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I can not prove anything. I hope that this hand-waving argument is satisfying for you. Hamiltonian dynamics is a continuous process. Two neighbouring points of the phase space can only move continuously away from each other. There is no sudden jump. The phase space can not be split in disjunct ensemble by a finite time time evolution because if this was ...


1

To calculate explicitly the curvature and geodesic equations for the conical spacetime you need an explicit metric. The metric $ds^{2}=dr^{2}+r^{2}d\phi^{2}$ describe a conical spacetime in the range of definition of the coordinates $(r,\phi)\in (0,\infty)\times [0,2\pi-\alpha)$. You can notice that this metric describe a flat spacetime in the domain of ...


4

This is a very good question. The same operator algebra does not imply that $H(J,h)$ and $H(h,J)$ have the same spectrum. As has been mentioned in Dominic's answer, even the ground state degeneracy is different under the interchange of $J$ and $h$ ($J\gg h$: symmetry-broken two-fold degeneracy, and $J\ll h$ unique ground state), therefore it is impossible to ...


1

1) In general, an algebra can have many representations. In this case, however, if you assume that there is a unique joint +1 eigenstate of the $\sigma_i$'s, that determines the representation uniquely. [All the other states can be found from this state by applying products of $\sigma_i^x$to it. And from the anti-commtation of $\sigma_i^x$ and ...


9

I was always told that to find whether or not a field is conservative, see if the curl is zero. This is almost always true, but not always true. I have now been told that just because the curl is zero does not necessarily mean it is conservative. Correct! To illustrate what's going on, let's do an example. Conside the following vector field: ...


3

Basically it means that in the case of OAM=0 the wave fronts make a structure similar to a stack of plates, and in the case of OAM=1 they make a helix-like structure, and 1 refers to the helix multiplicity (for a double helix it would be 2 and so on). One cannot be changed to the other continuously, so this is a topological feature. There are other ...


-1

Mantras like “the Bloch sphere contains all of the observable information about the system” are junk. This implicitly refers to the concept of a (pure) quantum state, and confuses it with an important concept of observable, but quantum states are a tricky stuff and the word “information” may become especially treacherous. Let us start from mathematics. ...



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