Quantum Mechanics on the h-deformed Quantum Plane

We find the covariant deformed Heisenberg algebra and the Laplace-Beltrami operator on the extended \(h\)-deformed quantum plane and solve the Schr\"odinger equations explicitly for some physical systems on the quantum plane. In the commutative limit the behaviour of a quantum particle on the quantum plane becomes that of the quantum particle on the Poincaré half-plane, a surface of constant negative Gaussian curvature. We show the bound state energy spectra for particles under specific potentials depend explicitly on the deformation parameter \(h\). Moreover, it is shown that bound states can survive on the quantum plane in a limiting case where bound states on the Poincaré half-plane disappear.