  We apply a many-body Wentzel-Kramers-Brillouin (WKB) approach to determine
the leading quantum corrections to the semiclassical dynamics of the Josephson
model, describing interacting bosons able to tunnel between two localized
states. The semiclassical dynamics is known to divide between regular
oscillations and self-trapped oscillations where the sign of the imbalance
remains fixed. In both cases, the WKB wave functions are matched to Airy
functions, yielding a modified Bohr-Sommerfeld quantization condition. At the
critical energy dividing normal and self-trapped oscillations, the WKB wave
functions should instead be matched to parabolic cylinder functions, leading to
a quantization formula that is not just the Bohr-Sommerfeld formula, and
recovering the known logarithmic quantum break times at this energy. This work
thus provides another illustration of the usefulness of the WKB approach in
certain many-body problems.
