a) Prove that the time evolution of the density operator rho (in the Schrodinger picture)?

Question

a) Prove that the time evolution of the density operator rho (in the Schrodinger picture) is given by – rho(t)=U(t,to) rho(to) hermitian conjugate of U(t,to).
b) Suppose 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 Schrodinger equation.

Answer 1

Dear Lindsay,
I notice that you have not bothered to acknowledge answers to questions that you asked 5 months ago. I have marked them now. Do you realise the amount of time that answers to this type of high level question takes? Some 2 hours! Do you think this is fair on people who have a Genuine Need for assistance in their Degree Courses.



This is a standard question in a higher Physics Degree Course.
I threw my notes away, but if no other person answers I'll try looking it up

RCat

I had to look it up in:
Richard C Tolman: The Principles of Statistical Mechanics
don't try to read this book end to end just use it for the bits of interest to you!!!!!!!!! CHAPTERS 7 8 & 9


PART A TOLMAN pp237 SECTION6.3
1) Use use the governence by Schroedinger's Equation to show that the time dependence of an operator F is determine by the commutator [H,F].
2) You work out the expectation value of F using U* and U.

Essentially it says that if an operator commutes with the hamiltonian operator then the physical quantity associated with it is conserved.



PART B
I assume that the Question asked is at roughly the same level in which case you have to deal with a state expansion in terms of Eigenvector and Eigenvalues, and their time evolution..
This is covered in Section 66b

If my assumptiion is wrong - have you covered Liouville's Theorem in QMech. Then if you are dealing with the time evolution of ensembles you will need Section 90 in the same Works!!!!!!!!!