  The path integral for a point particle in a Coulomb potential is solved in
momentum space. The solution permits us to give for the first time a negative
answer to an old question of quantum mechanics in curved spaces raised in 1957
by DeWitt, whether the Hamiltonian of a particle in a curved space contains an
additional term proportional to the curvature scalar $R$. We show that this
would cause experimentally wrong level spacings in the hydrogen atom. Our
solution also gives a first experimental confirmation of the correctness of the
measure of integration in path integrals in curved space implied by a recently
discovered nonholonomic mapping principle.
