Quantum mechanical streamlines. III. Idealized reactive atom–diatomic molecule collision

Abstract

An atom–diatomic molecule collision is simulated by considering an idealized potential energy surface which is a two‐dimensional duct with an adjustable potential in the corner region. This potential is symmetric with respect to an interchange of the x and y Cartesian coordinates. Explicit expressions for the wavefunctions are obtained which make use of this symmetry. Also analytical relations are obtained between the transmission and reflection coefficients and their phases. Quantum mechanical streamlines are computer graphed for a large number of energies and positive, negative, and zero values of the potential energy in the corner region. Special attention is given to the quantized vortices (surrounding wavefunction nodes) which appear in the streamlines. When only one energy channel is open, the streamlines are symmetric and the flux is antisymmetric. This occurs because the wavefunction is a linear combination (with complex coefficients) of two real solutions, one symmetric, the other antisymmetric. When two energy channels are open, the streamlines are no longer symmetric. This is because the two wavefunctions are now linear combinations (with complex coefficients) of four real solutions. Although the second wavefunction is expressed in terms of the first, their streamlines have quite different patterns. From the Argand diagrams, it is clear that the reflection, transmission, and Feshbach‐type resonances which occur correspond to the Breit–Wigner type. At a reflection resonance energy all of the streamlines disappear; at slightly higher energy, the streamlines reappear with the same vortices as before but rotating in the opposite direction. The probability density is only symmetric at transmission resonance energies. At energies higher than the Feshbach‐type resonance energy (where a second energy channel becomes open), the streamlines pattern ceases to be symmetric and the flux antisymmetric. The exact quantum mechanical wavefunction for this idealized model is the same as the exact semiclassical wavefunction. These wavefunctions are infinite series. It is found that for the low energies which we consider, an excellent approximation is obtained by truncating the series after the first two terms. This corresponds to a semiclassical wavefunction formed from two manifolds of classical trajectories.