扩张型正则量子化方案的核心内容是位置、动量以及哈密顿量同时量子化。通过分析悬链面上粒子的扩张型正则量子化方案，并且与薛定谔理论进行比较，发现内禀几何中二维悬链面给不出与薛定谔理论相一致的结果，而考虑将二维悬链面嵌入在三维欧氏空间之后，还需要将正则量子化方案进行扩张，可以得到体系的几何势能和几何动量，并与薛定谔理论相一致。%There are two approaches to investigating the quantum mechanics for a particle constrained on a curved hypersur-face, namely the Schrödinger formalism and the Dirac theory. 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, T =−~22m∇2+VS=−~22m [∇2+(M 2−K )], p=−i~(∇2+M n), 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. 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 [xi, xj]= i~Λij, [pi, pj]= i~Ωij, [xi, pj]= i~Θij, where Λij ,Ωij , and Θij are 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., [x, H]= i~O({x, HC}c) and [p, H]= i~O({p, HC}c), where{f, g}c denotes the Poisson or Dirac bracket in classical mechanics, and ˆO({f, g}c) means a construction of operator based on the resulting {f, g}c, and in general we have [f , g] ̸= O({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. 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 three-dimensional Euclidean space, the geometric momentum and geometric potential are then in agreement with those given by the Schrödinger theory.
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