6.1.1.2 Von Neumann Equation

For quantum statistical mechanics the Schrödinger equation gives rise to the von Neumann equation for the density operator $\rho$

$$\imath \hbar \frac{\partial \rho}{\partial t} = [ H, \rho ] ,$$ (6.12)

where the brackets denote the commutator

$$[A,B] = AB - BA$$ (6.13)

of operators $A,B$. The density operator $\rho$ is assumed self-adjoint, positive and of trace class. The density matrix $\rho(x_1,x_2)$ is defined by

$$\rho(x_1,x_2) = \langle x_1 \vert \rho \vert x_2 \rangle .$$ (6.14)

For a pure state $\psi$ the density operator is written in bracket notation as

$$\vert\psi\rangle\langle\psi\vert$$ (6.15)

and the corresponding matrix in position space is given by

$$\rho(x_1, x_2) = \psi(x_1) \psi^*(x_2) .$$

The probabilistic structure is now given by the trace operation ${\mathrm{tr} }$. The expectation value of an observable $A$ becomes

$$\bar{A} = {\mathrm{tr} (}A \rho) = {\mathrm{tr} (}\rho A) .$$ (6.16)