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arXiv:2204.02708

Exact WKB-like Formulae for the Energies by means of the Quantum Hamilton-Jacobi Equation

M. Fusco Girard

Published Apr 6, 2022• 0 citations
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Abstract

It is shown that by means of the approach based on the Quantum Hamilton-Jacobi equation, it is possible to modify the WKB expressions for the energy levels of quantum systems, when incorrect, obtaining exact WKB-like formulae. This extends the results found in previous papers, where it was demonstrated that the QHJ method provides exact WKB-like expressions for the wave functions.

DOI
10.48550/arxiv.2204.02708

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It is shown that by means of the approach based on the Quantum Hamilton-Jacobi equation, it is possible to modify the WKB expressions for the energy levels of quantum systems, when incorrect, obtaining exact WKB-like formulae. This extends the results found in previous papers, where it was demonstrated that the QHJ method provides exact WKB-like expressions for the wave functions.
By means of numerical solutions of the quantum Hamilton Jacobi equation, a general WKB-like representation for one-dimensional wave functions is obtained. This representation is unique in the classically forbidden regions, while in the allowed one, each wave function corresponds to a one parameter family of solutions of the QHJE. The method has been applied to various systems, with different energies and initial conditions. In all investigated cases, the wave functions so obtained accurately reproduce the solutions of the Schrödinger equation, analytically or numerically computed by other ways. Some results for harmonic oscillator and radial Coulomb motion are presented.
General analytical solutions of the Quantum Hamilton Jacobi Equation for conservative one-dimensional or reducible motion are presented and discussed. The quantum Hamilton's characteristic function and its derivative, i.e. the quantum momentum function, are obtained in general, and it is shown that any one-dimensional wave function can be exactly represented in a WKB-like form. The formalism is applied to the harmonic oscillator and to the electron's motion in the hydrogen atom, and the above mentioned functions are computed and discussed for different quantum numbers. It is analyzed how the quantum quantities investigated tend to the corresponding classical ones, in the limit ℏ→0\hbar\to 0ℏ→0. These results demonstrate that the Quantum Hamilton Jacobi Equation is not only completely equivalent to the Schrödinger Equation, but allows also to fully investigate the transition from quantum to classical mechanics.
In this paper, we develop quantum mechanics of quasi-one-dimensional systems upon the framework of the quantum-mechanical Hamilton–Jacobi theory. We will show that the Schrödinger point of view and the Hamilton–Jacobi point of view are fully equivalent in their description of physical systems, but differ in their descriptive manner. As a main result of this, a wavefunction in Hamilton–Jacobi theory can be decomposed into travelling waves in any point in space, not only asymptotically. Using the quasi-linearization technique, we derive quantum correction functions in every order of . The quantum correction functions will remove the turning-point singularity that plagues the WKB-series expansion already in zeroth order and thus provide an extremely good approximation to the full solution of the Schrödinger equation. In the language of quantum action it is also possible to elegantly solve the connection problem without asymptotic approximations. The use of quantum action further allows us to derive an equation by which the Maslov index is directly calculable without any approximations. Stationary quantum trajectories will also be considered and thoroughly discussed.
In this article, we develop quantum mechanics upon the framework of the quantum mechanical Hamilton-Jacobi theory. We will show, that the Schroedinger point of view and the Hamilton-Jacobi point of view are fully equivalent in their description of physical systems, but differ in their descriptive manner. As a main result, a wave function in Hamilton-Jacobi theory can be decomposed into travelling waves in any point in space, not only asymptotically. The well known WKB-theory will be a special result of the more general theory, we will develop below. By the example of the linear potential and the harmonic oscillator, we will discuss quantum mechanics from the Hamilton-Jacobi point of view. Soft boundary value problems as the connection problem can be solved exactely. Quantizised energies and Maslov-indices can be calculated directely without orthonormalizing wave-functions. Also, we will focus on trajectory themes, which, in contrast to the Schroedinger point of view, follow naturally from the quantum mechanical action function.
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.