#### Date of Award

1980

#### Degree Type

Dissertation

#### Degree Name

Ph.D.

#### Department

Physics

#### Keywords

Physics, Atomic.

#### Rights

CC BY-NC-ND 4.0

#### Abstract

The JWKB approximation is widely used in many problems in quantum mechanics for its simplicity and also because a closed-form solution often provides more insight than a numerical one. Unfortunately, like all approximations, the JWKB method has its limitations: Firstly, it may fail near or at a classical turning point, and secondly, it employs solutions for a potential which deviates from the actual potential W(x). In this work, a new method is proposed to avoid the limitations of the JWKB method while keeping its advantages. First, the singularity around the classical turning point is resolved not only for a linear potential as proposed by Langer, but also for any potential with a linear leading term through the turning point, and second, a solution is constructed in closed form which is an exact solution for the actual potential W(x). The new method consists in finding a substitute potential U(x) in such a way that, together with its (DELTA){U,x} it is equal to the actual W(x). The solution is then based on U(x) but actually is the solution for W(x) because W(x) = U(x) + (DELTA){U,x}. The new method is first applied to potentials with only two turning points. The eigenvalue problem is reduced to the solution of a linear equation and the energy eigenvalues are exactly given. Then, potentials with more than two turning points are studied and the eigenvalue problem is again reduced to the solution of a polynomial equation whose order depends upon the number of turning points. Then the method is applied to the hindered or restricted rotation. The energy levels are found to be associated with the N-th roots of 1 in the complex plane, i.e. on the unit circle. The method described here is of rather general nature and can be used to solve a number of problems in quantum mechanics, where a solution for the Schrodinger equation in closed form is needed. The applications provided in this work are only given as examples of the applicability and the relative simplicity of the method.Dept. of Physics. Paper copy at Leddy Library: Theses & Major Papers - Basement, West Bldg. / Call Number: Thesis1980 .S223. Source: Dissertation Abstracts International, Volume: 41-03, Section: B, page: 1002. Thesis (Ph.D.)--University of Windsor (Canada), 1980.

#### Recommended Citation

SAAD, NASR YOUSSEF., "A NEW APPROACH TO THE JWKB TREATMENT." (1980). *Electronic Theses and Dissertations*. 681.

https://scholar.uwindsor.ca/etd/681