Generalized momentum operators from Fourier transform correspondence

TL;DR

This paper introduces a Hermitian generalization of the quantum momentum operator derived from Fourier transform principles, leading to local flow dynamics and a deformed spectrum in quantum systems.

Contribution

It presents a novel Hermitian generalization of the momentum operator based on Fourier transform correspondence, expanding the algebraic framework in quantum mechanics.

Findings

The generalized operator generates local flows with position-dependent rescaling.

Explicit eigenfunctions are derived with well-defined limits to standard plane waves.

Application to the infinite square well yields a deformed spectrum with a smooth classical limit.

Abstract

In this work we take a closer look at the algebraic-operator correspondence between the momentum space and the position space which defines the form of the canonical momentum operator in position space in Quantum Mechanics (QM). Starting from the Fourier transform (FT) relationship, we present a Hermitian generalization of the canonical momentum operator in position space. The action of the generalized operator is found to generate a local flow accompanied by position-dependent rescaling, rather than a global translation. Explicit eigenfunctions are obtained for representative cases and are shown to possess a well-defined limit to the plane-wave solution in QM. As an illustration, the infinite square well problem is solved using the generalized operator, yielding a deformed spectrum that has a smooth limit to the standard QM result.