An operator for simultaneous unsharp measurement of coordinate and momentum

Abstract

Starting from a complete but not overcomplete set of coherent states defined on a lattice in the phase space, we construct the orthonormal basis using the procedure due to Lodwin, in which all states to be normalized enter the procedure on equal footing. We show that our normalized states may be interpreted as eigenstates of the operator of simultaneous unsharp measurement of coordinate and momentum. In a sense we followed the classical idea of von Neumann but without drawbacks which in his construction of such an operator were due to the use of the Gram-Schmidt orthonormalization procedure which is somewhat inappropriate for this purpose. We discuss the obtained results.