Dynamics of the particle on a catenoid and the quantization of the constrained system in the extended space

Abstract

There are two approaches to investigating the quantum mechanics for a particle constrained on a curved hypersurface, namely the Schrödinger formalism and the Dirac theory.#br#The Schrödinger formalism utilizes the confining potential technique to lead to a unique form of geometric kinetic energy T that contains the geometric potential VS and the geometric momentum p,#br#T=-ħ2/(2m)▽2+VS=-ħ2/(2m)[▽2+(M2-K)],p=-iħ(▽2+Mn),#br#where ▽2 is the gradient operator on the two-dimensional surface. Both the kinetic energy and momentum are geometric invariants. The geometric potential has been experimentally confirmed in two systems.#br#The Dirac's canonical quantization procedure assumes that the fundamental quantum conditions involve only the canonical position x and momentum p, which are in general given by#br#[xi,xj]=iħÂij,[pi,pj]=iħΩij,[xi,pj]=iħΘij#br#where Âij, Ωij, and Θijare all antisymmetric tensors. It does not always produce a unique form of momentum or Hamiltonian after quantization. An evident step is to further introduce more commutation relations than the fundamental ones, and what we are going to do is to add those between Hamiltonian and positions x, and between Hamiltonian and momenta p, i.e.,#br#[x,Ĥ]=iħÔ({x,HC}c) and [p,Ĥ]=iħÔ({p,HC}c)#br#where {f,g}c denotes the Poisson or Dirac bracket in classical mechanics, and Ô({f,g}c) means a construction of operator based on the resulting {f,g}c, and in general we have [f,ĝ]≠Ô({f,g}c). The association between these two sets of relations means that the operators {x,p,H must be simultaneously quantized. This is the basic framework of the so-called enlarged canonical quantization scheme.#br#For particles constrained on the minimum surface, momentum and kinetic energy are assumed to be dependent on purely intrinsic geometric quantity. Whether the intrinsic geometry offers a proper framework for the canonical quantization scheme is then an interesting issue. In the present paper, we take the catenoid to find whether the quantum theory can be established satisfactorily. Results show that the theory is not self-consistent. In contrast, in the threedimensional Euclidean space, the geometric momentum and geometric potential are then in agreement with those given by the Schrödinger theory.