Derivation of Sommerfeld-Dirac Fine-Structure Formula by WKB Method

Using the WKB approximation in the spirit of Miller and Good, we have obtained an expression which connects the energy eigenvalues of two parts: One is supposedly known and the other is to be solved. We use the Balmer energy formula of the one-electron bound state as the known part and solve for its relativistic counterpart, the Sommerfeld-Dirac fine-structure formula. An exact form of the relativistic energy-level formula can be obtained through this modified WKB expansion to order ${\hslash }^2$.