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Math. Comput. Appl. 2016, 21(4), 41; doi:10.3390/mca21040041

A Normal and Standard Form Analysis of the JWKB Asymptotic Matching Rule via the First Order Bessel’s Equation

Faculty of Engineering, Adnan Menderes University, Aytepe Central Campus-09100 Aydın, Turkey
Academic Editor: Fazal M. Mahomed
Received: 25 June 2016 / Revised: 29 August 2016 / Accepted: 27 September 2016 / Published: 19 October 2016
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Abstract

Traditional first order JWKB method ( = : ( J W K B ) 1 ) is a conventional semiclassical approximation method mainly used in quantum mechanical systems for accurate solutions. ( J W K B ) 1 general solution of the Time Independent Schrodinger’s Equation (TISE) involves application of the conventional asymptotic matching rules to give the accurate wavefunction in the Classically Inaccessible Region (CIR) of the related quantum mechanical system. In this work, Bessel Differential Equation of the first order ( = : ( B D E ) 1 ) is chosen as a mathematical model and its ( J W K B ) 1 solution is obtained by first transforming into the normal form via the change of independent variable. The ( J W K B ) 1 general solution for appropriately chosen initial values in both normal and standard form representations is analyzed via the generalized ( J W K B ) 1 asymptotic matching rules regarding the S ˜ i j matrix elements given in the literature. Instead of applying the common ( J W K B ) 1 asymptotic matching rules relying on the physical nature of the quantum mechanical system, i.e., a physically acceptable (normalizable) wavefunction, a pure semiclassical analysis is studied via the ( B D E ) 1 model mathematically. Finally, an application to a specific case of the exponential potential decorated quantum mechanical bound state problem is presented. View Full-Text
Keywords: WKB; JWKB; asymptotic modification; asymptotic matching; exponential potential decorated bound state problems; Schrodinger’s equation; standard form; normal form; Bessel functions WKB; JWKB; asymptotic modification; asymptotic matching; exponential potential decorated bound state problems; Schrodinger’s equation; standard form; normal form; Bessel functions
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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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Deniz, C. A Normal and Standard Form Analysis of the JWKB Asymptotic Matching Rule via the First Order Bessel’s Equation. Math. Comput. Appl. 2016, 21, 41.

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