Thin-layer Quantization Scheme And Its Applications In Corrugated Torus

The bending of a plane will cause a certain change in the geometry of the plane,which will cause the particle motion on the curved surface to be significantly different from the motion on the plane.For studying the motion of particles constrained on a two-dimensional surface,a thin-layer quantization scheme,also known as a confining potential approach,is an effective research scheme.The thin-layer quantization scheme is to introduce a squeezing potential so that the particles are restricted by the force perpendicular to the low-dimensional manifold,so that the particles can only be moved along the tangential direction of the low-dimensional manifold.This method separates the normal and tangential motions while also preserving the influence of the geometric properties of low-dimensional manifolds on tangential motions.In chapter 2 of this article,we systematically study the fundamental framework of the thin-layer quantization scheme.we review the related knowledge and concepts of differential geometry,and give the dynamic equations of a two-dimensional system defined in three-dimensional space.Using the thin-layer quantization scheme,by introducing squeezing potential,the original dynamic equation of the particles is analytically divided into the normal Schr¨dinger equation and the equivalent Schr¨dinger equation.Based on the thin-layer quantization scheme,we give the specific calculation procedures.In addition,we also apply the calculation of the thin-layer quantization scheme to a cylindrically symmetric surface,and give a specific example of deriving the geometric potential.In chapter 3,in the spirit of the thin-layer quantization scheme,we give the effective Shr¨dinger equation for a particle confined to a corrugated torus,in which the geometric potential is substantially changed by corrugation.We find the attractive wells reconstructed by the corrugation not being at identical depths,which is strikingly different from that of a corrugated nanotube,especially in the inner side of the torus.According to the transmission matrix method,by numerically calculating the transmission probability,we find that the resonant tunneling peaks and the transmission gaps are merged and broadened by the corrugation.Our research object in this chapter are the quarter corrugated torus,these results show that,the quarter corrugated torus can be used not only to connect two tubes with different radiuses in different directions,but also to filter the particles with particular incident energies,we can use its characteristics to design quantum electronic and photonic devices with specific structures in practical applications.In chapter 4,what we focus on are the mechanical properties of annular corrugated wires.Based on the previous chapter,we find that the geometrically induced potential is considerably influenced by corrugations.And for the quantum confinement system,it can be analytically divided into two one-dimensional(1D)quantum components:the component of annular corrugated wire and the component of z-axis.The effect of geometric potential on electronic dynamics is reflected in its motion on the annular corrugated wire.Therefore,in this chapter,we mainly focus on the mechanical properties of annular corrugated wires.By calculating numerically,we investigate the eigenenergies and the corresponding eigenstates of non-interacting electrons,and find that the transition energies can be sufficiently improved by adding corrugations and increasing the magnitude of corrugations.Particularly,the transition energy between the adjacent eigenstates corresponds to energy level differences based on the wavefunction of annular wire,and the number of the energy levels is equal to the number of corrugations.Furthermore,the larger magnitude of corrugations is capable of increasing the number of bound states.In addition,the distribution of ground state probability density is reconstructed by the corrugations,and the energy shift is generated.