The hydrogen atom in a uniform magnetic field — An example of chaos

Abstract

The hydrogen atom in a uniform magnetic field is discussed as a real and physical example of a simple nonintegrable system. The quantum mechanical spectrum shows a region of approximate separability which breaks down as we approach the classical escape threshold. Classical dynamics depends only on the scaled energy given as the true energy divided by the third root of the square of the field strength. The classical transition from regular motion below the escape threshold to chaos near the escape threshold is accompanied by a corresponding transition in statistical properties of the short ranged quantum spectral fluctuations. Spectral properties involving correlations on a longer range depend sensitively on system-specific nonuniversal properties such as the occurrence of prominent periodic classical orbits. Knowledge of the classical periodic orbits leads to a quantitative understanding of the low frequency properties of the quantum spectra as summarized in Gutzwiller's trace formula. These developments have led to a deeper understanding of the long known “quasi-Landau resonances” and other modulations in photoabsorption spectra.