Normal ordering of the su(1, 1) ladder operators for the quasi-number states of the Morse oscillator

Abstract

The Morse oscillator is an adequate zero-order model for describing the highly excited vibrational states and large-amplitude vibrational motion. The corresponding Schrödinger equations can be conveniently solved by algebraic methods using the so-called quasi-number states (QNS) resembling the true wave functions of the Morse oscillator. The associated QNS ladder operators are occupation number, creation and annihilation operators K0,K+,K− that satisfy the commutation relations [K0,K±]=±K± and [K+,K−]=−2K0 of the Lie algebra su(1, 1). The efficient applications of the corresponding algebraic techniques within the variational or perturbative approximations require reduction of arbitrary multiplicative combinations of the su(1,1) ladder operators to the linear combinations of the normalized products K0aK+bK−c. The closed form solution of this problem for a typical complex form of the operator product is derived and proven mathematically. The validity and the numerical stability of the derived expressions are demonstrated by the systematic numerical tests for the product of six-term multiplicative operators.