1303.0909 (D. M. Xun et al.)

A revisit of relation between Cartesian coordinates and canonical
quantization scheme for quantum motions on catenoid and helicoid    [PDF]

D. M. Xun, Q. H. Liu

For particles constrained on a curved surface, how to perform quantization within Dirac's canonical quantization scheme is a long-standing problem. On one hand, Dirac stressed that the Cartesian coordinate system has fundamental importance in passing from the classical Hamiltonian to its quantum mechanical form while preserving the classical algebraic structure between positions, momentum and Hamiltonian to the extent possible. On the other, on the curved surface, we have no exact Cartesian coordinate system within intrinsic geometry. These two facts imply that the three-dimensional Euclidean space in which the curved surface is embedded must be invoked otherwise no proper manner of quantization is attainable. Since the minimum surfaces, catenoid and helicoid studied in this paper, have vanishing mean curvature, we explore whether the intrinsic geometry offers a proper framework in which the quantum theory can be established in a self-consistent way. Results show that it does for quantum motions on catenoid and it does not for that on helicoid, but neither is compatible with Schr\"{o}dinger theory. In contrast, in three-dimensional Euclidean space, the geometric momentum and potential are then in agreement with those given by the Schr\"{o}dinger theory.

View original: http://arxiv.org/abs/1303.0909