  The isotropic harmonic oscillator in dimension 3 separates in several
different coordinate systems. Separating in a particular coordinate system
defines a system of three commuting operators, one of which is the Hamiltonian.
We show that the joint spectrum of the Hamilton operator, the $z$ component of
the angular momentum, and a quartic integral obtained from separation in
prolate spheroidal coordinates has quantum monodromy for sufficiently large
energies. This means that one cannot globally assign quantum numbers to the
joint spectrum. The effect can be classically explained by showing that the
corresponding Liouville integrable system has a non-degenerate focus-focus
point, and hence Hamiltonian monodromy.
