Step Momentum Operator

TL;DR

This paper introduces the concept of a quantum particle with step momentum, deriving energy eigenvalues and eigenfunctions within a generalized momentum framework, highlighting differences between Hermitian and -symmetric cases.

Contribution

It presents a novel quantum model with step momentum, extending the generalized momentum operator to analyze bound states in square wells.

Findings

Infinite bound states for Hermitian step momentum

Finite bound states for -symmetric momentum

Derivation of energy eigenvalues and eigenfunctions

Abstract

In the present study, the concept of a quantum particle with step momentum is introduced. The energy eigenvalues and eigenfunctions of such particles are obtained in the context of the generalized momentum operator, proposed recently in [13, 14]. While, the number of bound states with real energy for the particles with Hermitian step momentum inside a square well is infinite, it is finite for a particle with \mathcal{PT} -symmetric momentum.