11.2. THEEIGENVALUESOFANGULARMOMENTUM 175
11.2 The Eigenvalues of Angular Momentum
TheenergyeigenvaluesofH ̃,foraparticleinacentralpotential,aredegenerate,since
H ̃commuteswiththecomponentsofangularmomentum,whichdonotcommutewith
eachother. ThesamecanbesaidfortheeigenvaluesofL ̃^2. Theseeigenvaluesmust
bedegenerate,becauseL ̃^2 alsocommuteswithL ̃x, L ̃y, L ̃z,whichdocommutewith
eachother.BythecommutatortheoremL ̃^2 andL ̃xhaveacommonsetofeigenstates;
sodoL ̃^2 andL ̃x,andalsoL ̃^2 andL ̃z.But,againbythecommutatortheorem,these
mustbedifferentsetsofeigenstates.
DenotetheeigenstatesofL ̃^2 and,say,Lz,asφab. Angularmomentumhasunits
ofaction,i.e. energy×time,whichisthesameunitsasPlanck’sconstant ̄h. Sowe
mayaswellwritetheeigenvaluesofangularmomentuminunitsof ̄hwhere
L ̃^2 φab = a^2 ̄h^2 φab
L ̃zφab = bh ̄φab (11.20)whereaandbaredimensionlessnumbers.TheeigenvaluesofL ̃zmaybepositiveor
negative,buttheeigenvalues ofL ̃^2 arenecessarilypositive, because L ̃^2 isasumof
squaresofHermitianoperators. Therefore,
<φab|L ̃^2 φab> = <φab|L ̃^2 x|φab>+<φab|L^2 y|φab>+<φab|L^2 z|φab>
a^2 ̄h^2 <φab|φab> = <φab|L ̃^2 x|φab>+<φab|L ̃^2 y|φab>+ ̄h^2 b^2 <φab|φab>
a^2 ̄h^2 = <L ̃xφab|L ̃xφab>+<L ̃yφab|L ̃yφab>+ ̄h^2 b^2
a^2 ̄h^2 = ̄h^2 b^2 +|L ̃xφab|^2 +|L ̃yφab|^2 (11.21)whichestablishesthattheeigenvaluesofL ̃^2 arepositivesemi-definite,andalsothat
−a≤b≤a (11.22)ThefactthatL ̃^2 isasumofsquaresofHermitianoperators,whichcommutein
asimplewayamongthemselves(eq. (11.14)),suggeststryingthesametrickthatwe
usedfortheHarmonicoscillator. Applyingeq. (9.13)ofLecture9,wehave
L ̃^2 = (L ̃^2 x+L ̃^2 y)+L ̃^2 z
= (L ̃+L ̃−+i[L ̃x,L ̃y])+L ̃^2 z
= L ̃+L ̃−− ̄hLz+L ̃^2 z (11.23)where
L ̃+ ≡ L ̃x+iL ̃y
L ̃− ≡ L ̃x−iL ̃y (11.24)L ̃−istheHermitianconjugateofL ̃+.Equivalently,since
[L ̃+,L ̃−]=− 2 i[L ̃x,L ̃y]=2 ̄hL ̃z (11.25)