  In this article we give explicit solutions for the wave equations associated
to the modified Schr\"odinger operators with magnetic field on the disc and the
upper half plane models of the hyperbolic plane. We show that the modified
Schr\"odinger operator with magnetic field on the upper half plane model and
the Schr\"odinger operator with diatomic molecular Morse potential on $\R$ are
related by means of one-dimensional Fourier transform. Using this relation we
give the explicit forms of the wave kernels associated to the Schr\"odinger
operator with the diatomic molecular Morse potential on $\R$ in terms of the
two variables confluent hypergeometric function $\Phi_1$.
