  The numerical version of the Hamilton-Jacobi quantization method, recently
proposed, is applied to the one dimensional quartic oscillator. A suitable
quantization condition is formulated and various energy levels and wave
functions are computed. The results very well agree with those obtained by
means of the Schroedinger equation, and confirm that the Quantum Hamilton
Jacobi approach, which is the exact version of the semiclassical WKB scheme, is
a self-contained quantization procedure, equivalent and independent from the
Schoedinger's one but more general. Indeed, with respect to this latter, the
Quantum Hamilton-Jacobi equation can be used to investigate the limit h->0,
where the Schroedinger equation loses its significance, and explains how the
fundamental quantities of the classical mechanics, the Hamilton's
characteristic function and the classical momentum, are generated from the
corresponding quantum ones.
