  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
