[Paper Review] Analytical Solutions of the Quantum Hamilton-Jacobi Equation and Exact WKB-Like Representations of One-Dimensional Wave Functions

M. Fusco Girard|arXiv (Cornell University)|Dec 4, 2015

Quantum Mechanics and Non-Hermitian Physics3 citations

TL;DR

This paper presents analytical solutions to the quantum Hamilton-Jacobi equation (QHJE) for one-dimensional systems, deriving exact WKB-like representations of quantum wave functions. By solving the QHJE via Riccati equation techniques, it identifies complex, continuous quantum momentum and action functions that smoothly reduce to classical quantities in the $ackslash$hbar\to 0$ limit, clarifying the quantum-to-classical transition and enabling exact wave function representations not accessible through the Schr\textbackslash{}{"o}dinger equation alone.

ABSTRACT

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 $\hbar o 0$. 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.

Motivation & Objective

• To derive general analytical solutions of the quantum Hamilton-Jacobi equation (QHJE) for one-dimensional conservative systems.
• To identify quantum analogues of the classical Hamilton's characteristic function and momentum that remain continuous and complex in classically allowed regions.
• To demonstrate that quantum wave functions can be exactly represented in a WKB-like form using the real part of the quantum reduced action and its derivative.
• To clarify the quantum-to-classical transition by showing that smoothed quantum momentum and action functions converge to classical values as $\hbar \to 0$.
• To establish that the QHJE provides a fully equivalent, independent formulation of quantum mechanics, capable of recovering both energy levels and eigenfunctions without solving the Schr\textbackslash{}{"o}dinger equation.

Proposed method

• Solving the QHJE using general integrals derived from the theory of Riccati differential equations.
• Constructing complex, continuous solutions for the quantum momentum function $p_{F,G}(x,E)$ and quantum reduced action $W_{G}(x,E)$ in classically allowed regions.
• Ensuring that the imaginary part of the quantum momentum vanishes in the classical limit, while the real part converges to the classical momentum after coarse graining.
• Applying a coarse graining procedure to average quantum functions over small intervals to eliminate high-frequency oscillations and reveal classical behavior.
• Verifying that the real part of the quantum reduced action converges to the classical action $W_C(x)$ after smoothing, with oscillations diminishing in amplitude but increasing in frequency as $\hbar \to 0$.
• Confirming that the quantum momentum function's imaginary part is proportional to $\hbar$, ensuring it vanishes in the classical limit.

Experimental results

Research questions

• RQ1 How can general analytical solutions of the QHJE be derived for one-dimensional systems, particularly in classically allowed regions?
• RQ2 What is the nature of the quantum momentum function in the classically allowed region, and how does it relate to the classical momentum in the $\hbar \to 0$ limit?
• RQ3 Can every one-dimensional quantum wave function be exactly represented in a WKB-like form using the quantum reduced action?
• RQ4 How do quantum fluctuations in the momentum and action functions disappear in the classical limit, and what role does coarse graining play?
• RQ5 To what extent is the QHJE equivalent to the Schr\textbackslash{}{"o}dinger equation, and can it independently yield energy levels and eigenfunctions?

Key findings

• The quantum momentum function $p_{F,G}(x,E)$ is complex and continuous in classically allowed regions, with its real part oscillating around the classical momentum $p_C(x)$, and its imaginary part proportional to $\hbar$.
• As $\hbar \to 0$, the real part of the quantum reduced action $W_G(x,E)$ develops an infinite number of ripples but converges to the classical action $W_C(x)$ after coarse graining.
• The coarse-grained real part of $p_{F,G}(x)$ converges to the classical momentum $p_C(x)$, confirming that quantum fluctuations must be smoothed to recover classical behavior.
• The imaginary part of $p_{F,G}(x)$ fluctuates around zero with finite amplitude and vanishes in the classical limit, as required.
• Every one-dimensional wave function can be exactly represented in a WKB-like form using the real part of the quantum reduced action as the phase and its derivative as the amplitude regulator.
• The QHJE provides a fully equivalent, independent formulation of quantum mechanics that clarifies the quantum-to-classical transition and allows exact reconstruction of wave functions without solving the Schr\textbackslash{}{"o}dinger equation.