Physics & Astronomy ETDs

Publication Date

7-24-1970

Abstract

A new WKB-like quantization rule is developed by means of a transformation method. This method makes use of a combination of a Liouville-like substitution and the Prüfer transformation. The only quality required of the potential is that it be twice differentiable on the real axis. The new quantization rule is an exact rule. However, some terms in the rule must be approximated before it can be evaluated for any particular potential function. An interesting feature of the derivation is that no asymptotic expansions are used -- unlike the usual development of the 1st order WKB quantization rule (FWKBQR). The terms corresponding to those of the FWKBQR can be isolated in the new exact rule. This feature offers the possiblity of determining rigorous error bounds on the 1st order WKB quantum number. Error bounds are derived on the 1st order WKB quantum number for the potential V(x)=kxn with n = 2, 4, 6, 8, 12, and 16. It is shown that an upper bound on the 1st order quantum number error for this type of potential is ½ of a quantum number. This reaffirms a result of Titchmarsh. The lower error bounds are determined at "low" energies and they are more loose than the upper bound error. Graphs of these results are included. A new approximate quantization rule is derived from the exact rule. This new approximate rule can be evaluated by the same numerical procedures which are used to determine energy eigenvalues and 1st energy differences for the potential V(x)=kxn with n = 2, 4, 6, 8, 12, and 16. The new results are compared with the previously known exact and 1st order WKB results. In addition, the new rule is numerically tested on some oscillators which approximate a square well and comparisons are made with the exact square well energy values. Except for the case of the harmonic oscillator (where the FWKBQR gives exact results), the new rule is found to be generally superior to the FWKBQR both in its prediction of energies and of energy differences for the potentials tested. In addition, the new rule gives highly accurate results for the harmonic oscillator. Tables of these results are presented.

Degree Name

Physics

Level of Degree

Doctoral

Department Name

Physics & Astronomy

First Committee Member (Chair)

Charles Leroy Beckel

Second Committee Member

Bernard Epstein

Third Committee Member

James Daniel Finley III

Project Sponsors

N.A.S.A. Traineeships from 1967 to 1970

Language

English

Document Type

Dissertation

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