Stationary properties of approximate wave functions

Abstract

The accuracy of quantum mechanical wave functions is examined in terms of certain stationary properties. The most elementary of these, namely that displayed by the class of wave functions which yields a stationary value for the total energy of the system, is demonstrated to necessarily require few other stationary properties, and none of these appear to be particularly useful. However, the class of wave functions which yields both stationary energies and charge densities has very important stationary properties. A theorem is proven which states that any wave function in this class yields a stationary expectation value for any operator which can be expressed as a sum of one-particle operators. Since the Hartree–Fock wave function is known to possess these same stationary properties, this theorem demonstrates that the Hartree–Fock wave function is one of the infinitely many wave functions of the class. Methods for generating other wave functions in this class by modifying the Hartree–Fock wave function without changing its stationary properties are applied to the calculation of wave functions for the helium atom.