An alternative solution to the radial quantum condition for the hydrogen atom

The quantization method applied to the hydrogen atom involves the solution of the phase integral $\oint p_r\,dr=n_r h$, which was named by Sommerfeld as the \textit{radial quantum condition} and solved using complex integration. In this work, we present an alternative solution to the \textit{radial quantum condition} using standard integration methods as an accessible way for students of introductory quantum mechanics courses. In addtion, we show that in the Sommerfeld model the degeneracy of the energy levels is related to the geometric properties of the ellipse describing the electron motion around the nucleus.