Superintegrability of the Fock-Darwin system

Abstract

The Fock-Darwin system is analysed from the point of view of its symmetry properties in the quantum and classical frameworks. The quantum Fock-Darwin system is known to have two sets of ladder operators, a fact which guarantees its solvability. We show that for rational values of the quotient of two relevant frequencies, this system is superintegrable, the quantum symmetries being responsible for the degeneracy of the energy levels. These symmetries are of higher order and close a polynomial algebra. In the classical case, the ladder operators are replaced by ladder functions and the symmetries by constants of motion. We also prove that the rational classical system is superintegrable and its trajectories are closed. The constants of motion are also generators of symmetry transformations in the phase space that have been integrated for some special cases. These transformations connect different trajectories with the same energy. The coherent states of the quantum superintegrable system are found and they reproduce the closed trajectories of the classical one.

21 pages,16 figures

References in corpus (4)

• The Fock-Darwin States of Dirac Electrons in Graphene-based Artificial Atoms
• Factorizations of one dimensional classical systems
• The anisotropic oscillator on curved spaces: A new exactly solvable model
• Solving the Schrodinger Equation for a Charged Particle in a Magnetic Field using the Finite Difference Time Domain Method

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• Coherent states in magnetized anisotropic 2D Dirac materials
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• Coherent states in the symmetric gauge for graphene under a constant perpendicular magnetic field