Runge-Lenz operator in the momentum space

TL;DR

This paper introduces a new, simplified form of the Runge-Lenz operator in momentum space for the quantum Coulomb problem, revealing SO(4) symmetry and enabling more effective analysis in this domain.

Contribution

It presents the first formulation of the Runge-Lenz operator in momentum space, simplifying the mathematical treatment of the Coulomb problem in this setting.

Findings

Derived a differential equation with SO(4) symmetry in momentum space.

Established the relation of the new operator to the Fock sphere's rotation.

Demonstrated the operator's simplicity compared to the coordinate space version.

Abstract

The fundamental quantum Coulomb problem in the momentum space is considered. A differential equation with SO(4) simmetry has been obtained in the momentum space instead of the integral Fock equation. The corresponding equation in the coordinate space is the sum of the squares of the angular momentum and the Runge-Lenz operators.This approach is unknown in the momentum space where the Runge-Lenz operator is not applied in the existing theory. The Runge-Lenz operator obtained in the momentum space is simplier than that in the coordinate space and allows one to effectively consider the Coulomb problem in the momentum space. A relation of new operator to the infinitesimal rotation operator of the three-dimensional the Fock sphere has been determined.