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European Journal of Pure and Applied Mathematics

Quantization of Singular Systems using Fractional Calculus

Eyad Hasan Hasan

Published May 1, 2025• 0 citations
Publisher

Abstract

In this paper, we examined the theory of singular systems using fractional calculus. We quantized these systems using the fractional \textbf{WKB} approximation. We applied the Hamilton–Jacobi treatment for these systems. We obtained equations of motion. We constructed the fractional Hamilton–Jacobi partial differential equations (\textbf{FHJPDEs}) to obtain the action functions \textbf{S}. The action function enables us to obtain the wave function for these systems. We achieved that the quantum results agree with the classical results. Finally, we examined two mathematical examples to demonstrate the theory.

DOI
10.29020/nybg.ejpam.v18i2.5765

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In this paper, we examined the theory of singular systems using fractional calculus. We quantized these systems using the fractional \textbf{WKB} approximation. We applied the Hamilton–Jacobi treatment for these systems. We obtained equations of motion. We constructed the fractional Hamilton–Jacobi partial differential equations (\textbf{FHJPDEs}) to obtain the action functions \textbf{S}. The action function enables us to obtain the wave function for these systems. We achieved that the quantum results agree with the classical results. Finally, we examined two mathematical examples to demonstrate the theory.
This paper examined dissipative systems with second order Lagrangian in the framework of the Hamilton-Jacobi method. The principal function is determined using the method of separation of variables in the same manner as for regular time-independent Lagrangian. The equations of motions are obtained from the action function. By finding the action function, the wave function can be constructed and the quantization has been carried out using the WKB approximation. One example is given to illustrate our formalism.
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.
In this work time independent damping systems are studied using Lagrangian and Hamiltonian for time independent damping, which are present through the factor eλq. The Hamilton Jacobi equation is formulated to find the Hamilton Jacobi function S using separation of variables technique. We can form this function in compact form of two parts the first part as a function of coordinate q, and the second part as a function of time t. Finally, we find the ability of these systems to quantize through an illustrative example.