Recurrence relations and path representations of matrix elements of an SU(1,1) algebra

TL;DR

This paper develops a unified algebraic framework using SU(1,1) to describe various quantum systems, providing recurrence relations and diagrammatic representations for matrix elements, with applications to coherent and squeezed states.

Contribution

It introduces a normal ordered representation of unitary operators from SU(1,1) algebra, generalizing the Baker-Campbell-Hausdorff relation, and links matrix elements to recurrence relations and diagrammatic structures.

Findings

Functions satisfy recurrence relations

Diagrammatic representations similar to Pascal's triangle

Special cases include coherent and squeezed states

Abstract

It is shown that a SU(1,1) algebra may be used to provide a unified description of the simple hamonic oscillator and the angular momentum algebras and a class of other semi-infinite algebras. A normal ordered representation of a Unitary operator $U$ constructed from the generators of a SU(1,1) algebra, which is a generalisation of the Baker- Campbell - Hausdorff relation for Lie algebras, is given. It is shown that the normal ordered representatiion of $U$ may be used to calculate expectation values which are functions of the parameters used to construct the operator. The functions so constructed satisfy certain recurrence relations and the entire set of functions may be interpreted in terms of diagrams similar to the Pascal triangle for binomial coefficients. Coherent states, squeezed states and rotation matrices of the angular momentum algebra emerge as special cases.