Can a Pure Ensemble Evolve into a Mixed Ensemble?

[beans73]

Homework Statement

Hi there. just working on a problem from sakurai's modern quantum mechanics. it is:

A) Prove that the time evolution of the density operator ρ (in the Schrödinger picture) is given by
$ρ(t)=U(t,t_{0})ρ(t_{0})U^\dagger(t,t_{0})$

B) Suppose that we have a pure ensemble at t=0. Prove that it cannot evolve into a mixed ensemble as long as the time evolution is governed by the Schrödinger equation.

Homework Equations

The Attempt at a Solution

Working out:

part a) ok, so what I've done is simply say the state |α$^{i}>$ at some time t can be described as:

$|α^{i};t>=U(t)|α^{i};t_{0}>$

Knowing that:
$ρ(t)=\sum w_{i}|α^{i}><α^{i}|$

then
$ρ(t)=\sum w_{i}U(t)|α^{i};t_{0}><α^{i};t_{0}|U^\dagger(t)$
$ρ(t)=U(t,t_{0})ρ(t_{0})U^\dagger(t,t_{0})$

part b)
for this i looked at the trace of ρ$^{2}$

$tr(ρ^{2}))=tr(U(t)ρ(t_{0})U^\dagger(t)Uρ(t_{0})U^\dagger)$
$tr(ρ^{2}))=tr(ρ(t_{0})ρ(t_{0})U^\dagger(t)U(t)$
$tr(ρ^{2}))=tr(ρ^{2}(t_{0}))$all the other questions i have been given in this class have taken a couple of pages worth of working out, and that has made me paranoid that I'm over-simplifying this problem and possibly missing something. any feedback would be much appreciated.

cheers guys!

 

[TSny]

That all looks good to me. Alternatively, for part (b) you could try to show that the pure state condition $ρ^{2}=ρ$ holds at all times if it holds at $t = t_0$.