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

TL;DR

The paper compares the explanation of the Stark effect in hydrogen using the old quantum theory and Schrödinger's wave mechanics, highlighting how the latter resolved many limitations of the former and provided more accurate, assumption-free results.

Contribution

It demonstrates how Schrödinger's wave mechanics improved the explanation of the Stark effect over the old quantum theory by removing arbitrary assumptions and clarifying the nature of quantum states.

Findings

Wave mechanics provides more accurate line splittings.

It eliminates arbitrary assumptions required in old quantum theory.

Results align better with experimental data.

Abstract

The explanation of the first-order Stark effect in hydrogen by Epstein and Schwarzschild 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 the experimentally found line splittings, one had to make some arbitrary assumptions in addition to the basic quantum conditions to rule out certain orbits. The calculation of intensities of lines on the basis of Bohr's correspondence principle likewise required arbitrary additional assumptions. Finally, the actual orbits predicted by the old quantum theory depend on the coordinates chosen to impose the quantum conditions. Both Sommerfeld and Epstein recognized this problem but offered no solution for it. All these problems were solved in 1926 when Schr\"odinger and Epstein explained the Stark effect on the basis of the new wave mechanics. The calculations in the two theories are similar. In particular, both the Schr\"odinger equation in the new theory and the Hamilton-Jacobi equation in the old theory are separated in parabolic coordinates. The new quantum mechanics determines all allowed states and transitions without any additional assumptions. It also replaced the ambiguous guidelines based on the correspondence principle for calculating intensities by the straightforward prescription that intensities are given by the squares of the matrix elements of position, leading to results that agreed much better with the experimental data. Finally, the embarrassing non-uniqueness of orbits in the old quantum theory turned into the innocuous non-uniqueness of bases of eigenfunctions in wave mechanics. To this day, the Stark effect is remembered as one of the few qualified successes of the old quantum theory. We suspect that this is largely because after 1926 it became just one of the many unqualified successes of the new quantum theory.