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Some Recent Developments in WKB Approximation SOME RECENT DEVELOPMENTS in WKB APPROXIMATION

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Some Recent Developments in WKB Approximation SOME RECENT DEVELOPMENTS in WKB APPROXIMATION 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 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 quantization condition. First I discuss how the combination of super-symmetric quan- 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-Odelinwho 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. v 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 Ic . 70 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, VI(x)........................ 84 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 vii List of Tables 1.1 The six lowest even-parity bound state energies of the quartic potential 4 well, V (x) = x , taken from Ref. [24]. (WKB)q denotes the energy 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 SWKB exact zero. For these potentials, SWKB is exact, En = En . 29 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 2 1=6 Eq. 3.40 with R = 5(¯h =2mV0) . In the F-T method, the approximate phase loss of Fig. 3.9 is used. 55 viii 3.5 Numerical results showing that the value of I0 +Ic for the linear poten- 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 1 integrates backwards to construct the wave function. 67 ix List of Figures 1.1 Schematic drawing of the WKB wave function; oscillatory in the al- lowed zone, decaying in the forbidden zone, and diverging near the turning point, b .............................. 5 1.2 Illustrating the two connection formulae in two cases . 6 1.3 Schematic view of a barrier. 7 1.4 Splitting of the incoming wave eik0x, into reflected and transmitted waves at the potential steps in the segmentation method. 12 1.5 Schematic view of a quantum well V (x), and ground state wave func- tion, (x), of a particle confined between the turning points a; b. 13 2.1 The parter potential V2 (right) of the infinite square well V1 (Left). 25 2.2 The Ginocchio potential of Eq. 2.35 for two choices of parameters: a)λ = 0:5 and ν = 10:5. b)λ = 6:25 and ν = 5:5. 32 2.3 Comparison of the accuracy of WKB vs. SWKB energies for two choices of the parameters in the Ginocchio potential. a)λ = 0:5 and ν = 10:5. b)λ = 6:25 and ν = 5:5. 33 3.1 Sub-barrier reflection of a wave incident on a vertical step . 37 3.2 Reflection phase shift due to scattering from a step potential . 38 x 3.3 a) A barrier of Woods-Saxon shape. b) Phase shift in scattering from the smooth step barrier. 39 3.4 Comparison of two potential barriers with a smaller and a larger surface thickness. 41 3.5 A flat potential for x < 0 with a quadratic barrier for x > 0. 43 3.6 Phase loss φ of Eq. 3.24, for reflection from a quadratic potential shown in Fig. 3.5 . 45 3.7 The Woods-Saxon potential, Eq. 3.25, for V0 = 1, R = 25 and σ = 2:5 . 46 3.8 Phase loss in reflection from a Woods-Saxon potential step for different values of the parameter σ (Eq. 3.27) . 48 3.9 Comparison of the approximate phase loss (dashed line) obtained from Eq. 3.36 with the exact phase loss for the quartic barrier of Eq. 3.37 2 2 1=3 as function of = (mE=h¯ )[¯h =(2mV0)] , Ref. [29] . 54 3.10 The piecewise constant potential well in Eq. 3.41 . 56 3.11 Truncation of an arbitrary potential well at xL and xR, far away from turning points a and b . ......................... 58 3.12 The dynamical triangle relating P (x) and k(x) to the phase φ(x) . 62 3.13 Three wave functions at energies E = 1, E = E0 and E = 3, in the linear potential V x) = x, 0; x; . E = 2:33811 is the ground state 1 0 energy in units whereh ¯ = 2m = 1. The left boundary condition at x = 0 is satisfied, but for non-eigenenergies the other boundary condition at + is not. 64 1 xi 3.14 Illustration of three wave functions of the linear potential correspond- ing to the ground state energy, E0, and two non-bound energies. In contrast to Fig. 3.13, the wave functions satisfy the boundary condition at + . .................................. 66 1 3.15 Correction integrand in Cao's quantization condition for the third bound state of the quartic potential well, V (x) = x4 . 70 4.1 The exponential radial potential for α = 1, along with the first three eigenfunctions. 75 4.2 Relative error defined in Eq. 4.10 of the LO-WKB approximation ap- plied to the exponential potential, with ten bound states (α = 1 and a = 32). The upper line is for LO-WKB, while the lower line includes a second order WKB correction, written in Eq. 4.11. 77 4.3 The symmetric 1D exponential potential, and its first three bound state wave functions. 79 4.4 Relative error of WKB energies for a deep exponential potential with 32 bound states.
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