On integral representation of the translation operator

Abstract

The formulation in the explicit form of quantum expression of the one-dimensional translation operator as well as Hermitian operator of momentum and its eigenfunctions are presented. The interrelation between the momentum and the wave number has been generalized for the processes with a non-integer dimensionality. The proof of the fractional representation of the translation operator is considered. Some aspects of the translations in graduate spaces and their integral representation, as well as applications in physics are discussed. The integral representation of the translation operator is proposed.