Quantum Mechanics of the Hydrogen Atom

Abstract

In this chapter, we shall solve the Schrödinger equation of the hydrogen atom. For our calculations, we will not initially restrict ourselves to the Coulomb potential of the electron in the field of the nucleus of charge Z, V(r) = − Ze ²/(4πε₀ r), but rather will use a general potential V(r), which is symmetric with respect to a centre. As the reader may know from the study of classical mechanics, the angular momentum of a particle in a spherically symmetric potential field is conserved; this fact is expressed, for example, in Kepler’s law of areas for the motion of the planets in the solar system. In other words, we know that in classical physics, the angular momentum of a motion in a central potential is a constant as a function of time. This tempts us to ask whether in quantum mechanics the angular momentum is simultaneously measurable with the energy. As a criterion for simultaneous measurability, we know that the angular momentum operators must commute with the Hamiltonian. As we have already noted, the components l _x , l _y , and l _z of the angular momentum l are not simultaneously measurable; on the other hand, l _z and l ², for example, are simultaneously measurable. A long but straightforward calculation reveals that these two operators also commute with the Hamiltonian for a central-potential problem. Since the details of this calculation do not provide any new physical insights, we shall not repeat it here.