Introduction of the density operator: the pure case A. Description by a state vector Consider a system whose state vector at instant t is:

Dr. Wasserman in his text, when introducing quantum thermal Mathematical tools of crucial importance in quantum approach to thermal physics are the density operator op and the mixed state operator M. They are similar, but not identical. Dr. Wasserman in his text, when introducing quantum thermal physics, often “switches” from op to M or vice versa, and one has to be alert when reading and always know which operator the text is talking about at a given moment. I thought it would help if you could learn about the density operator not only from Dr. Wasserman’s text, but also from another source, and therefore I made a short “auxiliary” slide presentation about the density operator and its significance, based on another book (“Quantum Mechanics” by Cohen-Tannoudji et al.). The pages I used for preparing this presentation will be given to you as a handout. Cohen-Tannoudji uses a slightly different notation than Dr. Wasserman, but I decided not to change it.

B. Description by a density operator Relation (6) shows that the coefficients c(t) enter into the mean values through quadratic expressions of the type These are simply the matrix elements of the operator, the projector onto the ket as it easy to show using (3): It is therefore natural to introduce ther density operator ρ(t) defined by: The density operator is represented in the {|un} basis by a matrix called density matrix whose elements are:

We are going to show that the specification of ρ(t) suffices to characterize the quantum state of the system: that is, it enables us to obtain all the physical predictions that can be calculated from . To do this, let us write formulas (4), (6) and (7) in terms of the operator ρ(t). According to (10), relation (4) indicates that the sum of the diagonal elements of the density matrix is equal to 1: In addition, using (5) and (10), formula (6) becomes:

Finally, the time evolution of the ρ(t) operator can be deduced from the TDSE: SUMMARY