  We find the covariant deformed Heisenberg algebra and the Laplace-Beltrami
operator on the extended $h$-deformed quantum plane and solve the Schr\"odinger
equations explicitly for some physical systems on the quantum plane. In the
commutative limit the behaviour of a quantum particle on the quantum plane
becomes that of the quantum particle on the Poincar\'e half-plane, a surface of
constant negative Gaussian curvature. We show the bound state energy spectra
for particles under specific potentials depend explicitly on the deformation
parameter $h$. Moreover, it is shown that bound states can survive on the
quantum plane in a limiting case where bound states on the Poincar\'e
half-plane disappear.
