ABSTRACT
The problem of interacting particles can usually be reduced in quantum mechanics to that of one particle, as can be done in classical mechanics. Normally, this problem is one of an electron orbiting around, or being affected by, a positive atomic core or a scattering center with the interaction governed by the Coulomb potential. In general, this is a multi-dimensional problem, and not one of the simpler one-dimensional problems with which we have been concerned in the previous chapters. Once we begin to treat multiple dimensions, then state degeneracy begins to arise more frequently, and the most common problem treated is that of the hydrogen atom. These extra dimensions provide more degrees of freedom and more complexity. In this chapter, we want to discuss the motion of a charged particle in a centrally symmetric potential, and so will discuss the hydrogen atom. First, however, we want to begin to understand how the degeneracies arise and just what they mean. To facilitate this, we will treat first the two-dimensional harmonic oscillator motion for a central potential. We will then consider the manner in which the degeneracies are split by a magnetic field. Following this, we will be ready to discuss the hydrogen atom with its three-dimensional potential. Finally, we will briefly discuss the energy levels that arise in atoms more complex than the hydrogen atom with real, but noncoulombic potentials. Finally, we will discuss hydrogenic impurities in semiconductors.
