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 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation,broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionor informationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purposeofbeingenteredandexecutedonacomputersystem,forexclusiveusebythepurchaserofthe work. Duplication of this publication or parts thereof is permitted only under the provisions of theCopyrightLawofthePublisher’slocation,initscurrentversion,andpermissionforusemustalways beobtainedfromSpringer.PermissionsforusemaybeobtainedthroughRightsLinkattheCopyright ClearanceCenter.ViolationsareliabletoprosecutionundertherespectiveCopyrightLaw. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexempt fromtherelevantprotectivelawsandregulationsandthereforefreeforgeneraluse. While the advice and information in this book are believed to be true and accurate at the date of publication,neithertheauthorsnortheeditorsnorthepublishercanacceptanylegalresponsibilityfor anyerrorsoromissionsthatmaybemade.Thepublishermakesnowarranty,expressorimplied,with respecttothematerialcontainedherein. Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) 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