The Hydrogen Atom

Abstract

Based on the discussions in Chap. 4 we will now apply the quantum mechanical treatment to the simplest atom, the $\mathrm {H}$ atom, which consists of one proton and one electron moving in the spherical symmetric Coulomb potential of the proton. These one-electron systems, such as the hydrogen atom and the ions $\mathrm {He}^{+}$, $\mathrm {Li}^{++}$, $\mathrm {Be}^{+++}$, etc., are the only real systems for which the Schrödinger equation can be exactly (i.e., analytically) solved. For all other atoms or molecules approximations have to be made. Either the Schrödinger equation for these systems can be solved numerically (which offers a mathematical solution within the accuracy of the computer program, but generally gives little insight into the physical nature of the approximation), or the real atoms are described by approximate models that can be calculated analytically. In any case, for all multielectron systems, one has to live with approximations, either in the numerical solution of the exact atomic model or for the exact solution of the approximate model.