Bicovariant differential calculus on quantum groups and wave mechanics

The bicovariant differential calculus on quantum groups being defined by Woronowicz and later worked out explicitly by Carow-Watamura et at. and Juro for the real quantum groupsSU_q(N) andSO_q(N) through a systematic construction of the bicovariant bimodules of these quantum groups is reviewed forSU_q(2) andSO_q(N). The resulting vector fields build representations of the quantized universal enveloping algebras acting as covariant differential operators on the quantum groups and their associated quantum spaces. As an application a free particle stationary wave equation on quantum space is formulated and solved in terms of a complete set of energy eigenfunctions.Presented at the Colloquium on the Quantum Groups, Prague, 18–20 June 1992.