Variational Principle for a Particle in a Box

Abstract

The particle-in-a-box problem is reexamined, using different model wave functions, to illustrate the use of the variational principle applied to the simplest solvable quantum mechanical problem. The scheme presented here provides a useful paradigm for the LCAO approach used in atomic and molecular calculations. Suitable polynomial basis functions are introduced that provide a way for controlling both the accuracy and the number of independent variational parameters used in the energy calculations, when the coordinate origin is taken either at the left-hand edge or the center of the box. Links with the Bransden and Joachain calculations on the first two even parity states are made. The approximate ground state wave function, x(x - L), is also redefined by introducing a variation parameter in the form of a prescription based on the value of the position coordinate. Finally, a straightforward method is presented for determining an approximate energy for an arbitrary excited state, based on the use of any app...