  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.
