Hermitian operators and boundary conditions

Abstract

The case of the hermeticity of the operators representing the physical observable has received considerable attention in recent years. In this paper we work with a method developed by Morsy and Ata [1] for obtaining Hermitian differential operators independently of the values of the boundary conditions on wave functions. Once obtained these operators, called intrinsically Hermitic operators, we build the Hamiltonian for the harmonic oscillator, hydrogen atom and the potential well of infinite walls. In the first two cases we use the factorization method of ladder operators (also intrinsically Hermitic) and show that results obtained with conventional operators, based on the annulation of the wave functions on the boundaries, are preserved. For the infinite well we show that the version of the Hamiltonian intrinsically Hermitic provides a solution to a paradox that appears in a particular wave function.