Landau Problem in Noncommutative Quantum Mechanics

TL;DR

This paper investigates the Landau problem within noncommutative quantum mechanics, deriving energy levels and corrections due to space noncommutativity, and extends the analysis to noncommutative phase space with explicit solutions.

Contribution

It provides explicit solutions for Landau energy levels and eigenfunctions in noncommutative space and phase space, including energy corrections caused by noncommutativity.

Findings

Derived Landau energy levels in noncommutative space

Calculated energy corrections due to space-space noncommutativity

Explicit eigenfunctions for noncommutative phase space

Abstract

The Landau problem in non-commutative quantum mechanics (NCQM) is studied. First by solving the Schr$\ddot{o}$dinger equations on noncommutative(NC) space we obtain the Landau energy levels and the energy correction that is caused by space-space noncommutativity. Then we discuss the noncommutative phase space case, namely, space-space and momentum-momentum non-commutative case, and we get the explicit expression of the Hamiltonian as well as the corresponding eigenfunctions and eigenvalues.