Semiclassical wave packet propagation in atomic systems in magnetic fields

The theory of semiclassical wave packet propagation is developed for several dimensions in configuration space. Multiple classical reference trajectories guide Gaussian wave packets where the Hamiltonian is linearized about the classical orbit. Each classical reference trajectory corresponds to a contribution to the correlation function. Each contribution is added coherently to give the full correlation function. An investigation of the dynamics of the Wigner distribution of a Gaussian wave packet in a quadratic Hamiltonian is used to guide the development of the theory. The criteria of choosing appropriate reference trajectories is described, and the semiclassical wave function is derived using two approaches: a wave-equation approach and a Greens-function propagator approach. The wave packet propagation theory is applied to the problem of wave packets in hydrogen which travel on elliptical orbits. Expressions for the stability matrix for elliptical orbits are found analytically and applied to calculating autocorrelation functions in two dimensions. Spectra derived from the autocorrelation function agrees with the expected spectra derived from quantum and EBK calculations. The wave packet propagation theory is also applied to the problem of Kepler wave packets in a homogeneous magnetic field in two dimensions. Variational equations for the eccentricity, precession, and angular momentum of the Kepler ellipse are derived and used to obtain approximate expressions for the classical motion in the limit of low eccentricity and low magnetic field strength. The stability matrix is derived from these approximate analytical expressions for nearly circular orbits. The stability matrix is used to find the autocorrelation functions in two dimensions. Spectra derived from the autocorrelation function agrees with the expected spectra derived from an approximate quantum calculation.