- #1
soviet1100
- 50
- 16
For a state [itex] |\Psi(t)\rangle = \sum_{k}c_k e^{-iE_kt/\hbar}|E_k\rangle [/itex], the density matrix elements in the energy basis are
[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.
[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.