Hydrogen-Like Atoms

Abstract

In a history of quantum mechanics, it was first successfully applied to the motion of an electron in a hydrogen atom along with a harmonic oscillator. Unlike the case of a one-dimensional harmonic oscillator we dealt with in Chap. 2, however, with a hydrogen atom we have to consider three-dimensional motion of an electron. Accordingly, it takes somewhat elaborate calculations to constitute the Hamiltonian. The calculation procedures themselves, however, are worth following to understand underlying basic concepts of the quantum mechanics. At the same time, this chapter is a treasure of special functions. In Chap. 2, we have already encountered one of them, i.e., Hermite polynomials. Here we will deal with Legendre polynomials, associated Legendre polynomials, etc. These special functions arise when we deal with a physical system having, e.g., the spherical symmetry. In a hydrogen atom, an electron is moving in a spherically symmetric Coulomb potential field produced by a proton. This topic provides us with a good opportunity to study various special functions. The related Schrödinger equation can be separated into an angular part and a radial part. The solutions of angular parts are characterized by spherical (surface) harmonics. The (associated) Legendre functions are correlated to them. The solutions of the radial part are connected to the (associated) Laguerre polynomials. The exact solutions are obtained by the product of the (associated) Legendre functions and (associated) Laguerre polynomials accordingly. Thus, to study the characteristics of hydrogen-like atoms from the quantum-mechanical perspective is of fundamental importance.