Physical Problems Solved by the Phase-Integral Method

by Nanny Fröman , Per Olof Fröman

• ISBN13: 9780521812092
• ISBN10: 0521812097
• Format: Hardcover
• Copyright: 2002-06-24
• Publisher: Cambridge University Press

Summary

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.

Table of Contents

Preface	xi
Historical survey	1	(11)
Development from 1817 to 1926	1	(7)
Carlini's pioneering work	1	(2)
The work by Liouville and Green	3	(1)
Jacobi's contribution towards making Carlini's work known	4	(1)
Scheibner's alternative to Carlini's treatment of planetary motion	4	(1)
Publications 1895-1912	5	(1)
First traces of a connection formula	5	(1)
Publications 1915-1921	6	(1)
I Both connection formulas are derived in explicit form	7	(1)
The method is rediscovered in quantum mechanics	7	(1)
Development after 1926	8	(4)
Description of the phase-integral method	12	(47)
Form of the wave function and the q-equation	12	(1)
Phase-integral approximation generated from an unspecified base function	13	(8)
F-matrix method	21	(14)
Exact solution expressed in terms of the F-matrix	22	(3)
General relations satisfied by the F-matrix	25	(1)
F-matrix corresponding to the encircling of a simple zero of Q2(z)	26	(1)
Basic estimates	26	(2)
Stokes and anti-Stokes lines	28	(1)
Symbols facilitating the tracing of a wave function in the complex z-plane	29	(1)
Removal of a boundary condition from the real z-axis to an anti-Stokes line	30	(2)
Dependence of the F-matrix on the lower limit of integration in the phase integral	32	(1)
F-matrix expressed in terms of two linearly independent solutions of the differential equation	33	(2)
F-matrix connecting points on opposite sides of a well-isolated turning point, and expressions for the wave function in these regions	35	(11)
Symmetry relations and estimates of the F-matrix elements	36	(2)
Parameterization of the matrix F(x1, x2)	38	(2)
Changes of α, β and γ when x1 moves in the classically forbidden region	40	(1)
Changes of α, β and γ when x2 moves in the classically allowed region	41	(1)
Limiting values of α, β and γ	42	(1)
Wave function on opposite sides of a well-isolated turning point	43	(2)
Power and limitation of the parameterization method	45	(1)
Phase-integral connection formulas for a real, smooth, single-hump potential barrier	46	(13)
Exact expressions for the wave function on both sides of the barrier	48	(2)
Phase-integral connection formulas for a real barrier	50	(3)
Wave function given as an outgoing wave to the left of the barrier	53	(1)
Wave function given as a standing wave to the left of the barrier	54	(5)
Problems with solutions	59	(146)
Base function for the radial Schrodinger equation when the physical potential has at the most a Coulomb singularity at the origin	59	(2)
Base function and wave function close to the origin when the physical potential is repulsive and strongly singular at the origin	61	(1)
Reflectionless potential	62	(1)
Stokes and anti-Stokes lines	63	(3)
Properties of the phase-integral approximation along an anti-Stokes line	66	(1)
Properties of the phase-integral approximation along a path on which the absolute value of exp[iw(z)] is monotonic in the strict sense, in particular along a Stokes line	66	(3)
Determination of the Stokes constants associated with the three anti-Stokes lines that emerge from a well isolated, simple transition zero	69	(3)
Connection formula for tracing a phase-integral wave function from a Stokes line emerging from a simple transition zero t to the anti-Stokes line emerging from t in the opposite direction	72	(1)
Connection formula for tracing a phase-integral wave function from an anti-Stokes line emerging from a simple transition zero t to the Stokes line emerging from t in the opposite direction	73	(1)
Connection formula for tracing a phase-integral wave function from a classically forbidden to a classically allowed region	74	(3)
One-directional nature of the connection formula for tracing a phase-integral wave function from a classically forbidden to a classically allowed region	77	(2)
Connection formulas for tracing a phase-integral wave function from a classically allowed to a classically forbidden region	79	(2)
One-directional nature of the connection formulas for tracing a phase-integral wave function from a classically allowed to a classically forbidden region	81	(2)
Value at the turning point of the wave function associated with the connection formula for tracing a phase-integral wave function from the classically forbidden to the classically allowed region	83	(4)
Value at the turning point of the wave function associated with a connection formula for tracing the phase-integral Wave function from the classically allowed to the classically forbidden region	87	(1)
Illustration of the accuracy of the approximate formulas for the value of the wave function at a turning point	88	(3)
Expressions for the a-coefficients associated with the Airy functions	91	(5)
Expressions for the parameters α, β and γ when Q2(z) = R(z) = -z	96	(2)
Solutions of the Airy differential equation that at a fixed point on one side of the turning point are represented by a single, pure phase-integral function, and their representation on the other side of the turning point	98	(4)
Connection formulas and their one-directional nature demonstrated for the Airy differential equation	102	(3)
Dependence of the phase of the wave function in a classically allowed region on the value of the logarithmic derivative of the wave function at a fixed point x1 in an adjacent classically forbidden region	105	(2)
Phase of the wave function in the classically allowed regions adjacent to a real, symmetric potential barrier, when the logarithmic derivative of the wave function is given at the centre of the barrier	107	(8)
Eigenvalue problem for a quantal particle in a broad, symmetric potential well between two symmetric potential barriers of equal shape, with boundary conditions imposed in the middle of each barrier	115	(2)
Dependence of the phase of the wave function in a classically allowed region on the position of the point x, in an adjacent classically forbidden region where the boundary condition ψ(x1) = 0 is imposed	117	(4)
Phase-shift formula	121	(2)
Distance between near-lying energy levels in different types of physical systems, expressed either in terms of the frequency of classical oscillations in a potential well or in terms of the derivative of the energy with respect to a quantum number	123	(2)
Arbitrary-order quantization condition for a particle in a single-well potential, derived on the assumption that the classically allowed region is broad enough to allow the use of a connection formula	125	(2)
Arbitrary-order quantization condition for a particle in a single-well potential, derived without the assumption that the classically allowed region is broad	127	(3)
Displacement of the energy levels due to compression of an atom (simple treatment)	130	(3)
Displacement of the energy levels due to compression of an atom (alternative treatment)	133	(4)
Quantization condition for a particle in a smooth potential well, limited on one side by an impenetrable wall and on the other side by a smooth, infinitely thick potential barrier, and in particular for a particle in a uniform gravitational field limited from below by an impenetrable plane surface	137	(3)
Energy spectrum of a non-relativistic particle in a potential proportional to cot2(x/a0), where 0 < x/a0 < π x/0 and a0 is a quantity with the dimension of length, e.g. the Bohr radius	140	(2)
Determination of a one-dimensional, smooth, single-well potential from the energy spectrum of the bound states	142	(2)
Determination of a radial, smooth, single-well potential from the energy spectrum of the bound states	144	(3)
Determination of the radial, single-well potential, when the energy eigenvalues are -mZ2e4/[2h2(l + s + 1)2], where l is the angular momentum quantum number, and s is the radial quantum number	147	(3)
Exact formula for the normalization integral for the wave function pertaining to a bound state of a particle in a radial potential	150	(2)
Phase-integral formula for the normalized radial wave function pertaining to a bound state of a particle in a radial single-well potential	152	(3)
Radial wave function ψ(z) for an s-electron in a classically allowed region containing the origin, when the potential near the origin is dominated by a strong, attractive Coulomb singularity, and the normalization factor is chosen such that, when the radial variable z is dimensionless, ψ(z)/z tends to unity as z tends to zero	155	(5)
Quantization condition, and value of the normalized wave function at the origin expressed in terms of the level density, for an s-electron in a single-well potential with a strong attractive Coulomb singularity at the origin	160	(3)
Expectation value of an unspecified function f(z) for a non-relativistic particle in a bound state	163	(3)
Some cases in which the phase-integral expectation value formula yields the expectation value exactly in the first-order approximation	166	(1)
Expectation value of the kinetic energy of a non-relativistic particle in a bound state. Verification of the virial theorem	167	(2)
Phase-integral calculation of quantal matrix elements	169	(2)
Connection formula for a complex potential barrier	171	(10)
Connection formula for a real, single-hump potential barrier	181	(5)
Energy levels of a particle in a smooth double-well potential, when no symmetry requirement is imposed	186	(4)
Energy levels of a particle in a smooth, symmetric, double-well potential	190	(2)
Determination of the quasi-stationary energy levels of a particle in a radial potential with a thick single-hump barrier	192	(5)
Transmission coefficient for a particle penetrating a real single-hump potential barrier	197	(3)
Transmission coefficient for a particle penetrating a real, symmetric, superdense double-hump potential barrier	200	(5)
References	205	(4)
Author index	209	(2)
Subject index	211