Generalized momentum operators from Fourier transform correspondence

TL;DR

This work addresses how the canonical momentum operator in quantum mechanics can be generalized beyond the standard $q\leftrightarrow -i\frac{d}{dx}$ correspondence by exploiting the Fourier transform relation between momentum and position representations. The authors construct a Hermitian family of momentum operators parameterized by a Fourier scale $q$ and deformation parameters $a,b$, yielding a local-flow action rather than global translations, with eigenfunctions that smoothly approach plane waves as the parameters vanish. They provide explicit eigenfunctions in representative cases and solve the infinite square well using the generalized operator, obtaining a deformed spectrum that continuously recovers the familiar QM spectrum in the limit $a,b\to0$. The framework situates standard QM at the center of an open unit disk in the deformation space and offers a controlled interpolation to generalized momentum dynamics, potentially informing how nonstandard position-momentum relations might emerge in modified quantum theories.

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.