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The explanation of the first-order Stark effect in hydrogen by Epstein andSchwarzschild in 1916 was seen as a great success for the old quantum theory.Yet, it also revealed some serious limitations of the theory. To recover theexperimentally found line splittings, one had to make some arbitraryassumptions in addition to the basic quantum conditions to rule out certainorbits. The calculation of intensities of lines on the basis of Bohr'scorrespondence principle likewise required arbitrary additional assumptions.Finally, the actual orbits predicted by the old quantum theory depend on thecoordinates chosen to impose the quantum conditions. Both Sommerfeld andEpstein recognized this problem but offered no solution for it. All theseproblems were solved in 1926 when Schr\"odinger and Epstein explained the Starkeffect on the basis of the new wave mechanics. The calculations in the twotheories are similar. In particular, both the Schr\"odinger equation in the newtheory and the Hamilton-Jacobi equation in the old theory are separated inparabolic coordinates. The new quantum mechanics determines all allowed statesand transitions without any additional assumptions. It also replaced theambiguous guidelines based on the correspondence principle for calculatingintensities by the straightforward prescription that intensities are given bythe squares of the matrix elements of position, leading to results that agreedmuch better with the experimental data. Finally, the embarrassingnon-uniqueness of orbits in the old quantum theory turned into the innocuousnon-uniqueness of bases of eigenfunctions in wave mechanics. To this day, theStark effect is remembered as one of the few qualified successes of the oldquantum theory. We suspect that this is largely because after 1926 it becamejust one of the many unqualified successes of the new quantum theory.

The Stark effect in the Bohr-Sommerfeld theory and in Schr\"odinger's wave mechanics