Studies of the Potential-Curve Crossing Problem. I. Analysis of Stueckelberg's Method

A detailed critical analysis is made of Stueckelberg's treatment of inelastic transitions at a crossing of two potential curves. Using an asymptotic method analogous to the WKB approximation, Stueckelberg obtained the well-known Landau-Zener-Stueckelberg (LZS) formula for the inelastic transition probability. His method involved the determination of "connection formulas" linking amplitudes associated with his asymptotic approximants on either side of the crossing-point region. Here we show that (a) Stueckelberg's asymptotic approximants are just the WKB approximants for elastic scattering on the adiabatic (noncrossing) potential curves; (b) Stueckelberg's method for obtaining the connection formulas can be put on a rigorous footing, including sufficient conditions for its validity, using the classical trajectory equations derivable from a general semiclassical theory of inelastic atomic collisions; (c) there is an undetermined phase in the $S$ matrix, which Stueckelberg incorrectly assumed to be zero, and which has the value $\frac{1}{4}\pi$ in the distorted-wave approximation; (d) Stueckelberg's derivation is not valid whenever the inelastic transition probability is small, either in the rapid-passage (diabatic) case or the near-adiabatic limit; (e) for realistic model parameters, the conditions needed for Stueckelberg's derivation to be valid are almost never satisfied. Since the LZS formula is known from numerical computations to be valid under some conditions when the Stueckelberg derivation is not valid, we conclude that analysis via connection-formula methods is not a useful technique for treating the crossing problem. In an appendix we derive an analytical result for the Stokes's constants determining the Stueckelberg connection formulas. The result is an absolutely convergent, infinite series whose numerical evaluation would yield exactly the unknown phase associated with the LZS formula.