Kaori Tanaka

Date of Award


Degree Type


Degree Name

Doctor of Philosophy (PhD)




Rajat K. Bhaduri


The theme of this thesis is to understand global shell structure of a finite many-fermion system in connection with short periodic orbits of the corresponding classical system. It is the overall shell structure, or partly resolved quantum fluctuations in the density of states, that is often enough for describing various properties of a system of interacting particles. Through semiclassical periodic orbit theory, one can visualize quantum-mechanical phenomena in terms of simple classical orbits. It is particularly interesting to study this quantum-classical connection in the mesoscopic systems of simple metal clusters and quantum dots, as their size as well as the number of particles can be much larger than in such systems as atoms and nuclei. We first illustrate a direct connection between quantum shells and classical periodic orbits by means of a mathematical model of a cranked two-dimensional harmonic oscillator. The quantum spectrum exhibits intriguing features, forming the Farey fan pattern. Furthermore, there is an analogy between this cranked model and the system of charged particles in a uniform magnetic field. We then go on to examine the electronic shell structure of simple metal clusters and quantum dots under a homogeneous magnetic field, taking simple mean-field models for these systems. The so-called supershell structure, a long-range, beating modulation of the electronic shell structure of simple metal clusters, is a fascinating example which can be explained semiclassically in terms of short periodic orbits of high degeneracy. We study the effect of an external magnetic field on this supershell structure, assuming a spherical infinite well as a simple yet realistic mean-field potential for the valence electrons. It is found that there is little perceptible change in the supershells for experimentally feasible field strengths, and if yet stronger fields are assumed, the supershells get destroyed and new beat patterns appear. For semiclassical understanding of these phenomena, we apply a recently developed trace formula for broken symmetry to this system. The system of quantum dots is more interesting than metal clusters in that its size can be much larger and the effect of magnetic fields on the electronic shell structure is observable for readily available field strengths. We examine the magnetization and magnetic susceptibility of a circular quantum dot with the two limiting cases of mean-field potential for a small and large number of confined electrons. The shell structure rejected in these magnetic properties are compared in the two cases and are interpreted through short classical orbits. In particular, the Aharonov-Bohm oscillations that appear in the strong-field limit, superimposed on the de Haas-van Alphen oscillations in the magnetization, are explained in terms of the shortest orbits that go along the edge of the system.

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