  Axions and photons resonantly interconvert in regions where the plasma
frequency approximately equals the axion mass. This process is directly
analogous to an avoided level crossing in quantum mechanics, for which the
celebrated Landau-Zener (LZ) formula provides a simple, non-perturbative
solution for the conversion probability. The LZ formula is commonly used in
studies of axion-photon conversion; however, in this context, it relies on the
assumption that the magnetic field variation is small compared to variations of
the plasma frequency, which is frequently not the case in real plasmas. We
derive a generalised version of the LZ formula by allowing the boundaries to be
located at a scale that is similar to the inhomogeneities. We find that the LZ
formula fails when the oscillation range is small compared to the resonance
region. This failure is more severe in the adiabatic limit, when the plasma
frequency varies slowly, resulting in a conversion probability that is not
maximal as opposed to a naive application of the LZ formula. Moreover, we
consider circumstances where the generalised LZ formula does not apply and
present an alternative semi-classical approximation with complementary regime
of validity.
