The statistical wave function

Abstract

Statistical considerations are applied to quantum mechanical amplitudes. The physical motivation is the progress in the spectroscopy of highly excited states, The corresponding wave functions are “strongly mixed.” In terms of a basis set of eigenfunctions of a zeroth-order Hamiltonian with good quantum numbers, such wave functions have contributions from many basis states. The vector x is considered whose components are the expansion coefficients in that basis. Any amplitude can be written as a^†·x. It is argued that the components of x and hence other amplitudes can be regarded as random variables. The maximum entropy formalism is applied to determine the corresponding distribution function. Two amplitudes a^†·x and b^†·x are independently distributed if b^†·a=0. It is suggested that the theory of quantal measurements implies that, in general, one can one determine the distribution of amplitudes and not the amplitudes themselves.