Quantum Mechanics in Noncommutative space

TL;DR

This paper explores how noncommutative geometry modifies quantum mechanics predictions for the hydrogen atom, deriving exact solutions and revealing novel phenomena like bound states in repulsive potentials at high energies.

Contribution

It introduces a rotationally invariant noncommutative quantum mechanics framework and analytically solves the NC Schrödinger equation for the hydrogen atom, including bound and scattering states.

Findings

NC corrections vanish as λ -> 0, recovering standard QM

Existence of bound states for repulsive potentials at high energies

NC framework preserves key properties of Coulomb systems

Abstract

This paper provides an examination of how are prediction of standard quantum mechanic (QM) affected by introducing a noncommutative (NC) structure into the configuration space of the considered system (electron in the Coulomb potential in the present case). The parameter controlling the extent of modification is denoted as {\lambda}. The coordinates in the NC space are realized via creation and annihilation operators acting in an auxiliary Fock space, this one being chosen in such a way that the rotational invariance of the system remains intact also in NCQM. Analog of Schr\"odinger equation for hydrogen atom is found and analytically solved, both for bound states and scattering. The exact formulas for NC corrections are given. None of the NC predictions contradicts experimentally verified QM results, since in the correspondence limit {\lambda} -> 0 both QM and NCQM coincide. Highly surprising feature of the NC version is the existence of bound states for repulsive potential at ultra-high energies. However, these disappear from the Hilbert space in the mentioned limit. The whole problem is solved also using a method analogous to that of Pauli. Besides rotational invariance, the dynamical symmetry related to the conservation of NC analog of Laplace-Runge-Lenz vector is being used and the results obtained this way are in the full agreement with those given by "Schr\"odinger-like" approach. The presented NC deformation of QM preserves all those mysterious properties of the Coulomb system that made it a distinguished key-stone of the modern physics.