Volume157,number1 PHYSICSLETTERSA 15 July 1991

TunnellingthroughtheMorsebarrier

ZafarAhmedNuclearPhysicsDivision, BhabhaAtomicResearchCentre.Bombay400085,India

Received16 July 1990; revisedmanuscriptreceived16 January1991; acceptedfor publication17 May 1991Communicatedby J.P.Vigier

TheexactandtheWKB formsofthetransmissioncoefficientof theMorsebarrierhavebeenobtained.Comparingtheresultsobtainedfrom theexactandtheWKB analysison differenttypesof potentials,wepresentapossiblecriterionto ascertaintheperformanceof theWKB method.Extractionof boundstatesof a potential from the transmissioncoefficientof theinvertedpotentialis alsosuggested.

Thetunnellingthroughone-dimensionalpotential tivity towardsthe asymptoticbehaviourof the po-barriershas,sincethe infancyof quantummechan- tentials.TheboundstatesoftheMorseoscillatorhaveics,beenplayinganimportantrole in manybranches beenretrievedfrom the transmissioncoefficient.of physics [1,2]. However,thereare not many po- The Morsebarrierpotential (fig. 1)tentialswhich admita simpleform for transmissionamplitudet(k) andtransmissioncoefficient T(k). V(x)= V0[2 exp(x/a)—exp(2x/a)] (1)Moreover,thefactthat thesimplepolesin thek-planeoft(k) yield the boundstates,metastablestates,res- offersrepulsionto an incomingparticle (x<0) and

after a finite distance,it providesto the outgoingonancesandvirtual states[3] of a potentialaddstothe motivation of obtainingt(k) for a potential.In particlean asymptoticallydivergentattraction.Be-this Letter,the transmissionamplitudeof theMorse causeof this divergentnature,finding T(k) by nu-

mericalintegrationis renderedinfeasible.We insertbarrier (invertedMorseoscillatorpotential) is ob-

the Morsebarrierpotential (1) in the Schrodingertamed.The Morseoscillatorpotentialhaslongbeenusedto investigatethe anharmonicitiesof thevibra- equationtionalspectrain molecularandnuclearphysics. h

2 d2~x) + V0 [2 exp(x/a) — exp(2x/a)] 9’(x)

Morerecently,therehasbeena revivalof interest — ~

in the quantummechanicsof the Morse oscillator[4,5], duemainlyto anextensivestudyofthegroup- =EW(x) forE> 0. (2)theoreticalpropertiesof the correspondingSchrö-dingerequation.At this juncturewhenthe boundstates,the scatteringstates,thepropagatorandthesupersymmetricpropertieshavebeenwell discussed, vit seemsnaturaltodiscusstheMorsebarrierforwhichthe transmissioncoefficienthasremainedelusivein

Theexactandthe WKB expressionsfor thetrans-the past. ~ _______

mission coefficientof the Morsebameraregiven. x -.

TheMorsebarrierandthe Eckartbarrierareusedtoinfer a sharperconditionon thevalidity of the semi-classical WKB approximation[6] andits insensi- Fig. I. TheMorsebarrierpotential.

0375-9601/91/S03.50 © 1991 — ElsevierSciencePublishersB.V. (North-Holland)

Volume157, number1 PHYSICSLET1TERSA 15 July 1991

Substituting u =pexp(x/a) in (2) andeliminating~P(x,t)= (2au)~2 exp(xa/2)Mjp,ja(u)the first derivativeof W( u), we get

d2w(u)+(fl22fl2+ ~+a2~ Xexp(—iEtfll), (6b)du2 ~2 pu u2 )w(u)=O~ (3) or

where~P(u)=(1/,.J~)w(u).Inordertobring(3)to 1(x,t)=(2au)_2exp(7a/2)exp(_u/2)u2+I~~the standard Whittaker equation [8] form, we set x

1 F1(~+ ia — ifl, 1 +2ia; u) exp ( — iEt/h).p=2ifl anda

2=—i2a2to write (6c)

d2~v(u)+(!+~+ ~—i2a2\du2 4 u u2 ~ (4) Using

1F1(a, b; ~)=exp(~)1F1(b—a, b; —~) and1F1(a, b; O)=1 [7], we observethat asx—’—~x

wherewe have introduced/J=(2~tV0a2/h2)”2and ~P(x,t)’-.~exp(x/2a)

a= (2~Ea2/h2)”2or a=ka.Thereare eight forms for the solutionsof ~v(u) xexp{i[flexp(x/a)+kx—Et/h]} , (6d)

(w( u) = Z~i = 1, ...~ 8) and their propertieshave indicatingthateqs.(6) representa waveincidentonwidely beenworkedout [8]. UsingZ

7, onetime-de- thebarrierfromthe left. Anotherchoiceof thefunc-pendentsolutionof theSchrodingerequation(4) can tion çv(u) is Z2 which leadsto a wave functionbe written as

W(x, t)= (2au)~2 exp(—xa/2)Z

2x,t)=exp(xfl/2)u~

2Z7exp(—iEt/h) , (5a)

Xexp(—iEt/fz), (7a)where u = 2ifièxp (x/a) or in Whittaker’s notation(MK,m, Wrm) or

~P(x,t)=exp(xfl/2)[2iflexp(x/a)]”2 W(x,t)=(2au)~’2exp(—xa/2)

XMjp,_j~(—u)exp(—iEtfll), (7b)X W_jp,ja(2iflexp(x/a)) exp(—iEt/Fz). (5b)

which in terms of the confluenthypergeometricIn order to find the asymptotic behaviour of the

functions can be expressed aswavefunction (Sb), we use its confluent hypergeo-metric function representation [81 and write !P(x, t) = (2au) — 1/2 exp ( — xa/2) exp ( — u/2)

W(x,t)=exp(xfl/2)u~”2exp(u/2)(—u)”2~1’~ Xuh/2~~xiFi(~_ia_ifl,1_2ia;u)

xU(1+icr+i/3,1+2ia;—u)exp(—iEt/h). Xexp(—iEl/h). (7c)(5c)

Since (7b) can be obtained from (6b) by changingUsing the limit ~ U(a, b, ~ [8] as a to —a, one can easily check that Y’(x, 1) (7) de-x—~x,we get notes a wave travelling to the left after being re-

flected from the barrier. Henceforth, for conve-~P(x,t)—~exp(—x/2a) nience,the wavefunction W(x, t) in eqs. (5), (6)

xexp{i[flexp(x/a)—flx/a—Et/1l]}, (Sd) and (7) will bedenotedby !P, ~t~and!P1.respectively.

We note that ZT=Mip,_ja(—u)=Z4,which denotesa wave moving from left to right Z~=M_~,~( — u) = Z3 andZ~= Wjp, —ia(u) = Z6 or(transmitted)throughthe barrier. Z5. Next, we use [8] Z4=exp[—i~(~—ia)]Z2,

Anotherrelevantform of thesolutionçv(u) ofour Z3 = exp[— i~t(~+ ia) ]Z~and the Wronskianrela-interestis Z1, which gives tions [Z~, Z2] = —2ia, [Z5, Z7] =exp(—,~fl)to ob-

tam the following Wronskian relations for the~P(x,t)=(2au)~’2exp(xa/2)Z

1exp(—iEt/h), wavefunctions,(6a)

[~Pr,Y~]=i, [!1rl~~,lFr]=_i,or

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Volume 157,number1 PHYSICSLETTERSA 15 July 1991

[~P~’,!P] = i. (8) smallnessof which makesthe transmissionthroughthebarrierbehavesemi-classically,thereby resultingBy virtue of a standard connection formula amongin a good performanceof the WKB method.In the

ZI=MKm(U), Z2=Mic,_m(U) andZ7 W_pc,m(U)

[8] viz limit (4-+0)h—~0thetransmissioncoefficients(l2a)and (13) degenerateinto the classicallimit, i.e.

sin(2itm) ~= ( exp[—iit(~+m)]Z1 T(E)—~O(E—V0). Further,the curvatureof the topit ~ f’(~—m±K)F(1+2m) of the barrier,

exp[—ix(~—m)]Z2\ (9) c0—~ d2V/dx2 I

+F(i+m+K)T(l2m))~ — [1+(dV/d\213/21I J Ix=O

onecanwrite thedesiredlinear connectionamong for theMorsebarrierbecomes2 V0/a

2,suggestingthat

~, ~r and !P as theWKB will work well forwiderbarrierswith lesser

(10) topcurvature.The fully repulsive Eckart potential, V(x) =

with the transmissionamplitude t(k) and the re- V0sech

2(x/a),which convergeson bothsides (x—~flectionamplitude r(k) definedby ±cc), is anotheridealizationof potential barriers

which admitsan analyticform for the transmissiont(k)= (2a)”2 exp[ — 7t(a+fl)/2]coefficient [1] expressibleas

1(~—ia+ifl)(ha) cosh(2xa)—l

X I’(1—2ia) T(E)=cosh(2xa)+cos(2,tw)’

and(14)

r(k)=exp(—ita)For very small valuesof A, or E> 4 andalso V

0> 4F(~—ia+ifl)r(l+2ia) (llb) (unlike the Morse banier),T(E) degeneratesinto

X F(~+ia+ifl)F(l—2ia) (13) which onceagain turns out to be the WKB

transmissioncoefficient of the Eckart barrier. Therespectively.The currentdensitiesarecalculatedus-coincidenceof theWKB transmissioncoefficientfor

ing (8), subsequentlyweobtaintheMorseandthe Eckartbarrier(whichbehavedif-

— exp( — 4ica) (1 2a) ferently at x—’ — no) suggestsan insensitivityof theT(k)=t*(k)t(k)

= 1+exp[2~t(fl—a)] WKB approximation towards the asymptotic behav-iour of the potentials.

andFor the parabolicbarrier,V(x)= V0[l — (x/a)

2],exp( — 2ica)+exp(2itfl) purely incidentally, the WKB andthe exact trans-

R(k)=r~(k)r(k)= exp(2ita)+exp(2irfl) ‘ mission are the same i.e., T(E)={l+

exp[2x(V0—E)/(4V04)”

2]}’ [2]. On the othera=(E/4)’1’2, fl=(V

0/4)”2. (12b) hand,the Morse andthe Eckart barrierswhich are

In an interesting approximation when exp ( — 4ita) two different and generic instances, do render ais ignoredin (12a), we get sharpercondition for the validity of the WKB ap-

proximation.Herein the smallnessof 4 ensurestheT(E)= (1 +exp{2it [ ( V~/4)1/2 (E/A)”2 1 } ) — validity of the WKB approximation,or else when

(13) V0>4 the WKB methodwill work verywell at ener-

an expressionthat canotherwisebeobtainedby us- giesE> 4.ing the WKB method which is knownto work well In the one-dimensionalbarrierpenetrationmodelfor the slowly varying potentialprofiles at asymp- [2] of theheavy-ionfusionthecalculationof thefu-totically large energies. Therefore, the WKB approx- sion crosssection is done by obtaining the trans-imation is expectedto work very well for energies missioncoefficientoftheeffectivebarrierformedbyE>A (= h

2/2~ta2).4 is an interestingparameter,the short-rangenuclear attractionand Coulomb plus

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Volume 157, number1 PHYSICSLETTERSA 15 July 1991

centrifugalrepulsion.To thisend,oneeitherusesthe the numberof boundstatesfinite, the positivesignWKB approximationfor T(E) or parametrizesthe herehasto be discardedandn hasto be restrictedfusion interactionby a potentialbarrier (V0 andC0 as n<ô— ~. Finally, by settingE~= — h

2k~/2j~inor A are the parameters)which admitsan analytic (16) we againretrievetheboundstatesof theMorseform for T(E). The typical values of the barrier oscillatorpotentialas givenby (17).heights are given by V

0(MeV) Z1Z2/(A ~ + In fact, whenV0 is changedto — V0, similar polesA ~/3), whereZ andA refertotheatomicnumberand of the WKB transmissioncoefficient (13) shouldatomic massnumberof the colliding nuclei, respec- yield the semiclassicalboundstatesconsistentwithtively. The typical values of A are given by the Bohr—Sommerfeldquantizationlaw, which in-A(MeV)~4/V0. In the light of the presentdiscus- cidentally turn out to be the exact eigenvaluesassion, we would like to emphasizethat the smallness givenby (17).of A andthe largevaluesof V0 underliethe success For a potentialthathasa minimum, if thereexistof the WKB methodin heavy-ionfusion. A fusion two real turningpoints (rootsof E= V(x)) at anybarriershouldhavea Coulombictail; however,ow- energy(positive or negative),aninfinite numberofing to the insensitivityof theWKB methodtowards boundstateswill berealized(finite otherwise).Duethetail ofa potential,its parametrizationby thepar- totheverydefinitionof T(k), i.e. T(k) = t” (k) 1(k),

abolic [2] shapeworkssatisfactorily. it shouldbe clear that half the numberof polesofThephysicalinterpretationof the singularitiesof T(k) are redundantanddo not give correctbound

t ( k) in the complexk-planefacilitatesthe natureof states.Thereforethe conditionk~>0andtheknowl-the energyeigenspectrumof the potential [3]. For edgeof thecardinalityoftheboundstatesenableoneinstance,the simplepolesof 1(k) lying on the upper to rule out thoseredundantpoles. It may be inter-half of the imaginary line in the complex k-plane estingtocheckthatthisconstitutesa generalmethod(physicalsheet)areknownto representthepossible of obtainingthe eigenvaluesof theboundstatesofboundstatesof the potential [3]. By changingfi to a potential from the transmissioncoefficient of theiô (o=~JIV0/AI) andk to ik~(notethata=ka) in invertedpotential.eq. (11a) we get the transmissionamplitudefor the Lastly, it may be notedthat the wave functionsMorseoscillator potential t0 (k) as givenby eqs.(5)—(7) suggestcorrectionstothewave

I 2 functionsof the Morse barrieras obtainedby thet0(k)=(2ia~)/ exp[—jt(ia~+iô)]

complexcoordinatemethodin ref. [4].x1(~+a~—ô)/F(l+2a~), (15) In the light of the presentresults,the Morse bar-

rier comesin the classof thebarriermodelsthatad-which hassimplepoles (ku) in the k-plane, mit asimpleanalyticform for thetransmissioncoef-k~= [ö— (n+ ~)] Ia if ô> n+ ~ ficient. The otherinterestingpotentialbarriermodels

which havelately beensolved canbe foundin refs.(physical sheet) . (16) [3,9,10].

From these, the well-known bound statesof theMorseoscillatorare recovered(E~= —h

2k~J2fL)as I would like to thankDr. Asish KumarDharaandSudhirRanjanJam for discussions.

n=0,h,2,..., [(V0/4)”

2—fl . (17) References

Alternatively, by changingk to ik~and fi to iô(V

0—*— V0) in the transmissioncoefficient T(k) of [1] S. Bjørnholm andJ.E. Lynn, Rev. Mod. Phys.52 (1980)

eq. (12a), we canlocatethe possiblesimplepolesof 725.

T(k). Theyare [2] M. Beckerman,Rep.Prog.Phys.51(1988)1047;C.Y. Wong,Phys.Rev.Lett. 31(1973)766.

k~= [d± (n+~)]/a. (18) [3]J.N. Ginocchio,Ann.Phys.(NY) 152 (1984) 203.[4] A.O. Barut, A. InomataandR. Wilson, .1. Math.Phys.28

In orderto keepk~onthephysicalsheet(k~>0)and (1987)605.

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Volume157,number1 PHYSICSLETTERSA IS July 1991

[5] Y. Alhassid,F. Gurseyand F. Iachello,Ann. Phys.(NY) [7] M. AbrainowitzandL.A. Stegun,Handbookofmathematical148 (1987)346; functions(Dover,NewYork, 1970).M. BerrondoandA. Palma,J.Phys.A 13 (1980)773; [8] U. Slater,Confluenthypergeometncfunctions(CambridgeP.C.Ojha,J.Phys.A 21(1988)875. Univ. Press,Cambridge,1960).

[6] P. Froman and P.O. Froman, JWKB approximations: [9) W.M. Zheng,J.Phys.A 16 (1983)43.contributionsto thetheory (North-Holland,Amsterdam, [10]A.O. Barut,A. InomataandR. Wilson,J.Phys.A 20 (1987)1965). 4083.

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