A detailed proof of the von Neumann's Quantum Ergodic Theorem

We present a simplified proof of the von Neumann’s Quantum Ergodic Theorem. This important result was initially published in german by J. von Neumann in 1929. We are interested here in the time evolution ${\psi }_t$, $t\ge 0$, (for large times) under the Schrodinger equation associated to a given fixed Hamiltonian $H:\mathcal{H}\to \mathcal{H}$ and a general initial condition ${\psi }_0$. The dimension of the Hilbert space $\mathcal{H}$ is finite.