Applications to Optics and Wave Mechanics of the Criterion of Maximum Cramer-Rao Bound

Abstract

A wide class of physical problems requires the estimation of probability laws. Diffraction patterns and quantum mechanical probability laws on position are examples. Minimum Fisher information is one approach to estimating such laws; maximum entropy is another. In this paper, we show that the minimum Fisher information approach may be derived from a prior principle of maximum Cramer-Rao bound (MCRB). The MCRB-Fisher approach is then applied to some fundamental physical problems, including diffraction theory and quantum mechanics. It is found to give the correct physical solutions, that is the Helmholtz and Schrödinger wave equations respectively. By comparison, the maximum entropy approach gives incorrect solutions to these problems, except in the special (thermodynamic) case of a harmonic oscillator potential.