Ordered operator expansions by comparison

Abstract

Ordered operator expansions for operators forming physically important low-dimensional Lie algebras are derived in a simple unified way. Starting with the Zassenhaus formula for the disentangling of exponential operators, series expansions of both undisentangled and disentangled exponentials and comparison of the operator coefficients of equal powers of an ordering parameter alpha leads to ordered operator expansions. This 'comparison method' gives an alternative simple derivation of some already known formulae and a number of new formulae in the physical and chemical applications of the harmonic oscillator and for master equation problems with nearest-neighbour transition probabilities. The 'comparison method' cannot be applied to the angular momentum algebra directly. By a slight modification it can be used to derive from one matrix element or trace of JxkJylJzn all possible combinations k, l, n by simply comparing powers of ordering parameters.