Stochastic Theory of Line Shape: Generalization of the Kubo-Anderson Model

A general solution is presented for the line shape of radiation emitted by a system whose Hamiltonian jumps at random as a function of time between a finite number of possible forms $V_1, V_2, \cdots , V_n$. The solution is valid even if these forms do not commute with one another ($[V_i, V_j]\ne 0$), so that the Hamiltonian need not commute with itself at different times: $\left[\mathcal{H}\mathrm{}\right(t), \mathcal{H}(t^{\prime }\left)\right]\ne 0$. This is a generalization of the Kubo-Anderson model in which it is assumed that the Hamiltonian does commute at different times. The treatment given here thus extends this adiabatic or random-frequency-modulation theory to include the nonadiabatic effects of transitions induced by the fluctuating Hamiltonian. The solution involves the inversion of a matrix and is similar to Sack's solution of the Kubo-Anderson model. The matrix found in the present case is labeled by quantum-mechanical as well as stochastic indices, and it reduces to the form found by Sack when the possible forms of the Hamiltonian commute with one another. Numerical evaluation of line shapes can be accomplished easily with a computer, and in the simplest cases analytical expressions can be found. The applicability of the theory to NMR and Mössbauer line shapes and to perturbed angular correlations is discussed. A specific example of the NMR line shape of a spin-½ nucleus in a fixed magnetic field and a fluctuating field perpendicular to it is considered in detail as an illustration of the utility of the derived expressions. The solution uses the Liouville-operator notation and this is discussed in an Appendix.