[Paper Review] Quantum Hamilton - Jacobi study of wave functions and energy spectrum of solvable and quasi - exactly solvable models

K G Geojo|ArXiv.org|Oct 1, 2004

Quantum optics and atomic interactions3 citations

TL;DR

This thesis introduces a quantum Hamilton-Jacobi (QHJ) formalism to compute energy spectra and wave functions for exactly solvable and quasi-exactly solvable (QES) quantum systems. By deriving a non-linear differential equation for the quantum momentum function (QMF), the method determines energy levels via a contour integral quantization condition based on QMF singularities and residues, enabling exact energy spectrum calculations without solving the Schrödinger equation directly. The key contribution is a unified framework that derives QES conditions from the behavior of QMF at infinity, showing that quasi-exact solvability arises when QMF becomes a rational function after transformation.

ABSTRACT

In this thesis, the quantum Hamilton Jacobi (QHJ) formalism is used to study various exactly solvable (ES) and quasi -exactly solvable (QES) models. Using this method, we obtain the bound state eigenvalues and the eigenfunctions for the models studied. The central entity of this formalism in the logarithmic derivative of the wave function, known as the quantum momentum function (QMF).It is assumed that the point at infinity is an isolated singular point.The kowledge of the singularity structure of the QMF is used to arrive at the required solutions. We show that there are marked differences between the singularity structures of the ES and QES models.

Motivation & Objective

• To develop a systematic QHJ-based method for computing energy spectra and bound state wave functions in quantum mechanics.
• To establish a connection between the quantum momentum function (QMF) and the node structure of wave functions through exact quantization conditions.
• To derive the quasi-exact solvability (QES) condition for one-dimensional potentials from the asymptotic behavior of the QMF.
• To demonstrate that wave functions and energy levels for QES models can be computed using the same residue-based formalism as for exactly solvable models.
• To explore the complex-plane structure of QMF, particularly the location of poles and zeros, and their relation to bound state nodes and potential parameter phases.

Proposed method

• Formulate the quantum Hamilton-Jacobi (QHJ) equation as a non-linear differential equation for the quantum momentum function (QMF).
• Apply an exact quantization condition based on a contour integral of the QMF in the complex plane, with residues determined from Laurent series expansions.
• Use the classical limit ($\hbar \to 0$) as a boundary condition to select physically valid QMF solutions.
• Compute wave functions by reconstructing them from the singularities and residues of the QMF, using known residue techniques from prior work.
• Derive the QES condition by assuming that the QMF becomes a rational function at infinity after a suitable transformation.
• Verify the method on standard models (e.g., harmonic oscillator, Morse, P\

Experimental results

Research questions

• RQ1 How can the energy spectrum of exactly solvable quantum systems be computed without solving the Schr\

Key findings

• The QHJ formalism enables exact energy spectrum computation for solvable models via a contour integral quantization condition based on QMF singularities and residues, bypassing direct solution of the Schr\