Matrix elements, correspondence principles and line broadening

Abstract

Introduction: During the early development of quantum mechanics much use was made of what Bohr referred to as “a formal analogy between the quantum theory and the classical theory”. The conceptual foundation of this “formal analogy”, which he later called the “correspondence principle”, was based on the assumption that the quantum theory contains classical mechanics as a limiting case. With the advent of modern quantum mechanics, the applications of this correspondence have until recently been neglected, perhaps due to a feeling that they were not necessary or that their range of applicability was too limited. There are unfortunately many cases where the methods of quantum mechanics are too cumbersome to be used without approximations. Recent astrophysical investigations have for instance involved transitions with principal quantum numbers of up to 250. If the approximations that have to be made become too restrictive a better policy could be to use an approximate method which can be used to solve the problem exactly. This theoretical investigation is in two parts. The first summarizes the correspondence principle methods and discusses their range of validity; by applying the correspondence principle to problems whose quantum mechanical solutions are known in special cases, we can compare the analytic expressions obtained by each method. We shall show that the two results agree over a far wider range of values than is generally realised, and that the agreement can be considerably improved by adjusting free parameters that arise naturally in the correspondence principle. The second part considers the application of classical mechanics and the correspondence principle to the broadening of spectral lines. The observation of spectral lines involving very high principal quantum numbers has led to a resurgence of approximate methods because of the difficulty of applying quantum mechanics exactly. A survey of the existing literature showed wither very formal solutions to the line broadening problem which were difficult to apply, or detailed results which had very limited validity. We shall show that our calculations agree with these accepted results in their region of validity whilst describing the line shape in the intermediate region. In Chapter II we describe the correspondence principles we invoke. In Chapter II we obtain the solution of the motion of a particle in various potentials and calculate matrix elements and other quantities in these potentials. The potentials we consider are a harmonic potential, a Morse potential, and a Coulomb potential, and we compare the results of our calculation with the quantum mechanical expression where these are known. In Chapter IV we outline the main causes of spectral line broadening. We consider the limits in which various physical approximations can be made and examine in detail the work and conclusions of Lindholm, one of the foremost workers in the field of non-quantum mechanical broadening. In Chapter V we present our theory and compare it with other in various limits, and in Chapter VI and VII we obtain line shapes which we compare with the Lindholm shapes.