Angular Momentum - University of Notre Dame

1 Chapter 1 Angular MomentumUnderstanding the quantum mechanics of Angular Momentum is fundamental intheoretical studies of atomic structure and atomic transitions. Atomic energylevels are classi ed according to Angular Momentum and selection rules for ra-diative transitions between levels are governed by Angular - Momentum additionrules. Therefore, in this rst chapter, we review Angular - Momentum commu-tation relations, Angular - Momentum eigenstates, and the rules for combiningtwo Angular - Momentum eigenstates to nd a third. We make use of Angular - Momentum diagrams as useful mnemonic aids in practical atomic structure cal-culations. A more detailed version of much of the material in this chapter canbe found in Edmonds (1974).

2 Orbital Angular Momentum - SphericalHarmonicsClassically, the Angular Momentum of a particle is the cross product of its po-sition vectorr=(x; y; z) and its Momentum vectorp=(px;py;pz):L=r p:The quantum mechanical orbital Angular Momentum operator is de ned in thesame way withpreplaced by the Momentum operatorp! i hr. Thus, theCartesian components ofLareLx= hi y@@z z@@y ;Ly= hi z@@x x@@z ;Lz= hi x@@y y@@x :( )With the aid of the commutation relations betweenpandr:[px;x]= i h;[py;y]= i h;[pz;z]= i h;( )12 CHAPTER 1. Angular MOMENTUMone easily establishes the following commutation relations for the Cartesiancomponents of the quantum mechanical Angular Momentum operator:LxLy LyLx=i hLz;LyLz LzLy=i hLx;LzLx LxLz=i hLy:( )Since the components ofLdo not commute with each other, it is not possible to nd simultaneous eigenstates of any two of these three operators.

3 The operatorL2=L2x+L2y+L2z, however, commutes with each component ofL. It is, there-fore, possible to nd a simultaneous eigenstate ofL2and any one component ofL. It is conventional to seek eigenstates Quantum Mechanics of Angular MomentumMany of the important quantum mechanical properties of the Angular momen-tum operator are consequences of the commutation relations ( ) alone. Tostudy these properties, we introduce three abstract operatorsJx;Jy, andJzsatisfying the commutation relations,JxJy JyJx=iJz;JyJz JzJy=iJx;JzJx JxJz=iJy:( )The unit of Angular Momentum in Eq.( ) is chosen to be h, so the factor of hon the right-hand side of Eq.( ) does not appear in Eq.

4 ( ). The sum ofthe squares of the three operatorsJ2=J2x+J2y+J2zcan be shown to commutewith each of the three components. In particular,[J2;Jz]=0:( )The operatorsJ+=Jx+iJyandJ =Jx iJyalso commute with the angularmomentum squared:[J2;J ]=0:( )Moreover,J+andJ satisfy the following commutation relations withJz:[Jz;J ]= J :( )One can expressJ2in terms ofJ+,J andJzthrough the relationsJ2=J+J +J2z Jz;( )J2=J J++J2z+Jz:( )We introduce simultaneous eigenstatesj ; miof the two commuting opera-torsJ2andJz:J2j ; mi= j ; mi;( )Jzj ; mi=mj ; mi;( )and we note that the statesJ j ; miare also eigenstates ofJ2with eigenvalue . Moreover, with the aid of Eq.( ), one can establish thatJ+j ; miandJ j ; miare eigenstates ofJzwith eigenvaluesm 1, respectively:JzJ+j ; mi=(m+1)J+j ; mi;( )JzJ j ; mi=(m 1)J j ; mi:( ) ORBITAL Angular Momentum - SPHERICAL HARMONICS3 SinceJ+raises the eigenvaluemby one unit, andJ lowers it by one unit,these operators are referred to as raising and lowering operators, , sinceJ2x+J2yis a positive de nite hermitian operator, it followsthat m2:By repeated application ofJ to eigenstates ofJz, one can obtain states of ar-bitrarily small eigenvaluem, violating this bound, unless for some statej ; m1i,J j.

5 M1i=0:Similarly, repeated application ofJ+leads to arbitrarily large values ofm, unlessfor some statej ; m2iJ+j ; m2i=0:Sincem2is bounded, we infer the existence of the two statesj ; m1iandj ; from the statej ; m1iand applying the operatorJ+repeatedly, onemust eventually reach the statej ; m2i; otherwise the value ofmwould increaseinde nitely. It follows thatm2 m1=k;( )wherek 0 is the number of times thatJ+must be applied to the statej ; m1iin order to reach the statej ; m2i. One nds from Eqs.( , ) that j ; m1i=(m21 m1)j ; m1i; j ; m2i=(m22+m2)j ; m2i;leading to the identities =m21 m1=m22+m2;( )which can be rewritten(m2 m1+ 1)(m2+m1)=0:( )Since the rst term on the left of Eq.

6 ( ) is positive de nite, it follows thatm1= m2. The upper boundm2can be rewritten in terms of the integerkinEq.( ) asm2=k=2=j:The value ofjis either integer or half integer, depending on whetherkis evenor odd:j=0;12;1;32; :It follows from Eq.( ) that the eigenvalue ofJ2is =j(j+1):( )The number of possiblemeigenvalues for a given value ofjisk+1=2j+ possible values ofmarem=j; j 1;j 2; ; j:4 CHAPTER 1. Angular Momentum xyzr Figure : Transformation from rectangular to spherical =Jy+, it follows thatJ+j ; mi= j ; m+1i;J j ; m+1i= j ; mi:Evaluating the expectation ofJ2=J J++J2z+Jzin the statej ; mi, one ndsj j2=j(j+1) m(m+1):Choosing the phase of to be real and positive, leads to the relationsJ+j ; mi=p(j+m+ 1)(j m)j ; m+1i;( )J j ; mi=p(j m+ 1)(j+m)j ; m 1i:( ) Spherical Coordinates - Spherical HarmonicsLet us apply the general results derived in Section to the orbital angularmomentum operatorL.

7 For this purpose, it is most convenient to transformEqs.( ) to spherical coordinates (Fig. ):x=rsin cos ;y=rsin sin ;z=rcos ;r=px2+y2+z2; = arccosz=r; = arctany=x:In spherical coordinates, the components ofLareLx=i h sin @@ + cos cot @@ ;( )Ly=i h cos @@ + sin cot @@ ;( )Lz= i h@@ ;( ) ORBITAL Angular Momentum - SPHERICAL HARMONICS5and the square of the Angular Momentum isL2= h2 1sin @@ sin @@ +1sin2 @2@ 2 :( )Combining the equations forLxandLy, we obtain the following expressions forthe orbital Angular Momentum raising and lowering operators:L = he i @@ +icot @@ :( )The simultaneous eigenfunctions ofL2andLzare called spherical are designated byYlm( ; ).

8 We decomposeYlm( ; ) into a product of afunction of and a function of :Ylm( ; )= l;m( ) m( ):The eigenvalue equationLzYl;m( ; )= hmYl;m( ; ) leads to the equation id m( )d =m m( );( )for m( ). The single valued solution to this equation, normalized so thatZ2 0j m( )j2d =1;( )is m( )=1p2 eim ;( )wheremis an integer. The eigenvalue equationL2Yl;m( ; )= h2l(l+1)Yl;m( ; ) leads to the di erential equation 1sin dd sin dd m2sin2 +l(l+1) l;m( )=0;( )for the function l;m( ). The orbital Angular Momentum quantum numberlmust be an integer sincemis an can generate solutions to Eq.( ) by recurrence, starting with thesolution form= land stepping forward inmusing the raising operatorL+,or starting with the solution form=land stepping backward using the loweringoperatorL.

9 The function l; l( ) satis es the di erential equationL l; l( ) l( )= h l+1( ) dd +lcot l; l( )=0;which can be easily solved to give l; l( )=csinl , where c is an arbitraryconstant. Normalizing this solution so thatZ 0j l; l( )j2sin d =1;( )6 CHAPTER 1. Angular MOMENTUMone obtains l; l( )=12ll!r(2l+ 1)!2sinl :( )ApplyingLl+m+toYl; l( ; ), leads to the result l;m( )=( 1)l+m2ll!s(2l+ 1)(l m)!2(l+m)!sinm dl+mdcos l+msin2l :( )Form= 0, this equation reduces to l;0( )=( 1)l2ll!r2l+12dldcos lsin2l :( )This equation may be conveniently written in terms of Legendre polynomialsPl(cos )as l;0( )=r2l+12Pl(cos ):( )Here the Legendre polynomialPl(x) is de ned by Rodrigues' formulaPl(x)=12ll!

10 Dldxl(x2 1)l:( )Form=l, Eq.( ) gives l;l( )=( 1)l2ll!r(2l+ 1)!2sinl :( )Starting with this equation and stepping backwardl mtimes leads to analternate expression for l;m( ): l;m( )=( 1)l2ll!s(2l+ 1)(l+m)!2(l m)!sin m dl mdcos l msin2l :( )Comparing Eq.( ) with Eq.( ), one nds l; m( )=( 1)m l;m( ):( )We can restrict our attention to l;m( ) withm 0 and use ( ) to obtain l;m( ) form<0. For positive values ofm, Eq.( ) can be written l;m( )=( 1)ms(2l+ 1)(l m)!2(l+m)!Pml(cos );( )wherePml(x) is an associated Legendre functions of the rst kind, given inAbramowitz and Stegun (1964, chap. 8), with a di erent sign convention, de nedbyPml(x)=(1 x2)m=2dmdxmPl(x):( ) SPIN Angular MOMENTUM7 The general orthonormality relationshl; mjl0;m0i= ll0 mm0for Angular mo-mentum eigenstates takes the speci c formZ 0Z2 0sin d d Y l;m( ; )Yl0;m0( ; )= ll0 mm0;( )for spherical harmonics.