The Absorption of Radiation by Multiply Periodic Orbits, and its Relation to the Correspondence Principle and the Rayleigh-Jeans Law. Part I. Some Extensions of the Correspondence Principle

Abstract

This part deals with the quantum theory aspects of the problem. In the absence of external radiation fields the distortion in the shape of the orbit is essentially the same in both the classical and quantum theories provided in the former we retain only one particular term ${\ensuremath{\tau}}_{1}$, ${\ensuremath{\tau}}_{2}$, ${\ensuremath{\tau}}_{3}$ in the multiple Fourier expansion of the force $\frac{2{e}^{2}\stackrel{\ifmmode\ddot\else\textasciidieresis\fi{}}{v}}{3{c}^{3}}$ on the electron due to its own radiation. The term to be retained is, of course, the combination overtone asymptotically connected to the particular quantum transition under consideration. Then the changes $\ensuremath{\Delta}{J}_{1}$, $\ensuremath{\Delta}{J}_{2}$, $\ensuremath{\Delta}{J}_{3}$ in the momenta ${J}_{k}$ which fix the orbits and which in the stationary states satisfy the relations ${J}_{k}={n}_{k}h$, are in the ratios of the integers ${\ensuremath{\tau}}_{1}$, ${\ensuremath{\tau}}_{2}$, ${\ensuremath{\tau}}_{3}$ in both the classical and quantum theories, making the character of the distortion the same in both even though the speed of the alterations may differ. One particular term in the classical radiation force is thus competent to bring an orbit from one stationary state to another.The correspondence principle is then extended so as to include absorption as well as the spontaneous emission ordinarily considered. Commencing always with a given orbit it is possible to pair together the upward and downward transitions in such a way that in each pair the upward and downward optical frequencies (determined by the $h\ensuremath{\nu}$ relation) are nearly equal for large quantum numbers (usually long wave-lengths). That is, if $s$ denotes the initial orbit there exist levels $r$ and $t$ such that the ratio $\frac{({W}_{r}\ensuremath{-}{W}_{s})}{({W}_{s}\ensuremath{-}{W}_{t})}$ or $\frac{{\ensuremath{\nu}}_{\mathrm{rs}}}{{\ensuremath{\nu}}_{\mathrm{st}}}$ approaches unity when the quantum numbers become large. We shall define as the differential absorption the excess of positive absorption due to the upward transition $s\ensuremath{\rightarrow}r$, over the negative absorption (induced emission) for the corresponding downward transition $s\ensuremath{\rightarrow}t$. It is proved that for large quantum numbers the classical theory value for the ratio of absorption to emission approaches asymptotically the quantum theory expression for the ratio of the differential absorption to the spontaneous emission. Consequently a correspondence principle which makes the numerical values of the emission in the two theories agree asymptotically, of necessity achieves a similar connection for the absorption.The correspondence principle basis for a dispersion formula proposed by Kramers, which assumes the dispersion to be due not to the actual orbits but to Slater's "virtual" or "ghost" oscillators having the spectroscopic rather than orbital frequencies, is then presented. Kramers' formula has both positive and negative terms and the differential dispersion may be defined in a manner analogous to the differential absorption. It is shown that the quantum differential dispersion approaches asymptotically the dispersion which on the classical theory would come from the actual multiply periodic orbit found in the stationary states. This asymptotic connection for the general non-degenerate multiply periodic orbit must be regarded as an important argument for Kramers' formula.