Thisbookprovidesa pedagogicalintroductionto theformalism,foundationsandapplicationsofquantummechanicsThisbookisintendedforuseasatextbookforbeginning graduateandadvanced undergraduatecourse,6
6 This book provides a pedagogical introduction to the formalism, foundations and applications of quantum mechanics. This book is intended for use as a textbook for beginning graduate and advanced undergraduate course
2978Scattering theory2978.1Aims andproblems of scattering theory302Integral equation for scattering problems8.2305The Born series and potentials of the Rollnik class8.33078.4Partial wave expansion3108.5The Levinson theorem3148.6Scattering from singular potentials3178.7Resonances3208.8Separable potential model3238.9Bound states in the completeness relationship3248.10Excitable potential model3278.11Unitarity of the Moller operator3288.12Quantum decay and survival amplitude3358.13Problems
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Scattering theory3108.5 The Levinson theoremHere we depart from theprevious restriction to the continuous spectrumand deal with poeiblediscrete states.For convenience,only bound stateswith E< 0 are considered, although discrete normalizable states withE>0 arepoesible (Simon 1969).Allof these satisfythehomogeneounLippmann-Schwingerequation(Ma198)（0)Consider the stationary Schrodinger equation for a central potentialinR3,berewrittenintheform (with()correspondingto Pl,e(t)in(8.4.20))CMa1+-da-V()()=0,(8.5.1)where=2,X=+V()=U(r),and the potenitial takm(30)to satisfytheconditionsI"-iv()]ldr<00,(30) (8.5.2)(Ma(92s)(Ma19s)V(1)dr<00,(36) (8.5.3)(36)which are automatically satisfied by potentials withcompact support andeverywhere finite.If the logarithmic derivative of y(r)at ro is continuousat a definite energy E <O,there is a bound state with this energy.(M198）（2）Moreover, the logarithmic derivative is monotonic with respect to theenergy (see below),and bencethecontinuity of the logarithmic derivativeat zero energy determines whether there are bound states or not. On theother hand, the logarithmic derivative for E ≥>O determines the phaseshift at zero energy S:(O),The Levinson theorem shows the link betweenG;(0) and the number nof bound states (Levinson 1953),(Ma 1985) (2)inourproofweassume,forsimplicity,thatcondition (8.5.3)isfulflledby apotential withcompact support:(2)V()=0Vr≥ro.(ha)(8.5.4)Finat, we introduceareal parameerfor whhoe can wie C)V(.)=aV(r)(aeer)(8.5.5)The idea is that, as μranges from 0 to 1, therescaled potential V(r.a)(Ma 198) (12)ranges from O to the original value V(r).The radial equation (8.5.1) ishence replaced by82-[%+-二2-v0 b, -0(8.5.6)F28
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Scattering theory3108.5TheLevinsontheorem3118.5 The Levinson theoremWenow.consider Eq. (8.5.6) for two different values, e.g.k (with solutionHere wedepart from theprevious restriction to the continuous spectrumk,)andE (withsolutionyx.),andmultiplytheequationsforye,(r)and deal with possible discrete states,For convenience, only bound statesand yh.a(r,i)by u.a(r,) andk,a(f,u),respectively.On taking thewith E< 0 are considered, although discrete normalizable states withdifference of the resulting equations, one findsE>0arepoesible(Simon 1969).Allofthesesatisfythebomogeneoun818.8()+()0.Lippmann-Schwingerequation.(8.5.7)Consider the stationary Schrodinger oquation for a central potentislSince,byregularity,both yk,and yu.haveavanishinglimit as→inR3,berewritten intheform (with()correspondingto Pl,s(t)inthe integration of Eq. (8.5.7) over the interval [0,rol yields(8.4.20))(8.rE(8.n0)Ma10+2-da[(()一()(]-V(-)(t)=0,Sh Ll(8.5.1)F2+(F-) /x(,)m()dr0,(8.5.8）应便元where=2,X=+V()=U(r),snd thepotenitial takato satisfy the conditionswhererg denotes limeo(rg-e).SinceF+kbyhypothesis, we can(Maltr)multiplybothsido (8.5.8)by(pwwhichyields(30) (8.5.2)"-iv(r)]dr<00,四号(Mat9es)1018一)[(0同器显(。一(0用尿统,(]赶今馄限PiV(r)dr<00(36) (8.5.3)manndwhich are automatically satisfiedby potentials with compact support and(8.5.9)e,(rosp)e(ro)everywhere finite.If thelogarithmic derivative of y(r)at ro is continuousat s definite energy E <O, there is a bound state with this energy.Atthisstage,on takingthelimitfboth sidesof (8.5.9)as-k,onefindsMoreover, the logarithmic derivative is monotonic with respect to theenergy (see below),and bencethecontinuity of the logarithmic derivative18a[a((at zero energy determines whether there are bound states or not. On theother hand, the logarithmic derivative for E >0 determines the phaseshift at zero energy &i(O). The Levinson theorem shows the link between--VEa(ro.m) / Ea(f,m)dr <0.(8.5.10)&r(0)and theumber ng of bound states (Levinson1953),inourproofweassume,for simplicity,thatcondition (8.5.3)isfulfledwhrthesubscripfory(r)hasbeenreplacedbywhihbyapotentialwith compactsupport:(2)is more convenient from now on.Simillarly,onefindsV(t)=0Vr≥ro.(Ma (90s)(8.5.4)11aOE[vEA(rA)OB,A()Fin, we introducealprmerfrwhoewie()V(r.n)-av(r)(Ma(9pr)(8.5.5)=(ro) ,(r,)dr >0.(8.5.11)The idea is that, as μranges from 0 to 1, therescaled potential V(r.)ranges from O to the original value V(r).The radial equation (8.5.1) iswherertmeans-limg-o(ro+e).Equations (8.5.10) and (8.5.11)provehence replaced bythat, as the energy increases, the logarithmic derivative of the radialfunction atro decreases monotonically,whereas that at rtincreases82.[%+-些2-vm m -0monotonicaliy.This expresesthe Sturm-Liouyille theorem (cf.Sturm(8.5.6)r21836).9
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312Scattering theoryThematching condition at ro for the logarithmic derivative of thefanctionis,ofcourse,(Ma l9hs)(M1911a[,A(Ax(E,M)=JrmraCryX18[VE, ()B,A(rm)](13) 4 ()-taSome limiting cases ofEq.(8.5.6)areeasilydealtwith,Forexampksfreeperticewhichcorespondstoaanishingvalueofj,onefinbeing the Bessel function of first kind and order )(8.5)wna(r.0)-VerJa(kr),(Ma(90s)(2)when E>0,and(Ma9ps)(VEX(r.0)=-AN/nrJa(ixr),(8.5.136)withkmE,ifE≤0In the interval re,oo[thepotentialvanishes,andfor poeitivealofEtwooscillatingsolutionsofEq.(8.5.6)exist,sothatthegen(Ma 19t)solution reads(98)0VE,a(n,)=/mkr[a(er)co ba(8, ) -Na(ker)sinba(k,1)], (84)where &a(k,p) is thephaseshift and N, is theNeumann function oford.Thematching condition (8.5.12) leads toa very useful formulafoc(Ma (9r)phase shift, upon using the result (8.5.14), Le.98CH(kro) [A(E,m)-会(时-]tanda(k,u)m(8.5.25MA( A(E.n)--)CIsaEquation (8.5.15)providesthekeytool forproving theLevinsontrem,jointlywitha careful analysis ofmatching couditions.Here wesrequire that the phase shift is determined with respect to the phase slinBa(k,O) for afreeparticle, where, bydefinition, onechooses &a(k,0)=With such a convention,the phase shift is determined completelyincreasesfrom0to1.(Ma (98s) (18)Ir E≤0, the only square-integrable solution of Eq.(8.5.6) is(Malay(a)Ve.AMa-nDTKTH(iKr),(8.5.10)10
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