First correction to the quasiclassical scattering amplitude

We consider the first correction (of order $\hslash$) to the quasiclassical scattering amplitude in nonrelativistic quantum mechanics. When represented by two-loop diagrams, this correction contains twofold integrals. In the present paper, the sum of the contributions of the two-loop diagrams is transformed by means of the classical equations of motion into an expression that contains only a single integral. A characteristic feature of the result is that the integrand does not have singularities at the focal points of the classical trajectory. In the one-dimensional case, the general formula simplifies considerably.