Some Recent Developments in WKB Approximation

121 Pages·2013·2.04 MB·English

by  Akbar Safari

Some recent developments in WKB approximation
SOME RECENT DEVELOPMENTS IN WKB APPROXIMATION
BY
AKBAR SAFARI, M.Sc.
a thesis
submitted to the department of Physics & Astronomy
and the school of graduate studies
of mcmaster university
in partial fulfilment of the requirements
for the degree of
Master of Science
c Copyright by Akbar Safari, August 2013
(cid:13)
All Rights Reserved
Master of Science (2013) McMaster University
(Physics & Astronomy) Hamilton, Ontario, Canada
TITLE: Some recent developments in WKB approximation
AUTHOR: Akbar Safari
M.Sc., (Condensed Matter Physics)
IUST, Tehran, Iran
SUPERVISOR: Dr. D.W.L. Sprung
NUMBER OF PAGES: xiii, 107
ii
To my wife,
Mobina
Abstract
WKB theory provides a plausible link between classical mechanics and quantum me-
chanics in its semi-classical limit. Connecting the WKB wave function across the
turning points in a quantum well, leads to the WKB quantization condition. In this
work, I focus on some improvements and recent developments related to the WKB
quantizationcondition. FirstIdiscusshowthecombinationofsuper-symmetricquan-
tum mechanics and WKB, gives the SWKB quantization condition which is exact for
a large class of potentials called shape invariant potentials. Next I turn to the fact
that there is always a probability of reflection when the potential is not constant
and the phase of the wave function should account for this reflection. WKB theory
ignores reflection except at turning points. I explain the work of Friedrich and Trost
who showed that by including the correct “reflection phase” at a turning point, the
WKB quantization condition can be made to give exact bound state energies. Next I
discuss the work of Cao and collaborators which takes reflection of the wave function
into account everywhere. We show that Cao’s method provides a way to compute the
F-T reflection phase. Finally I discuss a paper of Fabre and Gu´ery-Odelin who used
the exponential potential to study the accuracy of WKB. In their results the accuracy
deteriorates as the energy increases, which is inconsistent with Bohr’s correspondence
principle. Using the Friedrich and Trost method, we resolved this problem.
iv
Acknowledgements
Foremost, I would like to express my sincere gratitude to my supervisor Prof. Donald
Sprung for providing me with this unique opportunity. For always being available
to help me with my academic and non-academic concerns. He was a vast source of
wisdom, knowledge, support and guidance throughout the past two years. I would
also like to thank the department of Physics and Astronomy for the financial support
without which this work would not have been possible.
Last but not the least, I would like to thank my family. My parents, for their
understanding and supporting me throughout my entire academic career and also my
wife for her great encouragement.
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Contents
Abstract iv
Acknowledgements v
1 Introduction to the semi-classical or WKB approximation 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Analogy to optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.3 Derivation of WKB from an optical analogy . . . . . . . . . . . . . . 11
1.4 WKB quantization condition . . . . . . . . . . . . . . . . . . . . . . . 12
1.5 Langer modification . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.6 WKB asymptotic series . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2 Super-Symmetric WKB 21
2.1 Super-symmetric quantum mechanics . . . . . . . . . . . . . . . . . . 21
2.2 Reflectionless potentials . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.3 Shape invariant potentials . . . . . . . . . . . . . . . . . . . . . . . . 27
2.4 Super-symmetric WKB . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.5 Barclay’s insight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
vi
3 Phase Shift Modification 36
3.1 Reflection phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.2 Friedrich and Trost method . . . . . . . . . . . . . . . . . . . . . . . 41
3.3 Approximating the phase loss . . . . . . . . . . . . . . . . . . . . . . 49
3.4 Cao’s quantization condition . . . . . . . . . . . . . . . . . . . . . . . 55
3.5 Segmentation method . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.6 Another approach to Cao’s condition . . . . . . . . . . . . . . . . . . 60
3.7 General remarks about the correction term I . . . . . . . . . . . . . 70
c
4 Exponential Potential 74
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.2 Symmetric 1D exponential potential . . . . . . . . . . . . . . . . . . . 78
4.3 Apparent violation of the correspondence principle . . . . . . . . . . . 80
4.4 GT asymptotic series for the Bessel function . . . . . . . . . . . . . . 81
4.5 The radial potential, V (x) . . . . . . . . . . . . . . . . . . . . . . . . 84
I
4.6 The symmetric 1D exponential potential, even-parity states . . . . . . 91
5 Summary 96
5.1 Final comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
5.2 Suggestions for further research . . . . . . . . . . . . . . . . . . . . . 98
A Asymptotic series 100
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List of Tables
1.1 The six lowest even-parity bound state energies of the quartic potential
well, V(x) = x4, taken from Ref. [24]. (WKB) denotes the energy
q
computed in qth order WKB. The number in brackets is the best rela-
tive accuracy achieved for that level, by truncating the calculation at
order q. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.1 List of shape invariant potentials with translation, taken from Ref.[32].
The potentials, V(x), are shifted to place the ground state energy at
zero. For these potentials, SWKB is exact, ESWKB = Eexact. . . . . . 29
n n
3.1 Bound state energies for the Woods-Saxon potential well of Eq. 3.25
with parameters R = 25 and σ = 2.5 . . . . . . . . . . . . . . . . . . . 48
3.2 Bound state energies for the Woods-Saxon potential well of Eq. 3.25
with parameters R = 25 and σ = 0.5 . . . . . . . . . . . . . . . . . . . 49
3.3 Comparison of the F-T method (Eq. 3.36) with LO-WKB for ground
states of SIPs. (Taken from Ref. [32]) . . . . . . . . . . . . . . . . . . 52
3.4 The first eight bound state energies of the stretched quartic well in
Eq.3.40withR = 5(h¯2/2mV )1/6. IntheF-Tmethod,theapproximate
0
phase loss of Fig. 3.9 is used. . . . . . . . . . . . . . . . . . . . . . . . 55
viii
3.5 Numerical results showing that the value of I +I for the linear poten-
0 c
tial, satisfies Eq.3.62 for any energy E, if the integration starts from
the boundary at x = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.6 Numerical results showing that Eq.3.62 confirms the bound state en-
ergies for the linear potential, providing that one starts from + and
∞
integrates backwards to construct the wave function. . . . . . . . . . 67
ix