Extension of the caldirola procedure in space

Abstract

Based on results known until now of Caldirola’s chronon theory, we present an extension of this theory to the space co-ordinates with the use ofq-representation. First the new momentum operators are defined. These depend on a constant with dimension of length and their commutators with the position operator satisfy noncanonical commutation relations. The eigenfunctions and eigenvalues of the operators are found. We also study with this new extension the problems of free particles and the discrete time-dependent and time-independent Schrodinger equations are found. We next determine the coherent states of the new annihilation operators and finally calculate the Bloch density matrices for both cases&—Caldirola-Montaldi procedure and its new extension. These are noncanonical.