Studies in Semiclassical Mechanics

This thesis consists of the study of three different problems in the semiclassical mechanics of integrable systems. The first is a semiclassical approximation for matrix elements with respect to eigenstates of an integrable Hamiltonian. This is the "Heisenberg correspondence" between matrix elements and Fourier components of the classical motion. The approximation is shown to be valid through first order in hbar . This result is used to prove that an asymptotic approximation for Clebsch-Gordan coefficients is valid through first order in hbar and is also used to demonstrate the consistency between classical and quantum-mechanical (Rayleigh-Schrodinger) perturbation theory through the first two orders in hbar . The second part of the thesis is a generalization of the Langer modification. The Langer modification is the replacement of hbar^2l( l + 1) by hbar^2(l + 1/2)^2 in the centrifugal potential of the radial Schrodinger equation. This modification provides an improvement in the WKB analysis. In particular, the modified WKB wavefunctions have the exact limiting behavior for small radius. The modification is generalized from the Schrodinger operator (kinetic plus potential) to any radial operator. It is shown that the modified WKB wavefunctions have the exact limiting behavior for small radius in all cases. The final part of the thesis is a semiclassical analysis of the Dirac equation. A new WKB method for vector wave equations is applied to the radial Dirac equation and to the full Dirac equation. For the radial Dirac equation a new quantization condition is found. The full Dirac equation is analyzed for the special case of a central potential. In the classical mechanics, the Hamiltonian is the relativistic Hamiltonian for a scalar particle, with the effect of the spin as a correction to the symplectic form. A covariant treatment is then presented, in which the Hamiltonian is the covariant scalar Hamiltonian. The correction to the symplectic form is one-half of the form describing the classical Thomas precession.