Study On The Geometric Potential Of Dirac Fermion On A Twodimensional Curved Surface Of Revolution

In recent years,quantum mechanics for a particle that moves non-relativistically on a curved surface attracts much attention.Geometric momentum and geometric potential,both depending on the intrinsic and extrinsic curvature of the surface,have been well-defined and experimentally confirmed.Since surface state is significantly influenced by the geometric potential,and the quantum states in topological insulator usually show off in form of surface states.Thus,whether the geometric potential exists can be inferred from the behaviors of the surface states,which poses a question that must be answered in both theoretical and experimental sides.The theoretical and experimental explorations of the topological insulator is a study under intensive study,from which the equivalent fermions move relativistically.No evidence can be found to indicate the possible existence of the geometric potential.All implies that there is no geometric potential for a free particle constrained to surface moves relativistically.The present dissertation is devoted to a study of the problem.First,we propose a dynamical quantum condition within the Dirac quantization scheme.It constitutes the new set of quantum conditions in conjunction with the fundamental ones,in which the momentum is the generally covariant geometric momentum.Secondly,we prove a theorem,which states for a Dirac fermion on a twodimensional surface of revolution,no geometric potential is admissible.Lastly,we explicitly treat some typical systems.Present dissertation is organized in following.The first part briefly introduces the geometric momentum and geometric potential energy and the Dirac equation of relativistic quantum mechanics under non-relativistic conditions.In order to be able to deal with particles containing spin,the geometric momentum needs to be generalized to generalized covariant geometric momentum.There are two ways to obtain this momentum.Section one is Introduction.The geometric momentum and geometric potential for non-relativistic particle on the curved surface are present.Besides,the Dirac equation in relativistic quantum mechanics is briefly mentioned.In order to dealing with the particle with spin,it is necessary to extend the geometric momentum to the generally covariant one.There are two methods to achieve it.Section two mainly proves a theorem.For a general two-dimensional surface of revolution,we can assume the most general form of the geometric potential in the Hamiltonian.Utilization of the dynamical quantum condition directly leads to the vanishing geometric potential.Section three demonstrates the theorem by carrying out explicit calculations for three typical systems,i.e.,Dirac fermions on the torus,catenoid,and oblate ellipsoid.No appearance of the geometric potential is identified.For a relativistic particle constrained to remain on the curved surface,whether curvature-induced potential can exist remains far from fully understanding.It is challenging in both theoretical and experimental areas.Nevertheless,once a Dirac fermion on the two-dimensional surface of revolution,there is no geometric potential,in contrast to the situation where the particle moves non-relativistically.