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The old quantum theory and Schr\"odinger's wave mechanics (and other forms ofquantum mechanics) give the same results for the line splittings in thefirst-order Stark effect in hydrogen, the leading terms in the splitting of thespectral lines emitted by a hydrogen atom in an external electric field. Weexamine the account of the effect in the old quantum theory, which was hailedas 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 oneruns 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 thequantum conditions are imposed which electron orbits are allowed in a hydrogenatom. The allowed energy levels and hence the line splittings are independentof the coordinates used but the size and eccentricity of the orbits are not. Inthe new quantum theory, this worrisome non-uniqueness of orbits turns into theperfectly innocuous non-uniqueness of bases in Hilbert space. Second, we reviewhow the so-called WKB (Wentzel-Kramers-Brillouin) approximation method forsolving the Schr\"odinger equation reproduces the quantum conditions of the oldquantum theory amended by some additional half-integer terms. These extra termsremove the need for some arbitrary extra restrictions on the allowed orbitsthat the old quantum theory required over and above the basic quantumconditions

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