Quantization of a one-dimensional system by means of the quantum Hamilton-Jacobi equation

TL;DR

This paper demonstrates that the quantum Hamilton-Jacobi method provides a self-contained, accurate, and more general quantization approach for one-dimensional systems, aligning with Schrödinger results and extending to classical limits.

Contribution

It applies a numerical quantum Hamilton-Jacobi method to the quartic oscillator, confirming its equivalence to Schrödinger quantization and its ability to explore classical limits.

Findings

Results agree with Schrödinger equation energy levels and wave functions.

The method is more general and can analyze the classical limit h->0.

Confirms the quantum Hamilton-Jacobi approach as a valid quantization procedure.

Abstract

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.