  The quasilinearization method (QLM) of solving nonlinear differential
equations is applied to the quantum mechanics by casting the Schr\"{o}dinger
equation in the nonlinear Riccati form. The method, whose mathematical basis in
physics was discussed recently by one of the present authors (VBM), approaches
the solution of a nonlinear differential equation by approximating the
nonlinear terms by a sequence of the linear ones, and is not based on the
existence of some kind of a small parameter. It is shown that the
quasilinearization method gives excellent results when applied to computation
of ground and excited bound state energies and wave functions for a variety of
the potentials in quantum mechanics most of which are not treatable with the
help of the perturbation theory or the 1/N expansion scheme. The convergence of
the QLM expansion of both energies and wave functions for all states is very
fast and already the first few iterations yield extremely precise results. The
precison of the wave function is typically only one digit inferior to that of
the energy. In addition it is verified that the QLM approximations, unlike the
asymptotic series in the perturbation theory and the 1/N expansions are not
divergent at higher orders.
