Exactly solvable problems in the momentum space with a minimum uncertainty in position

TL;DR

This paper introduces a new method for solving simple quantum mechanics problems in deformed space with minimal length, providing exact solutions for delta potentials and revisiting Coulomb-like potentials.

Contribution

It generalizes the Schrödinger equation in momentum space for deformed Heisenberg algebra with minimal length, offering exact solutions and addressing operator hermicity issues.

Findings

Exact solutions for delta and double delta potentials.

Revisited Coulomb-like potential in deformed space.

Resolved inverse coordinate operator hermicity problem.

Abstract

A new approach in solution of simple quantum mechanical problems in deformed space with minimal length is presented. We propose the generalization of Schro\"edinger equation in momentum representation on the case of deformed Heisenberg algebra with minimal length. Assuming that the kernel of potential energy operator do not change in the case of deformation, we obtain exact solution of eigenproblem of a particle in delta potential as well as double delta potential. Particle in Coulomb like potential is revisited and the problem of inversibility and hermicity of inverse coordinate operator is solved.