An Extension of Bohr's Correspondence Principle to Apply to Small Quantum Numbers

Abstract

Generalized correspondence principle for the Bohr atom.---It is shown that when an electron spirals in from an orbit of quantum number ${n}_{2}$ to an orbit of quantum number ${n}_{1}$, the frequency of the radiation emitted is equal to an integer (${n}_{2}\ensuremath{-}{n}_{1}$) times the mean frequency of rotations in its orbit during the spiraling, assuming that the number of rotations made for a change of quantum level $\mathrm{dn}$ is proportional to $\mathrm{dn}$. That is, the mean is defined as the ratio to (${n}_{2}\ensuremath{-}{n}_{1}$) of the integral from ${n}_{1}$ to ${n}_{2}$ of ${\ensuremath{\nu}}_{\mathrm{c}} \mathrm{dn}$ where ${\ensuremath{\nu}}_{c}$ is the classical frequency of rotation corresponding to any value of $n$. No explanation is offered as to why radiation begins and stops at integral values of $n$, nor why the radiation emitted in the manner assumed is monochromatic.