Generally Geometric Momentum Of Particle On Hypersurface

The quantum mechanical problem of geometrically constrained systems has attracted extensive attention since the 1990 s.In pursuing this goal,it is necessary to overcome the habitual thinking generated by the classical results,that is,dynamics only depends on the intrinsic geometry of the constraint.On the contrary,researches have shown that quantum mechanics is very sensitive to extrinsic geometry,which appears in the effective constrained dynamics by means of coupling with gauge fields and geometric potentials.The framework of Dirac's canonical quantization for constraint system provides a conventional solution to the quantization of constraint systems,but there are still some deep contradictions in this method.For the c onstrained motion on a surface,superfluous results appear in the processes of Dirac's canonical quantization theory,and there is neither physical nor mathematical principle that one can obey to make a reasonable choice.Confining potential theory may lea d to the geometric potential and the geometric momentum caused by constraints,however,it is unable explain why the Schr(?)dinger equation cannot be directly formulated on the constrained curved surface without considering any embedding.Based on the Dirac's canonical quantization theory for constraint system,we propose the following scheme to analyze the quantum motion of the constrained particles on a curved surface.We consider the embedding of the constrained surface in a higher-dimensional flat space,and insist that symmetry is on the top priority in the whole process of quantization.Space embedding reveals the influence of extrinsic geometry on the system,and we specifical ly discuss the surface embedding in a higher-dimensional flat space,which is called hypersurface.Symmetry precedence means that the commutation relations are possibly preserved after the quantization if a system has a classical correspondence.The Hamiltonian operator is not naturally determined after the quantization of coordinat es and momentums.Instead,it is simultaneously quantized with the position and momentum.This thesis mainly consists of five parts.In the first part,we describe the progress of the research on quantization for constrained systems and introduce the quantum motion of constrained particles on two-dimensional hypersurfaces with confining potential theory.Then we review the Dirac's canonical quantization theory for constraint system,and finally give the conditions of dynamics quantization.In the second part,we investigate the motion of a particle on arbitrary-dimensional hypersphere.The geometric momentum and orbital angular momentum are reconstructed by adding the gauge potential.We find the gauge structure is rooted in acoupling between the generators of rotational symmetry group and curvature and the generally geometric momentum and the generally angular momentum satisfy closed so(N,1)algebra.In the third part,we study the motion of a relativistic particle constrained on hypersurface and obtain the generally covariant geometric momentum of the particle.The results illustrate that once the geometric momentum becomes generally covariant as to be applicable to spin particles on the hypersurface,and the spin connection in it can be interpreted as a gauge potential.We propose a general framework of quantum conditions for a spin particle on the hypersurface,and establish a general formalism between generally covariant geometric momentum and quantization conditions.The Dirac fermions constrained on two-dimensional hyperspheres are discussed in detail,and taking the geometry as the origin we show the generalized angular momentum by using the fundamental quantization conditions and the generalized covariant geometric momentum.Moreover,we prove that there is no curvature-induced geometric potential for spin half particle.In the fourth part,we respectively discuss the quantum mechanics of Dirac fermions constrained on two-dimensional pseudospheres and helical surfaces as two applications of the general formalism for the quantization of spin particles constrained on hypersurfaces,and get the generally covariant geometric momentum of the particle.We show that there is no curvature-induced geometric potential for Dirac fermions constrained on pseudospheres,and the geometric potential for Dirac fermions constrained on helical surfaces is a constant matrix independent of parameters,which is composed of the z-direction Pauli matrix and the identity matrix.We also verify that the dynamic quantization conditions are effective in dealing with Dirac fermions on constrained hypersurfaces.Although it is currently not possible to obtain the general result of the constrained particles on two-dimensional hypersurfaces,we can solve the generally covariant geometric momentum and geometric potential of the constrained particles on surfaces with definite parametric equations.In the fifth part,we show the Dupin indicatrix of two-dimensional curved surfaces and extend Dupin scalar to hypersurfaces in higher dimensions to analyze the commutation relation between momentum components of constrained particles.Based on the Dirac's canonical quantization theory for constraint system,the commutation relation between the momentum components can be easily established.The structure of commutation relation is very complicated because of the order of op erators.For a nonrelativistic particle that is constrained on an N-1(N ?2)-dimensional hypersurface,two different components of the Cartesian momentum of the particle are not mutually commutative,and explicitly the commutation relations depend on the products of positions and momenta in uncontrollable ways.The generalized Dupin indicatrix of the hypersurface offers a local perspective to explore the geometric effect of the bending.The noncommutativity of two different components of the Cartesian momentum of the particle on a local point of the hypersurface is due to the local curvature of the surface,and in quantum mechanics it leads to a geometrically infinitesimal translation operator.