Coherent states and the role of the affine group in the quantum mechanics of the Morse potential

The coherent states of the Morse potential that have been obtained earlier from supersymmetric quantum mechanics, are shown to be connected with the r epresentations of the affine group of the real line and some of its extensi ons. This relation is similar to the one between the Heisenberg-Weyl group and the coherent states of the harmonic oscillator. The states that minimiz e the uncertainty product of the generators of the affine Lie algebra are s hown to contain all the coherent states of the Morse oscillator plus the in telligent states of the Morse Hamiltonians with different shape parameter s . The representations of the central extension of the affine group denoted by G(o) and its further extension (G) over tilde (o) will be shown to defin e the phase space relevant to the problem by choosing an appropriate orbit of the coadjoint representation of (G) over tilde (o). This allows one to c onstruct a generalized Wigner function on this phase space, which is again essentially in the same relation with the affine group, as the ordinary Wig ner function with the Heisenberg-Weyl group.