Quantum Mechanical Motions over the Group Manifolds and Related Potentials

Schrödinger equations for many potentials are solved in terms of the special functions. Almost all of these special functions are the matrix elements of the representations of the Lie groups, with their arguments being the group parameters [1,2]. Even the simplest “special functions” namely the elementary transcendentals are related to the Lie groups, i.e., the one parameter Abelian Lie groups. The connection between the group representations and the special functions explains the mystrious properties of them, such as the recurance relations and addition theorems.