The doublet separation of the balmer lines

Abstract

According to the Bohr Theory the hydrogen atom consists of an electron in circular or elliptical motion about a positively charged nucleus. The steady states are defined by discrete values of the angular momentum and, in the case of elliptical motion, the eccentricities of the ellipses are limited to certain definite values. Sommerfeld, by an extension of Bohr's theory involving quantising both angular and radial momentra, has established the formula γ = 2π 2 m E 2 e 2 / h 3 [ 1/(T' 1 + T' 2 ) 2 - 1/ (T 1 + T 2 ) 2 ] for the spectral series emitted by a system consisting of a nucleus with charge + E and an electron with charge - e . This is the familiar formula in which terms are included corresponding to elliptical as well as to circular motion for the revolving electron. By the theory of relativity it can be shown that the mass of an electron in the elliptical orbit is not the same as its mass in the circular orbit, but that it depends on the velocity v , thus m = m 0 (√1 - β 2 ) -1/2 , where β = v / c and m 0 is the mass of the slow-moving electron. From a mathematical standpoint for a slow-moving electron we may consider the path as an ellipse with slowly moving perihelion.