The Hydrogen Spectrum in the New Quantum Theory

The Born and Jordan matrices representing the "coordinates" of the hydrogen atom are calculated. The model is that of an electron revolving without rotation about a positive nucleus. The computation is based on a method developed independently by the author and by E. Schroedinger. Half quantum numbers appear in the formulas for the matrices. It is stated without proof, that they also enter into the formula for the relativistic fine structure. This will necessitate the consideration of a more elaborate model for the hydrogen atom (the rotating electron of Uhlenbeck and Goudsmit). The intensity ratio of the two brighter components of ${\mathrm{H}}_{\alpha }$ is calculated and found to be 10:2.1 in contradiction with the observed 10:8. For ${\mathrm{H}}_{\beta }$ the calculated ratio is 10:3.6, and the observed 10:9. The discrepancy is too great to be accounted for by the incompleteness of the model, and points to a difficulty in the Born and Jordan theory.