WKB Approximation in Atomic Physics

179 Pages·2012·2.4 MB·English

by  Boris Mikhailovich Karnakov

WKB Approximation in Atomic Physics
Boris Mikhailovich Karnakov
Vladimir Pavlovich Krainov
WKB Approximation
in Atomic Physics
123
BorisMikhailovich Karnakov Vladimir Pavlovich Krainov
MoscowEngineering PhysicalInstitute MoscowInstituteof Physics
Moscow and Technology
Russia Dolgoprudny
Russia
ISBN 978-3-642-31557-2 ISBN 978-3-642-31558-9 (eBook)
DOI 10.1007/978-3-642-31558-9
SpringerHeidelbergNewYorkDordrechtLondon
LibraryofCongressControlNumber:2012944377
(cid:2)Springer-VerlagBerlinHeidelberg2013
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Preface
This book has evolved from lectures of authors devoted to applications of the
WKB(orquasi-classical)approximationandofthemethodof1/N—expansionfor
solvingofvariousproblemsinatomicandnuclearphysics.Theintentofthisbook
is to help students and investigators in this field to extend and to make contem-
porary their knowledge of these important calculational methods of quantum
mechanics. It is envisioned that both advanced students and active researchers in
this field will find it useful. Much material is contained herein that is not to be
found elsewhere. WKB approximation, while constituting a fundamental area in
atomic physics, has not been the focus of many books.
Anovelmethodhasbeenadoptedforthepresentationofthesubjectmatter.The
material is presented as a succession of problems. These problems are stated
succinctly, solved using basic principles of quantum mechanics, and then the
results are discussed in detail. It has been our experience that a possible initial
discomfort with this unfamiliar structure gives way to an appreciation of its
important advantages: First, different aspects of a single topic are treated in sep-
arate problems, which makes possible a progressive deepening of the under-
standingofthesubject.Second,byconsideringlimitedcasesofageneraltopic,it
is possible to simplify the underlying mathematics so as to highlight the funda-
mental concepts. Third, although some of these problems build progressively on
the results of those that precede it, there is also the possibility to enter into the
subject matter at any point. Fourth, a very important feature is that the problem/
solution format reinforces in the reader the ability to analyze the content of a
physical problem and to apply the suitable mathematical and physical tools to
solve it. Finally, the qualitative discussion of the outcome of the solution aids in
the development of physical intuition.
It is presumed that the reader is already acquainted with the basics and mathe-
matical apparatus of quantum mechanics in the frames of standard university
coursesdescribed,forexample,inthewell-knownbookofLandauandLifshitz[1].
v
vi Preface
Detailedinvestigationofthechapter‘‘Quasi-ClassicalApproximation’’inthisbook
is desirable.
The authors wish to thank Professors V. D. Mur and V. S. Popov for many
valuable discussions.
Contents
1 WKB Approximation in Quantum Mechanics. . . . . . . . . . . . . . . . 1
1.1 One-Dimensional Motion. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 WKB-Approximation for a Particle in Central Field.
Langer Transformation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2 1/N-Expansion in Quantum Mechanics. . . . . . . . . . . . . . . . . . . . . 31
2.1 1/N Expansion for Energy Levels of Binding States . . . . . . . . . 35
2.2 Wave Functions of 1/n Expansion. . . . . . . . . . . . . . . . . . . . . . 49
Reference. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3 Rydberg States of Atomic Systems . . . . . . . . . . . . . . . . . . . . . . . . 57
3.1 Unperturbed Rydberg States of Atoms. . . . . . . . . . . . . . . . . . . 59
3.2 Interaction Between a Rydberg Electron
and an Electromagnetic Radiation . . . . . . . . . . . . . . . . . . . . . . 75
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
4 Penetrability of Potential Barriers and Quasi-Stationary States. . . 105
4.1 Quasi-Stationary States of One-Dimensional Systems . . . . . . . . 107
4.2 Quasi-Stationary States and Above-Barrier Reflection . . . . . . . . 115
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
5 Transitions and Ionization in Quantum Systems . . . . . . . . . . . . . . 155
5.1 Adiabatic Transitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
5.2 Ionization of Quantum Systems. . . . . . . . . . . . . . . . . . . . . . . . 161
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
vii
Chapter 1
WKB Approximation in Quantum Mechanics
Wentzel–Kramers–Brillouin(WKB)orquasi-classicalapproximationisoneofthe
most important approximation methods in quantum mechanics and mathematical
physics.Thisisexplainedbyseveralfactors.
Unliketheperturbationtheorythisapproximationdoesnotrequiresmallnessof
theperturbationpotential.Therefore,WKBapproximationallowstoinvestigateboth
qualitativeandquantitativepropertiesofquantum–mechanicalsystemswithstrong
interactionbetweenparticles.
WKBapproximationgives,asarule,sufficientlysimple,obviousfromthephys-
icalpointofview,analyticalexpressionsforinvestigatedphysicalquantities(energy
levels,wavefunctions,transitionprobabilities,andsoon).Thisisausefuladditional
approachtothenumericalcalculationsusingmoderncomputers.
ThoughformallyWKBapproximationisvalidforhigh-excitedstatesofquantum
systems,usuallyitsapplicabilityisextendeduptothevaluesofquantumnumbers
n ∼1(includingeventhegroundstate)inthecaseofsmoothpotentials.
Finally, WKB approximation allows to find the so-called correspondence
principlebetweenquantum–mechanicalandclassicalquantitiesandrelations.
1.1 One-DimensionalMotion
WKBWaveFunctionsofStationaryStates
LetusrememberthegeneralstatementsofWKBapproximation(seeRefs.[1–3]).
Stationary one-dimensional Schrödinger equation for the wave function is of the
form:
(cid:2)2
− ψ(cid:3)(cid:3)(x)+U(x)ψ (x)= Eψ (x), (1.1)
2m E E E
or
B.M.KarnakovandV.P.Krainov,WKBApproximationinAtomicPhysics, 1
DOI:10.1007/978-3-642-31558-9_1,©Springer-VerlagBerlinHeidelberg2013
2 1 WKB-ApproximationinQuantumMechanics
1
ψ(cid:3)(cid:3)(x)+ p2(x)ψ (x)=0, (1.2)
E (cid:2)2 E
wherethelinearmomentumofaparticleis
(cid:2)
p(x)= 2m(E −U(x)).
Here, E isthetotalenergy,andU(x)isthepotentialenergy,misthemassofthe
particle. Let us introduce the quantity S (it is similar to classical action), which is
connectedwiththewavefunctionbyrelation:
(cid:3) (cid:4)
i
ψ (x)≡exp S(x,E) .
E (cid:2)
Weobtainfrom(1.2)thenonlineardifferentialequationforthisquantity:
(cid:5) (cid:6)
S(cid:3)(x) 2− p2(x)−i(cid:2)S(cid:3)(cid:3)(x)=0.
The solution of this equation is presented as a series in powers of the Planck
constant(cid:2)(pureclassicalcasecorrespondstothelimit(cid:2)→0):
S = S +(cid:2)S +(cid:2)2S +··· (1.3)
0 1 2
Thefirsttwotermsofthisexpansionareofthesimpleform
(cid:7)x
i
S (x)=± p(x(cid:3))dx(cid:3); S (x)= lnp(x). (1.4)
0 1
2
Hence,inWKBapproximation,twolinearlyindependentsolutionsofEq.(1.1)are
⎡ ⎤
(cid:7)x
ψ±(x)= √ 1 exp⎣±i p(x(cid:3))dx(cid:3)⎦. (1.5)
E p(x) (cid:2)
x0
Intheclassicallyforbiddenregionthemomentum p(x)isanimaginaryquantity
sothatoneofthesolutions(1.5)increasesexponentially,whiletheotherdecreases
exponentially.
Thesolutions(1.5)areapplicableundertheWKBcondition
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
(cid:12)(cid:12)(cid:12)dλ(cid:12)(cid:12)(cid:12)=(cid:12)(cid:12)(cid:12) d (cid:2) (cid:12)(cid:12)(cid:12)=(cid:12)(cid:12)(cid:12)(cid:2)p(cid:3)(x)(cid:12)(cid:12)(cid:12)=m(cid:2)(cid:12)(cid:12)(cid:12)U(cid:3)(x)(cid:12)(cid:12)(cid:12)(cid:7)1. (1.6)
dx dx p(x) p2(x) p3(x)
ThegeneralsolutionoftheSchrödingerequation(1.1)isthesuperpositionoftwo
WKBsolutions(1.5):
1.1 One-DimensionalMotion 3
Fig.1.1 Thesimplestpoten- (a) (b)
tialenergies
U(x)
U(x)
E
E
x
x
a b
ψ (x)=C ψ+(x)+C ψ−(x). (1.7)
E 1 E 2 E
However, usually the WKB condition (1.6) is violated in some regions of the
variablex,forexample,neartheclassicalturningpoints.Inthiscasetheproblemof
matchingofWKBsolutionsonbothsidesofsuchregionsshouldbesolved.Thus,
wemustconnectthecoefficientsC ,C inEq.(1.7)forthedifferentWKBregions
1 2
oftheparticlemotiononbothsidesoftheturningpoint.
MatchingofWKBSolutions.KramersRelations
WecanusematchingconditionsofWKBsolutionsforthesimplestpotentialenergies
depicted in Fig.1.1a, b. The linear Taylor expansion of these potential energies is
assumedtobevalidinthevicinityofbothclassicalturningpointsa,b:
U(x)≈U(x )+U(cid:3)(x )(x −x ). (1.8)
0 0 0
Here, x = a (Fig.1.1a),orb (Fig.1.1b).Weassumealsothatthisexpansionis
0
validuptosuchvaluesofx onbothsidesfromx ,atwhichWKBcondition(1.6)is
0
applicable.
If we start from the exponentially decreasing solution at x → −∞ in the left
classicallyforbiddenregioninFig.1.1a
⎡ ⎤
(cid:7)a
(cid:12) (cid:12)
ψ (x)= √C exp⎣−1 (cid:12)p(x(cid:3))(cid:12)dx(cid:3)⎦, x <a, (1.9)
E 2 |p(x)| (cid:2)
x
thenthewavefunctioninclassicalregionoftheparticle’smotionshouldbewritten
intheform
⎡ ⎤
(cid:7)x
ψ (x)= √C sin⎣1 p(x(cid:3))dx(cid:3)+ π⎦, x >a. (1.10)
E p(x) (cid:2) 4
a
4 1 WKB-ApproximationinQuantumMechanics
Fig.1.2 Thepotentialwell
U(x)
E
n
x
a b
Let us underline that both the above functions describe the same solution of
Schrödinger equation (1.1), but for different values of the independent variable x
(andnottooclosetotheclassicalturningpointx = a ).Equations(1.9)and(1.10)
arecalledKramersrelations[4].
Analogously,ifwestartfromexponentiallydecreasingsolutionat x → +∞in
therightclassicallyforbiddenregioninFig.1.1b
⎡ ⎤
(cid:7)x
(cid:3) (cid:12) (cid:12)
ψ (x)= √C exp⎣−1 (cid:12)p(x(cid:3))(cid:12)dx(cid:3)⎦, x >b, (1.11)
E 2 |p(x)| (cid:2)
b
thenthewavefunctioninclassicalregionoftheparticle’smotioncanbewrittenin
theform ⎡ ⎤
(cid:7)b
(cid:3)
ψ (x)= √C sin⎣1 p(x(cid:3))dx(cid:3)+ π⎦, x <b. (1.12)
E p(x) (cid:2) 4
x
Bohr-SommerfeldQuantizationRule
NowwesuggestthattheparticleisfoundinthepotentialwelldepictedinFig.1.2.
Thenthewavefunctions(1.10)and(1.12)correspondtothesamebindingstate
oftheparticlewiththeenergy E.Hence,theymustcoincidewitheachother.Thus,
thesumofphasesofsinefunctionsshouldbeequalto(n+1)πwherenistheinteger
number(thisisso-calledquantumnumber n;itcoincideswiththenumberofzeros
inthewavefunctionanditdeterminestheordinalnumberofthelevel).Thisresults
intheso-calledwell-knownBohr-Sommerfeldquantizationrule:
(cid:7)b (cid:7)b(cid:2) (cid:13) (cid:14)
1 1 1
p (x)dx ≡ 2m(E −U(x))dx =π n+ ; (1.13)
(cid:2) n (cid:2) n 2
a a
herewehave