Thin-layer Quantization Scheme And Applications

The thin-layer quantization scheme is probably severe and breaks with the natural limits set by the uncertainty principle, because of the introduction of the squeezing po-tential, but it is useful and effective to quantize a particle confined on a two-dimensional curved surface. In the presence of electromagnetic fields, the fundamental framework of the thin-layer quantization procedure is essential to the expectant results. It is re-gretful that the fundamental framework was not clearly given in the original papers.In chapter 2, we clearly refine the fundamental framework of the thin-layer quanti-zation procedure:(1) Dynamical equation is originally defined in the three-dimensional space; (2) By introducing a new wave function that can be analytically separated into normal part and surface part, in terms of metric tensor Gij defined in three-dimensional space, one can expand all derivatives in the original dynamical equation; (3) To sepa-rate the original dynamical equation into normal and surface components by limiting q3â†'0. In order to make convenience for readers, we briefly review the mathematical formula and concepts that will be employed in this thesis. And we briefly review the results of a particle confined on a curved surface and particles bounded on a curved surface.In chapter 3, in the presence of electromagnetic fields, we reconsider a charged particle constrained on a curved surface. According to the fundamental framework of the thin-layer quantization procedure, we accomplish the decoupling of electromag-netic fields and the surface curvature, the separability of normal dynamical equation and surface dynamical equation. In order to consider the influence of the surface thick-ness, we extend the thin-layer quantization scheme by adding a step, to take the proper terms of degree one in q3 (q3 denotes the curvilinear coordinate variable perpendicular to curved surface) back into the surface quantum equation. The well-known geometric potential and kinetic term are modified by the surface thickness. Applying the devel-oped formalism to a toroidal system, we obtain the modification for the kinetic term and the modified geometric potential including the influence of the surface thickness.In chapter 4, we derive the Pauli equation for a charged spin particle confined to move on a spatially curved surface S in an electromagnetic field. Using the thin-layer quantization scheme to constrain the particle on S, and in the transformed spinor representations, we obtain the well-known geometric potential Vg and the presence of e-iφ, which can generate additive spin connection geometric potentials by the curvi-linear coordinate derivatives, and we find that the two fundamental evidences are still valid in the present system without source current perpendicular to S. One is that there is no coupling between the magnetic field and the curvature of the surface, indepen-dently of the shape of the surface, of the electromagnetic fields and of the gauge. The other is that with a proper choice of the gauge the dynamics on the surface and the normal dynamics are decoupled. Finally, we apply the surface Pauli equation to spher-ical, cylindrical, and toroidal surfaces, in which we obtain the geometric potentials and new spin connection geometric potentials, and find that only the normal Pauli matrix appears in these equations.In chapter 5, a model including a periodically corrugated thin layer with GaAs substrate is employed to investigate the effects of the corrugations on the transmission probability of the nanostructure. We find that transmission gaps and resonant tunneling domains emerge from the corrugations, in the tunneling domains the tunneling peaks and valleys result from the boundaries between adjacent regions in which electron has different effective masses, and can be slightly modified by the layer thickness. These results can provide an access to design a curvature-tunable filter.