Von Neumann equation

The Von- Neumann equation (after John von Neumann ) is the quantum mechanical analogue of the Liouville equation of classical statistical mechanics; they describe the time evolution of the density operator in the Schrödinger picture:

Is the system of the Hamiltonian and a commutator. The density operator. This is the probability of measuring the pure state in a mixture, if the states are orthogonal. The trace of the density operator returns 1 since.

Discussion

The general solution of the von Neumann equation, where the time evolution operator and its adjoint operator are used:

The density operator is stationary, if this reversed with the Hamiltonian.

Using the von Neumann equation, one can show that the trace of the square density operator is constant in time:

Here, the cyclic invariance of the trace has been exploited in the penultimate step. Because with equality if and only if a pure state describes, it follows that pure states remain pure and mixed mixed.

Expectation values ​​of operators are expressed by. The time dependence of the expected value

In the stationary case is equal to:

The expectation value of a measurement of time-independent observables in the stationary case is time-independent.

Derivation

The von Neumann equation can be derived from the Schrödinger equation.

It is the partial derivative of the statistical operator where one takes into account the product rule:

The Schrödinger equation is of Hilbert space vectors ( Ket )

And for dual Hilbert space vectors (Bra)

This one is above:

Simplify supplies the von Neumann equation: