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Proceedings of the National Academy of Sciences

A New Kind of Test of the Correspondence Principle Based on the Prediction of the Absolute Intensities of Spectral Lines

Richard C. Tolman, Richard M. Badger

Published Mar 1, 1926• 0 citations
Publisher

Abstract

The correspondence principle postulates a relation between the average rate at which energy is actually being discontinuously emitted by quantum transitions and the continuous rate of emission which would be calculated from the classical theory. The principle has hitherto been used to predict the relative intensities of spectral lines of similar origin. Since, however, the classical theory permits a claculation of absolute rates of energy emission, it should be possible to calculate from the correspondence principle the absolute intensities of spectral lines. In the present article, the correspondence principle is subjected for the first time to this new kind of absolute test, using Czerny's data on the intensities of the far infra-red lines Nos. 8, 9, and 10 in the pure rotational spectrum of the dipole HCl. Tests of this kind have not previously been easy to make owing to lack of information as to the experimental values of absolute intensities or lack of knowledge of the the atomic models involved. The agreement between experimental and calculated intensities is close enough greatly to increase our confidence in the correspondence principle, justify us in subjecting it to further tests, and lead us to hope that an exact quantitative formulation of the principle may be found.

DOI
10.1073/pnas.12.3.173

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The correspondence principle postulates a relation between the average rate at which energy is actually being discontinuously emitted by quantum transitions and the continuous rate of emission which would be calculated from the classical theory. The principle has hitherto been used to predict the relative intensities of spectral lines of similar origin. Since, however, the classical theory permits a claculation of absolute rates of energy emission, it should be possible to calculate from the correspondence principle the absolute intensities of spectral lines. In the present article, the correspondence principle is subjected for the first time to this new kind of absolute test, using Czerny's data on the intensities of the far infra-red lines Nos. 8, 9, and 10 in the pure rotational spectrum of the dipole HCl. Tests of this kind have not previously been easy to make owing to lack of information as to the experimental values of absolute intensities or lack of knowledge of the the atomic models involved. The agreement between experimental and calculated intensities is close enough greatly to increase our confidence in the correspondence principle, justify us in subjecting it to further tests, and lead us to hope that an exact quantitative formulation of the principle may be found.
Abstract 1. The procedure usually adopted in the Quantum Theory for the estimation of the intensities of a spectrum line includes two stages : (1) the electric moment, p, of the distribution responsible for the spectrum line is calculated by means of Schrodinger’s integrals or some equivalent method ; (2) the rate of loss of energy due to radiation is calculated by the classical formula due to H. Hertz, viz., 2p2/3c3, just as if the distribution were concentrated in a dipole of precisely the same moment p. The first stage involves no difficulties beyond those met with in the evaluation of the integrals or matrices with which we are concerned, the only principles needed being those of the particular quantum theory adopted. But the second stage is open to grave objections and can only be justified on the ground of necessity owing to the absence of any completely satisfactory quantum theory of the electromagnetic field. Sometimes an appeal is made to Bohr’s Correspondence Principle, but this itself involves a somewhat hazardous extrapolation from infinitely large down to small quantum numbers and cannot take the place of a rigorous calculation. A logically consistent method is possible, based on Maxwell’s theory as represented by his electromagnetic equations and their solutions in terms of the classical retarded potential integrals. It may be objected that we do not know how far Maxwell’s equations can be applied to the interiors of atoms, in particular to distributions like those of Schrodinger, but precisely the same objection applies to the customary use of the classical expression for the radiation from an oscillating dipole. Moreover, the attempts made by Heisenberg and Pauli, Dirac and others to formulate a quantum theory of the electromagnetic field have all been based on the assumption that such a theory must be consistent with Maxwell’s equations. The direct application of the methods of classical electromagnetic theory to Schrodinger’s distributions can be justified at least as a rigorous attempt to determine their electromagnetic field and the radiation from them, and the validity of the procedure will have to be judged by the agreement, or otherwise, of the results obtained with experience. In this way we avoid the assumption that a spatially extended distribution can be treated as if its electric moment were concentrated in a point; in fact, the results obtained in this paper for the more important spectrum line series show definitely that the assumption is false, because it leads to results in many cases quite inconsistent with a rigorous application of Maxwell’s theory.
The process of simultaneous emission of two photons (of continuous probability distribution in energy) in the transition from one discrete quantum level to another is here analyzed in the correspondence principle limit in which the quantum numbers of the two states are large and the change in quantum number is small by comparison. In this limit, it proves possible to define a classical dynamical quantity [Eq. (19)], the frequency analysis of whose classical variation with time in a definite orbit determines in the given approximation the absolute transition probability and intensity distribution in the continuous spectrum in question.
Lux aeterna luceat eis. Let there be light. Take the rest of the week off. Atomic energy levels deduced from optical spectra comprise one of the most precisely known sets of physical measurements that exist. However, the precision of the determinations of the relative oscillator strengths of these spectral lines from the relative intensities of spectral lines is much less precise. Fortunately, time-dependent methods for the study of the dynamics of the emission process exist (and are being developed) that permit the transition probability rates, oscillator strengths, and reaction rates to be determined with ever-increasing precision. In most elementary quantum mechanics textbooks, the section on the emission of radiation is the least satisfactory section of the book. Whereas the development of relationships among various spatial overlap integrals between state vectors for various operators (such as the electrostatic dipole moment) can be formulated in a very elegant ab initio manner, the connection of these matrix elements to the time dependence of the system often seems driven by a posteriori assumptions that are extended beyond their justifiable range of applicability. As Fermi observed in stating his Golden Rule , “the transition probability and energy perturbation can be calculated with the help of perturbation theory ( i.e. , there is no better way known).” The Weisskopf–Wigner approximation offers a scheme for making precise calculations, but does not provide the rigor that has characterized so many other areas of quantum mechanics.
These results suggest that the natural width of a spectral line may be less than the value to be expected from the classical theory of damping by radiation.The subject of width of spectral line is one of considerable importance.If the lines are really much narrower than 0.12 X-Unit the radiation can- not come from a damped oscillating electron.The mechanism must be such as to maintain a pure harmonic oscillation of constant amplitude until the quantum of energy is completely emitted.Such a train of waves would need to have a great number of elements and so have considerable length.An alternate hypothesis would be that a quantum is an entity (the word "pulse" is avoided) that may be resolved into a train of waves by the crystal grating.In this case the width of a spectral line would de- pend on the degree of perfection of the crystal.The quantum theory of crystal grating action advanced by Duane8 might also give a narrow spec- tral line, whose width would be a property of the crystal grating and not of the radiation.
Abstract In a recent paper under the same title, G. A. Schott finds that the usual method of calculating the intensities of spectrum lines is liable to lead to gross errors. Although one would expect from general reasons that any improvement of the usual method would only lead to small corrections and that, therefore, Schott’s conclusion cannot be free from error, it may not be devoid of interest to show that a correct calculation starting with Schott’s formulæ leads, in contradiction to Schott’s conclusion, to the generally accepted classical result. The aim of Schott’s calculations is to obtain a quantum mechanical expression for the energy radiated by an atom, which is classically given by R=2/3C3{р(t-r/c)}2, (1) p(t) being the electric moment of a dipole representing the atom. Schott considers the special case of a hydrogen atom emitting a radiation corresponding to a transition from the centrosymmetrical state 1=Af(r) (2)
Correspondence theorem for frequencies.---It is well known that the frequency emitted by a hydrogen atom as it falls from one of its quantized states to another may be expressed as a mean value of the frequency of motion of the electron (or overtone thereof) when averaged in the proper manner over orbits intermediate between the initial and final states. In the present paper it is proved that, similarly, the Compton shift is a properly taken mean of the classical Doppler shift. The quantum frequency actually scattered is thus a properly taken average of the frequency which would be scattered on the classical theory as the electron is accelerated from its state of rest to its final recoil condition.Correspondence principle for intensities.---In like manner, the amount of light scattered in various directions may be determined if it is assumed that the intensity in the quantum theory equals a proper average of the intensities scattered according to the classical theory. A comparison is made with observed data on the scattering of $\ensuremath{\gamma}$-rays.The characteristic feature of the present paper is that the corresponding classical electron is assumed to have the same direction of motion as the scattered quantum, whereas an actual classical electron would from symmetry recoil straight forward in the direction of the incident beam, as in Compton's and Woo's theories of intensities. This new point of view eliminates the difficulty of a constant correction-factor which has been encountered by Compton and Woo in their explanation of intensity relations.