Lorentz Trial Function for the Hydrogen Atom: A Simple, Elegant Exercise

Abstract

The quantum semester of a typical two-semester physical chemistry course is divided into two parts. The initial focus is on quantum mechanics and simple model systems for which the Schrödinger equation can be solved in closed form, but it then shifts in the second half to atoms and molecules, for which no closed solutions exist. The underlying principle that bridges this chasm is the variational method, which is easily the most important guiding principle for the construction of approximate wavefunctions including molecular orbitals. Students frequently encounter difficulties crossing this bridge, even when they did well in the first half. One contributing factor is the dilemma of the variational method that variational problems tend to be either too trivial or mathematically too involved to clearly show the concepts at work. Here, a Lorentz trial function for the hydrogen atom is discussed, a very elegant, yet straightforward example that somewhat alleviates this dilemma. Together with the standard example of a Gauss trial function and the exact solution, the Lorentz function provides an opportunity for an in-depth study of the variational principle before applying it to heavier atoms and molecules.