On the Dynamical Invariants and the Geometric Phases for a General Spin System in a Changin

AliMostafazadeh∗

DepartmentofMathematics,Ko¸cUniversity,RumelifeneriYolu,80910Sariyer,Istanbul,Turkey

Abstract

WeconsideraclassofgeneralspinHamiltoniansoftheformHs(t)=H0(t)+H′(t)whereH0(t)andH′(t)describethedipoleinteractionofthespinswithanarbitrarytime-dependentmagneticfieldandtheinternalinteractionofthespins,respectively.WeshowthatifH′(t)isrotationallyinvariant,thenHs(t)admitsthesamedynamicalinvariantasH0(t).Adirectapplicationofthisobservationisastraightforwardred-erivationoftheresultsofYanetal[Phys.Lett.A251(1999)2and259(1999)207]ontheHeisenbergspinsysteminachangingmagneticfield.

1Introduction

InRef.[1],theauthorsconstructadynamicalinvariant[2]fortheHeisenbergspinsysteminachangingmagneticfield.1Thisinvariantinvolvestwoauxiliaryfunctionsthatsatisfyasystemofcoupledfirstorderdifferentialequations.Thesesamedifferentialequationsariseintheconstructionofadynamicalinvariantforthedipoleinteractionofasinglespininachangingmagneticfield.ThisobservationtogetherwiththemorerecentresultsofRef.[4]onthecharacterizationofthequantumsystemsadmittingthesamedynamicalinvariantarethemainmotivationforthepresentstudy.

TheHamiltonianHHeisenbergfortheHeisenbergspinsysteminachangingmagneticfieldB

󰀢(t)isaspecialcaseofthespinHamiltoniansoftheformHs(t)=H0(t)+H′(t),

(1)

whereH0(t)isthedipoleinteractionHamiltoniangivenby

󰀢(t)·H0(t):=B

N󰀃i=1

󰀢i=S

N󰀃3󰀃i=1α=1

Bα(t)Siα,

(2)

󰀢i=(S1,S2,S3)isthespinoperatorforthei-thparticle,Nisthenumberofparticles,andSiiiH′(t)istheHamiltoniancorrespondingtotheinternalspininteractionofthesystem.ForHHeisenberg(t),H′(t)isatime-independentHamiltoniangivenby

′

HHeisenberg

=−AH1,:=

󰀉

H1:=

1α2Qαi1i2

δα1,α2ifi1andi2labelparticlesthatarenearestneighbors0

otherwise,

󰀃󰀃

α1α2α1α2

QiSi1Si2,1,i2

(3)(4)

i1,i2α1,α2

whereAisaconstantandδa,bdenotestheKroneckerdeltafunction.

ThepurposeofthisarticleistoshowthatanyHamiltonianoftheform(1)admitsadynamicalinvariantthatisalsoadynamicalinvariantofthedipoleHamiltonian(2)providedthatH′(t)hasrotationalinvariance.

First,werecallthatbydefinitionadynamicalinvariantI(t)foraHamiltonianH(t)isa(nontrivial)solutionoftheLiouville-von-Neumannequation:

d

theevolutionoperatorUi(t)generatedbyH(t)satisfiesUi(t)=Wi(t)Zi(t)whereZi(t)isaunitaryoperatorcommutingwithSi3,[4].TheseresultscanbedirectlyemployedfortheHamiltonianH0(t)ofEq.(2).Specifically,thisHamiltonianadmitstheinvariant

󰀁N󰀂󰀁N󰀂NN󰀃󰀃󰀃󰀃†††󰀢(t)·󰀢i=W0(t)I(t)=Ii(t)=RSS3W0(t)=U0(t)S3U0(t)=U0(t)I(0)U0,

i

i

i=1

i=1

i=1

i=1

(7)

whereW0(t)=

󰀇N

i=1Wi(t)andU0(t)=

󰀇N

i=1

Ui(t)and

N󰀃i=1

I(0)=

Si3.

(8)

NotethatbecauseSiαwithdifferentvaluesoficommute,[Wi(t),Wj(t)]=[Ui(t),Uj(t)]=0.󰀢i,andU0(t)=eiM(t)whereρFurthermore,wehaveUi(t)=eiMi(t),Mi(t):=ρ󰀢(t)·S󰀢(t)are

󰀅

󰀢(t)andM(t):=ρ󰀢determinedintermsofR󰀢(t)·Ni=1Si.invariant,i.e.,forallα∈{1,2,3},

[H(t),

Then

[H0(t),H′(t)]=0,[I(t),H′(t)]=0.

(10)(11)

′

N󰀃i=1

Now,consideraspinHamiltonianoftheform(1)andsupposethatH′(t)isrotationally

Siα]=0.

(9)

InviewofEqs.(1)and(11)andthefactthatI(t)isadynamicalinvariantforH0(t),wehave

d

Aswementionedearlier,theHeisenbergspinHamiltonianHHeisenbergconsideredinRefs.[1,

′

3]isaspecialcaseofHs(t).ItcanalsobeeasilycheckedthatHHeisenbergisrotationallyin-

variant.HenceHHeisenbergalsoadmitstheinvariantI(t)ofEq.(7).TheinvariantconstructedinRef.[1]differsfromtheinvariant(7)byaconstanttermthatcommuteswiththeHamil-tonian.Therefore,thistermdropsfrombothsidesofthedefiningequation(5).Theonlyeffectofthisadditionaltermistochangethedegeneracyoftheeigenvaluesoftheinvariant.Infact,themostgeneraldynamicalinvariantfortheHamiltonianHs(t)(andinparticularforHHeisenberg)isgivenby

††

Is(t)=Us(t)I(0)Us(t)=U0(t)U′(t)I(0)U′(t)U0(t),

†

whereI(0)isaconstantHermitianoperator.Theinvariant(7)correspondstothechoice(8)forI(0).TheinvariantconstructedinRef.[1]correspondstothechoice

I(0)=

N󰀃i=1

Si3+H1.

(13)

AsmentionedinRef.[1],theeigenvectorsoftheinvariantcorrespondingto(13)arenotknown.Thismeansthatanexplicitsolutionofthetime-dependentSchr¨odingerequationusingthisinvariantisnotpossible.Unlikethisinvarianttheeigenvectorsoftheinvariant(7)areeasilycalculated;theyare(tensor)productsoftheeigenvectorsofIi(t),i.e.,Wi(t)|ni󰀋where|ni󰀋aretheeigenvectorsofSi3witheigenvalueni=±1/2.Wewishtoconcludethisarticlewiththefollowingremarks.

1.Becauseitisthedynamicalinvariantsthatdeterminethegeometricphases[7,6],thegeometricphasesobtainedfortheHamiltonianHs(t)coincidewiththoseofthedipoleHamiltonianH0(t).ThisisthereasonwhytheexpressionobtainedinRef.[1]forthegeometricphasesofaHeisenbergspinsysteminachangingmagneticfieldisessentiallythesameastheoneforthegeometricphasesofasinglespininthesamemagneticfield.

2.WecanconstructmoregeneralinternalinteractionHamiltoniansH′(t)thatarerota-tionallyinvariantandthusthecorrespondingtotalHamiltonianHs(t)admitsthesame

4

dynamicalinvariantasH0(t).Forexample,wecanset

H(t)=Hn:=

′

󰀃

n

λn(t)Hn,

3󰀃

α1···α2nα1α2n

QiS···S,···iii12n12n

(14)(15)

i1,···,i2n=1α1,···,α2n=1

α1···α2n

wherentakespositiveintegervalues,λnarereal-valuedfunctionsoft,andQi

1···i2n

N󰀃

arerealcouplingconstantssatisfyingcertainsymmetryconditions.Inordertostatetheseconditions,weintroducethefollowingabbreviatednotation

˜µ,ν:=Qα1···αj···αk···α2n

Qi1···ij···ik···i2nijik

withαj=µ,αk=ν.

Thentheabovementionedconditionstaketheform

forij=ik:forij=ik:

˜µν=Q˜νµ,Qijikijik

󰀄󰀆󰀄󰀆βννβµββµ˜˜˜˜δµ,γQijik+Qikij+δν,γQijik+Qikij−󰀄󰀆󰀄󰀆γννγµγγµ˜˜˜˜δµ,βQijik+Qikij−δν,βQijik+Qikij=0.

(16)

(17)

Theserelationsmustbesatisfiedforallpossiblevaluesofthelabelsj,k,ij,ik,β,γ,µ,andν.TheyfollowfromtherequirementthatH1commuteswiththetotalspinoper-󰀅α

atorsNi=1Siandtheidentities

SiαSiβ

=

1

2

3󰀃γ=1

ǫαβγSiγ,

ββα

SiαSj=SjSi

fori=j.

ItisnotdifficulttogeneralizetheconditionsobtainedforH1toHn.

1α2

Perhapsthesimplestnontrivialexamplethatfulfilsconditions(16)and(17)isQαi1i2

ofEq.(4).

Acknowledgment

IwouldliketothankProfessorAldenMeadforhismostinvaluablecommentsandsugges-tions.

5

References

[1]F.Yan,L.Yang,andB.Li,Phys.Lett.A259,207-211(1999).[2]H.R.LewisJr.andW.B.Riesenfeld,J.Math.Phys.10,1458(1969).[3]F.Yan,L.Yang,andB.Li,Phys.Lett.A251,2-293(1999).

[4]A.Mostafazadeh,‘Geometricphases,symmetriesofdynamicalinvariants,andexact

solutionoftheSchr¨odingerequation,’quant-ph/0101010.[5]X.-C.Gao,J.-B.Xu,T.-Z.Qian,Phys.Lett.A152,449(1991);

Y.-Z.Lai,J.-Q.Liang,W.M¨uller-Kirsten,andJ.-G.Zhou,J.Phys.A:Math.Gen.29,1773(1996).

[6]A.Mostafazadeh,DynamicalInvariants,AdiabaticApproximation,andtheGeometric

Phase,NovaSciencePubl.,inpress.

[7]A.Mostafazadeh,J.Phys.A:Math.Gen.31,9975(1998)and32,8157(1999).

6