Phase Space Formulation of Quantum Mechanics and the Two Dimensional Electron Gas in a Magnetic Field.

Abstract

In this dissertation, the theory of the phase-space formulation of quantum mechanics (PSFQM) is discussed and applied to investigations of various low-temperature properties of both two- and three-dimensional electron systems. The origin of an apparent discrepancy recently reported by others between results obtained by the PSFQM and conventional (Schrodinger) quantum mechanics is discussed. A complete agreement is arrived at here by demonstrating the correct way to use the PSFQM. A general formulation is developed, through use of the PSFQM, for determining the free energy of a Fermi gas contained in an arbitrary smooth external potential and in a weak magnetic field, in the low-temperature limit. Explicit formulae are given, which enable one to compute surface and temperature effects on various physical properties (susceptibility, specific heat, etc.) of the system. As an illustration that the general formalism presented can be applied to other kinds of Fermi gas (for example, nucleons) contained in an external potential, the modified Thomas-Fermi theory is extended to include temperature effects. Expressions are derived for the magnetic susceptibility and Fermi energy of a non-interacting two-dimensional electron gas (2DEG) in the strong-magnetic-field limit and for non-zero temperatures. The effect of level broadening on the steady part of the magnetic moment and the specific heat is calculated by deriving an expression for the free energy of a 2DEG in a uniform magnetic field, with an arbitrary Landau level broadening and a finite temperature. Systematic expansions, in powers of 1/B, for the free energy and the density of states, are derived for a degenerate 2DEG in the presence of a strong magnetic field and an arbitrary potential. They are then applied to a system involving random impurities. Level broadenings, induced by long range electron-impurity scatterings alone, are shown to be independent of the magnetic field in the strong magnetic field limit. Broadened Landau levels can have a large variety of shapes. This theory leads to good agreement with the recent experiment on the de Haas-van Alphen effect in Br$\sb2$-graphite intercalation compounds.