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## 7.2.1 Separation of Neumann Equation

For the Hamiltonian 6.2 the von Neumann equation for the density matrix reads (7.11)

To regain the Schrödinger equation from the von Neumann equation we make a separation ansatz for the density : (7.12)

We get (omitting in the function arguments) (7.13)

Factoring out and gives (7.14)

Division by separates the equation and we get (with as separation constant) (7.15) (7.16)

If is a solution to Equation 7.15 with separation constant , then is a solution to Equation 7.16 with complex conjugate separation constant and the density matrix is of the form corresponding to a pure state.

We get an additional term in the separated equation, which is at first surprising since we are not in the stationary case. However, the von Neumann equation is invariant under a change of to , while the Schrödinger equation is not.

By separation of the transient von Neumann equation we get the Schrödinger equation with an additional term . This shifts the value of in the stationary Schrödinger equation. But varying the set of solutions stays the same. So we can set without loss of generality.