Classical‐Limit Quantum Mechanics and the Theory of Molecular Collisions

A previously developed theory in which exact solutions of the classical equations of motion for a complex collision system (i.e., numerically computed trajectories) can be used to generate the classical limit of the quantum mechanical S matrix (the “classical S matrix”) for the scattering process is applied to rigid rotor–atom collisions (rotational excitation). Comparison with essentially exact quantum results shows that transition probabilities (the square modulus of an S‐matrix element) between individual quantum states are given reasonably accurately by classical dynamics provided the interference terms are properly accounted for; a strictly classical approach (neglect of interference) gives poor agreement with the quantum values. For averaged collision properties, however, it is found that interference and tunneling effects are rapidly quenched. The linear atom–diatom system (vibrational excitation) and the rigid rotor–atom system are both investigated with regard to this question, namely, how much averaging is necessary to quench these quantum effects. Results indicate that even summation over a few quantum states is often sufficient to make a completely classical treatment appropriate.

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Classical S Matrix for Rotational Excitation; Quenching of Quantum Effects in Molecular Collisions

A semiclassical S‐matrix theory is developed and applied to spectral line broadening in linear molecules perturbed by atoms. This theory uses curved classical trajectories determined by the isotropic part of the atom–molecule interaction and the S‐matrix is treated to all orders in the interaction. Numerical calculations can be made rather easily even for high quantum numbers. The theory is least accurate for very low quantum numbers, but even then calculations agree to within 10% with close coupling results where comparisons could be made. Comparisons were also made with other theoretical approaches using model potentials and with experiment using ab initio potential surfaces.

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A semiclassical theory for spectral line broadening in molecules

Collision theory of Rydberg atoms with neutral and charged particles

Heisenberg's form of the correspondence principle for non-relativistic matrix elements has been used to evaluate matrix elements for various potentials. These are compared with available quantum mechanical results to check the general validity of the use of classical mechanics and correspondence principles in the derivation of quantum mechanical expressions. Using a one dimensional harmonic oscillator potential matrix elements of the form (n mod qm mod n+s) and (n mod pm mod n+s) are worked out. A Morse potential is then considered and the matrix elements for position, momentum, and kinetic energy are evaluated. Using a Coulomb potential a similar procedure is used for functions of various position coordinates, from which dipole and quadrupole moments from transitions between n, l states are calculated. The agreement with quantum mechanics is found to be generally good, and in some cases identical results are obtained.

http://iopscience.iop.org/article/10.1088/0022-3700/5/7/008/pdf

Matrix elements and correspondence principles.

Heisenberg's correspondence principle for non-relativistic matrix elements has been generalized: quantal transition amplitudes between strongly coupled states are expressed as Fourier components of integrals over classical trajectories. The new theory reduces to Heisenberg's correspondence principle in particular cases. The theory is applicable whenever the change in quantum number is small compared with the initial quantum number. In these situations it is more comprehensive than both quantal first order perturbation theory and the sudden approximation. Furthermore, compared with standard quantal methods, the evaluation of amplitudes is particularly simple. The theory is worked out for one- and three-dimensional separable systems and the generalization to a system of arbitrary dimension is indicated.

I C Percival

Papers

A correspondence principle for strongly coupled states