von Neumann's hypothesis concerning coherent states

An orthonormal basis of modified coherent states is constructed. Each member of the basis is an infinite sum of coherent states on a von Neumann lattice. A single state is assigned to each unit cell of area h (Planck constant) in the phase plane. The uncertainties of the coordinate x and the square of the momentum p(2) for these states are shown to be similar to those for the usual coherent states. Expansions in the newly established set are discussed and it is shown that any function in the kq-representation can be written as a sum of two fixed kq-functions. Approximate commuting operators for x and p(2) are defined on a lattice in phase plane according to von Neumann's prescription.