Comment on the accuracy of the uniform WKB theory of inelastic collisions

In a recent paper by Eu and Tsien on the uniform WKB theory of inelastic collision, the Magnus-approximation formula for the transition probability was applied to the ${\mathrm{He}}^{+}$-Ne inelastic-scattering process. In this Comment, we report that for the system studied, i.e., the ${\mathrm{He}}^{+}$-Ne system, the uniform WKB Magnus approximation underestimates the cross sections by about 50% and thus overestimates the coupling constant $a_4$ by about 30% above what it should be, while the distorted-wave theory underestimates the coupling constant by about 5%. The uniform-WKB Magnus-approximation formula for the transition probability is further assessed by making comparisons with the results of the exact theory, uniform-WKB numerical-solution method, and the Landau-Zener-Stückelberg theory, and is found to behave correctly in the strong-coupling regime as $a_4$ increases. The uniform-WKB Magnus approximation therefore appears to be complementary to the distorted-wave theory which is better in the weak-coupling regime, but breaks down in the strong-coupling regime. The uniform-WKB Magnus approximation is generally better behaved than the Landau-Zener-Stückelberg theory over all values of $a_4$ for the system studied.