Exponential Approximation for Transition Probability

Abstract

An approximate solution of the time-dependent Schrödinger's equation in the form of the exponential representation is discussed. A method is proposed for making calculations feasible in the exponential representation by means of diagonalization of the time-dependent matrix. The method is applied to a problem of transition which involves a continuous state. We propose a set of orthogonal functions with which we can construct the unitary transformation matrix which diagonalizes the continuous matrix in question. A formula for the transition probability of predissociation or autoionization is obtained. The error estimation for the approximate values and a rough criterion for the validity of the exponential representation are given.