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[itex] \rho_{ab}(t) = c_a c^*_be^{-it(E_a -E_b)/\hbar} [/itex]

How is it that in the long time limit, this reduces to [itex] \rho_{ab}(t) \approx |c_a|^2 \delta_{ab} [/itex]?

Is there some characteristic time scale here? Or has the density matrix been

*time averaged*to get rid of the oscillatory terms (off diagonal coherences) ?

I'm studying the quantum harmonic oscillator, if that helps. Thanks!

EDIT: The Hamiltonian for the system described by [itex] |\Psi(t)\rangle [/itex] is just the standard harmonic oscillator hamiltonian. No interaction terms are present, so the problem is that of an isolated simple harmonic oscillator.