von Neumann-Landau equation for wave functions, wave-particle duality and collapses of wave functions

It is shown that von Neumann-Landau equation for wave functions can present a mathematical formalism of motion of quantum mechanics. The wave functions of von Neumann-Landau equation for a single particle are `bipartite', in which the associated Schr\"{o}dinger's wave functions correspond to those `bipartite' wave functions of product forms. This formalism establishes a mathematical expression of wave-particle duality and that von Neumann's entropy is a quantitative measure of complementarity between wave-like and particle-like behaviors. Furthermore, this extension of Schr\"{o}dinger's form suggests that collapses of Schr\"{o}dinger's wave functions can be regarded as the simultaneous transition of the particle from many levels to one.