Numerical Evaluation of Barrier Penetration and Resonance Effects on Phase Shifts

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Abstract
Quantum‐mechanical calculations for a Lennard‐Jones (12, 6) potential are presented showing the dependence of the reduced phase shift η* on the quantum parameter Λ* for a fixed reduced effective potential and various reduced energies E*. The observed oscillatory behavior of η* (Λ*) is due primarily to the inclusion of the physically unimportant contribution of Mπ to the phase shift, where M is the number of quasibound (virtual) states of energy less than E*. A modified reduced phase shift η*, defined by excluding this contribution, displays only the sharp inflections associated with barrier penetration under resonance conditions. Except for the resonance contribution, the phase shifts may be accurately reproduced by a second‐order JWKB procedure. This method also accurately predicts the resonance energies (i.e., the energies of the quasibound states). In the region of a resonance, the scattering cross section varies rapidly with energy in a typical resonance manner. If they do not seriously overlap, the resonance ``lines'' should be observable as perturbations on an otherwise smooth background in low‐energy beam‐scattering experiments. The first‐order JWKB treatment of the barrier penetration problem by Ford, Hill, Wakano, and Wheeler suffices for the purpose of estimating the level widths and lifetimes of the virtual states as well as the main features of the resonant phase shifts, but does not accurately reproduce the quantal calculations.

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