I was told that quantum chaos is just a system whose Hamiltonian's classical version shows chaotic behavior. However, I just wondering

  1. what happens when one eigenstate of this Hamiltonian evolves?

  2. what the chaotic (nonlinear, of course) Hamiltonian looks like in quantum version (for Hamiltonian should be a linear one)?

I know that mean-field theory could reduce the linear Hamiltonian into a nonlinear form, but it is not an exact form.

Maybe a good start point is the BEC equation and GP equation. I hope you can help me here.

For the second question, after a whole year, I think the many-body interaction will directly introduce the nonlinearity. However I still have no idea of how the eigenstate will evolve under "quantum chaotic" Hamiltonian.

  1. Just apply the normal rules. If the Hamiltonian is time-independent, the eigenstate gains an overall phase related to its energy. If the Hamiltonian is time-dependent, then the time evolution operator can be obtained by exponentiation of the Hamiltonian, and that determines the evolution of the eigenstate.

  2. It depends on the Hamiltonian. The coordinates in a classical Hamiltonian have corresponding operators. If all of the operators involved commute with each other, then you can just directly replace the classical coordinates with operators. If there are some non-commuting operators, you must re-derive the Hamiltonian, paying attention this time to the non-commutativity of the operators (whereas in the classical version, all coordinates commute).

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