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PHYSICAL PROBLEMS SOLVED BYTHE PHASE-INTEGRAL METHOD

NANNY FR OM AN AND PE R OL OF FR OM ANUniversity of Uppsala, Sweden

CAMBRIDGEUNIVERSITY PRESS


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Contents

Preface page xi1 Historical survey 11.1 Development from 1817 to 1926 1

1.1.1 Ca rlini's pioneering work 11.1.2 The work by Liou ville and Green 31.1.3 Jaco bi's contribution towards making Carlini's

work known 41.1.4 Sch eibner's alternative to Ca rlini's treatmentof planetary mo tion 41.1.5 Pub lications 1895-1 912 51.1.6 First traces of a connection formula 51.1.7 Pub lications 191 5-19 21 61.1.8 Both connection formulas are derived in explicit form 71.1.9 The method is rediscovered in quantum mechanics 7

1.2 Development after 1926 82 Description of the phase-integral method 122.1 Form of the wave function and the ^-equa tion 12

2.2 Phase-integral approxim ation generated from an unspecifiedbase function 13

2.3 F-m atrix method 212.3.1 Exact solution expressed in terms of the f-m atr ix 222.3.2 General relations satisfied by the F-m atrix 252.3.3 F-m atrix corresponding to the encircling of a simplezero of Q 2(z) 262.3.4 Basic estimates 262.3.5 Stokes and anti-Stokes lines 282.3.6 Sym bols facilitating the tracing of a wave function

in the complex z-plane 29


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vi Contents2.3.7 Rem oval of a boundary condition from the real z-axis

to an anti-Stokes line 302.3.8 Dependence of the F-m atrix on the lower limit of

integration in the phase integral 322.3.9 F-m atrix expressed in terms of two linearly independentsolutions of the differential equation 33

2.4 F-m atrix connecting points on opposite sides of a well-isolatedturning point, and expressions for the wave functionin these regions 35

2.4.1 Symm etry relations and estimates of the F-m atrixelements 36

2.4.2 Parameterization of the matrix F (x i, xi) 382.4.2.1 Changes of a, /3 and y when x\ moves in theclassically forbidden region 402.4.2.2 Changes of a, f5 and y when x-i moves in the

classically allowed region 412.4.2.3 Lim iting values of a, fi and y 42

2.4.3 Wave function on opposite sides of a well-isolatedturning point 43

2.4.4 Pow er and limitation of the parameterization method 452.5 Phase-integral connection formulas for a real, smooth,

single-hump potential barrier 462.5.1 Exact expressions for the wave function on both sides

of the barrier 482.5.2 Phase-integral connection formulas for a real barrier 50

2.5.2.1 Wave function given as an outgoing waveto the left of the barrier 53

2.5.2.2 Wave function given as a standing waveto the left of the barrier 543 Problems with solutions 59

3.1 Base function for the radial Schrodinger equation whenthe physical potential has at the most a Coulombsingularity at the origin 59

3.2 Base function and wave function close to the origin whenthe physical potential is repulsive and strongly singularat the origin ' 613.3 Reflectionless potential 62

3.4 Stokes and anti-Stokes lines 633.5 Properties of the phase-integral approximation along

an anti-Stokes line 66


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Contents vii3.6 Properties of the phase-integral approximation along a path on

which the absolute value of exp[zu;(z)] is monotonicin the strict sense, in particular along a Stokes line 66

3.7 Determ ination of the Stokes constants associated with the threeanti-Stokes lines that emerge from a well isolated, simpletransition zero 69

3.8 Connection formula for tracing a phase-integral wave functionfrom a Stokes line emerging from a simple transition zero / tothe anti-Stokes line emerging from t in the opposite direction 72

3.9 Connection formula for tracing a phase-integral wave functionfrom an anti-Stokes line emerging from a simple transitionzero t to the Stokes line emerging from t in the oppositedirection 733.10 Connection formula for tracing a phase-integral wave functionfrom a classically forbidden to a classically allowed region 74

3.11 One-directional nature of the'conn ection formula for tracinga phase-integral wave function from a classically forbiddento a classically allowed region 77

3.12 Connection formulas for tracing a phase-integral wave functionfrom a classically allowed to a classically forbidden region 79

3.13 One-directional nature of the connec tion formulas for tracinga phase-integral wave function from a classically allowedto a classically forbidden region 81

3.14 Value at the turning poin t of the wave function associatedwith the connection formula for tracing a phase-integral wavefunction from the classically forbidden to the classicallyallowed region 83

3.15 Value at the turning point of the wave function associated with aconnection formula for tracing the phase-integral wave functionfrom the classically allowed to the classically forbidden region 87

3.16 Illustration of the accuracy of the approxim ate formulas forthe value of the wave function at a turning point 88

3.17 Expressions for the a-coefficients associated withthe Airy functions 91

3.18 Exp ressions for the param eters a, fi and y whenQ\z) = R(Z) = -z ' 963.19 Solu tions of the Airy differential equation that at a fixed pointon one side of the turning point are represented by a single,pure phase-integral function, and their representation onthe other side of the turning point 98


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viii Contents3.20 Connection formulas and their one-directional nature

dem onstrated for the Airy differential equation 1023.21 Depend ence of the phase of the wave function in a classically

allowed region on the value of the logarithmic derivativeof the wave function at a fixed point x\ in an adjacentclassically forbidden region 105

3.22 Phase of the wave function in the classically allowed regionsadjacent to a real, symmetric potential barrier, whenthe logarithmic derivative of the wave function is givenat the centre of the barrie r 107

3.23 Eigenvalue problem for a quantal particle in a broad, symmetricpotential well between two symmetric potential barriers ofequal shape, with boundary cond itions imposed in themiddle of each barrier 115

3.24 Dependence of the phase of the wave function in a classicallyallowed region on the position of the point x\ in an adjacentclassically forbidden region where the boundary conditionijf(xi) = 0 is imposed 117

3.25 Phase-shift formula 1213.26 Distan ce betw een near-lying energy levels in different types

of physical systems, expressed either in terms of thefrequency of classical oscillations in a potential w ellor in terms of the derivative of the energy with respec t toa quantum number 123

3.27 Arbitrary-order quantization condition for a particle ina single-well potential, derived on the assumptionthat the classically allowed region is broad enoughto allow the use of a conn ection formula 1253.28 Arb itrary-order quantization condition for a particle ina single-well potential, derived without the assumptionthat the classically allowed region is broad 127

3.29 Displacement of the energy levels due to compressionof an atom (simple treatment) 130

3.30 Displacement of the energy levels due to compressionof an atom (alternative treatment) 133

3.31 Quantization cond ition for a paVticle in a smooth potential we ll,limited on one side by an impenetrable wall and on the otherside by a smooth, infinitely thick poten tial barrier, and inparticular for a particle in a uniform gravitational fieldlimited from below by an impenetrable plane surface 137


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x Contents2A1 Energy levels of a particle in a smooth, symmetric,

double-well potential 1903.48 Determination of the quasi-stationary energy levels

of a particle in a radial potential with a thicksingle-hump barrier 1923.49 Transm ission coefficient for a particle penetrating a realsingle-hump potential barrier 197

3.50 Transmission coefficient for a particle penetrating a real,symmetric, superdense double-hum p potential barrier 200

References 205Author index 209Subject index 211

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