Exactly solvable potentials of nonrelativistic quantum mechanics are known to
be shape invariant. For these potentials, eigenvalues and eigenvectors can be
derived using well known methods of supersymmetric quantum mechanics. The
majority of these potentials have also been shown to possess a potential
algebra, and hence are also solvable by group theoretical techniques. In this
paper, for a subset of solvable problems, we establish a connection between the
two methods and show that they are indeed equivalent.Comment: Latex File, 10 pages, One figure available on request. Appeared in
  the proceedings of the workshop on "Supersymmetric Quantum Mechanics and
  Integrable Models" held at University of Illinois, June 12-14, 1997; Ed. H.
  Aratyn et a

A.O. Barut

D. Barclay

F. Cooper

G. A. Natanzon

J. Wu

L. E. Gendenshtein

M.J. Englefield

R.D. Tangerman

V. P. Spiridonov

Y. Alhassid

English

arXiv

Exactly solvable potentials of nonrelativistic quantum mechanics are known to be shape invariant. For these potentials, eigenvalues and eigenvectors can be derived using well known methods of supersymmetric quantum mechanics. The majority of these potentials have also been shown to possess a potential algebra, and hence are also solvable by group theoretical techniques. In this paper, for a subset of solvable problems, we establish a connection between the two methods and show that they are indeed equivalent

Gangopadhyaya, Asim

Mallow, Jeffrey

Sukhatme, Uday P.

Loyola eCommons

arXiv:quant-ph/9805042v1  14 May 1998Novem ber21,2016Shape Invariance and Its C onnection to PotentialA lgebraAsim Gangopadhyayaa;1,Jery V.M allow a;2 and Uday P.Sukhatm eb;3.a) Departm entofPhysics,Loyola University Chicago,Chicago,IL 60626;b) Departm entofPhysics,University ofIllinoisatChicago,(m /c 273)845 W .TaylorStreet,Chicago,IL 60607-7059.A bstractExactly solvablepotentialsofnonrelativisticquantum m echanicsareknown to beshapeinvariant.Forthesepotentials,eigenvaluesand eigenvectorscan bederived usingwellknownm ethodsofsupersym m etricquantum m echanics.Them ajority ofthesepotentialshavealsobeen shown to possessa potentialalgebra,and hencearealso solvableby group theoreticaltechniques. In this paper,for a subset ofsolvable problem s,we establish a connectionbetween the two m ethodsand show thatthey are indeed equivalent.1e-m ail:agangop@ luc.edu,asim @ uic.edu2e-m ail:jm allow@ luc.edu3e-m ail:sukhatm e@ uic.eduI.IntroductionItiswellknown thatm ostoftheexactly solvable potentialsofnonrelativistic quantumm echanics fallunderthe Natanzon class ([1]) where the Schrodinger equation reduces ei-ther to the hypergeom etric or the conuent hypergeom etric dierentialequations. A fewexceptions are known ([2,3]),where solvable potentials are given asa series,and can notbewritten in closed form in general.W ith theexception ofG innochio potential,allexactlysolvablepotentialsareknown to beshapeinvariant([4,5]);i.e.theirsupersym m etricpart-ners are ofthe sam e shape,and their spectra can be determ ined entirely by an algebraicprocedure,akin to thatofthe one dim ensionalharm onic oscillator,withouteverreferringto the underlying dierentialequations([6]).Severalofthese exactly solvable system s are also known to possess what is generallyreferred to asa potentialalgebra ([7,8,9,10,12,11]). The Ham iltonian ofthese system scan bewritten astheCasim irofan underlying SO (2,1)algebra,and allthequantum statesofthesesystem scan bedeterm ined by group theoreticalm ethods.Thus,there appear to be two seem ingly independentalgebraic m ethods for obtainingthe com plete spectrum ofthese Ham iltonians. In this paper,we analyze this ostensiblecoincidence. For a category ofsolvable potentials,we nd that these two approaches areindeed related.In the next section,we briey describe supersym m etric quantum m echanics (SUSY-Q M ),and discusshow theconstraintofshapeinvariancesucesto determ inethespectrumofa shape invariant potential(SIP).In sec. 3,we judiciously construct som e algebraicoperatorsand show thatthe shape invariance constraintcan be expressed asan algebraiccondition.Forasetofshapeinvariantpotentials,wendthattheshapeinvarianceconditionleads to the presence ofa SO (2,1) potentialalgebra,and we thus establish a connectionbetween the two algebraic m ethods.In sec. 4,forcom pleteness,we provide a briefreviewofSO (2,1)representation theory.In sec.5,we derive the spectrum ofa classofpotentialsand explicitly show thatboth m ethodsindeed give identicalspectrum .1II.SU SY -Q M and Shape InvarianceA quantum m echanicalsystem specied by a potentialV  (x)can alternatively be de-scribed by itsground state wavefunction  (  )0 . Apartfrom a constant(chosen suitably tom ake the ground state energy zero),itfollowsfrom the Schrodingerequation thatthe po-tentialcan be written asV  (x)= 000 0,where prim e denotesdierentiation with respectto x. In SUSY-Q M ,it is custom ary to express the system in term s ofthe superpotentialW (x)=   00 0ratherthan thepotential,and theground statewavefunction isthen givenby  0  exp Rxx0W (x)dx,where x0 is an arbitrarily chosen reference point. W e areusing unitswith h and 2m = 1.TheHam iltonian H   can now bewritten asH   =  d2dx2+ V  (x)!=  d2dx2+ W 2(x) dW (x)dx!: (1)However,asweshallsee,thereisanotherHam iltonian H + with potentialV+ (x)=W 2(x)+dW (x)dx,thatisalm ostiso-spectralwith the originalpotentialV  (x).In particular,the eigenvaluesE +n ofH + (x)satisfy E+n = E n+ 1,where E n are eigenvaluesofH   (x)and n = 0;1;2;   ,i.e. exceptthe ground state allotherstates ofH   are in one-to-one correspondence withstatesofH + .ThepotentialsV  (x)and V+ (x)are known assupersym m etricpartners.In analogywith theharm onicoscillator,wenow denetwooperators:A ddx+ W (x),and and its Herm itian conjugate A +   ddx+ W (x). Ham iltonians H   and its super-partnerH + aregiven by operatorsA+ A and AA + respectively.Now weshallexplicitly establish theiso-spectralrelationship between statesofH + andH   . Letusdenote the eigenfunctionsofH  thatcorrespond to eigenvalues En ,by  ( )n .Forn = 1;2;   ,H +A (  )n= AA +A (  )n= AA+A (  )n= AH   (  )n= E  nA (  )n: (2)Hence, excepting the ground state which obeys A (  )0= 0, for any state  (  )n of H  there exists a state A (  )n ofH + with exactly the sam e energy,i.e. E+n  1 = E n ,wheren = 1;2;   , i.e. A (  )n /  (+ )n  1. Conversely, one also has A+  (+ )n /  (  )n+ 1. Thus,iftheeigenvaluesand theeigenfunctionsofH   wereknown,one would autom atically obtain2the eigenvalues and the eigenfunctions ofH + ,which is in generala com pletely dierentHam iltonian.Now,let us consider the specialcase where V  (x) is a SIP.This im plies that V  (x)and V+ (x) have the sam e functional form ; they only dier in values of other discreteparam eters and possibly an additive constant. To be explicit,let us assum e that in ad-dition to the continuous variable x, the potentialV  (x) also depends upon a constantparam eter a0; i.e., V   V  (x;a0). The ground state of the system of H   is givenby  0(x;a0)  exp Rxx0W (x;a0)dx:Now, for a shape invariant V  (x;a0), one has,V+ (x;a0) = V  (x;a1)+ R(a0) ;where R(a0) is the additive constant m entioned above.Since potentials V+ (x;a0)and V  (x;a1)dieronly by R(a0),their com m on ground stateis given by  0(x;a1)  exp Rxx0W (x;a1)dx. Now using SUSY-Q M algebra,the rstexcited state ofH   (x;a0) is given by A+ (x;a0) (  )0 (x;a1). Its energy is E(  )1 ,which isequalto E(+ )0.ButsinceE(  )0= 0,E(+ )0m ustbeR(a0).Continuing up theladderofseriesofpotentials V  (x;ai),we can obtain the entire spectrum ofH   by algebraic m ethodsofSUSY-Q M .Theeigenvaluesare given byE(  )0= 0; and E (  )n =n  1Xk= 0R(ak) forn > 0;and then-th eigenstate isgiven by (  )n+ 1(x;a0) A+ (a0)A+ (a1)   A+ (an  1) (  )0(x;an  1):(Toavoid notationalcom plexity,wehavesuppressed thex-dependenceofoperatorsA(x;a0)and A + (x;a0).)III.Shape Invariance and PotentialA lgebraLetusconsiderthespecialcaseofapotentialV  (x;a0)with an additiveshapeinvariance;i.e. V+ (x;a0)= V  (x;a1)+ R(a0),where an = an  1 +  = a0 + n,where  isa constant.M ostSIP’sfallinto thiscategory.Forthe superpotentialW (x;am ) W (x;m ),the shapeinvariance condition im pliesW2(x;m )+ W 0(x;m )= W 2(x;m + 1)  W 0(x;m + 1)+ R(m ) (3)As described in the last section,this constraint suces to determ ine the entire spectrum3ofthe potentialV  (x;m ). In thissection,we shallexplore the possible connection ofthism ethod with thepotentialalgebra discussed by severalauthors([7,8,9,10,12,11]).Since for a SIP,the param eter m is changed by a constant am ount each tim e as onegoes from the potentialV  (x;m ) to its superpartner,it is naturalto ask whether such atask can beform ally accom plished by theaction ofa ladder-type operator.W ith thatin m ind,werstdenean operatorJ3 =   i@@,analogoustothez-com ponentofthe angularm om entum operator. Itactsupon functionsin the space described by twocoordinatesx and ,and itseigenvaluesm play the role ofthe param eterofthe potential.W e also denetwo m ore operators,J  and itsHerm itian conjugate J+ byJ = e i@@x  Wx;  i@@12: (4)Thefactorse i in J ensurethatthey indeed operateasladderoperatorsforthequantumnum ber m . O perators J are basically ofthe sam e form as the A  operators describedearlierin sec.2,exceptthattheparam eterm ofthesuperpotentialisreplaced by operatorsJ3 12.W ith explicitcom putation we ndJ3;J=  J ; (5)and hence operators J change the eigenvalues of the J3 operator by unity, sim ilar tothe ladderoperatorsofangularm om entum (SU (2)). Now letusdeterm ine the rem ainingcom m utator[J+ ;J  ].TheproductJ+ J  isgiven byJ+J  = ei@@x  Wx;J3 +12e  i @@x  Wx;J3  12=" @2@x2+ W 2x;J3  12  W 0x;J3  12#(6)Sim ilarly,J J+ =" @2@x2+ W 2x;J3 +12+ W 0x;J3 +12#: (7)Hence the com m utatorofoperatorsJ+ and J  isgiven byJ+;J =" @2@x2+ W 2x;J3  12  W 0x;J3  12# " @2@x2+ W 2x;J3 +12+ W 0x;J3 +12#=   RJ3 +12; (8)4wherewehaveused theconstraintofshapeinvariance,i.e.V  (x;J3  12)  V+ (x;J3 +12)=  R(J3 +12).Thus,we see thatShapeInvariance enablesusto close the algebra ofJ3 andJ toJ3;J=  J ;J+;J =   RJ3 +12: (9)Now,ifthefunction R(J3)werelinearin J3,thealgebra ofeq.(9)would reduceto thatofaSO (3)orSO (2,1).SeveralSIP’sareofthistype,am ongthem aretheM orse,theRosen-M orse and the Poschl-Teller Iand IIpotentials. For these potentials,RJ3 +12= 2 J3,and eq.(9)reducestoan SO (2,1)algebra and thusestablishestheconnection between shapeinvariance and potentialalgebra. Even though there is m uch sim ilarity between SO (2,1)and SO (3) algebras,there are som e im portant dierences between their representations.Hence,forcom pleteness,wewillbriey describetheunitary representationsofSO (2,1)andreferthe readerto [13]fora m oredetailed presentation.IV .U nitary R epresentations ofSO (2,1) A lgebraIn thissection,weshallbrieyreview theSO (2,1)algebraand itsunitaryrepresentations(unireps). This description is prim arily based upon a review article by B.G .Adam s,J.Cizeka and J.Paldus(1987).ThegeneratorsoftheSO (2,1)algebra satisfyJ3;J=  J ;[J+ ;J  ]=   2J3 ; (10)whereJ arerelated to theirCartessian counterpartsby J = J1  J2.(Forthefam iliarSO (3)case,one has[J+ ;J  ]= + 2J3).TheCasim iroftheSO (2,1)algebra isJ2 =   J+ J  + J23   J3 =   J J+ + J23 + J3 : (11)In analogy to therepresentation ofangularm om entum algebra,onecan choose J2 andone ofthe Ji’sastwo com m uting observables. However,unlike the SO (3)case,each suchchoice ofa pairgeneratesa dierentsetofinequivalentrepresentations.Forbound states,wechoosethefam iliarrepresentation spaceofstatesjj;m ion which theoperatorsfJ2;J3gare diagonal: J2jj;m i= j(j+ 1)jj;m i,J3jj;m i= m jj;m i. O peratorsJ actupon jj;m istatesasladderoperators:J jj;m i= [  (j m )(j m + 1)]12 jj;m + 1i.Since the quan-tum num berm increasesin unitstepsfora given j,the generalvalue form isofthe form5m 0 + n,where n is an integer and m 0 isa realnum ber. There is also another constrainton the quantum num bersm and j. In unitary representations,J+ and J  are Herm itianconjugates ofeach other,and J+ J  and J  J+ are therefore positive operators. Thisim -plies [  (j m )(j m + 1)] =  j+ 122 m + 122 0. These constraints canbeillustrated on a two dim ensionalplanardiagram [Fig.1]depicting theallowed valuesofm and j.O nly theopen triangularareasDFB,HEG and thesquareAEFC aretheallowedregions.Thevaluesofjm jareno longerbounded by j,and dependingon them 0 (thestart-ing value ofm ),representationsm ultipletsare eithersem i-innite (bounded from below orabove)orcom pletely unbounded.Thusthereisno nite(nontrivial)unitary representationofSO (2,1).In general,thereare fourclassesofunireps.D + (j)Bounded from below(j;m 0)lie alongthe segm entAB8><>:m =   j+ n; n = 0;1;2;   ;j< 0;D   (j)Bounded from above(j;m 0)lie alongthe segm entAG8><>:m = j+ n; n = 0;  1;  2;   ;j< 0;D s(j;m 0)(j;m 0)lie inthe squarearea8>>>><>>>>:m = m 0 + n; n = 0; 1; 2;   ;j(j+ 1)< (jm 0j  1)jm 0j;  12< m 0 <  12;D p(j;m 0)Unbounded andcom plex j8>>>><>>>>:m = m 0 + n; n = 0; 1; 2;   ;  12< m 0 <  12;j=   12+ i:Here we willbe interested in representations that are bounded from either below orabove.Such representationsfallin triangularareasDFB and HEG .For the D + representation,the starting value ofm can be anywhere on the darkenedpartofthelineAB;otherallowed valuesofm arethen obtained by theaction oftheladderoperatorJ+ .O wing to the equivalence ofD + (j)and D + (  j  1),they correspond to the6sam evalueofj(j+ 1).O necould haveequivalently started anywhereon thesegm entCD aswelland used D + (  j  1).Both areequivalentand each isunique.Sim ilarly,forcom pleteD   (j)(D   (  j  1))representation,onestartsfrom AG (G H)and generatesallotherstatesby the action ofthe J  operator.V .Exam pleAs a concrete exam ple,we willexam ine the Scarfpotentialwhich can be related tothe Poschl-Teller IIpotentialby a redenition ofthe independentvariable. W e willshowthattheshapeinvarianceoftheScarfpotentialautom atically leadsto itspotentialalgebra:SO (2,1). (Exactly sim ilar analysis can be carried out for the M orse, the Rosen-M orse,and the Poschl-Teller potentials.) The Scarfpotentialis described by its superpotentialW (x;a0;B )= a0tanhx+ B sechx.ThepotentialV  (x;a0;B )= W2(x;a0;B )  W0(x;a0;B )isthen given byV  (x;a0;B )=hB2   a0(a0 + 1)isech2x + B (2a0 + 1)sechx tanhx + a20 : (12)Theeigenvaluesofthissystem are given by ([6])E n = a20   (a0   n)2: (13)ThepartnerpotentialV+ (x;a0;B )= W2(x;a0;B )+ W0(x;a0;B )isgiven byV+ (x;a0;B ) =hB2   a0(a0   1)isech2x + B (2a0   1)sechx tanhx + a20 := V  (x;a1;B )+ a20   a21 ; (14)wherea1 = a0   1.Thus,R(a0)forthiscase isa20   a21 = 2a0   1,linearin a0.Now,following the m echanism ofthe sec. 2,consider a set ofoperators J which isgiven byJ = e i@@x   i@@12tanhx + B sechx: (15)Note the sim ilarity between the operators J and operators A  dened in sec. 2. Sinceonly the param etera0 changesin the shape invariance condition,itisreplaced by J3 12.Com m utatorsoftheseoperatorswith J3 =   i@@can beshown to closeon J ,asdiscussedin generalin Sec. 2. Now,from eq.(9)and (14),the com m utatorofJ operatorsisgiven7by   2J3,thusform ing a closed SO (2,1)algebra. M oreover,the operatorJ+ J  ,acting onthe basisjj;m igives:J+J  B2  m2  14sech2x+ B2m  12+ 1sechx tanhx +m  122: (16)which isjusttheH scarfx;m   12;B,i.e.theScarfHam iltonian with a0 replaced by m  12.Thus the energy eigenvalues ofthe Ham iltonian willbe the sam e as that ofthe operatorJ+ J  = J23   J3   J2.Hence,theenergy isgiven by E = m 2   m   j(j+ 1).Substitutingj= n   m ,one getsE n = m2   n   (n   m )2= (m  12)2  n   (m  12)2: (17)which isthesam easeq.(13),with a0 replaced bym   12.Thusforthispotential,aswellasforthe otherthree potentialsm entioned above,there are actually an innite num berofpotentials characterised by allallowed values ofthe param eter m that correspond to thesam e value ofj and hence to the sam e energy E . Hence the nam e \potentialalgebra"([7,12]).Conclusion: The algebra ofShape Invariance plays an im portantrole in the solvability ofm ostexactly solvableproblem sin quantum m echanics.Theirspectrum can beeasily gener-ated sim ply by algebraic m eans.M any ofthese system salso have been shown to possessapotentialalgebra,which providesan alternate algebraic m ethod to determ inetheeigenval-uesand eigenfunctions. An obviousquestion iswhetherthese are two unrelated algebraicm ethodsorthere isa link between them .Fora subsetofexactly solvable potentials,thosewith R(a0)linearin param etera0,wehaveshown theequivalenceoftheirshapeinvarianceproperty to an SO (2,1)potentialalgebra.Asa concreteexam ple,westarted with theScarfpotentialand showed explicitly how shapeinvariance translatesinto the SO (2,1)potentialalgebra. W e determ ined the spectra using the algebra ofSO (2,1)and showed them to bethe sam easthatobtained from shapeinvariance.However,we only worked with solvable m odelsforwhich R(J3)is a linear function ofJ3. There are m any system sforwhich the above isnottrue. Also there were new Shape8Invariantproblem sdiscovered in 1992 ([3])forwhich itisnotpossibleto writethepotentialin closed form . It willbe interesting to know whether there are potentialalgebras thatdescribe these system ,and whether they are connected to their Shape Invariance. Theseare open problem sand are currently underinvestigation.O neofus(AG )would liketo thank thePhysicsDepartm entoftheUniversity ofIllinoisforwarm hospitality.W e would also like to thank Dr.Prsanta Panigrahiform any relateddiscussion.R eferences[1] Natanzon,G .A.(1979):Teor.M at.Fiz.38,219.[2] Spiridonov,V.P.(1992):Phys.Rev.Lett.69,298.[3] Barclay,D.,Dutt,R.,G angopadhyaya,A.,K hare,A.,Pagnam enta,A.and Sukhatm e,U.(1993):Phys.Rev.A 48 2786.[4] G endenshtein,L.E.(1983):JETP Lett.38,356;[5] G endenshtein,L.E.,K rive,I.V.(1985):Sov.Phys.Usp.28,645.[6] Cooper,F.,K hare,A.,and Sukhatm e,U.(1995):Phys.Rep.251,268 and referencestherein.[7] Alhassid,Y.,G ursey,F.and Iachello,F.(1983):Ann.Phys.148,346.[8] Barut,A.O .,Inom ata,A.,W ilson,R.(1987):J.Phys.A:M ath.G en.20 4075;J.Phys.A:M ath.G en.20 4083.[9] Engleeld,M .J.,Q uesne,C.(1987):J.Phys.A:M ath.G en.24 827(1987).[10] Engleeld,M .J.(1987):J.Phys.A:M ath.G en.28 3557.[11] Tangerm an R.D.,Tjon,J.A.(1993):Phys.Rev.A 48 1089.[12] W u J.,Alhassid,Y.(1990):Phys.Rev.A 31 557.9[13] Adam s,B.G ,Cizeka,J.,Paldus,J.(1987):Lie Algebraic M ethods And TheirApplica-tions to Sim ple Quantum System s.(From the Advancesin Q uantum Chem istry,Vol.19,Edited by Per-O lov Lowdin.Academ ic Press).10FIG URE CAPTIO N:FIG 1.Two dim ension plotshowing theallowed region form and j.11

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