The semi-classical expansion for a charged particle on a curved space background

We give the semi-classical expansion, with remainder to any order in ℏ, for the wave function of a nonrelativistic quantum particle in a classical external magnetic field on a curved space background. The basic assumption is of a ‘‘no caustics condition’’ on the underlying classical mechanics, at least up to the time in question. The gauge invariance of the result is emphasized together with a discussion of the geometric meaning of the classical mechanical quantities involved.