point on its circumference wouldn't travel so fast anymore (because of the reduced radius), so I'm guessing that the maximum speed of such a point is lower than speed of light.</p>
<p>I'm pretty sure I am way off the tracks as far as relativity is concerned, but I made some calculations back then, and using some help from Wolfram|Alpha I came to the conclusion that if my assumptions are correct (that the disk would shrink, and high-school geometry still applies), the maximum speed of a point on the circumference of the disk would be exactly <strong>c/2</strong>, before requiring an infinite amount energy to increase its speed any further.</p>
<p>So... what <em>would in fact happen</em>?</p>
| 306 |
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] |
<p>This is a question I've been mulling over for a while and I'm hoping someone here can point me in the right direction. Sorry if it's a bit of a novice question. For the record, I don't fully know GR, but don't let that stop you from using it in the answer.</p>
<p>Since the universe is expanding - that is, the spacetime metric is expanding by way of a near-exponentially increasing scale factor - we can say that the distance between any two non-bound objects is increasing over time. Herein lays my dilemma; if there were two objects separated by a large distance that had no relative velocities initially, after a long time, the effects of expansion would cause them to have large apparent velocities away from each other. Given that there hasn't been any acceleration to cause these velocities, are there still relativistic effects in play? That is, is there time dilation between the two frames?</p>
<p>Furthermore, given long enough time, the rate of increasing distance between the two objects could place them outside of their visible horizon (ie they are travelling away from each other at superluminal velocities). Since there was still no acceleration to achieve this feat, what can one say about the relativistic effects in this case?</p>
<p>At first I thought this was an easy question. I thought of course there would be relativistic effects and when the objects go superluminal, the visible horizon is there to ensure there can never be causal contact and thus preserve physics. But then I thought what if spacetime stopped expanding abruptly (seems crazy but as far as I know, nothing makes this completely impossible)? Since there was no initial relative velocities, wouldn't the two objects return to being in the same inertial frame? And seeing as none of them experienced any sort of acceleration, how then could we describe their two final states? By which I mean, if we were to assume there were relativistic effects during transit, how would we overcome such simple paradoxes like the twin paradox, or other relevant ones?</p>
<p>At this point, I'm stumped. I even attended a lecture by <a href="http://en.wikipedia.org/wiki/Miguel_Alcubierre" rel="nofollow">Miguel Alcubierre</a> since he would have had to consider these types of effects in his design... No help. Equations are great to illustrate a point, but I'm really going to need a conceptual answer as well to fully understand this.</p>
| 9,347 |
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<p>Suppose we have 1 kg of wood and 1 kg of uranium and if we need to find out how much energy would each of the substance give, we'd have to use Einstein's mass-energy relation as follows:</p>
<p>In the case of wood, $E_{wood} = 1 × (3×10^8)^2 = 9 × 10^{16} J$</p>
<p>In case of Uranium, $E_{uranium} = 1 × (3×10^8)^2 = 9 × 10^{16} J$</p>
<p><em>Question:</em> According to the equation, both give the same amount of energy. But in reality, 1 kg of uranium will give a lot of energy when compared to 1 kg of wood. What have I missed? Any explanation regarding the $E=mc^2$ <em>problem</em> would be helpful...</p>
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<p>I'm reading often that a possible reason to explain why the Nobel committee is coping out from making the physics Nobel related to the higgs could be among other things the fact that the spin of the new particle has not yet been definitively determined, it could still be 0 or 2.</p>
<p>This makes me wonder if the spin would (very very surprisingly!) finally be discovered to be 2, this then necessarily would mean that the particle has to be a graviton? Or could there hypothetically be other spin-2 particles? If not, why not and if there indeed exist other possibilities what would they be?</p>
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] |
<p>I found this part in the book of Bernard Schutz on GR. (Sec 12.2: General Relativistic Cosmological Models). </p>
<blockquote>
<p>Robertson-Walker metrics: We shall adopt the following assumptions about the universe: (i) spacetime can be sliced into hypersurfaces of constant time which are perfectly homogeneous and isotropic; and (ii) the mean rest frame of the galaxies agrees with the definition of simultaneity.</p>
</blockquote>
<p>Can someone kindly explain the second assumption, in particular what is a "mean rest frame" for a galaxy.</p>
| 9,350 |
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<p>if the potential is bounded below $ V(x)=V(-x) \ge a $ for some real number a</p>
<p>and we can be sure that as $ x\rightarrow \infty $ we know that $ V(x) \ge 0 $ then does it mean that the energies for our one dimensional system will be positive ??</p>
<p>my idea WKB quntization $ N(E)= 2\int_{0}^{a} \sqrt {E_{n}-V(x)} $ here $ V(a)=E $ is a turnign point so if the potential is positive for big 'x' then the energies should be also positive otherwise the epxresion inside the integral would be complex am i right ?</p>
| 9,351 |
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<p>We know the Schrodinger equation for free Hamiltonian is : </p>
<p>$$
i\hbar\frac{\partial\psi}{\partial t} = H_f \psi
$$
the wave function could be written as
$$
\psi(x,t)=\hat{S}(t) \psi(x,0)
$$
$$
\hat{S}(t)=\exp\left(-\frac{i}{ \hbar }H_ft\right ) .....(1)
$$
I want to know that how exponential term arisen in the equation (1) and how can we write the equation below . Please show me the matrix manipulation for the below equation .</p>
<p>$$
\hat{S}(t) = \tau_3 S^\dagger \tau_3
$$</p>
<p>Thanks .</p>
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