Algebraic properties of energy raising and lowering ladder operators for the finite and infinite spectrum potential systems

摘要

An enquiry is made into the Lie algebraic property of the energy raising and lowering ladder operators for a discrete (bound state) spectrum. Starting with the most general construction of these operators in the energy representation we investigate the necessary and sufficient conditions for these operators to form a realization of su(1,1) or su(2) algebra. We formally establish the truth of the undeclared conjecture that a finite spectrum bound state system carries a realization of su(2) algebra and that an infinite spectrum bound state system carries a realization of the su(1,1) algebra. We have discussed the position space realization of these operators for some well known solvable potential systems.