A Study Of The Quantization Difficulty Of Particle Constrained On Surfaces

Quantization difficulty is a fundamental problem in quantum physics.Dirac's canonical quantization hypothesis provides an effective rule for the system without operator ordering problems.However,a complex operator ordering problem appears when a particle is constrained on a two-dimensional surface.As a result,how to carry out the quantization becomes a very serious problem.In this paper,we study the quantization of a particle moving on a two-dimensional surface.For the quantum mechanics of a particle moving on a two-dimensional surface,there is a mature scheme shown as follows.First,one assumes that the surface is a thin layer with certain thickness instead of a geometric surface,and then writes down the Schrodinger equation about this thin layer.Next,supposing that the thickness tends to zero,it is possible to establish an effective theory of quantum mechanics of a particle moving on a surface.By this time,we can find that the momentum is a geometric momentum,while there is an additional geometric potential in the particle's Hamiltonian.Geometric potential has been verified experimentally.The next question is how to use Dirac's canonical quantization hypothesis to directly give a reasonable quantization result on the surface.This study mainly consists of three parts.In the first part,we briefly review the quantification of a constrained system and its research history.We firstly introduce Dirac's canonical quantization scheme,confining potential technique,and the derivation of geometric momentum and geometric potential.The theoretical framework,basic contents and basic properties of Enlarged Canonical Quantization Scheme(i.e.,ECQS)are also present.ECQS expands the three fundamental commutation relations and simultaneously achieves the quantization of location X,momentum P and Hamilton H.In the second part,we introduce the origin of dummy factor technique.Two different dummy factor techniques are brought in by taking the cylinder as an example.In the framework of ECQS,we continue to use this technique to further deal with the torus,quadric surface and general two-dimensional surface,and obtain the closed form solutions.In addition,the geometric momentum and the geometric potential are coexistent in the general two-dimensional surface.In the third part,we study the quantization problem on the implicit function surface which is expressed by F(x,y,z)= 0.The geometric momentum and the quantization of momentum motion equation of the implicit function surface in the Cartesian coordinates are investigated,and the second type dummy factor technique is used to analyze two kinds of higher-order surface:funnel surface and double button surface.We find that it is possible to obtain closed form solutions on some particular higher-order surfaces.Specifically,we obtain the exact solutions in three directions on the funnel surface as well as that in the x direction on the double button surface,respectively.The corresponding partial differential equations in theother two directions on the double button surface are also present.This study shows that dummy factor technique provides a universal solution for the quantization of a particle moving on a two-dimensional surface.In the meanwhile,this scheme is not perfect,which implies there being a better way to solve this problem.