The trouble with orbits: the Stark effect in the old and the new quantum theory

TL;DR

This paper compares the old quantum theory and Schrödinger's wave mechanics in explaining the Stark effect in hydrogen, highlighting how wave mechanics resolves fundamental issues related to orbit non-uniqueness and reproduces old quantum conditions with added precision.

Contribution

It demonstrates how wave mechanics addresses the non-uniqueness problem of orbits in the old quantum theory and connects the quantum conditions to the WKB approximation.

Findings

Wave mechanics solves the non-uniqueness of orbits problem.

WKB approximation reproduces old quantum conditions with additional terms.

Both theories agree on spectral line splittings in the Stark effect.

Abstract

The old quantum theory and Schr\"odinger's wave mechanics (and other forms of quantum mechanics) give the same results for the line splittings in the first-order Stark effect in hydrogen, the leading terms in the splitting of the spectral lines emitted by a hydrogen atom in an external electric field. We examine the account of the effect in the old quantum theory, which was hailed as a major success of that theory, from the point of view of wave mechanics. First, we show how the new quantum mechanics solves a fundamental problem one runs into in the old quantum theory with the Stark effect. It turns out that, even without an external field, it depends on the coordinates in which the quantum conditions are imposed which electron orbits are allowed in a hydrogen atom. The allowed energy levels and hence the line splittings are independent of the coordinates used but the size and eccentricity of the orbits are not. In the new quantum theory, this worrisome non-uniqueness of orbits turns into the perfectly innocuous non-uniqueness of bases in Hilbert space. Second, we review how the so-called WKB (Wentzel-Kramers-Brillouin) approximation method for solving the Schr\"odinger equation reproduces the quantum conditions of the old quantum theory amended by some additional half-integer terms. These extra terms remove the need for some arbitrary extra restrictions on the allowed orbits that the old quantum theory required over and above the basic quantum conditions