Reduced matrix elements

J S Briggs

Reduced matrix elements of a tensorial product of n tensor operators in the basis of N-particle angular-momentum eigenstates are expanded into a sum of products of n 1-particle reduced matrix elements and a single (N + n)-particle recoupling coefficient. Application of the formula is illustrated by specific examples. The method leaves the coupling schemes of N-particle states undisturbed, which allows summation over intermediate states in a product of matrix elements to be made for any number of factors. A formula is given for the sum of products of two matrix elements and the extension to a greater number of matrix elements is illustrated by an example which is reduced to a form suitable for numerical evaluation. Such summations over products of matrix elements occur in the perturbation theory of configuration interaction and have hitherto been discussed in terms of effective operators. The connection of the method used here with the effective-operator approach is demonstrated.

Rodney P. Meyer

Relative dipole matrix elements are measured by means of a new spectroscopic technique called polarization interference spectroscopy. It is a two-color two-photon nonresonant process that maps the matrix elements into the energy domain by means of quantum mechanical interference. The technique measures the relative magnitude of the reduced matrix elements, particularly those involving excited states, to accuracies that are an order of magnitude better than previously available. Also, for the first time the relative phase of the reduced matrix elements is measured. The ratio of reduced matrix elements for the transitions 3s 2S1/2 →3P 2P3/2 →5s 2S1/2 to 3s 2S1/2 →3P 2P1/2 →5s 2S1/2 in sodium is +1.0012(12). This result is in agreement with the result of +1 for J independent matrices.

C. Lee

Adopting the so-called genealogical construction, the eigenstates of collective operators can be expressed corresponding to a specified mode for an N-atom system in terms of those for an (N-1)-atom system. Matrix element of a collective operator of an arbitrary mode is presented which can be written as the product of an m-dependent factor and an m-independent reduced matrix element (RME). A set of recursion formulas for the RME was obtained. A graphical representation of the RME on the branching diagram for binary irreducible representations of permutation groups was then introduced. This gave a simple and systematic way of calculating the RME. Results show explicitly the geometry dependence of superradiance and the relative importance of r-conserving and r-nonconserving processes and clears up the chief difficulty encounted in the problem of N two-level atoms, spread over large regions, interacting with a multimode radiation field.