| Abstract: | This article introduces a set of localized orthonormal functions to serve as basis functions for quantum calculations. They are defined to be eigenfunctions of the position operator in a function space. Their properties, including their variances, for a one-dimensional system are developed. The application to simple harmonic motion is considered as an example and, in particular, the time evolution of an initially localized function is calculated and shown to be periodic. The theory can be interpreted as producing a discrete quantization of space with Hamiltonian interactions that are predominantly between nearest neighbors. These functions can also be used in approximate calculations. To illustrate their accuracy, the example of a Morse oscillator treated as a perturbation of a harmonic oscillator is reconsidered. It is shown that the localized functions in a variational calculation lead to a result that is a good approximation for the lowest states. Furthermore, the use of a wave function that is defined only at discrete points can be justified as the first approximation to this, so that its accuracy can also be discussed. © 1995 John Wiley & Sons, Inc. |
