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

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 mechanics 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 quantum 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 Guéry-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 relative 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 potential, 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 energies for the linear potential, providing that one starts from + and ∞ integrates backwards to construct the wave function...... 67

ix List of Figures

1.1 Schematic drawing of the WKB wave function; oscillatory in the allowed 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 function, ψ(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 ∞ 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 ∞

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 ∞ 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 applied 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...... 80 4.5 Illustration of the problem in Eq. 4.35. A centrifugal barrier on the left, and a hard wall at y = a ...... 85 4.6 Energies obtained from solving Eq. 4.37. The upper line uses ν everywhere, and the lower line replaces ν by b everywhere Eq. 4.9 . . . . . 86 4.7 Approximate binding energies. Upper pair of lines are from Fig. 4.2. Lower pair result from the approximation given in Eq. 4.43 with and without the second order WKB correction...... 88

xii 4.8 Comparison of the quantisation condition using only b (dashed line), or a combination of b and ν, (solid line). A horizontal line at (n+3/4)π/a intersects one of them at the value corresponding to a solution of Eq. 4.9 or Eq. 4.43. Note that the horizontal axis to the left of the origin, represents behaviour on the imaginary axis...... 90 4.9 Illustration of how the repulsive centrifugal potential ν2/y2 becomes attractive when b < 1/2 (ν iν¯)...... 91 → 4.10 Relative error in the location of the even-parity bound states of the 1D exponential potential. The three lines are for three diﬀerent approximations to the derivative, as explained in text...... 94

xiii Chapter 1

Introduction to the semi-classical or WKB approximation

1.1 Introduction

In quantum mechanics, three approximate methods are commonly applied: pertur- bation theory, (which produces a series expansion for quantities of interest); the variational method, (which allows a best estimate from a trial solution); and the semiclassical approximation which supposes thath ¯ is small compared to the action function in the corresponding classical problem. We will mostly be concerned with WKB which is a semiclassical approximation invented in the 19th century and then applied to the Schr¨odingerequation by Wentzel [1], Kramers [2] and Brillouin [3] independently at the advent of quantum mechanics. In the UK it is usually called JWKB where the ‘J’ stands for Jeﬀreys [4]. Though the WKB approximation is as old as quantum mechanics, it is still an area of interest, both conceptually and practically.

1 M.Sc. Thesis - Akbar Safari McMaster University - Physics and Astronomy

In the Schr¨odingerequation

d2ψ + k2(x)ψ(x) = 0 (1.1) dx2

where

2m k(x) = 2 (E V (x)) , (1.2) r h¯ −

the wave function ψ(x), can be written in phase-amplitude form as

i S(x) ψ(x) = A(x)e h¯ (1.3)

For a constant potential, the wave function is a plane wave, S(x) = hkx¯ , and A(x) is ± a constant. When the potential varies slowly in space, it is reasonable to expect that the action S(x) will increase at the rate δS(x) hk¯ (x)dx, and that the amplitude ∼ A(x) will vary more slowly than S(x). A(x) can be included in the exponent, as the imaginary part of S(x). In the classical limit,h ¯ 0. → Following Dunham [5], we assume that S(x) can be expanded in a convergent (or at least asymptotic) series in powers ofh ¯ as follows :

h¯2 S(x) = S (x) +h ¯ S (x) + S (x) + ..., (1.4) 0 1 2 2

and substitute Eqs. 1.4 and 1.3 into the Schr¨odinger equation (Eq. 1.1). Re-arranging

2 M.Sc. Thesis - Akbar Safari McMaster University - Physics and Astronomy

in terms of powers ofh ¯, one obtains (up to second order inh ¯)

1 dS 2 h¯ dS dS i d2S 0 = 0 + (V (x) E) + 0 1 0 + 2m dx − m dx dx − 2 dx2 h¯2 dS dS dS 2 d2S 0 2 + 1 i 1 + ... (1.5) 2m dx dx dx − dx2 " #

Since the wave equation must be satisﬁed independent of the value ofh ¯, the coeﬃcient of each power ofh ¯ must be equal to zero. This leads to a sequence of equations

1 dS 2 0 + V (x) E = 0 (1.6) 2m dx −

dS dS i d2S 0 1 0 = 0 (1.7) dx dx − 2 dx2

dS dS dS 2 d2S 0 2 + 1 i 1 = 0 (1.8) dx dx dx − dx2 ···

Solving in sequence for S0, S1 and S2 one ﬁnds

x x S = 2m (E V (x)) dx = hk¯ (x) dx (1.9) 0 ± − ± Z p Z

i dS S = ln 0 (1.10) 1 2 dx

1 m(dV/dx) 1 m2(dV/dx)2 S = dx (1.11) 2 2 (¯hk)3 − 4 (¯hk)5 ··· Z

3 M.Sc. Thesis - Akbar Safari McMaster University - Physics and Astronomy

In general, S1 is not negligible compared to S0, but S2 will be small providing that dV/dx is small and E V (x) is non-zero, i.e. not close to a classical turning point. − The lowest order WKB approximation (LO-WKB) neglects the terms involving S with n 2 in the phase of the wave function. Therefore the criterion for the n ≥ lowest order WKB to be valid becomes

m dV/dx 1 (1.12) h¯2 k3(x)

Hence, WKB should be applicable for slow varying potentials in regions far from classical turning points.

Keeping only S0 and S1 in the Eq. 1.4 produces the LO-WKB approximate wave function

A x B x WKB i b k(x) dx i b k(x) dx ψ(x) = e + e− (1.13) k(x) R k(x) R p p where b is an arbitrary point that aﬀects only the normalization factors A and B. In the classically forbidden region, where V (x) > E, the wave number k(x) → WKB iκ(x), becomes imaginary and ψ(x) varies exponentially. In the classically allowed region, the WKB wave function is oscillatory (Fig. 1.1). As can be seen from Fig. 1.1 and Eq. 1.13, the WKB approximation cannot be used near a turning point, x = b, deﬁned by V (b) = E, because near such a point the condition for WKB applicability breaks down (see Eq. 1.12) and the amplitude of the WKB wave function diverges. Therefore the main problem in WKB is how to connect the sinusoidal solution in the allowed zone to the exponential type solution in the neighbouring classically forbidden zone across a classical turning point.

4 M.Sc. Thesis - Akbar Safari McMaster University - Physics and Astronomy

ψ(x)

V (x)

Figure 1.1: Schematic drawing of the WKB wave function; oscillatory in the allowed zone, decaying in the forbidden zone, and diverging near the turning point, b

In most cases, close enough to a turning point, the potential can be approximated by a straight line. In that case, the Schr¨odinger equation has a solution called the Airy function, which can be expressed in terms of Bessel functions of order 1/3 [6](page 446). Far from the turning point where WKB is applicable, the asymptotic form of the Airy function can be connected smoothly to the WKB wave function. In this way the constants A and B of Eq. 1.13 on the classically allowed side can be related to the corresponding coeﬃcients of the two exponentially varying waves on the forbidden side. I will not go through the details of this procedure here, but merely quote the results. Fig. 1.2 shows how four connection formulae connect sinusoidal wave functions to exponential ones and vice versa. There is another method to treat the connection problem; the complex variable method, where the two regions across the turning point are joined by a path in the complex z-plane. The method was invented by Stokes [7], and applied to quantum mechanics by Zwaan [8] and later summarized and discussed by Heading [9]. Many authors have dwelt on the danger of using symbol )* in connection formulae. It is

1 M.Sc. Thesis - Akbar Safari McMaster University - Physics and Astronomy

Case I: Barrier on the right of the turning point

V (x)

E • • a b

1 a π 1 a π 1 x κdx sin kdx = cos kdx + e a (C.1) √k x 4 √k x 4 √κ − − R R R V (x)

E • • a b

1 a π 1 a π 1 x κdx cos kdx = sin kdx + e a (C.2) √k x 4 √k x 4 2√κ − − R R R

Case II: Barrier on the left of the turning point

V (x)

E • • a b 1 1 b κdx 1 x π 1 x π e x cos kdx = sin kdx + (C.3) 2√κ − √k b 4 √k b 4 R − R R V (x)

E • • a1 b

1 b κdx 1 x π 1 x π e x sin kdx = cos kdx + (C.4) √κ √k b 4 √k b 4 R − − R R

Figure 1.2: Illustrating the two connection formulae in two cases

6 1 M.Sc. Thesis - Akbar Safari McMasterψ(x) University - Physics and Astronomy

V (x)

E Region III

Region I Region II

x = a x = b

Figure 1.3: Schematic view of a barrier. not legitimate to run the arrows in both ways. We are allowed to use the connection formulae in the direction in which the exponential wave function is growing [10]; In the top formula, C.1, from the allowed region to the barrier, and in the second formula, C.2, from the exponential to the sinusoidal wave function. This is because they are asymptotic formulae, and a decaying exponential is negligible when any amount of a growing exponential is present. The review article of Berry and Mount [11] summarized the subject as of 1970. An example will clarify how to use the connection formulae.

Example 1 - Penetration probability through a barrier. A travelling sinusoidal wave, representing a non localized particle, is coming from the left to the barrier shown in Fig. 1.3. Energy of the particle is supposed to be greater than the potential in the classically allowed regions I and III, E > V1(x) and

E > V3(x) respectively. We expect the wave function inside the barrier to decay exponentially to the right.

1 M.Sc. Thesis - Akbar Safari McMaster University - Physics and Astronomy

Therefore following Landau we start from region III with the transmitted wave

C x WKB i( b k3(x) dx + π/4) ψIII = e k3(x) R C x x = cos k3(x) dx + π/4 + i sinp k3(x) dx + π/4 (1.14) k (x) b b 3 Z Z p where k3(x) is the wave number in region III and a, b are the turning points. The phase π/4 is for convenience and could be absorbed in the constant C. According to the second connection formula, C.2, the cosine function connects to an exponential function that grows to the left, and the sine function connects to the exponential function that decays to the left. Then under the barrier,

C b i C b WKB x κ(x) dx x κ(x) dx ψII = e + e− κ(x) R " 2 κ(x) R #

C b κ(x) dx px κ(x) dx i C pb κ(x) dx + x κ(x) dx = e a − a + e− a a (1.15) κ(x) R R " 2 κ(x) R R # p p Here, κ = ik2 where k2 is the wave number inside the barrier. The terms in large square brackets arise from using the connection formula in the wrong way. But we are going to carry them along, because the resulting expressions will conserve the ﬂux of probability, as noted by Merzbacher page 126 [12]. Now, we use the last connection formula, C.4, from region II to region I to ﬁnd the wave function in region I with wave number k1

a WKB 2C θ ψI = sin k1(x) dx + π/4 k (x) x 1 Z piC a + cos k1(x) dx + π/4 (1.16) 2θ k (x) x " 1 Z # p 8 M.Sc. Thesis - Akbar Safari McMaster University - Physics and Astronomy

For brevity we have written θ exp[ b κ(x) dx], the barrier penetration amplitude. ≡ a WKB To ﬁnd the transmission and reﬂectionR amplitudes, we write ψI in terms of left

going and right going waves denoted by e→ and e← respectively:

WKB Cθ i C ψI = (e← e→) + (e← + e→) i k1(x) − " 4θ k1(x) # iCp 1 iCp 1 = θ e← + + θ e→ (1.17) 4θ − 4θ k1(x) k1(x) p p where

a i( k1dx+π/4) e← e x (1.18) ≡ R

a i( k1dx+π/4) e→ e− x (1.19) ≡ R

Therefore the transmission and reﬂection amplitudes, t and r respectively, would be

k 1 t = 1 (1.20) k θ + 1 r 3 4θ

1 θ 4θ r = i − 1 (1.21) − θ + 4θ

In obtaining this result we violated the rules, by using the connection formulae in both directions. However, probability is conserved because t 2 + r 2 = 1 if k = k . | | | | 1 3 Landau and Lifshitz [13] give a diﬀerent argument, which does not violate the rules. As a result, their solution does not involve the terms enclosed in square brackets

9 M.Sc. Thesis - Akbar Safari McMaster University - Physics and Astronomy

in Eqs. 1.15 to 1.17. They find r = i and t = 1/θ for the case k = k which − 1 3 however does not conserve probability, to order 1/θ2. So in practice, we can in some circumstances run the arrows both ways, even though a rigorous justification of such a procedure may be rather difficult [10]. Miller and Good [14], also Hecht and Mayer [15], used a more involved argument. By mapping the barrier penetration problem onto an inverted harmonic oscillator potential, they managed to justify the result in Eq. 1.20, 1.21.

1.2 Analogy to optics

A discontinuity in a potential in quantum mechanics corresponds to a discontinuity in the index of refraction in optics. Photons are reflected by such a discontinuity, just as quantum particles are. The same analogy is true in the opposite limit of slowly changing potentials which correspond to a slowly changing index of refraction. In a medium where the index of refraction changes in a continuous way, light is little reflected, although its path may be curved as a result of refraction [10]. A great deal of reflection occurs when the wave length λ, changes significantly within one wavelength:

dλ δλ = λ λ (1.22) dx ∼

dλ where dx λ is the change in wavelength in a distance δx = λ. Since λ is given by the de Broglie wavelength λ = h/p, the condition for no reﬂection is

dλ h dp = 1 (1.23) dx p2 dx

10 M.Sc. Thesis - Akbar Safari McMaster University - Physics and Astronomy

By substituting p2 = 2m(E V ), one gets −

hmdV/dx¯ 1 (1.24) [2m(E V )]3/2 − which is exactly the condition for WKB to be valid (see Eq. 1.12).

1.3 Derivation of WKB from an optical analogy

WKB is also applicable in optics for the ordinary diﬀerential equation

d2u + k2(x)u(x) = 0 (1.25) dx2 which describes the behaviour of a plane wave propagating in a medium with refractive index n(x) = k(x)/k0, where k0 is the wave number in vacuum. By considering propagation of light through an inhomogeneous medium, Brem- mer [16] obtained the WKB wave function as the first term of an infinite series, where each term represents a wave produced by a particular number of reflections inside the medium. He replaced the inhomogeneous medium by a countable set of discrete homogeneous layers (segmentation method) (Fig. 1.4).

When the incoming wave, u = eik0x, arrives from the left, hits the ﬁrst step, and

ik0x ik1x splits into reﬂected and transmitted waves, r1e− and t1e respectively. At the

ik2x next potential step the transmitted wave would be t2t1e and this procedure of splitting is repeated at each succeeding step. Bremmer made the thickness of the layers inﬁnitesimally small, and obtained the following formula for the transmitted

11 M.Sc. Thesis - Akbar Safari McMaster University - Physics and Astronomy

t2 r3

r2 eik0x

Figure 1.4: Splitting of the incoming wave eik0x, into reﬂected and transmitted waves at the potential steps in the segmentation method. wave:

k0 i x k(x)dx u0(x) = e 0 (1.26) sk(x) R where zero index for u indicates the lowest order approximation that takes no reﬂec- tions into account. This expression is just the WKB approximation. Therefore, in this sense, WKB supposes that the potential changes so slowly that no reﬂected wave is generated inside the active medium. .

1.4 WKB quantization condition

For a particle confined in a quantum well schematically shown in Fig. 1.5, the WKB approximation can be applied to estimate the energy eigenvalues. The particle is confined classically between the two turning points labelled as a and b. The wave function decays in the forbidden region as sketched in the figure. Starting from either forbidden region, the connection formulae lead to two slightly

1 M.Sc. Thesis - Akbar Safari McMaster University - Physics and Astronomy

V (x)

ψ(x) E

Region I Region II Region III

x = a x = b

Figure 1.5: Schematic view of a quantum well V (x), and ground state wave function, ψ(x), of a particle conﬁned between the turning points a, b. diﬀerent expressions for the wave function in the allowed zone:

x 1 a 1 WKB x κdx WKB ψI = e− ψII = sin kdx + π/4 (1.27) 2√κ R −→ √k a Z and

b A x A WKB b κdx WKB ψIII = e− ψII = sin kdx + π/4 (1.28) 2√κ R −→ √k x Z where again κ is related to the wave number in the forbidden zones and A is a constant to be determined. A smooth match of the two expressions in the allowed zone requires

x b sin kdx + π/4 = A sin kdx + π/4 (1.29) Za Zx

With

b b η kdx and ζ kdx + π/4 (1.30) ≡ ≡ Za Zx

1 M.Sc. Thesis - Akbar Safari McMaster University - Physics and Astronomy

the continuity condition becomes

sin (η + π/2 ζ) = A sin ζ (1.31) − or, equivalently,

sin (η + π/2) cos ζ cos (η + π/2) sin ζ = A sin ζ (1.32) −

This equation is solved by setting

π η + = (n + 1)π (1.33) 2 and the corresponding values of A are ( 1)n. This implies the condition −

b kdx = (n + 1/2) π (n = 0, 1, 2, ) (1.34) ··· Za which is called the WKB quantization condition. In old quantum mechanics, the Bohr-Sommerfeld quantization condition rule said

pdx = Nh = 2πNh,¯ (N = 1, 2, 3, ) (1.35) ··· I where p =hk ¯ is the classical momentum and the integral is evaluated over one whole period of classical motion, from a to b and back. Taking into account that N = n+1, the WKB quantization condition agrees with the Bohr-Sommerfeld quantization condition, except that in WKB the wave function is not completely conﬁned between the turning points. We can say that the phase π/2 has leaked out of the classically

14 M.Sc. Thesis - Akbar Safari McMaster University - Physics and Astronomy

allowed region into the classically forbidden regions, i.e. π/4 to each forbidden region. Thus the interpretation of the WKB quantization condition is that the phase of the wave function changes by an integer multiple of π in a half cycle. The WKB phase integral between the turning points is just (n+1/2)π, and the π/2 which leaked out is additional. (Friedrich and Trost consider that it is part of the reﬂection phase which occurs at the turning point.)

Example 2 - Centrifugal barrier The radial Schr¨odingerequation for a particle of mass m, energy E and angular momentum quantum number l in a spherically symmetric potential is

2 d ψl(r) 2m(E V (r)) l(l + 1) + − ψl(r) = 0 (1.36) dx2 h¯2 − r2

Consider, for instance, the Coulomb potential

e2 V (r) = (1.37) − r

To ﬁnd the WKB eigen-energies, we must evaluate the integral

b 2m(E V (r)) l(l + 1) − dr (1.38) h¯2 − r2 Za r and equate it to (n + 1/2)π to solve for the energy E. The result is [17]

4 WKB me Enr = − 2 (nr = 0, 1, 2, 3, ...) (1.39) 2 1 2¯h nr + 2 + l(l + 1) h p i

15 M.Sc. Thesis - Akbar Safari McMaster University - Physics and Astronomy

compared to the exact energy

4 exact me Enr = 2 − 2 (nr = 0, 1, 2, 3, ...) (1.40) 2¯h [nr + l + 1]

This discrepancy is resolved by replacing l(l + 1) by (l + 1/2) which is called the “Langer modiﬁcation”, explained in morep detail below.

Example 3 - Harmonic Oscillator The Schr¨odingerequation for a simple harmonic oscillator is

d2ψ 2m 1 + E mω2x2 ψ = 0 (1.41) dx2 h¯2 − 2

With

2mE x2 2E k2 1 and x2 (1.42) ≡ h¯2 − x2 0 ≡ mω2 0 the classical turning points are at x and the phase integral would be ± 0

2mE x0 x2 1/2 2 1 2 dx = x0 π/2 (1.43) h¯ x0 − x0 r Z−

Therefore the energy levels follow from

2mEn 2En 2 2 π/2 = (n + 1/2)π (1.44) r h¯ rmω

16 M.Sc. Thesis - Akbar Safari McMaster University - Physics and Astronomy

En = (n + 1/2)¯hω (1.45)

As is well-known, WKB gives the exact bound state energies for the harmonic oscillator. Apparently, Eq. 1.12 determines only the validity of the WKB wave function. Unfortunately there is no way to predict whether the WKB quantization condition works reasonably in a particular example or not. In chapter 3, we will have more to say about this.

1.5 Langer modiﬁcation

There are two problems with the application of WKB for a spherically symmetric potential. From example 2 it is apparent that we do not obtain the exact bound state energies except in the limit of large l values. Secondly, the WKB method fails to give the correct form of the wave function near the origin where the centrifugal potential is dominant. The behavior of the WKB wave function near the origin is

WKB √l(l+1)+1/2 lim ψl = constant r , (1.46) r 0 → × whereas the exact behaviour is known to be

exact l+1 lim ψl = constant r (1.47) r 0 → ×

17 M.Sc. Thesis - Akbar Safari McMaster University - Physics and Astronomy

and the exact energies are

me4 Eexact = (1.48) nr 2 2 −2¯h (nr + 1 + l)

The way to improve the WKB approximation for the angular momentum barrier was devised by Kramers [2]. He immediately realized that these two defects can be eliminated by replacing l(l + 1) by (l + 1/2)2. The first full analysis of the situation was by Langer [18, 19] and consequently the above replacement is called the Langer modification. Langer’s argument was based on the idea that the truncation of the radial space to the range 0 r < is the source of the problem. He mapped ≤ ∞ the half space of r onto < x < by setting r = ex. He then applied the −∞ ∞ usual WKB approximation to the one-dimensional x-space problem, requiring that the wave function vanish as x . When the phase integral is transcribed back → −∞ into the r-space, the (l + 1/2)2 appears where it is needed. Seetharaman and Vasan [20] showed that if one uses higher order WKB approximations, a different replacement for l(l+1) is required in each order to get the correct behaviour near the origin, and exact bound state energies. They also found that by summing up all orders of WKB terms, the two defects are resolved and no replacement is required. In fact, the Langer modification is a detour which effectively adds up all higher order terms in the WKB series. More recently it has been found by David Barclay [21] that a similar modification of potential parameters leads to exact energies for many other potentials.

18 M.Sc. Thesis - Akbar Safari McMaster University - Physics and Astronomy

1.6 WKB asymptotic series

Higher order WKB approximations are obtained by keeping more terms in Eq. 1.4. For

instance keeping the term involving S2 leads to the second order WKB quantization condition [22, 23]

b 1 ∂ b d2V (x)/dx2 kdx dx = (n + 1/2)π (1.49) − 24 ∂E k(x) Za Za

In this way, one finds the WKB series in which, the first q terms are obtained by keeping the first 2q terms in Eq. 1.4. That is because only even orders in Eq. 1.4 contribute to a calculation of the eigen-values [24]. The WKB series in Eq. 1.4 is an asymptotic series and does not generally con- verge1, except in a few cases (see section 2.5). For an asymptotic series, the best result is obtained by truncating the series at q terms, when the (q + 1)’th term is larger than the q’th term in magnitude. Table 1.1 (from Bender and Orszag’s book [24]) compares the accuracy of the WKB quantization condition going up to order 12, for the quartic potential, V (x) = x4. For the ground state, LO-WKB is particularly

bad with 20% error. Including the second order term, WKB2, reduces the error to 10%, which is the best result they could get. For the ﬁrst excited state the best result

5 has an error of 10− , at second order. The accuracy improves as you go up in the spectrum, as higher orders of the approximation can be included.

1See the appendix for a short discussion of asymptotic series.

19 M.Sc. Thesis - Akbar Safari McMaster University - Physics and Astronomy

Table 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 relative accuracy achieved for that level, by truncating the calculation at order q.

E0(exact) = 1.06 E6(exact) = 26.528 471 183 (WKB)0 0.87 (WKB)0 26.506 335 511 1 (WKB)2 0.98 (10− ) (WKB)2 26.528 512 552 (WKB)4 0.95 (WKB)4 26.528 471 873 (WKB)6 0.78 (WKB)6 26.528 471 147 (WKB)8 1.13 (WKB)8 26.528 471 179 11 (WKB)10 1.40 (WKB)10 26.528 471 182 (10− ) (WKB)12 1.64 (WKB)12 26.528 471 181 E2(exact) = 7.455 6 E8(exact) = 37.923 001 027 033 (WKB)0 7.414 0 (WKB)0 37.904 471 845 068 5 (WKB)2 7.455 8 (10− ) (WKB)2 37.923 021 140 528 (WKB)4 7.455 3 (WKB)4 37.923 001 229 358 (WKB)6 7.455 2 (WKB)6 37.923 001 021 414 (WKB)8 7.455 2 (WKB)8 37.923 001 026 832 13 (WKB)10 7.455 2 (WKB)10 37.923 001 027 043 (10− ) (WKB)12 7.455 2 (WKB)12 37.923 001 027 030 E4(exact) = 16.261 826 0 E10(exact) = 50.256 254 516 682 91 (WKB)0 16.233 614 7 (WKB)0 50.240 152 319 172 36 (WKB)2 16.261 936 7 (WKB)2 50.256 265 932 002 07 7 (WKB)4 16.261 828 6 (10− ) (WKB)4 50.256 254 592 948 49 (WKB)6 16.261 824 5 (WKB)6 50.256 254 515 324 64 (WKB)8 16.261 824 9 (WKB)8 50.256 254 516 650 43 (WKB)10 16.261 825 0 (WKB)10 50.256 254 516 684 34 15 (WKB)12 16.261 825 0 (WKB)12 50.256 254 516 682 99 (10− )

20 Chapter 2

Super-Symmetric WKB

2.1 Super-symmetric quantum mechanics

The advent of super-symmetric (SUSY) quantum mechanics led to the discovery of many analytically soluble potential models. Assuming that the potential energy function is a static local function of position < x V x0 >= V (x)δ(x x0), given the | | − ground state wave function, (or indeed, the wave function at any single energy), the potential is determined up to a constant energy shift. Without loss of generality, one can set the ground state energy to be zero. Then the Schr¨odingerequation for the ground state wave function, ψ0(x) says

h¯2 d2ψ (x) 0 + V (x)ψ (x) = 0 (2.1) −2m dx2 1 0 from which

2 h¯ ψ000(x) V1(x) = (2.2) 2m ψ0(x)

21 M.Sc. Thesis - Akbar Safari McMaster University - Physics and Astronomy

In SUSY quantum mechanics, one deﬁnes the superpotential by

h¯ ψ (x) W (x) 0 (2.3) ≡ −√2m ψ0(x) from which

2 h¯ V1(x) = W (x) W 0(x) (2.4) − √2m

This expression for V1(x), tempts one to deﬁne a new potential, V2(x), by adding rather than subtracting the derivative term:

2 h¯ V2(x) = W (x) + W 0(x) (2.5) √2m

It turns out that V2(x) has the same spectrum as V1(x), except for the ground state.

To that end we deﬁne operators A and A† by

h¯ d h¯ d A + W (x) A† − + W (x) . (2.6) ≡ √2m dx ≡ √2m dx

These allow one to write the ﬁrst Hamiltonian H1, corresponding to V1(x), as

h¯2 d2 H = + V (x) = A†A (2.7) 1 −2m dx2 1

and H2, corresponding to V2(x)

h¯2 d2 H = + V (x) = AA† (2.8) 2 −2m dx2 2

22 M.Sc. Thesis - Akbar Safari McMaster University - Physics and Astronomy

It can easily be shown that the eigenvalues and eigenfunctions of the two Hamil-

tonians H1 and H2 are related by (n = 0, 1, 2, ...)

(1) E0 = 0 ;

(2) (1) En = En+1 ; (2.9) while

(2) 1 (1) ψn = Aψn+1 (2.10) (1) En+1 q

(1) 1 (2) ψn+1 = A†ψn . (2.11) (2) En q

Like the annihilation and creation operators of the harmonic oscillator, A and A† destroy and create an extra node in the wave function, respectively.

The potentials V1(x) and V2(x) are called supersymmetric partner-potentials.

Equation 2.9 says that they have exactly the same spectra except that V2(x) lacks the ground state of H1.

Example 1 - Inﬁnite square well. The inﬁnite square well potential is

0 0 x L V (x) =  ≤ ≤ (2.12)  otherwise .  ∞ 

23 M.Sc. Thesis - Akbar Safari McMaster University - Physics and Astronomy

The ground state wave function is well known

(1) 2 πx ψ0 = sin( ) (0 x L) (2.13) rL L ≤ ≤ and the ground state energy is

h¯2π2 E = . (2.14) 0 2mL2

We shift the potential by E , so that H = H E and the energy eigenvalues for 0 1 − 0

V1(x) become

n(n + 2) E(1) = h¯2π2 (n = 0, 1, 2, ...) (2.15) n 2mL2

The superpotential is easily obtained using Eq. 2.3

h¯ π πx W (x) = cot( ) (2.16) −√2m L L

and the partner potential is

h¯2π2 πx V (x) = 2 csc2( ) 1 (2.17) 2 2mL2 L − h i

These relations are illustrated in Fig. 2.1. The partner potential has the same eigenen-

(1) ergies (except for E0 ), and their eigenfunctions are related by Eqs. 2.10 and 2.11. When the two partner potentials have continuous spectra, (unlike the example above), supersymmetry says that they have the same transmission and reﬂection

24 M.Sc. Thesis - Akbar Safari McMaster University - Physics and Astronomy

2 V1(x) = 0 V2(x) = 2 csc (x)

sin(x) sin(2x) sin(3x) E = 9

sin2(x) sin(2x) E = 4

sin(x) E = 1

0π 0 π x x

Figure 2.1: The parter potential V2 (right) of the inﬁnite square well V1 (Left). probabilities, namely

r 2 = r 2 , t 2 = t 2 . (2.18) | 1| | 2| | 1| | 2|

Hence it is clear that the partner potential of a zero (constant) potential is necessarily reﬂectionless.

2.2 Reﬂectionless potentials

To find a reflectionless potential we may start from a null potential and find its super- symmetric partner potential which would necessarily be reflectionless. But the issue is that a constant potential has no bound1 state and consequently no superpotential.

25 M.Sc. Thesis - Akbar Safari McMaster University - Physics and Astronomy

Nevertheless, starting from the superpotential

W (x) = B tanh(αx) (2.19) the two partner potentials are

2 αh¯ 2 V1(x) = B B B + sech (αx) (2.20) − √2m

2 αh¯ 2 V2(x) = B B B sech (αx) (2.21) − − √2m

By setting B = αh/¯ √2m, the second potential becomes a constant potential and consequently its partner potential

α2h¯2 α2h¯2 V (x) = sech2(αx) (2.22) 1 2m − m

will be reflectionless. Note that V1(x) depends onh ¯. In fact, it can be shown that reflectionless potentials are necessarilyh ¯-dependant [25]. A more general way to find reflectionless potentials is by solving the equation

2 h¯ V1,2(x) = W (x) W 0(x) = constant (2.23) ± √2m

for W (x). Three solutions are easily obtained

W (x) = constant, (2.24)

26 M.Sc. Thesis - Akbar Safari McMaster University - Physics and Astronomy

h¯ 1 W (x) = , and (2.25) ±√2m x

W (x) = B tanh(αx) (2.26)

The ﬁrst W (x) leads to a constant potential which is reﬂectionless but trivial. The second one leads to

h¯2 1 V (x) = (2.27) 1,2 ± m x2 which is another reflectionless potential. The third W (x) was already investigated in Eq. 2.19. For more information on this method see Ref. [26]. In summary, if there is an exactly soluble potential with at least one bound state, then SUSY finds its partner potential which is exactly soluble as well. The two partner potentials have the same bound state energies except for one, and in the continuum states, they have the same reflection and transmission probabilities.

2.3 Shape invariant potentials

An important class of super-symmetric potentials is called “shape invariant”. When the two partner potentials are described by the same formula, and diﬀer only in some parameter values, then they are called shape invariant potentials (SIPs). More explicitly, they are related by

V2(x, a2) = V1(x, a1) + R(a2) (2.28)

27 M.Sc. Thesis - Akbar Safari McMaster University - Physics and Astronomy

where a1 is a set of parameters and the set a2 is a functional of a1. R(a2) is a function of the parameters independent of x, which shifts the energy scale.

There are two main classes of SIPs. In the ﬁrst class, the parameter a2 is related

to a1 by translation (a2 = a1 + α). In the second class, a1 and a2 are related by scaling (a2 = λa1). What distinguishes SIPs from the others is that they are all exactly soluble and there are some ansätzesto find new SIPs (see section 2.5 and Ref. [25]). Table 2.1 lists some translational SIPs. We see that most well-known soluble potentials found in textbooks belong to this class. So far, no SIP with scaling has been found in a closed form; they are expressed as Taylor series in x. Shape invariance is a sufficient condition for exact solvability, but it is not necessary. In chapter 4 we discuss the exponential potential which is soluble, but not shape invariant.

28 M.Sc. Thesis - Akbar Safari McMaster University - Physics and Astronomy ) ) ) 2 2 2 ) Aα Aα Aα ) ) 2 nα nα nα ) 2 2 α ) C L α + − − 2 Aα 2 2 α α + + − D D − − 2 2 2 − nα − ) + 2 2 2 α α α − A A − A − − − − 1 2 2 A ( ( ( 1 2 ω A − ( 2 2 ω 2 α + + / / − / ( − D + A 2 2 2 A ) ), are shifted C C 2) nα nα A l l A / ( + x + √ B B B √ √ p ( ) Aα Aα Aα − − 2 + + 1) + p ) WKB n − n + + + ) V + − 2 n 2 L L nα C + 1 − − + n 2 E l l 2 n 2 2 ) . ( ) ) ) 2 2 2 n √ √ √ C C + 2 √ α 1 2 1 2 1 2 ( A A A 4 4 e L ≡ ≡ ≡ + + + (1 + 2 + exact + √ √ √ n − − (1 + 2 √ α L L L (1 + 2 n E n n n α A L L / / / ( 2 ( ( ( ( 2 2 2 − 2 2 2 + = B B B α α α − − − + − − SWKB n 2 ) ) E ) (2 ) (2 2 2 2 ) ) ) Aα 2 2 nα L 2 2 2 ) ) nα nα nα 2 2 2 αB αB αB + 1) (2 ω B B B ) ) ) − exact ω − + + − n l − 2 A A A 2 nα nα ( 2) + + − E Aα ( ( ( 2) 2 l A / nα nα nα Aα Aα α 1 ) +1) B / ( l +1) − − 1 l ( + = − − − p l + + − + − + + − 2 AB AB AB nα A A ω + 3 n 2 2 + 1 ( ( 2 2 2 4 2 2 ( A A A 4 l A Aα Aα Aα e 2 2 2 ) + ( ( A A − n A + − − 2 − − − B B B α − ≡ ≡ ≡ ( 2 2 2 +( − − SWKB n A A A A ≡ ( + nω E D D D + + + 2 C A ( ( )( )( 2 2 αx αr ) αx αx αr Aα Aα /r ) ) ) 2 2 αx 2 αx αx αr L 2) αx L − + − αx αr 2 2 2 2 Aα Aα Aα αx ) +1) 2 2 r l l + 1) +( tan 2 + 1) coth π/ tanh ( ) l + ) cot A 4 l Be + B − − x l sec ) coth x 2 ) tanh e ( ( 2 l α cot ( 2 2 2 csch l sech l α coth αx αr α tanh − + − + αx C ω 4( | ≤ A A A V C C 1 4 B − 2 2 2 p 2 + ( ( ( B α − r B + e + + sec / / / 2 ω 2 A B αx + + A csch 2 2 2 | A sech r A − + L ( − − ( 2 ( D ≡ ≡ D D A − A A ω A − ( − 4 1 + + C C + + ( ) +2 π ) ) +2 r αx )( )( +1) αr B 2 2 l αr αx αx αx B αx B αx L αx αr L ( − ≤ ) r +1) l x ( ( − ωx Be sec cot B/A B/A B/A αx csch sech coth 2 1 coth W tanh tanh tanh ωr − − A B A > B A < B B B 2 A B > A B < A 2 1 ≤ +1) e l ( A A A A − − − A − 2( Morse I metric) ( (trigono Eckart + (hyperbolic)P¨oschl- + Name of potential Shifted oscillator 3D oscillator Coulomb Morse Rosen- Rosen- Scarf II Generalized TellerScarf I ( Morse II + (trigonometric) (0 (hyperbolic) ( to place the ground state energy at zero. For these potentials, SWKB is exact, Table 2.1: List of shape invariant potentials with translation, taken from Ref.[32]. The potentials,

29 M.Sc. Thesis - Akbar Safari McMaster University - Physics and Astronomy

2.4 Super-symmetric WKB

Comtet, Bandrauk and Campbell [27] combined the idea of SUSY with the WKB method and obtained a modiﬁed quantization condition; their approach is called SWKB. As in the WKB quantization condition, accuracy of SWKB improves when the quantum number n, increases. SWKB is exact for the ground state (n = 0) which is no surprise, because, as we will see, SWKB is based on the knowledge of the ground state wave function. Although the SWKB method is not always superior to the usual WKB method, it is interesting to outline the basis of the method.

The usual LO-WKB quantization condition (Eq. 1.34) for the potential V1(x) in equation 2.4 is

b 2m (1) 2 h¯ En W (x) + W 0(x)dx = (n + 1/2)π (2.29) h¯2 − √2m r Za s

Comtet et al. argue that the second piece of the potential, because it has a factor h¯, should be treated separately from the W 2 part, Expansion of the left hand side in powers ofh ¯ gives

b0 b0 (1) 2 h¯ W 0(x) 1 2m En W (x) dx + dx + = (n + )¯hπ(2.30) a0 − 2 a0 (1) 2 ··· 2 Z r Z En W (x) − q

2 2 (1) Here the new turning points, a0 and b0, are deﬁned by W (a0) = W (b0) = En . The second integral results in

b0 b0 h¯ dW h¯ 1 W (x) hπ¯ = sin− = (2.31) 2 a0 (1) 2 2  (1)  a0 2 Z En W En

−   q q

30 M.Sc. Thesis - Akbar Safari McMaster University - Physics and Astronomy

Therefore the SWKB quantization condition, to leading order inh ¯, is obtained

b0 2m (1) 2 2 En W (x)dx = nπ (n = 0, 1, 2, ) (2.32) h¯ 0 − ··· r Za q

There are three changes in SWKB compared to the WKB quantization condition:

2 2 first the potential enters as W instead of W hW¯ 0/√2m. Consequently, the turning − points are redefined, and finally the phase integral must give an integer number times π rather than a half integer multiple. For the ground state, the two turning points

(1) coincide and E0 = 0. What is more surprising is that in many cases all energy levels are given correctly in LO-SWKB. In the SWKB approach for spherically symmetric potentials, the Langer correction is not required. This is because near the origin [25]

(l + 1) lim √E W 2 ih¯ (2.33) r 0 → − ∼ − r

so the SWKB wave function [27]

i r √E W 2dr r 0 l+1 ψ(r) e h¯ − → r (2.34) ∼ R ∼

behaves correctly near the origin without any Langer-like correction. This corrects one of our complaints about the WKB wave function. It has been proved that the lowest order SWKB quantization condition (Eq. 2.32) gives the exact eigenenergies for all SIPs with translation [25]. When the ground state wave function is not known, SWKB is not applicable and WKB is preferable. There are still a few soluble potentials which are not shape invariant. For that kind

31 M.Sc. Thesis - Akbar Safari McMaster University - Physics and Astronomy

a) b) V(x) V(x) x x -4 -2 2 4 -0.3 -0.2 -0.1 0.1 0.2 0.3 -200 -5

-400 -10 -600 -15 -800

-20 -1000

-25 -1200

-1400 -30

Figure 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. of potential, both WKB and SWKB are applicable and it would be interesting to compare the accuracy of these two methods. One example of a soluble non-SIP was proposed by Ginocchio [28]

(1 λ2) V (x) = (1 y2) λ2ν(ν + 1) + − 2 (6 λ2)y2 + 5(1 λ2)y4 (2.35) − − 4 − − − where λ and ν are parameters that measure the depth and shape of the potential and y is related to the independent variable x by 1 1

1 1 1 x = tanh− y √1 λ2 tanh− y√1 λ2 (2.36) λ2 − − − h i

This potential is shown in Fig. 2.2 for the two sets of parameters used by Friedrich and Trost [29]: λ = 0.5, ν = 10.5; λ = 6.25, ν = 5.5 . The corresponding superpotential

32 M.Sc. Thesis - Akbar Safari McMaster University - Physics and Astronomy

1.000 0.0100 WKB 0.500 WKB 0.0070 a)SWKB b) SWKB 0.0050 0.100 100

100 0.050 ×

0.0030 × E δE E 0.0020 δE 0.010 0.0015 0.005 0.0010 0.001 0 1 2 3 4 5 0 1 2 3 4 5 n n

Figure 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.

was given by Ginocchio as well

(1 λ2)y(y2 1) W (x) = − − + µ λ2y (2.37) 2 0

where

µ λ2 = λ2 (ν + 1/2)2 + (1 λ2)(n + 1/2)2 (n + 1/2) (2.38) n − − q 1 1 and the bound state energies are

E = µ2 λ4 (n = 0, 1, 2, ...) (2.39) n − n

Figure 2.3 compares the accuracy of SWKB with the WKB quantization condition for the two sets of parameters used in Fig. 2.2. As expected, SWKB is more accurate

for deeply bound states. But as energy1 increases, it is beaten by WKB.

33 M.Sc. Thesis - Akbar Safari McMaster University - Physics and Astronomy

2.5 Barclay’s insight

David Barclay wrote an important but cryptic paper [21] which aims to give a uniﬁed picture of all SIPs with translation. A second aim is to show that what can be done using SWKB can also be done in ordinary WKB, at the cost of a generalized Langer modiﬁcation of the potential. His starting point is to write the all-order WKB quantization condition in the form

b 2m(E V (x))dx = (n + 1/2)πh¯ + πf(¯h, E) (2.40) a − Z p This defines a function f(¯h, E) which represents the effect of everything beyond lowest order WKB. If the WKB expansion were convergent, f(¯h, E) could be determined by adding up all the WKB terms of orderh ¯2 and higher. On the other hand, if one knows the exact spectrum of V (x), then f(¯h, E) is known by subtraction. What separates Translational Shape Invariant Potentials (TSIPs) from the others is that for the former, f is independent of energy. That being so, all energy levels will be given exactly if the ground state is correct. As examples, the harmonic oscillator, Morse potential and the Coulomb potential with the Langer modification, where the WKB quantization condition works perfectly, one has f = 0. Barclay and Maxwell [30], set the minimum of the potential at zero and defined u2(x) V (x), where u(x) plays a role similar to the superpotential of SUSY, but for ≡ ordinary potentials. They proposed two classes of potentials for which one or other of the following equations holds:

du = a + bu2 + cu (Class I) (2.41) dx

34 M.Sc. Thesis - Akbar Safari McMaster University - Physics and Astronomy

du = a + bu2 + cu√a + bu2 (Class II) (2.42) dx where a, b and c are constants so far as x is concerned. They claimed that one or other of these two equations is satisfied by every TSIP. They also showed that all potentials that fall in these two classes, are necessarily shape invariant. Therefore one or other of these two conditions is necessary and sufficient for shape invariance [30]. In Barclay’s time, SIPs with translation were the only known SIPs, so whenever he mentions SIPs, Barclay means TSIPs. Therefore, to find a TSIP, one just needs to verify one of Eqs. 2.41 or 2.42 with appropriate parameters a, b and c. In this way, Barclay could find all known SIPs. Most well known soluble potentials fall in the first class (Eq. 2.41), including the harmonic oscillator, Morse potential and Rosen-Morse potential. The radial equation of the three dimensional harmonic oscillator belongs to the second class. According to Barclay, the WKB series has a finite radius of convergence for shape invariant potentials which means that one can sum all terms in the WKB quantization condition and arrive at the exact bound state energies. It is a Taylor series rather than an asymptotic series and SWKB is an effective way to sum this series.

35 Chapter 3

Phase Shift Modiﬁcation

In section 1.3 we saw that the WKB quantization condition does not take reflection of the wave function into account, except at the classical turning points. Two questions then arise; how much does the phase of a wave function change in a reflection? and how may we include this phase shift in the WKB quantization condition? In this chapter we focus on this matter and explain some improvements to the WKB quantization condition based on the reflection phase shift.

3.1 Reﬂection phase

In classical optics, we know that the phase of a wave changes by π in reflection from a hard wall. In quantum mechanics, due to the penetration of the wave function into the forbidden region, the reflection phase shift is not exactly π. The following examples clarify what we mean by phase shift due to reflection and how to calculate it.

36 M.Sc. Thesis - Akbar Safari McMaster University - Physics and Astronomy

ikx V = V0 re− E eikx Region I Region II x

Figure 3.1: Sub-barrier reﬂection of a wave incident on a vertical step

Example 1 - Reﬂection by a step potential.

Fig. 3.1 shows a wave travelling towards a sharp vertical step of height V0. We

suppose the energy of the wave, E, to be less than V0. The wave functions in regions I and II are

ikx ikx κx ψI(x) = e + re− , ψII(x) = te− (3.1)

where k = √2mE/h¯ and κ = 2m(V E)/h¯ is measured from top of the step. 0 − The reﬂection coeﬃcient r, canp be obtained from continuity conditions on the wave function and its derivative at x = 0 by solving

1 + r = t (3.2)

ik ikr = κt (3.3) − −

Then 1

ik + κ k iκ r = = − eiδ (3.4) ik κ k + iκ ≡ −

37 M.Sc. Thesis - Akbar Safari McMaster University - Physics and Astronomy

- Π δ 2

-Π 0 V0 2 V0 E

Figure 3.2: Reﬂection phase shift due to scattering from a step potential where

1 κ δ = 2 tan− (3.5) − k

For small energies, δ π, consistent with the convention that repulsive forces give → − negative phase shift. Conversely, at the top of the barrier, κ 0 and r 1. Here, → → δ indicates the difference between the phase of the incoming wave and the reflected wave. Fig. 3.2 shows δ vs. the energy of the incoming wave. At low energies, the step potential resembles a hard wall for the particle and the wave function cannot penetrate into the barrier, so the phase shift π is justified.

38 1 M.Sc. Thesis - Akbar Safari McMaster University - Physics and Astronomy

Π a) b)

Π V (x) 2

V0 δ 0

Π V0 - 2 2

-Π -4 -2 2 4 x 0 V0 2 V0 E

Figure 3.3: a) A barrier of Woods-Saxon shape. b) Phase shift in scattering from the smooth step barrier.

Example 2 - Smooth potential step. The exact solution of the Schr¨odingerequation for the potential step of Woods- Saxon (or Fermi function) shape

V0 V (x) = x/σ (3.6) 1 + e− is discussed in section 25 of [13]. Fig. 3.3a shows this potential step. The step is centered at the origin and has a width 2σ.

ik1x If the incoming wave is a plane wave of the form e with energy E > V0, then

ik1x the asymptotic form of the reflected wave, as x , will be re− where the → −∞ reflection coefficient r, is given by 1

Γ (2ik σ)Γ( i(k + k )σ)Γ( i(k + k )σ + 1) r = 1 − 1 2 − 1 2 (3.7) Γ( 2ik σ)Γ(i(k k )σ)Γ(i(k k )σ + 1) − 1 1 − 2 1 − 2

39 M.Sc. Thesis - Akbar Safari McMaster University - Physics and Astronomy

k and k are the asymptotic wave numbers as x and x + respectively 1 2 → −∞ → ∞

√2m k = √E (3.8) 1 h¯ √2m k = E V (3.9) 2 h¯ − 0 p

For energies less than V0, the above equation for the reﬂection coeﬃcient can be used providing k2 is replaced by iκ where

√2m κ = V E (3.10) h¯ 0 − p In any case, r is complex and has the form r eiδ. In this way, δ is the phase shift | | we were looking for;

Γ (2ik σ) Γ (( ik + κ)σ) Γ (( ik σ + 1 + κσ)) δ = arg(r) = 1 − 1 − 1 (3.11) Γ( 2ik σ) Γ ((ik + κ)σ) Γ ((ik σ + 1 + κσ)) − 1 1 1

Fig. 3.3b shows this phase shift vs. energy. At very low energies, the phase shift is π, as expected. As the energy increases, the phase shift varies and has a minimum because two eﬀects are in competition; energy and slope of the potential. In steep potentials, the wave function in the forbidden zone decays rapidly compared to a slowly varying potential. Consequently, in a steep potential, the total phase shift is higher because less phase leaks into the forbidden region. For the same reason, the lower the energy, the larger the phase shift. Note that to evaluate the phase leakage and the phase shift, the entire sloping region of the potential should be considered, not only the local slope at the turning point. For instance Fig. 3.4 shows two potential steps with the same slope at the

40 M.Sc. Thesis - Akbar Safari McMaster University - Physics and Astronomy

x x

Figure 3.4: Comparison of two potential barriers with a smaller and a larger surface thickness.

energy E, in the middle of the steps. The total phase shift due to reﬂection in the potential on the left is higher, because its phase leakage is less than that in the potential on the right. The next two sections are devoted to the two main developments in WKB quantization condition which deal with phase shifts under reﬂection.

3.2 Friedrich and Trost method

In order to improve WKB by taking into account reﬂection phase shift introduced in the previous section, Friedrich and Trost [29, 31] wrote the WKB quantization

condition (Eq. 1.34) in the form 1

b φ φ kdx 1 2 = nπ (3.12) − 2 − 2 Za

They called φ1 and φ2 the ‘phase loss’ due to reﬂection at the left turning point a,

and the right turning point b, respectively. Before explaining how to ﬁnd φ1 and

φ2, we should note that there is a diﬀerence between the phase loss φ and the phase shift δ introduced in the previous section. The WKB wave function in the classically

41 M.Sc. Thesis - Akbar Safari McMaster University - Physics and Astronomy

allowed region has a sinusoidal form

x iφ/2 WKB e− i x kdx iφ i x kdx ψ (x) cos kdx φ/2 = e b + e e− b (3.13) ∼ b − 2 R R Z

i x kdx i x kdx which is divided into the right going e b , and the left going e− b waves, with R R phase diﬀerence φ. In this sense, the phase loss φ is the phase diﬀerence between the incoming plane wave and the left going WKB wave function, while δ is the phase

ikx iδ ikx diﬀerence between the incoming plane wave e , and the reﬂected plane wave e e− . In the WKB quantization condition, the phase accumulated during half of a classical orbit in phase space (from a to b) is calculated. Therefore half of φ1 and φ2 are considered in Eq. 3.12. By introducing the Maslov index in the form

φ + φ µ 1 2 (3.14) ≡ π/2 the WKB quantization condition can be written as

b µ kdx = n + π (3.15) a 4 Z where µ = 2 in usual application of WKB. The following examples show that if one knows the exact phase loss and uses the appropriate (in general non-integral) Maslov index, then the WKB quantization condition in the form of Eq. 3.15 will give the exact bound state energies.

42 M.Sc. Thesis - Akbar Safari McMaster University - Physics and Astronomy

V (x) x2 ∼ E

x = 0 x = b

Figure 3.5: A ﬂat potential for x < 0 with a quadratic barrier for x > 0.

Example 3. Quadratic barrier potential. Consider the potential sketched in Fig. 3.5:

0 x < 0 V (x) = (3.16)  2  mω x2 x 0  2 ≥  The solution to the Schrödingerequation for x > 0 is given by parabolic cylinder functions [6] U( E/hω,¯ y), where y x 2mω/h¯. By matching this solution to a − ≡ ik0x ik0x superposition of incoming and reflected planep waves, e +re− , at x = 0, Friedrich and Trost obtained the reflection coefficient

1 iδ 1 iβ 1 r = e = − = exp[ 2i tan− β] , 1 + iβ − 1 δ = 2 tan− β , (3.17) −

where

2¯hω Γ (3/4 E/(2¯hω)) β − (3.18) ≡ E Γ (1/4 E/(2¯hω)) r −

43 M.Sc. Thesis - Akbar Safari McMaster University - Physics and Astronomy

The total phase shift due to reflection, δ, is the argument of the reflection amplitude r. In order to find the phase loss φ, to be used in the WKB method, we should find the WKB wave function in free space (x < 0) and equate its phase to the argument of r in Eq. 3.17;

x 2E ψWKB(x) cos kdx + φ/2 b = (3.19) ∼ mω2 Zb r ! where

√2m √E k x < 0 ¯h ≡ 0  k(x) =  (3.20)   2 √2m E mω x2 x 0 ¯h − 2 ≥   q  Then

x √2m 0 mω2 x Eπ kdx = E x2dx + k dx = − + k x . (3.21) h¯ − 2 0 2¯hω 0 Zb Zb r Z0

Therefore

Eπ φ ψWKB(x) cos k x + (3.22) ∼ 0 − 2¯hω 2

Comparing to the exact wave function in free space

ik0x iδ ik0x iδ/2 δ ψ(x) e + e e− e cos k x (3.23) ∼ ∼ 0 − 2

44 M.Sc. Thesis - Akbar Safari McMaster University - Physics and Astronomy

Figure 3.6: Phase loss φ of Eq. 3.24, for reﬂection from a quadratic potential shown in Fig. 3.5 . results in

Eπ 1 2¯hω Γ (3/4 E/(2¯hω)) Eπ φ = δ + = 2 tan− − + (3.24) − hω¯ E Γ (1/4 E/(2¯hω)) hω¯ r − !

The phase loss for this example is shown in Fig. 3.6. The beauty of the method by Friedrich and Trost (F-T) is that the two phase

losses φ1 and φ2 in Eq. 3.12 are independent, i.e. the phase loss in Eq. 3.24 can be used for any potential well which is a combination of a quadratic potential on one side and some other potential on the other side. For instance in a harmonic

oscillator, φ1 and φ2 are both equal to the φ in Eq. 3.24. Fig. 3.6 shows that for energies E = (n + 1/2)¯hω, the phase loss φ = π/2. This explains why LO-WKB gives the exact bound state energies for a harmonic oscillator; because the phase loss of LO-WKB, π/2, is equal to the correct phase loss φ, at these energies. The phase

45 M.Sc. Thesis - Akbar Safari McMaster University - Physics and Astronomy

1.0 V (x)

0.5

x 10 20 30 40 50

-0.5

-1.0

Figure 3.7: The Woods-Saxon potential, Eq. 3.25, for V0 = 1, R = 25 and σ = 2.5 . loss in Eq. 3.24 can be used in the stretched harmonic oscillator consisting of two half parabolas with a constant potential in between. In this case, the bound state energies are not (n+1/2)¯hω, but the WKB quantization condition with Maslov index µ = 2φ/(π/2) gives the exact energies, if one uses the phase loss φ given in Eq. 3.24.

Example 4. Woods-Saxon or Fermi function as a radial potential. The Woods-Saxon potential is widely used in nuclear physics, to describe neutrons bound in a spherical nucleus. For ` = 0 waves it consists of a smooth step potential and a hard wall at the origin (Fig. 3.7), namely

V0 1+e−(x−R)/σ x > 0  V (x) =  (3.25)   1 x 0 . ∞ ≤    This potential for x > 0 is the potential discussed in example 2, but shifted down by

V0 and to the right by R.

46 M.Sc. Thesis - Akbar Safari McMaster University - Physics and Astronomy

Shifting the potential does not change its reflection coefficient. So, the total phase shift due to reflection obtained in example 2 can be used here to find the phase loss. To this end, we split the potential well into a hard wall on the left, and a smooth potential well that extends to in x. As for the hard wall, the WKB wave function to the −∞ right in the flat potential bottom is a plane wave, so there is no difference between

the total phase shift and the phase loss due to reﬂection by a hard wall, i.e. φ1 = π. The Woods-Saxon potential was already studied in example 2. We just need to ﬁnd the asymptotic WKB wave function when x and equate it to the → −∞ asymptotic form of the exact wave function on the left

δ lim ψ(x) cos k1x (3.26) x ∼ − 2 →−∞

where δ is given in Eq. 3.11. The phase loss turns out to be

2 κ κ 1 k1 φ = δ + 2k σ 2 ln 2 ln 1 + 2 tan− (3.27) 2 − 1 − k2 − k κ 1 1

Here k1 and κ are the asymptotic wave numbers

√2m √2m k = E + V , κ = √ E (3.28) 1 h¯ 0 h¯ − p and E < 0. Fig. 3.8 shows this phase loss for energies inside the well. We have calculated the energy eigenvalues with the F-T method (Table 3.1) for the Woods-Saxon potential shown in Fig. 3.7. The results of the LO-WKB and the

5 exact energies are compared in the table as well. There is an error of order 10− or

47 M.Sc. Thesis - Akbar Safari McMaster University - Physics and Astronomy

Π σ = 2.5 σ = 1.0 σ = 0.5 σ = 0.2 Π φ2 2

0 0.0 0.2 0.4 0.6 0.8 1.0 E/V0

Figure 3.8: Phase loss in reﬂection from a Woods-Saxon potential step for diﬀerent values of the parameter σ (Eq. 3.27) .

4 10− in the F-T method. Friedrich and Trost did not discuss the source of this error. They just said that their method improves the WKB quantization condition greatly. We think the F-T method should give the exact bound state energies, if one uses the right Maslov index to take into account the correct phase shift due to reﬂection. The source of the error in Table 3.1 is that the potential in example 2 for which we

Table 3.1: Bound state energies for the Woods-Saxon potential well of Eq. 3.25 with parameters R = 25 and σ = 2.5 . n Exact F-T (%Error) LO-WKB (%Error) 5 7 -0.119708811 -0.119708929 (9.7 10− ) -0.117824097 (1.57) × 5 6 -0.261470786 -0.261470955 (6.4 10− ) -0.259684907 (0.68) × 5 5 -0.410600393 -0.410600624 (5.6 10− ) -0.409214573 (0.34) × 5 4 -0.555674393 -0.555674721 (5.9 10− ) -0.554818858 (0.15) × 5 3 -0.689682711 -0.689683184 (6.8 101− ) -0.689410278 (0.04) × 5 2 -0.807247338 -0.807248103 (9.4 10− ) -0.807569114 (0.04) × 4 1 -0.903337293 -0.903338714 (1.5 10− ) -0.904251355 (0.10) × 4 0 -0.971800911 -0.971804169 (3.3 10− ) -0.973396776 (0.16) ×

48 M.Sc. Thesis - Akbar Safari McMaster University - Physics and Astronomy

Table 3.2: Bound state energies for the Woods-Saxon potential well of Eq. 3.25 with parameters R = 25 and σ = 0.5 . n Exact F-T (%Error) LO-WKB (%Error) 5 7 -0.087462803 -0.087462818 (1.6 10− ) -0.073207111 (16.3) × 6 6 -0.284636273 -0.284636283 (3.6 10− ) -0.275000853 (3.38) × 6 5 -0.466774621 -0.466774628 (1.4 10− ) -0.461683924 (1.09) × 7 4 -0.625741817 -0.625741821 (6.4 10− ) -0.624517000 (0.19) × 7 3 -0.758495608 -0.758495611 (4.9 10− ) -0.760393419 (0.25) × 8 2 -0.863294758 -0.863294758 (1.3 10− ) -0.867447282 (0.48) × 7 1 -0.938966550 -0.938966551 (1.8 10− ) -0.944177944 (0.55) × 7 0 -0.984699682 -0.984699683 (1.6 10− ) -0.988977911 (0.43) ×

have calculated the phase loss φ2 (Eq. 3.27), is truncated at the origin in the Woods- Saxon potential well. The phase φ in Eq. 3.27 is the phase loss from x = to 2 −∞ x = + , but in the Woods-Saxon potential we should have calculated the phase loss ∞ only from x = 0 to x = + . To confirm this claim, Table 3.2 shows the energy ∞ eigenvalues for the Woods-Saxon potential well with smaller diffuseness parameter σ. Reducing σ has the effect of making the bottom of the well flatter, near the origin, and thus closer to the assumed free space condition. Here we see that the error in the F-T method with the phase loss φ2 (Eq. 3.27) is smaller, because the potential in Table 3.2 for x < 0, compared to the potential in Table 3.1, is flatter and has less contribution in the phase loss.

3.3 Approximating the phase loss

The two potentials studied in the previous section were soluble analytically and the phase loss was found by comparing the WKB wave function to the exact one. When the potential is not soluble analytically, Friedrich and Trost suggested to use the Green function, to calculate the phase loss approximately.

49 M.Sc. Thesis - Akbar Safari McMaster University - Physics and Astronomy

They wrote the Schr¨odingerequation as

h¯2 d2 ψ(x) = (V (x) E) ψ(x) (3.29) 2m dx2 −

The Green function is the solution to the inhomogeneous equation

h¯2 d2 G(x, x0) = δ(x x0) (3.30) 2m dx2 −

giving

2m G(x, x0) = (x0 x)Θ(x0 x) (3.31) h¯2 − −

where Θ(x0 x) is the step function, 0 for x0 < x and 1 for x0 x. The right − ≥ hand side of Eq. 3.29 accounts for the inhomogeneous part. Then the solution to the Schr¨odingerequation is

2m ∞ ψ(x) = (x0 x)Θ(x0 x)[V (x0) E] ψ(x0)dx0 h¯2 − − − Z−∞ 2m ∞ = (x0 x)[V (x0) E] ψ(x0)dx0 (3.32) h¯2 − − Zx

Therefore the wave function and its derivative at the turning point b are

2m ∞ ψ(b) = (x b)[V (x) V (b)] ψ(x)dx (3.33) h¯2 − − Zb

2m ∞ ψ0(b) = [V (x) V (b)] ψ(x)dx (3.34) − h¯2 − Zb

50 M.Sc. Thesis - Akbar Safari McMaster University - Physics and Astronomy

So far no approximation has been made. Now Friedrich and Trost suppose that the potential in the classically allowed zone is zero, so that the wave function in the allowed zone is proportional to cos(k x b φ/2) with k = √2mE/h¯. The 0| − | − 0 logarithmic derivative of this wave function at the classical turning point, should

match to ψ0(b)/ψ(b) from Eqs. 3.34 and 3.35

φ ψ0(b) k tan − = (3.35) 0 2 ψ(b) or

1 1 ψ0(b) φ = 2 tan− (3.36) − k ψ(b) 0

This is the phase loss we were looking for, because the potential in the classically allowed zone is supposed to vanish identically. Then there is no diﬀerence between the total reﬂection phase and the phase loss (section 3.2). Calculating the phase loss from Eq. 3.36 using Eq. 3.33 and 3.34 requires knowledge of the wave function in the classically forbidden region. Friedrich and Trost suggested to use the WKB wave function instead of ψ(x) in Eqs. 3.33 and 3.34. Since the potential is supposed to be zero in the allowed region, we expect this approximation to work for low energies in potentials which grow rapidly. To examine the accuracy of Eq. 3.36, it was used in Ref. [32] to calculate the approximate phase loss for all SIPs included in Table 2.1. In Table 3.3, taken from Ref. [32], ground state energies calculated by the F-T method, are compared with LO-WKB results.

51 M.Sc. Thesis - Akbar Safari McMaster University - Physics and Astronomy

Table 3.3: Comparison of the F-T method (Eq. 3.36) with LO-WKB for ground states of SIPs. (Taken from Ref. [32]) exact WKB F T Name of Choice of V (x) E0 E0 E0 − potential parameters (error) (error) 1 2 2 1 1 1 Shifted 4 ω x 2 ω 2 ω 2 ω oscillator (0%) (0%) √2 2 3D ω = 2 ( r r) 2.1716 2 2.158 oscillator l = 1− (8.07%) (0.61%) 1 1 2 Coulomb e = 1 2 r 4 0.0625 0.5677 0.0558 l = 1− (9.17%) (10.7%) α = 2 2x 2 Morse A = 1 (2 e− ) 3 3 2.7587 B = 1− (0%) (8.04%) Rosen- α = 1 1 2 Morse II A = 2 6 (1 + 6 tanh x) 1.9167 2.1030 1.8549 (hyperbolic) B = 1 (9.72%) (3.22%) α = 1 Eckart A = 2 2(coth r 3)2 7 6.51 6.1540 B = 6− (7.00%) (12.1%) Rosen- α = 1 Morse I A = 2 2(1 + cot x)2 3 2.5727 2.9356 (trigonometric) B = 2 (14.25%) (2.14%) 1 √ Scarf II α = 1 2 ( 10 1) −2 (hyperbolic) A = 1 sech x 1.0811 1.1925 1.0298 B = 1 +3sech− x tanh x (10.31%) (4.74%) 11 √ Generalized α = 1 2 10 − 2 P¨oschl- A = 1 +11csch r 1.3377 1.2790 1.2576 Teller B = 3 9cschr coth r (4.39%) (5.99%) − 7 √ Scarf I α = 1 2 6 (trigono A = 3− +7 sec− 2 x 3.050 2.689 3.1285 metric) B = 1 5 sec x tan x (11.8%) (2.56%) −

52 M.Sc. Thesis - Akbar Safari McMaster University - Physics and Astronomy

Example 5. Asymmetric quartic barrier potential Consider the potential

4 V0 x x 0 V (x) =  ≥ (3.37)  0 x < 0

 with V0 > 0. Deﬁning the length ζ

1/6 h¯2 ζ (3.38) ≡ 2mV 0 the turning point occurs at

E 1/4 b = = ζ k ζ , (3.39) V 0 0 p where k0 = √2mE/h¯. In Fig. 3.9 we compare the phase loss obtained from Eq. 3.36 to the exact value. The exact phase loss is calculated by comparing the exact wave function to the WKB wave function in the zero potential region. As expected, the approximate phase loss is acceptable at low energies (long wavelengths). As the energy increases, the approximate phase loss becomes increasingly inaccurate; it does not converge to the correct phase loss, π/2, in the short wave length (high energy) limit. The approximate phase loss shown in Fig. 3.9 can be used to calculate the bound

53 M.Sc. Thesis - Akbar Safari McMaster University - Physics and Astronomy

Figure 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 as function of 2 2 1/3 = (mE/h¯ )[¯h /(2mV0)] , Ref. [29] . state energies of a stretched quartic potential well of the form

(x + R)4 x R ≤ −  V (x) =  0 R < x < R (3.40)  −  (x R)4 R x − ≤    The ﬁrst eight bound state energies of this potential were calculated using the approximate phase loss and are shown in Table 3.4. For deeply bound states, the F-T method with the approximate phase loss is more accurate than LO-WKB, but further up in the spectrum the approximate phase loss deteriorates, and starting from the seventh state from bottom of the well, LO-WKB gives a better result.

54 M.Sc. Thesis - Akbar Safari McMaster University - Physics and Astronomy

Table 3.4: The ﬁrst eight bound state energies of the stretched quartic well in Eq. 3.40 2 1/6 with R = 5(¯h /2mV0) . In the F-T method, the approximate phase loss of Fig. 3.9 is used. n Exact F-T (%Error) LO-WKB (%Error) 7 1.8093647 1.7992174 (0.56) 1.8027518 (0.37) 6 1.3963910 1.3943704 (0.14) 1.3879559 (0.60) 5 1.0331110 1.0369163 (0.37) 1.1984827 (16.0) 4 0.7217078 0.7286595 (0.96) 0.8567424 (18.7) 3 0.4641405 0.4713903 (1.56) 0.5667067 (22.1) 2 0.2620643 0.2677489 (2.17) 0.3313643 (26.4) 1 0.1167859 0.1197302 (2.52) 0.1544972 (32.3) 0 0.0292434 0.0300133 (2.63) 0.0412557 (41.1)

3.4 Cao’s quantization condition

As mentioned in section 3.2, the phase integral of WKB theory, k(x)dx, is generally not the total change in the phase of the wave function. FriedrichR and Trost suppose that the remaining change in the phase occurs at the turning points. They introduced a non-integral Maslov index to take it into account. In contrast, Cao et al. [33] claim that the phase loss is always π at each turning point, but an additional shift occurs due to scattered subwaves reﬂecting from the non-constant potential between the turning points. We will see that it is more a matter of the interpretation, not the amount of the net correction. To illustrate this eﬀect Cao et al. studied the potential shown in Fig. 3.10,

VL x < 0  V 0 < x < d  1 1 V (x) =  (3.41)   V2 d1 < x < d1 + d2 V d + d < x  R 1 2   

55 M.Sc. Thesis - Akbar Safari McMaster University - Physics and Astronomy

VL VR

d1 V1 x 0 d1 d1 + d2

Figure 3.10: The piecewise constant potential well in Eq. 3.41 .

For energies V1 < V2 < E < VR < VL, the wave function is

κLx ALe x < 0  A eik1x + B e ik1x 0 < x < d  1 1 − 1 V (x) =  (3.42)  ik2x ik2x  A2e + B2e− d1 < x < d1 + d2 A e κRx d + d < x  R − 1 2    where

2m 2m κL = 2 VL E , κR = 2 VR E, r h¯ − r h¯ − 2pm 2mp k1 = 2 E V1 and k2 = 2 E V2 (3.43) r h¯ − r h¯ − p p

They showed that matching the wave function1 and its derivative at the boundaries, leads to the quantisation condition [33]

1 κL 1 κR k d + k d tan− tan− + Φ = nπ (3.44) 1 1 2 2 − k − k s 1 2

56 M.Sc. Thesis - Akbar Safari McMaster University - Physics and Astronomy

where

1 P2 1 P2 Φ = tan− tan− (3.45) s k − k 2 1 with

1 κR P = k tan tan− k d . (3.46) 2 2 k − 2 2 2

At any potential step the log-derivative of ψ(x) is continuous. Eq. 3.44 is a quantization condition that can be solved for the energy E, by varying E until a solution with decaying waves in both left and right regions x < 0 and x > d1 + d2 is obtained.

Evidently, k1d1 + k2d2 is the usual WKB phase-integral accumulated by the wave travelling in the ﬂat parts. The third and fourth terms in Eq. 3.44 are the phase shifts at the left and right turning points, respectively. If we set V1 = V2, then Φs = 0.

Therefore Φs can interpreted as the phase contribution picked up by the scattering from the the step. If there were more such discontinuities, there would be additional contributions of type Φs.

3.5 Segmentation method

One approach to solving the wave equation for a general continuous potential, is to approximate V (x) by a sequence of constant potentials or segments. To make this approach practical, the potential should be truncated at xL and xR, far enough away from the turning points as suggested in Fig. 3.11.

The approximate potential is set to constant values VL for x < xL, and VR for

57 M.Sc. Thesis - Akbar Safari McMaster University - Physics and Astronomy

x xL ba xR

Figure 3.11: Truncation of an arbitrary potential well at xL and xR, far away from turning points a and b .

x > xR. Within the three intervals [xL, a],[a, b] and [b, xR], V (x) is divided into a number l, f and g of constant segments respectively, with the same width, d (for simplicity). As suggested by Kalotas and Lee [34], Vj within each segment can be chosen to make E V d equal to the WKB phase integral in that segment for the − j selected trial energy.p (Or one may simply take Vj to be some other average value of the true potential in the segment.) Cao and his colleagues used the transfer matrix method to propagate the solution across each constant potential segment. They call their procedure the “Analytical Transfer Matrix Method” (ATMM). Generalizing Eq. 3.44, they arrived at the quantization condition

l+f 1 Pl 1 1 Pl+f+1 kjd tan− tan− + Φs = nπ (3.47) − kl+1 − kl+f jX=l+1 where kj is the wavenumber in the j’th segment

2m kj = 2 E Vj (3.48) r h¯ − p 58 M.Sc. Thesis - Akbar Safari McMaster University - Physics and Astronomy

and

l+f 1 − 1 Pj+1 1 Pj+1 Φs = tan− tan− (3.49) kj+1 − kj jX=l+1

where

1 Pj+1 P = k tan tan− k d (for j = l + 1, l + 2, ..., l + f) (3.50) j j k − j j

Since Pj in Eq. 3.50 depends on the next segment (Pj+1), to solve Eq. 3.47 numerically,

one must start from the last Pj which is Pl+f+g+1 and move backwards, while

2m Pl+f+g+1 = Ps = 2 Vs E (3.51) r h¯ − p

and in the interval [b, xR], Pj is given by

Pj+1 sinh(κjd) + cosh(κjd) κj Pj = κj (for j = l + f + 1, l + f + g) (3.52) Pj+1 cosh(κjd) + sinh(κjd) ··· κj where in a classically forbidden zone

κ = 2m(V E)/h¯ . (3.53) j j − q

If one lets d 0, then the wavenumber at each turning point, k and k in → l+1 l+f Eq. 3.47, will vanish. Then

1 Pl 1 Pl+f+1 π tan− = tan− = (3.54) k k 2 l+1 l+f

59 M.Sc. Thesis - Akbar Safari McMaster University - Physics and Astronomy

and Eq. 3.47 reduces to

l+f

kjd + Φs = (n + 1)π (3.55) jX=l+1

Comparing Eq. 3.55 and Eq. 3.47 to Eq. 3.44, makes it appear that the phase shift at a turning point is π/2 rather than π/4 as in usual LO-WKB approximation.

However there is an addition phase Φs accumulated as the wave propagates across the potential between the turning points. In cases where LO-WKB does give correct binding energies, the combination of the two eﬀects works out to give the usual π/4 phase. Cao and collaborators have applied their ATMM method to many examples, with about 5000 segments to get accurate numerical results. Since the transfer matrix for a constant potential segment is a simple combination of sine and cosines, the calculations are straightforward [35].

3.6 Another approach to Cao’s condition

To solve

2 ψ00 + k (x)ψ(x) = 0 (3.56)

60 M.Sc. Thesis - Akbar Safari McMaster University - Physics and Astronomy

on the range a to b, one can follow Infeld and Hull [36] in factorising the operator

2 2 2 2 [D +k ]ψ (D P )(D+P )ψ = [D +P 0 P ]ψ to obtain three ﬁrst order equations: ≡ − −

2 2 P 0 P (x) = k (x) − (D + P (x))ψ(x) = χ(x)

(D P (x))χ(x) = 0 . (3.57) −

The boundary conditions will be imposed one at x = a and the other at x = b. For example if they are

ψ0(a) + αψ(a) = A

ψ0(b) + βψ(b) = B (3.58) then we need P (a) = α and χ(a) = A. Whatever P (a) and χ(a) are, they are sufficient to start the solutions of their first-order differential equations. One can then integrate the first and third equations in 3.57 from x = a to x = b. Since we then know χ(x) and P (x) for a x b, we can determine the necessary initial value ≤ ≤ of ψ(b) from 3.57. One then integrates the middle equation in 3.57 for ψ(x) back to x = a and both boundary conditions will be satisfied. You have to solve three first order equations to apply the two boundary conditions at opposite ends of the range. Numerically it is safe because the first order equation for ψ(x) does not suffer from a spurious growing component when the function is actually decaying. When the boundary conditions are homogeneous, as they generally are for the Schrödingerequation, A = B = 0. Then the auxiliary function χ(x) will vanish

61 M.Sc. Thesis - Akbar Safari McMaster University - Physics and Astronomy

2 P + 2 k √ = P 0 P √

φ ) k

Figure 3.12: The dynamical triangle relating P (x) and k(x) to the phase φ(x)

identically. Only the ﬁrst two of Eq. 3.57 required, and P (x) is minus the log- derivative of the wave function. Of course, the boundary values P (a) and P (b) depend on the exterior potential outside the region where we are solving, and on the assumed energy E. Only when E is an eigenvalue will a satisfactory wave function be obtained, with P (b) = β. − The right-angled triangle shown in Fig. 3.12, provides another way to obtain Cao’s quantization condition, Eq. 3.55. The legs are k(x) and P (x), and the hypotenuse is the square root of P 0:

2 dP (x) ψ00(x) ψ0(x) 2 2 P 0(x) = = + = k (x) + P (x) (3.59) dx − ψ(x) ψ(x) 1

Knowing P (x) and k(x) is equivalent to knowing the wave function ψ(x). The angle φ can replace one of them. From

1 P φ = tan− (3.60) k

62 M.Sc. Thesis - Akbar Safari McMaster University - Physics and Astronomy

we have

d 1 P 1 kP 0 P k0 P tan− = − = k k0 (3.61) dx k 2 k2 − P 1 + (P/k) 0

Integrating both sides of Eq. 3.61 from a to b gives

b b b 1 P P tan− = kdx k0 dx I + I = (n + 1)π (3.62) k − P ≡ 0 c a Za Za 0

which is Cao’s quantization condition, Eq. 3.55 . Since the second integral, Ic, gives the correction to the LO-WKB, we call it the “correction term”. The value (n+1)π is

obtained as follows: In case of the ground state, at the left turning point a, P = ψ0/ψ − is negative, because ψ(a) and ψ0(a) must have the same sign for a decaying wave function to the left. But the local wavenumber vanishes at a, so the inverse tangent is π/2. At the upper limit b, the relative signs are opposite and φ takes an odd − multiple of +π/2. For excited states, the wave function ψ(x), will have n nodes, at each of which P (x) has a simple pole of unit residue. The Riccati diﬀerential equation,

Eq. 3.59, shows that P 0 is always positive, therefore P and consequently P/k increases

1 monotonically as x increases. This shows that the angle φ = tan− (P/k) increases by π between each pair of nodes of ψ(x). It is like the angle of classical action-angle variables. It is important to realize that Cao’s relation, though it is exact, is not a quantisation condition in the usual WKB sense. In LO-WKB, one wants the phase integral to take the value (n + 1/2)π. You simply guess E, and integrate the local wave number between the turning points until the desired phase integral I0 = (n + 1/2)π is obtained. Raising E raises I0 and lowering E lowers the value, allowing the eigenvalue

63 M.Sc. Thesis - Akbar Safari McMaster University - Physics and Astronomy

V (x)

E = 3 E = E0 E = 1

Figure 3.13: Three wave functions at energies E = 1, E = E0 and E = 3, in the linear potential V x) = x, 0, x, . E0 = 2.33811 is the ground state 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. ∞

to be bracketed. In contrast, the argument based on Eq. 3.62 says that the sum of

I0 + Ic is an integer multiple of π at any energy. As an example to clarify how Eq. 3.62 works, consider the linear potential shown in Fig. 3.13, truncated by a hard wall at the origin. The ground state energy is

E0 = 2.33811 and the ground state wave function is the solid curve in the ﬁgure. If

one chooses an energy less than E0, say E = 1, and starts from the origin to construct

the wave function numerically, then the1 result is a wave function diverging to + on ∞ the right (the dashed curve in Fig. 3.13). At the left turning point, x = a = 0, the

1 slope of the wave function is positive and ψ 0+, so φ = tan− (P (a)/k(a)) = π/2. → a − At the right turning point, b, the slope of the wave function is still positive and k 0, → 1 then φ = tan− (P (b)/k(b)) = π/2. Therefore I + I = φ φ = 0. The same b − 0 c b − a argument is true for higher energies up to E 1.98 at which the slope of the wave ∼ function at the right turning point, ψ0(b), turns out to be zero. For energies higher

64 M.Sc. Thesis - Akbar Safari McMaster University - Physics and Astronomy

than E 1.98 one gets I + I = π, until we again ﬁnd ψ0(b) = 0. Table 3.5 shows ∼ 0 c

the value of I0 + Ic for several energies near the bottom of the well. In summary, it

is seen that I0 + Ic does not change continuously, it jumps by π whenever the sign of

ψ0(x = b) changes. The results are completely diﬀerent if one satisﬁes the boundary condition at + and integrates backwards to construct the wave function. In this case, the ∞

wave functions for the energies E = 1, E = E0 and E = 3 are shown in Fig. 3.14.

Table 3.6 shows that there are no jumps in the sum I0 + Ic in this case. The sum changes continuously and coincides with (n + 1)π at the bound state energies. We have imposed a decaying wave function under the barrier at the right. At the left boundary k(a) is non-zero, and P (a) varies continuously as the energy increases, as does φ(a). The only objection is that we have had to solve the problem exactly for the eigenvalue, in order to verify Cao’s relation. In a reﬂection symmetric potential, V (x) = x , the same approach can be used, | | because one can ﬁnd the even and odd-parity solutions independently of each other by considering only the right side x > 0. For the odd-parity states we require a node at

Table 3.5: Numerical results showing that the value of I0 + Ic for the linear potential, satisﬁes Eq.3.62 for any energy E, if the integration starts from the boundary at x = 0. EI0 Ic I0 + Ic P (b) ψ0(b) 3.0 3.464 -0.322 π -1.360 -0.694 2.5 2.635 0.506 π 1.921 -0.509 E0 2.383 0.758 π 0.729 -0.369 2.0 1.885 1.255 π 0.016 -0.014 1.5 1.224 -1.224 0 -0.448 0.497 1.0 0.666 -0.666 0 -0.913 0.838 0.5 0.235 -0.235 0 -1.979 0.979

65 M.Sc. Thesis - Akbar Safari McMaster University - Physics and Astronomy

V (xx) 5 H L E = 3

4 E = E0 E = 1

x x -1 0 1 2 3 4 5

Figure 3.14: Illustration of three wave functions of the linear potential corresponding 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 + . ∞

the origin, which is tantamount to putting a hard wall at x 0. For the even parity ≤ states, one requires ψ0(0) = 0. Putting the spectrum together, all the eigenvalues are found by the condition (I0 + Ic) = (n + 1)π/2, where I0 and Ic are computed only on the x > 0 side. Generally, in a non-symmetric smoothly varying potential, Cao’s formula (Eq. 3.62) is satisfied at any energy, and does not single out the bound state energies. The function P (x) satisfies a Ricatti equation, so one must know the boundary condition at 1 one of the two turning points to start the integration for P (x). But that requires knowledge of the outside potential at either x b or x a. Whatever value is chosen ≥ ≤ at say x = b, one will arrive at the other end with a non-zero P (a), and an infinite

ratio of P/k at both turning points. As a result, I0 + Ic is an integer multiple of π for any chosen energy. WKB holds that the eigenvalue can be found by looking only at the potential between the turning points. When one arrives at b with P (b) = 0,

66 M.Sc. Thesis - Akbar Safari McMaster University - Physics and Astronomy

Table 3.6: Numerical results showing that Eq.3.62 conﬁrms the bound state energies for the linear potential, providing that one starts from + and integrates backwards to construct the wave function. ∞ EI0 Ic I0 + Ic P (b) 3.0 3.464 0.801 4.265 0.729 2.5 2.635 0.762 3.397 0.729 E0 2.383 0.758 π 0.729 2.0 1.885 0.776 2.661 0.729 1.5 1.224 0.844 2.068 0.729 1.0 0.666 0.885 1.551 0.729 0.5 0.235 0.789 1.025 0.729

one possibility is that the potential is cut oﬀ at a constant value V (b) to the right. If that were so, the assumed trial energy E would be a threshold bound state for the truncated force. Alternatively if P (b) diverges, one has a node of ψ(b = 0), and E would correspond to a bound state in a potential truncated by a hard wall at x = b. All energies between these two cases correspond to some truncation of the potential beyond x = b. For example you could take V (x > b) = β(x b), and vary 0 < β < . − ∞ If you want to use only information between the two turning points, you cannot ex- clude any choice of β outside. That is why you always get (n + 1)π for Cao’s relation at any E.

Example 6 - Harmonic Oscillator. To illustrate the exactness of Eq. 3.62, consider the harmonic oscillator. The Schr¨odingerequation

2 2 h¯ mω 2 ψ00 + E x ψ(x) = 0 (3.63) 2m − 2

67 M.Sc. Thesis - Akbar Safari McMaster University - Physics and Astronomy

can be written as

d2ψ + λ2 X2 ψ(X) = 0 (3.64) dX2 − where

hω¯ mω E = λ2 and b2 = (3.65) 2 h¯

In these units, the eigenvalues are λ2 = 2n + 1, and the solution which vanishes at inﬁnity is

X2/2 ψ(X) = Hn(X)e− (3.66)

where HN (X) is a Hermite polynomial. To evaluate the correction term, Ic, in Eq. 3.62 we need to know P (X). We have

X2/2 ψ0(X) = [H0 (X) XH (X)] e− (3.67) n − n

P (X) = X H0 (X)/H (X) (3.68) − n n

k2(X) = λ2 X2 (3.69) −

X X k0(X) = = (3.70) −√λ2 X2 −k(X) −

68 M.Sc. Thesis - Akbar Safari McMaster University - Physics and Astronomy

Therefore the integrand in the correction term is

P X X Hn0 (X)/Hn(X) k0 = − (3.71) 2 2 − P 0 −k(X) k + [X H (X)/H (X)] − n0 n

The prime in Eqs. 3.67 to 3.71 denotes derivative with respect to X. The simplest

case is n = 0 with H0(X) = 1. The WKB phase integral between the turning points λ = 1, takes the value π/2 and the correction term the same value: ± ±

1 1 π k(X)dX = √1 X2dX = cos2 θdθ = π/2 (3.72) 1 1 − 0 Z− Z− Z and

1 P 1 X2 k0 dX = dX 2 − 1 P 0 1 √1 X Z− Z− − I = π/2 (3.73) ≡ c0

So Eq. 3.57 reduces to the usual WKB quantization condition for the ground state and gives the exact energy. In the case n = 1, H (X) = 2X, P (X) = X 1/X and 1 − the correction term Ic1, becomes

λ λ 2 2 P X X 1 2 Ic1 = k0 dx = − dX = π/2 (λ = 3) (3.74) 2 2 2 − λ P 0 λ √λ X X + 1 Z− Z− −

which again gives the exact energy. The claim of Cao et al. [37] is that the correction

term Icn = π/2 for all n. That is correct, from the known fact that WKB reproduces all the harmonic oscillator states, but it is not easy to prove it directly by evaluating

Ic.

69 M.Sc. Thesis - Akbar Safari McMaster University - Physics and Astronomy

P k0 − P 0

0.2

0.1

x -1.5 -1.0 -0.5 0.5 1.0 1.5

-0.1

Figure 3.15: Correction integrand in Cao’s quantization condition for the third bound state of the quartic potential well, V (x) = x4

3.7 General remarks about the correction term Ic

Fig. 3.15 shows the correction integrand, k0P/P 0, for the third bound state of the − quartic potential V (x) = x4. This is the general shape of the correction integrand for a simple quantum well with one basin; it has an oscillatory behaviour with 2n 1 − local extrema for the nth state, and the amplitude diverging to inﬁnity at the turning points. Generally, the correction integrand can be written as

P k0 P k 0 k = − 2 2 (3.75) − P 0 k P + k

We know that

1 xy 1 (3.76) −2 ≤ x2 + y2 ≤ 2 1

70 M.Sc. Thesis - Akbar Safari McMaster University - Physics and Astronomy

for any x and y. Therefore P k/(P 2 + k2) can be replaced by (sin θ)/2. In fact θ = 2φ (see Fig. 3.12), because

P k sin 2φ = sin φ cos φ = (3.77) √P 2 + k2 √P 2 + k2 2

Hence

P k0 sin 2φ k0 = − (3.78) − P 0 k 2

The sinusoidal part is responsible for the oscillations and the amplitude grows to inﬁnity since k 0 at turning points. → Having the variational method in mind, we considered approximations to the correction term Ic. In the variational method, it is known that the ground state energy is not very sensitive to the ground state wave function. For instance, for an inﬁnite square well potential with width 2L, the trial wave function ψtrial(x) = L2 x2 − produces the ground state energy with only 1% error [38]. In a simple symmetric potential well, the ground state wave function can often be approximated by a Gaussian wave function

trial x2/β ψ (x) = Ae− (3.79) where constants A and β are amplitude and width of the function respectively. These two parameters should be chosen to ﬁnd the best ﬁt for the wave function. The beauty of the Gaussian wave function is that P (x)/P 0(x) = x, independent of β and

71 M.Sc. Thesis - Akbar Safari McMaster University - Physics and Astronomy

A. Then

b P b I = k0 dx = xk0dx (3.80) c − P − Za 0 Za

But

b dk b b b x dx = xk(x) kdx = kdx (3.81) dx − − Za a Za Za

and

b Ic = I0 = kdx (3.82) Za

Therefore with a Gaussian wave function, the correction term for the ground state would take the same value as the phase integral and Cao’s quantization condition changes to the WKB quantization condition. As an example let’s consider the linear potential V (x) = x . Choosing units in | | whichh ¯ = 2m = 1, the exact ground state wave function is

ψexact(x) = Ai ( x E ) (3.83) | | − 0

with the ground state energy E0 = 1.018793. As the trial wave function, we use a Gaussian with β = 3

trial x2/3 ψ (x) e− (3.84) ∼

72 M.Sc. Thesis - Akbar Safari McMaster University - Physics and Astronomy

As stated above, the WKB phase integral, I0, and correction term Ic, are equal:

E0 2m 4 2m 3/2 I0 = E0 x dx = E (3.85) h¯2 − | | 3 h¯2 0 r E0 r Z− p

E0 E0 2m x 4 2m 3/2 Ic = xk0dx = 2 dx = 2 E0 (3.86) E0 h¯ 0 √E0 x 3 h¯ Z− r Z − r

Since WKB gives the exact ground state energy of a simple harmonic oscillator and its ground state wave function is Gaussian, naturally, for a harmonic oscillator, Cao’s quantization condition with a Gaussian trial wave function is equivalent to the WKB quantization condition. But we see that this is not true only for the simple harmonic oscillator. For the ground state of any potential, the Gaussian trial wave function converts Cao’s quantization condition to the WKB quantization condition.

73 Chapter 4

Exponential Potential

4.1 Introduction

Many analytically soluble potentials have been found so far by super-symmetric quantum mechanics. For the class of “shape invariant” potentials, SWKB gives exact binding energies for all states. Meanwhile, there exist other potentials that are not shape invariant but still exactly soluble. The exponential potential is one of them. We begin this chapter by describing the work of Fabre and Gu´ery-Odelinwho exam- ined the accuracy of WKB for the exponential potential. Then we describe the work published by me as a Comment on their article, in American Journal of Physics. Consider the exponential as a radial potential in three dimensions:

αx V e− x 0 − 0 ≥ VI(x) =  (4.1)  x < 0  ∞  with parameters V0 > 0 and α > 0. This potential is sketched in Fig. 4.1

74 M.Sc. Thesis - Akbar Safari McMaster University - Physics and Astronomy

x 0 1 2 3 4

V − 0

Figure 4.1: The exponential radial potential for α = 1, along with the ﬁrst three eigenfunctions.

In terms of the dimensionless variable X = αx and dimensionless parameters

√8mV √ 8mE a = 0 and b = − (4.2) hα¯ hα¯ the Schr¨odingerequation

2 2 h¯ d ψ(x) αx + E + V e− ψ(x) = 0 (0 x < ) (4.3) 2m dx2 0 ≤ ∞ takes the form

2 d ψ 1 2 X 2 + a e− b ψ(X) = 0 (4.4) dX2 4 − 1 The parameter b measures the binding energy from top of the well and a is a measure

X/2 of the well depth. Next, one can change the variable X to y = ae− . The result is

75 M.Sc. Thesis - Akbar Safari McMaster University - Physics and Astronomy

Bessel’s equation of order b in the variable y:

d2ψ dψ y2 + y + [y2 b2]ψ(y) = 0 (4.5) dy2 dy −

Therefore the bound state wave functions are regular Bessel functions ψb(x) = Jb(y) (see Fig. 4.1). Note that the positive x-axis is mapped onto a segment of the y-axis 0 < y < a; x corresponds to y 0, and x = 0 corresponds to y = a. The → ∞ →

boundary condition at the origin, ψb(x = 0) = 0, implies that y = a must be a node

of the Bessel function Jb(y), i.e. the bound state energies can be found by solving the

equation Jb(a) = 0 for the appropriate order b. In a recent paper, Fabre and Gu´ery-Odelin(FG) [39] compared the accuracy of WKB and SWKB quantization conditions, taking the exponential potential as an example. Applying the WKB quantization condition directly to the Schr¨odinger equation in X-space, Eq. 4.4, leads to the quantisation condition

X2 1 2 X 2 3 a e− b dX = n + π (4.6) 2 X1 − 4 Z p

where the left turning point is X1 = 0 and the right turning point X2 is the solution

2 X 2 to a e− = b . Since we have a hard wall at the origin, the term 3/4 appears on

X/2 the r.h.s. of Eq. 4.6, instead of usual 1/2. In terms of the new variable y = ae− , Eq. 4.6 becomes

b b2 3 1 dy = n + π (4.7) − y2 4 Za s

76 M.Sc. Thesis - Akbar Safari McMaster University - Physics and Astronomy

0.500 à à 0.300 æ 0.200 à æ 0.150 à

100 æ à 0.100 à à × à à à 0.070 æ æ E δE 0.050 æ æ æ 0.030 æ æ

à WKB 0.020 æ 0 0.015 WKB2 0.010 0 2 4 6 8 n

Figure 4.2: Relative error deﬁned in Eq. 4.10 of the LO-WKB approximation applied 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.

The value of the indeﬁnite phase integral is

2 b 2 2 1 b 1 dy = y b + b sin− (4.8) s − y2 − y Z p Finally the WKB quantization condition reads

2 3 π b b 1 b π b n + = 1 + sin− F (b/a) (4.9) 4 a − a a a − 2 a ≡ s which can be solved for b and consequently for the energy E. The relative error of the energies obtained from Eq. 4.9 1

δE EWKB Eexact = − 100 (4.10) E Eexact ×

is shown by the upper line in Fig. 4.2

77 M.Sc. Thesis - Akbar Safari McMaster University - Physics and Astronomy

Since the results are not perfect, FG added the second order WKB quantization correction to Eq. 1.49 and obtained

1 3 π F (b/a) = n + (4.11) − 2 4 a 12a2 1 b − a q Fig. 4.2 compares the result of this equation with the LO-WKB (Eq. 4.9). As expected, the accuracy is improved for the entire spectrum. Note however, that after reaching a minimum around the ﬁfth bound state, the error begins to increase. We will return to this later.

4.2 Symmetric 1D exponential potential

The symmetric 1D exponential potential is shown in Fig. 4.3:

α x V (x) = V e− | | ; < x < . (4.12) II − 0 −∞ ∞

Since VII(x) is reflection symmetric, its eigenfunctions will have well defined parity. It is sufficient to solve the Schrödingerequation, Eq. 4.3, on the half space, but with new boundary conditions: ψ(x = 0) = 0 for the odd-parity solutions and ψ0(x = 0) = 0 for the even-parity solutions. The odd-parity solutions are antisymmetric, so satisfy

Jb(a) = 0 (odd parity states) (4.13)

We need only take over the solutions obtained above, and reﬂect the wave function in the origin, ψ( x) = ψ(x) − −

78 M.Sc. Thesis - Akbar Safari McMaster University - Physics and Astronomy

x -4 -2 2 4

V − 0

Figure 4.3: The symmetric 1D exponential potential, and its ﬁrst three bound state wave functions.

The even-parity states are reﬂection symmetric, ψ( x) = +ψ(x) and satisfy −

dJb(y) J 0(a) = = 0 (even parity states) (4.14) b dy y=a

Application of the LO-WKB quantization condition to the even-parity states of VII(x) leads to

1 π F (b/a) = n + (4.15) 2 2a

Since there is no hard wall and the potential VII(x) is continuous, the factor (n + 1/2) applies here, instead of (n + 3/4). The energies obtained from Eq. 4.13 for odd values of n, coincide with the LO-WKB energies for VI(x). 1

79 M.Sc. Thesis - Akbar Safari McMaster University - Physics and Astronomy

1.00 à 0.50

0.20 à 100

× 0.10 à E δE 0.05 à à à à à à 0.02 à à à à à à à à 0.01 à à à à à à à à à à à à à à à 5 10 15 20 25 30 n

Figure 4.4: Relative error of WKB energies for a deep exponential potential with 32 bound states. 4.3 Apparent violation of the correspondence prin-

ciple

As already mentioned, Fig. 4.2 shows that the accuracy of LO-WKB deteriorates as n, the number of nodes of the wave function, increases. This is an apparent violation of Niels Bohr’s correspondence principle, which states that the quantum energies should approach the classical energies as the number of quanta of excitation diverges. WKB is a semiclassical approximation valid in the short wavelength limit, so it would be legitimate to expect the WKB energies to approach the exact energies as the number of nodes increases. Possibly the relative error shown in Fig. 4.2 might reduce again for much higher energies in a very deep potential with1 many bound states. The example shown in Fig. 4.4 indicates that this is unlikely to be the case.. In a comment on the paper by Fabre and Gu´ery-Odelin[40], we showed that the LO-WKB result is greatly improved by a small but signiﬁcant change suggested by

80 M.Sc. Thesis - Akbar Safari McMaster University - Physics and Astronomy

the work of Friedrich and Trost [29]. The rest of this chapter explains our analysis. First we ﬁnd a proper expansion for the Bessel function that improves upon the WKB result. Then by comparing this expression with the WKB result, an appropriate phase loss is found. Such an expansion was developed by Goldstein and Thaler [41], and is explained in the coming section.

4.4 GT asymptotic series for the Bessel function

Goldstein and Thaler (GT) [41], carefully studied the problem of obtaining an accurate asymptotic expression for the Bessel functions built upon the WKB approximation. They used the phase-amplitude method and wrote the solution to the Bessel equation Eq. 4.5 in the form

iΦb(y) Jb(y) + iNb(y) = Ab(y)e (4.16)

Jb(y) = Ab(y) cos Φb(y) (4.17)

Changing the variable in Bessel’s equation to t 1/y, and deﬁning B (t) as ≡ b

πt Bb(t) = Ab(t) (4.18) r 2 leads to the coupled pair of equations

2 d Bb(t) 1 2 2 3 + 1 ν t B (t) = [B (t)]− (4.19) dt2 t4 − b b 81 M.Sc. Thesis - Akbar Safari McMaster University - Physics and Astronomy

dΦb(t) 2 = [B (t)]− (4.20) dt − b

where

ν2 b2 1/4 (4.21) ≡ −

If these could be solved exactly, no approximation is entailed. (Neglecting the B00(t)

on the l.h.s. of Eq. 4.19 would produce the WKB approximation to Jb(x).) Instead,

Goldstein and Thaler expanded both Bb(t) and Φb(t) in powers of t, and determined the coeﬃcients by the Frobenius method up to order t10. Their solution was written in the form

∞ 2k+1 3 5 7 9 11 Bb(t) = t + a2kt = t + a2t + a4t + a6t + a8t + a10t + ... (4.22) Xk=1 where

1 a = ν2 (4.23) 2 4

5 3 a = ν4 ν2 (4.24) 4 32 − 8

15 37 15 a = ν6 ν4 + ν2 (4.25) 6 128 − 32 8

195 611 1821 315 a = ν8 ν6 + ν4 ν2 (4.26) 8 2048 − 256 128 − 16

82 M.Sc. Thesis - Akbar Safari McMaster University - Physics and Astronomy

663 4199 29811 2223 2835 a = ν10 ν8 + ν6 ν4 + ν2 (4.27) 10 8192 − 1024 512 − 8 8 and

∞ 1 π 2k 1 Φ (t) = b + + b t − b − 2 2 2k Xk=1 1 π 1 = b + + + b t + b t3 + b t5 + b t7 + b t9 + ... (4.28) − 2 2 t 2 4 6 8 10 where1

1 b = ν2 (4.29) 2 2

1 1 b = ν4 ν2 (4.30) 4 24 − 4

1 7 3 b = ν6 ν4 + ν2 (4.31) 6 80 − 20 4

10 95 807 315 b = ν8 ν6 + ν4 ν2 (4.32) 8 1792 − 224 224 − 56

7 35 1975 315 b = ν10 ν8 + ν6 58ν4 + ν2 (4.33) 10 2304 − 72 192 − 4

The constant phase (b + 1/2)π/2 in Eq. 4.28 was imposed by GT in order to have the correct phase of the regular Bessel function as y , which is known from → ∞ 1In Eq. 4.32 a typo is corrected from 1/1792 to 10/1792.

83 M.Sc. Thesis - Akbar Safari McMaster University - Physics and Astronomy

Hankel’s asymptotic series as given, for example, in Watson (Ch.VII) [42]. Any other choice would produce a linear combination of Jb(y) and Nb(y). For instance, if the phase shifts were increased by an amount δ, the real part of Eq. 4.16 would produce an asymptotic approximation to the linear combination cos(δ)Jb(y) + sin(δ)Nb(y).

Solutions for the real functions Ab(y) and Φb(y) in Eqs. 4.22 and 4.28, are asymptotic series in powers of t = 1/y, valid as y . → ∞

4.5 The radial potential, VI(x)

In order to resolve the issue regarding the correspondence principle, we introduce Ψ(y) such that

2 ψ(y) = J (y) Ψ(y) (4.34) b ≡ πy r

Then the ﬁrst derivative term in Eq. 4.5 is removed

d2Ψ ν2 + 1 Ψ(y) = 0 (4.35) dy2 − y2 where again

ν2 b2 1/4 (4.36) ≡ −

Equation 4.35 would represent the radial wave equation for a free particle in 3D, if ν2 = `(` + 1), for integer `. The potential in Eq. 4.35 is shown in Fig. 4.5. The hard wall at y = a represents the hard wall at x = 0 in x-space.

The energy of the new problem in Eq. 4.35 is known, E0 = 1. The task is to ﬁnd ν

84 M.Sc. Thesis - Akbar Safari McMaster University - Physics and Astronomy

ν2/y2

E0 = 1

y y = ν y = a

Figure 4.5: Illustration of the problem in Eq. 4.35. A centrifugal barrier on the left, and a hard wall at y = a .

such that the energy E0 = 1 becomes a bound state energy, as illustrated in Fig. 4.5. Then the energy E, of the original problem can be found through Eqs. 4.2 and 4.36. Application of the LO-WKB quantization condition to Eq. 4.35 gives

2 3 π ν ν 1 ν π ν n + = 1 + sin− (4.37) 4 a − a a a − 2 a r

which is Eq. 4.9 with b replaced by ν everywhere. The results depicted in Fig. 4.6 show that not only is the problem of violation of the correspondence principle not resolved, but also the accuracy of the eigenvalues is worse over the entire spectrum. Equation 4.35 represents a centrifugal problem and one may think of using the Langer modiﬁcation. But the Langer modiﬁcation replaces `(` + 1) = ν2 with (` + 1 1/2)2 = b2, and takes us back to Eq. 4.9. To resolve the problem, we use the Friedrich and Trost approach, and apply the

85 M.Sc. Thesis - Akbar Safari McMaster University - Physics and Astronomy

é Eq. 4.37 10.0 5.0 à WKB0 é

100 1.0 é × 0.5 é à E é δE à é é é é à à é à à 0.1 à à à à

0 2 4 6 8 n

Figure 4.6: Energies obtained from solving Eq. 4.37. The upper line uses ν everywhere, and the lower line replaces ν by b everywhere Eq. 4.9 .

LO-WKB quantization condition to Eq. 4.35. The phase integral would be

a 2 ν φ 2 2 1 ν π φ 1 dy = y ν + ν sin− ν − y2 − 2 − y − 2 − 2 Zν s p 1 ν2 1 ν4 π φ y + + + ... ν (4.38) ∼ 2 y 24 y3 − 2 − 2

Here we introduced the phase φ that shifts the WKB phase integral to match better with the Bessel function. In other words, φ is the phase loss in the sense of Friedrich and Trost.

On the other hand, we realize that adding up the leading order terms of Bb(t) in Eq. 4.22 (involving only the ﬁrst terms on the r.h.s. of each of Eqs. 4.23 to 4.27), gives the WKB amplitude for the Bessel function1

2 1 Ab(y) (4.39) ∼ π (y2 ν2)1/4 r −

Also adding up the leading order terms in Eq. 4.28 for Φb(t), gives the WKB phase

86 M.Sc. Thesis - Akbar Safari McMaster University - Physics and Astronomy

for the Bessel function

1 ν2 1 ν4 π π Φ (y) y + + + ... b (4.40) b ∼ 2 y 24 y3 − 2 − 4

Substituting Eqs. 4.39 and 4.40 into Eq. 4.17 leads to the lowest order approximation to the Bessel function

Jb(y) = Ab(y) cos (Φb(y))

2 1 2 2 1 ν 1 π 1/4 cos y ν + ν sin− b + (4.41) ∼ rπ (y2 ν2) − y − 2 2 − p If we simply apply the WKB theory to Eq. 4.35 and use Eq. 4.34 to ﬁnd the WKB

approximation for Jb(y), we end up with Eq. 4.41 except that the constant phase takes the value (ν + 1/2)π/2 instead of (b + 1/2)π/2. Comparing Eq. 4.40 with Eq. 4.38 shows that the phase loss φ should be

1 φ = b ν + π (4.42) − 2

Using this value of the phase loss, the WKB quantization condition becomes

2 3 π ν ν 1 ν π b n + = 1 + sin− (4.43) 4 a − a a a − 2 a r

Now we have ν everywhere except in the constant phase (the last term). This equation

can also be obtained by setting the phase Φb(a) in Eq. 4.41 equal to (n + 1/2)π to satisfy the boundary condition at y = a. We have now seen three versions of the quantization condition. The ﬁrst (Eq. 4.9) involved only the ratio b/a. Then, there was the (rejected) possibility of replacing

87 M.Sc. Thesis - Akbar Safari McMaster University - Physics and Astronomy

0.500 à à æòõ 0.200 à æ à æõ à à 0.100 ò à à à à æ æ 0.050 òõ æ æ 100 æ æ æ òõ

× 0.020 òõ òõ E

δE 0.010 õ à WKB0 ò 0.005 òõ æ WKB2 òõ 0.002 õ modiﬁed WKB0 0.001 ò modiﬁed WKB2 òõ 0 2 4 6 8 n

Figure 4.7: Approximate binding energies. Upper pair of lines are from Fig. 4.2. Lower pair result from the approximation given in Eq. 4.43 with and without the second order WKB correction. b/a everywhere by ν/a. Finally, we have Eq. 4.43 in which the last term involves b/a, while all the other terms involve ν/a. In Fig. 4.7 we compare the relative accuracy δE/E, obtained from Eq. 4.43, to that of Eq. 4.9 as used in Ref.[39]. The errors are similar for states at the bottom of the well, but our approximation improves steadily as n increases, and the correspondence principle will be satisﬁed in the classical (large n) limit. The phase loss φ in Eq. 4.42 can be used in Eq. 4.11 to obtain the modiﬁed second order WKB approximation. Both new results are shown in Fig. 4.7 Further support for Eq. 4.43 is provided by the fact that when the potential strength makes a = Nπ, one exact solution of the quantization condition is b = 1/2. 1 The Bessel function J1/2(a) is proportional to sin a, whose nodes lie at a = Nπ. The quantization condition reduces to the statement that the wave function has n nodes. These states occur near threshold.

88 M.Sc. Thesis - Akbar Safari McMaster University - Physics and Astronomy

As another illustration of the accuracy of Eq. 4.43 consider the state at the threshold. When b 0, Eq. 4.43 reads →

2 π 3 1 1 1 1 n + = 1 + sin− a 4 2a − 2a 2a s 1 1 1 + + . (4.44) ≈ − 8a2 384a4 ···

Using this as an approximation to the nodes a = an of J0(a) gives

3 1 1 an (n + )π + 3 + (4.45) ∼ 4 8an − 384an ···

For the first node one obtains a 2.4079 compared to the exact 2.4048. For the 0 ∼ second node, a 5.5204 vs. 5.5201, and for the third a 8.6538 vs. 8.6537. The 1 ∼ 2 ∼ non-WKB pieces of the GT series further improve the results. In Fig. 4.8, the r.h.s. of Eq. 4.9 vs. b and the r.h.s. of Eq. 4.43 vs. ν are drawn. For deeply bound states, there is little difference between b and ν, and Eq. 4.9 and Eq. 4.43 give similar results (see Fig. 4.7). For weakly bound states, near the threshold region where the correspondence principle applies, the difference between using b and ν is evident. When b is below 1/2, ν becomes imaginary, making a huge difference in the results. This drawing also illustrates the well-known notion that when a state first becomes bound, its energy drops rapidly as the potential becomes stronger. Then its progress slows down before adjusting to an average rate. In the improved WKB quantization condition (Eq. 4.43), b = 0 implies ν i/2. → The arc-sine is replaced by a (negative) arc-sinh and the centrifugal barrier in Fig. 4.5 turns turtle. Therefore, for states in the near threshold region (b < 1/2), there is no

89 M.Sc. Thesis - Akbar Safari McMaster University - Physics and Astronomy

1.00

0.95

0.90

Total phase 0.85

0.80

i/2 0 1 2 3 4 5 b or ν

Figure 4.8: Comparison of the quantisation condition using only b (dashed line), or a combination of b and ν, (solid line). A horizontal line at (n + 3/4)π/a intersects one of them at the value corresponding to a solution of Eq. 4.9 or Eq. 4.43. Note that the horizontal axis to the left of the origin, represents behaviour on the imaginary axis. left turning point on the real y-axis. As illustrated in Fig. 4.9, as b 0 the left → turning point moves onto the imaginary axis ν iν¯. In eﬀect, the WKB phase → integral runs from iν¯ to a point on the y axis giving a node of the Bessel function. This is possibly the simplest example of a situation where a WKB phase integral involves an imaginary turning point. The importance of complex turning points was emphasized by Balian et al. [43]. Chebotarev [44] has recently discussed the case of a double well potential, U(x) = (x2 L2)2 which has two wells separated by a central barrier. Application of WKB − to the deep-lying bound states involves four classical turning points. A very readable account of the solution may be found in the older1 editions of Park’s text book [45]. When the energy increases, the two turning points surrounding the barrier approach each other and seem to disappear when E exceeds the height of the central barrier. However this is not the case, the turning points merely move into the complex plane

90 M.Sc. Thesis - Akbar Safari McMaster University - Physics and Astronomy

Im or V (x) ν2/y2 ν = i/2 è (b = 0)

è è y ν = 0 ν a (b = 1/2) (b > 1/2)

Figure 4.9: Illustration of how the repulsive centrifugal potential ν2/y2 becomes attractive when b < 1/2 (ν iν¯). → and their contribution remains signiﬁcant. Perhaps even more surprising, if we let L 0, the ground state of the quartic oscillator is notoriously badly reproduced → by LO-WKB. The complex turning points still exist, at iE1/4, and including the ± trajectory connecting them gives a huge improvement to the ground state binding energy, as veriﬁed by Chebotarev [44].

4.6 The symmetric 1D exponential potential, even-

parity states

α x In the even potential V (x) = V e− | |, the boundary condition for odd states is II − 0 1 still Jb(y = a) = 0 and the corresponding energies are given by Eq. 4.43. For the even-parity states, the quantization condition becomes Jb0(y = a) = 0. Setting to zero

91 M.Sc. Thesis - Akbar Safari McMaster University - Physics and Astronomy

the derivative of Jb(y) in Eq. 4.17 leads to

Ab0 0 = cos Φb Φb0 sin Φb (4.46) Ab −

In LO-WKB theory, Φb0 = k(y) and we deﬁne the angle γ such that

ν2 Φb0 (y) = k(y) = 1 cos γ . (4.47) s − y2 ∼

From the WKB theory we also have (see Eq. 4.39 or 4.41)

1 1 Ab(y) = = (4.48) (y2 ν2)1/4 yk(y) − p where k(y) is the wave number in Eq. 4.35, namely

ν2 k(y) = 1 (4.49) s − y2

Then

2 A0 1 1 k0 1 ν − b = + = k2 + A 2 y k 2yk2 y2 b 1 = sin γ (4.50) 2yk2 ∼

Therefore the quantization condition in Eq. 4.46 takes the form

sin (Φb(a) + γ(a)) = 0 (4.51)

92 M.Sc. Thesis - Akbar Safari McMaster University - Physics and Astronomy

where

1 tan γ(a) = (4.52) 2ak3(a)

Signs are chosen to make γ a positive angle, which varies from π/2 at ν a (deeply ≈ 1 bound states) to tan− [1/(2a)] as ν/a 0 (weakly bound states). Note that if we → did not include the derivative k0(y) in diﬀerentiating the amplitude, tan γ would take the value 1/[2ak(a)]. (It is often claimed that one should neglect k0 compared to k in LO-WKB.)

The quantization condition for even parity states is therefore Φb(a) + γ = nπ for n = 0, 1, 2, ..., which leads to

2 1 π ν ν 1 ν π b 1 1 1 n + = 1 + sin− + tan− (4.53) 4 a − a a a − 2 a a 2akp(a) r

According to Eq. 4.52, p = 3 in the arc-tan function; we tried taking p 0, 1, 2, 3 → and found the best results using p = 2. Solving Eq. 4.53 is only a little more com- plicated than for the odd-parity states. The approximation used by FG for the even- parity states is equivalent to dropping the last term in Eq. 4.53 and using ν = b in the other (variable phase) terms. In eﬀect, they neglect the variation of the amplitude (γ = 0) in taking the derivative. For the exact bound state energies, one has to solve for nodes of the derivative

Jb0(a). From the recurrence relations of the regular Bessel functions (see Watson[42], p.17), it is seen that nodes of Jb0(a) coincide with points where Jb+1(a) = Jb 1(a). − Comparing the two functions easily leads to the desired solutions. In Fig. 4.10, we compare the relative accuracy δE/E obtained from Eq. 4.53, and

93 M.Sc. Thesis - Akbar Safari McMaster University - Physics and Astronomy

àòó 0 à à æ à à ó à à à à à -2 ò ó ò ó æ ò ó ò ó ò ó -4 ò ó

100 æ ò ó æ ò × ó -6 æ ò E δE à WKB0 æ -8 ó p = 0 æ -10 ò p = 1 æ æ p = 2 æ -12 æ 0 2 4 6 8 10 12 14 16 18 q = 2n

Figure 4.10: Relative error in the location of the even-parity bound states of the 1D exponential potential. The three lines are for three diﬀerent approximations to the derivative, as explained in text.

by following the method of FG [39] (topmost line in Fig. 4.10). The abscissa is q = 2n, the number of nodes of the complete 1D wave function after the right side is reﬂected in the origin. The even parity states with even indices are interleaved by odd parity states (not shown) with odd indices q = 2n + 1. The other three lines correspond to using p = 0, 1, 2 in Eq. 4.53. The results are best for p = 2, while they deteriorate for p = 3 (not shown, but similar to p = 1). Taking p = 0 means using a constant value of γ, which already improves the results when compared to neglecting γ altogether. That the energies are further improved by taking p = 1 shows that some state dependence of γ gives better spacing of the nodes. Further study is required to understand why p = 2 is best. Empirically, 1 the nodal spacing is improved by the additional state dependence. Taking p = 3 degrades the results because the stronger state dependence pushes the nodes past their correct positions. As noted, p = 1 is suggested by the LO-WKB lore that one should neglect k0 compared to k. The choice p = 2 might possibly be justiﬁed by

94 M.Sc. Thesis - Akbar Safari McMaster University - Physics and Astronomy

considering higher-order eﬀects.

95 Chapter 5

Summary

5.1 Final comments

In many cases, computers are able to calculate wave functions and bound state energies with acceptable accuracy. Nevertheless, WKB approximation is still useful not only for its simplicity in practical applications, but also for its conceptual nature: the way it connects quantum mechanics to classical ideas. There is a criterion (Eq. 1.12) that can be used to predict validity of the WKB wave function. But unfortunately, accuracy of the WKB quantization condition is less predictable. If we do not know the exact bound state spectrum, it is impossible to know how reliable the WKB energies are. The LO-WKB quantization condition without any modifications, gives exact bound state energies only for a few cases, which include the simple harmonic oscillator and the Morse potential. If one applies the Langer modification, then the LO-WKB quantization condition works perfectly for the 3D harmonic oscillator and the Coulomb potential. It has been shown for several systems that LO-WKB with the Langer modification is equivalent to adding

96 M.Sc. Thesis - Akbar Safari McMaster University - Physics and Astronomy

up all order terms in the WKB series. Super-symmetry helped to discover many new analytically soluble potentials. A super-symmetry inspired WKB gives exact energies for shape invariant potentials with translation. In order to use the SWKB quantization condition, one needs to know the ground state wave function. By design, SWKB gives exact ground state energies. Therefore it is no surprise that for deeply bound states, SWKB is usually more accurate than LO-WKB. In most cases the WKB series does not converge, rather it is an asymptotic series. As such, there is an optimal cut-off in how many terms should be included, to have the most accurate result. An example, taken from Bender and Orszag, is given in Table 1.1 for the quartic oscillator. Barclay showed that for SIPs, for which SWKB is exact, the WKB series converges and gives exact energies as well. In contrast to classical mechanics, in a quantum well, the wave function penetrates into the confining barriers and the phase of the wave function requires an additive contribution that takes this penetration into account. In this sense, the WKB quantization condition is the same as the Bohr-Sommerfeld quantization condition in old quantum mechanics except that one should consider this phase leakage. The leaked out phase varies from 0 to π/2, depending on the strength of the barrier. LO-WKB takes this phase to be π/4. Friedrich and Trost claimed that the LO-WKB quantization condition can give exact bound state energies, if one considers the correct reflection phase at the turning points. According to Cao and collaborators, the WKB phase integral does not give the total phase shift between turning points. By taking into account the phase shift due to scattering of subwaves in a non-constant potential, they obtained an exact formula, Eq. 3.62. According to them, the phase leaking

97 M.Sc. Thesis - Akbar Safari McMaster University - Physics and Astronomy

out into the forbidden region is always π/2, but this is reduced by the contribution from scattered subwaves. In fact, Eq. 3.62 is not a quantization condition, because the equation is satisfied at any energy E. What it does establish is precisely what reflection phase should be applied in the Friedrich and Trost scheme. In 2011, Fabre and Guéry-Odelinused the exponential potential as a soluble but not shape invariant potential, to test the accuracy of SWKB and the WKB quantization condition. Their result was inconsistent with Bohr’s correspondence principle. We resolved this problem by using an appropriate phase loss in the sense of Friedrich and Trost.

5.2 Suggestions for further research

In section 4.5 we mapped the exponential potential on to the y-axis and converted the problem into a particle trapped between a centrifugal barrier and a hard wall. We saw that near the threshold, the turning point on the y-axis takes on an imaginary value. This is not the only case that requires imaginary turning points to get more accurate results. Another example is the quartic double well potential considered by Chebotarev. Further work is desirable to understand the role of imaginary turning points in real bound state energies. In addition to bound state energies, WKB theory can also be used to calculate the tunnelling probability, approximately. These two aspects are not independent. They are related by analytic continuation. Poles of the transmission coeﬃcient in the barrier scattering problem become energy eigen-values in bound states [12, 46]. Sil et al. [47] studied the symmetric case of the Morse-Feshbach potential which they call the Eckart potential. Their aim was to show the superiority of the SWKB

98 M.Sc. Thesis - Akbar Safari McMaster University - Physics and Astronomy

method in connecting the transmission resonances of the Eckart barrier to the bound states of the Eckart well. They claim to get an improvement on WKB. However, this improvement depends on two factors. First, they write the transmission coeﬃcient as

1 t = (5.1) 1 + e2K

2K b instead of the usual WKB recipe t = e− , where K = 2m(V (x) E)/hdx¯ is a − the phase integral between the classical turning points insideR p the barrier. Eq. 5.1 was obtained by Kemble [48, 49]. The second factor leading to improved results is the SWKB value for K. Finally, they conclude that although SWKB result for t is not exact, its poles give exact bound state spectra for the Eckart well, under the inversion procedure. Application of SWKB in tunnelling is contentious and their argument is ambiguous. More eﬀort is needed to understand and justify their method.

99 Appendix A

Asymptotic series

Following Poincar´e[50], we say that F (z) has an asymptotic series

∞ A F (z) n (A.1) → zn n=0 X in some unbounded region of the complex plane if the modulus of the error incurred by truncating the series after N terms is smaller than the next omitted term, as z . More precisely, | | → ∞

N A lim zN F (z) n = 0. (A.2) z − zn | |→∞ n=0 X

This is diﬀerent from a convergent series, where one adds more and more terms to the series within a region of analyticity and demands convergence of the sum to f(z). Instead, with an asymptotic series one demands that a small number of terms gives an answer with a bounded error, providing z is large enough. | | The prototype for an asymptotic series is Stirling’s approximation to the Gamma

100 M.Sc. Thesis - Akbar Safari McMaster University - Physics and Astronomy

function:

z 0.5 z 1 1 Γ(z) √2πz − e− 1 + + (A.3) → 12z 288z2 − · · · which is asymptotic in the right half z plane for z > 2. (See Abramowitz [6] page | | 257.) Although the coeﬃcients appear to be getting smaller, this is not true. They are related to the Bernoulli numbers and soon become huge after the tenth term. In WKB theory, 1/h¯ is the large parameter and we suppose that the series

N n 1 S h¯ − is asymptotic in the classical limith ¯ 0. In Lowest Order WKB, n=0 n →

Pone keeps just S0 and S1. The quartic potential studied by Bender and Orszag (Table 1.1) gives evidence of asymptoticity, in that for levels with higher n, keeping more terms of the WKB series gives an answer that falls closer to the exact energy. On the other hand, for the ground state the result deteriorates after passing the optimal number of terms. Presumably, ifh ¯ were smaller, more terms could be included. Unfortunately, in the real world,h ¯ is ﬁxed.

101 Bibliography

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104 M.Sc. Thesis - Akbar Safari McMaster University - Physics and Astronomy

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106 M.Sc. Thesis - Akbar Safari McMaster University - Physics and Astronomy

[45] D. A. Park. “Introduction to the Quantum Theory”. McGraw-Hill, 1974.

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