Schrödinger equation for a particle on a curved space and superintegrability. (English) Zbl 1326.81083

Bracken, Paul

J. Geom. Symmetry Phys. 38, 25-37 (2015).

Summary: A formulation of quantum mechanics on spaces of constant curvature is studied by quantizing the Noether momenta and using these to form the quantum Hamiltonian. This approach gives the opportunity of studying a superintegrable quantum system. It is shown there are three different ways of obtaining a Hilbert space of common eigenstates. Three different orthogonal coordinate systems are determined, one for each case. It is shown how the Schrödinger equation can be rendered separable in each of the cases.

MSC:

81Q35	Quantum mechanics on special spaces: manifolds, fractals, graphs, lattices
81S05	Commutation relations and statistics as related to quantum mechanics (general)
51P05	Classical or axiomatic geometry and physics
35P05	General topics in linear spectral theory for PDEs
81Q80	Special quantum systems, such as solvable systems

Keywords:

curvature; vector field; Hamiltonian; quantization; metric; canonical transformation