Quantum kinetic equations incorporating the Fano collision operator: The generalized Hess method of describing line shapes

Abstract

A Laplace-transformed quantum kinetic equation, quadratic in the singlet density matrix, is derived for gas mixtures in which, embedded as the collision term, the Fano relaxation tetradic allows for off-energy-shell scattering, i.e., incomplete collisions. A sufficient condition for the derivation is a stosszahl ansatz which is weaker at low frequencies than the one usually employed to derive Botlzmann-type equations. At high frequencies or, conversely, short times, it seems rather more stringent. The generalized Hess method, which is a quantum version of the Bhatnagar–Gross–Krook approximation, is used to solve it approximately, yielding a solution that describes the main features of collision broadening and Dicke narrowing. The relaxation tetradics that appear in the generalized Hess method, replace the collision term and are expressed in terms of collision integrals that are defined for finite concentration of optically active molecules. This means that self and resonant broadening and quenching are also included to some degree. The scattering operators in these collision integrals are expanded in partial waves—assuming that gas is composed of diatomic molecules—and recombined in the total angular momentum representation. Extensions to other representations seem straightforward. The reduction to the standard ‘‘impact approximation’’ or Shafer–Gordon theory is indicated as well as the symmetry effects of nuclear spin.