Physical Problems Solved by the Phase-Integral Method

A mathematical approximation method important for many branches of theoretical physics, applied mathematics and engineering.

Nanny Fröman (Author), Per Olof Fröman (Author)

9780521675765, Cambridge University Press

Paperback, published 14 July 2005

232 pages, 30 b/w illus. 50 exercises
24.4 x 17 x 1.2 cm, 0.38 kg

"Thorough" Quarterly of Applied Mathematics

This book provides a thorough introduction to one of the most efficient approximation methods for the analysis and solution of problems in theoretical physics and applied mathematics. It is written with practical needs in mind and contains a discussion of 50 problems with solutions, of varying degrees of difficulty. The problems are taken from quantum mechanics, but the method has important applications in any field of science involving second order ordinary differential equations. The power of the asymptotic solution of second order differential equations is demonstrated, and in each case the authors clearly indicate which concepts and results of the general theory are needed to solve a particular problem. This book will be ideal as a manual for users of the phase-integral method, as well as a valuable reference text for experienced research workers and graduate students.

Part I. Historical Survey: 1. History of an approximation method of wide importance in various branches of physics
Part II. Description of the Phase-Integral Method: 2. Form of the wave function and the q-equation
3. Phase-integral approximation generated from an unspecified base function
4. F-matrix method
5. F-matrix connecting points on opposite sides of a well isolated turning point, and expressions for the wave function in these regions
6. Phase-integral connection formulas for a real, smooth, single-hump potential barrier
Part III. Problems With Solutions: 1. Determination of a convenient base function
2. Determination of a phase-integral function satisfying the Schrödinger equation exactly
3. Properties of the phase-integral approximation along certain paths
4. Stokes constants and connection formulas
5. Airy's differential equation
6. Change of phase of the wave function in a classically allowed region due to the change of a boundary condition imposed in an adjacent classically forbidden region
7. Phase shift
8. Nearlying energy levels
9. Quantization conditions
10. Determination of the potential from the energy spectrum
11. Formulas for the normalization integral, not involving the wave function
12. Potential with a strong attractive Coulomb singularity at the origin
13. Formulas for expectation values and matrix elements, not involving the wave function
14. Potential barriers
References.

Subject Areas: Physics [PH], Applied mathematics [PBW]