  We reconsider the problem of quantising a particle on the $D$-dimensional
sphere. Adopting a Lagrangian method of reducing the degrees of freedom, the
quantum Hamiltonian is found to be the usual Schr\"odinger operator without any
boundary term. The equivalence with the Dirac Hamiltonian approach is
demonstrated, either in the cartesian or in the curvilinear basis. We also
briefly comment on the path integral approach.
