  We determine the Lie point symmetries of the Schr\"{o}dinger and the Klein
Gordon equations in a general Riemannian space. It is shown that these
symmetries are related with the homothetic and the conformal algebra of the
metric of the space respectively. We consider the kinematic metric defined by
the classical Lagrangian and show how the Lie point symmetries of the
Schr\"{o}dinger equation and the Klein Gordon equation are related with the
Noether point symmetries of this Lagrangian. The general results are applied to
two practical problems a. The classification of all two and three dimensional
potentials in a Euclidian space for which the Schr\"{o}dinger equation and the
Klein Gordon equation admit Lie point symmetries and b. The application of Lie
point symmetries of the Klein Gordon equation in the exterior Schwarzschild
spacetime and the determination of the metric by means of conformally related
Lagrangians.
