The General Principle of Relativity Applied to the Rutherford-Bohr Atom-Model

Application of the general principle of relativity to the Rutherford-Bohr atom.—In this atom the possible electron orbits are determined according to mechanical principles, then the actual orbits are selected by the quantum conditions. In computing the possible orbits, Bohr assumed classical mechanics, Sommerfeld applied the special theory of relativity with marked success, and now his assumptions are further generalized. In the expression for the world line element $\mathrm{ds}$ in polar coördinates, the potential functions $\gamma$ and ${\gamma }^{\prime }$ are put equal to ($1+\frac{a\kappa }{r}+\frac{b{\kappa }^2}{r^2}$) and ($1+\frac{a^{\prime }\kappa }{r}$) respectively, where $\kappa$ is the ratio of potential energy to mass. The quantum assumptions are the same as Sommerfeld's, with generalized $\mathrm{ds}$, with an additional alternative assumption that the radial expression involves ${\gamma }^{\prime }$. These lead to expressions for the energy in which all the constants are known from experimental results, except $B$ which Sommerfeld puts equal to 1/4 but which has not yet been experimentally determined. The method here used for the atom is the same as Eddington used for a macrocosmos.