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(adsbygoogle = window.adsbygoogle || []).push({});

[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 beentime averagedto 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.

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Decoherence in the long time limit of density matrix element

**Physics Forums | Science Articles, Homework Help, Discussion**