FOUNDATIONS OF QUANTUM MECHANICS

54 CHAPTER III. THE POSTULATES
which yields for (III. 55)
W (t) =
M∑
i=1
∑n i
j=1
w i U (t − t 0 ) W i,j (t 0 ) U † (t − t 0 ), (III. 57)
and therefore
W (t) = U (t − t 0 ) W (t 0 ) U † (t − t 0 ). (III. 58)
With (III. 11) we find
i d dt W (t) = [H, W (t 0)], (III. 59)
which is the analogue of the Liouville equation of motion, (III. 21), describing the time evolution of
the states ρ. Equation (III. 59) is called the Liouville - Von Neumann equation, it is the generalization
of the Schrödinger equation to an equation for mixed states.
The extensions of the Schrödinger postulate and the projection postulate for mixed states can now
be formulated.
5 ′ Generalized Schrödinger postulate. If no measurements are made on the physical system, the
time evolution of the state of the system is described by a unitary transformation,
W (t) = U (t − t 0 ) W (t 0 ) U † (t − t 0 ). (III. 60)
6 ′ Generalized projection postulate, discrete case. If the system is in a state W when a measurement
is made on a physical quantity A corresponding to an operator A having a discrete spectrum,
and the outcome of the measurement is the eigenvalue a i ∈ R, the system is, directly
after the measurement, in the eigenspace corresponding to the eigenvalue a i ,
W P a i
W P ai
Tr P ai W P ai
. (III. 61)
◃ Remark
Remember that, in general, the projectors P ai do not have to be 1 - dimensional. ▹
Finally, we give a theorem concerning the generalized Schrödinger postulate which is important
for the measurement problem.
VON NEUMANN’S THEOREM A:
The properties ‘pure’ and ‘mixed’ are invariant under a unitary time evolution.