A uniform approximation for one-dimensional matrix elements

A general expression for one-dimensional perturbation integrals is derived, which requires no knowledge of the radial wavefunctions. It is shown to generalize and to unify all available semi-classical approximations, and, further, to reduce to a number of known results for the curve-crossing problem and to the Jackson-Mott for the Landau-Teller vibrational relaxation model.