it's about quantum physics: evolution of pure ensemble to mixed ensemble.....help please!?

Question

why does a pure ensemble at t=0 cannot evolve into a mixed ensemble as long as the time evolution is governed by the Schrodinger equation?

(if it is proven mathematically, please show me...)

Answer 1

For an ensemble of quantum systems, you need to use a density matrix, which is a self-adjoint (or Hermitian) positive-semidefinite matrix, of trace one, that describes the statistical state of a complex quantum system.

This is the quantum-mechanical analogue to a phase-space probability measure (probability distribution of position and momentum) in classical statistical mechanics. The need for a statistical description via density matrices arises when one considers either an ensemble of systems, or one system when its preparation history is uncertain.

Now at t=0, we have a "pure ensemble", which cannot be decomposed as a convex combination of different ensembles, because in quantum mechanics, a pure density matrix is one of the form |φ><φ|. A pure ensemble corresponds to having many copies of the same quantum state. Hence, the time evolution of a pure ensemble is another pure ensemble.