Derivation of a time dependent Schr\"odinger equation as quantum mechanical Landau-Lifshitz-Bloch equation

The derivation of the time dependent Schr"odinger equation with transversal and longitudinal relaxation, as the quantum mechanical analog of the classical Landau-Lifshitz-Bloch equation, has been described. Starting from the classical Landau-Lifshitz-Bloch equation the transition to quantum mechanics has been performed and the corresponding von-Neumann equation deduced. In a second step the time Schr"odinger equation has been derived. Analytical proofs and computer simulations show the correctness and applicability of the derived Schr"odinger equation.