QMGreensite_merged

11.3. THEANGULARMOMENTUMCONES 179

whichimpliesthat:

bav = 0

bmax =

n

2

bmin = −

n

2

a^2 =

n

2

(n

2

+ 1

)

(11.49)

Itiscustomarytousethenotation

l ≡

n

2

m ≡ b (11.50)

andtorelabeleigenstates φab asφlm. Inthisnotation,the eigenvaluesofangular
momentumare

L ̃^2 φlm = l(l+1) ̄h^2 φlm

L ̃zφlm = m ̄hφlm

l = 0 ,

1

2

, 1 ,

3

2

, 2 ,

5

2

, 3 ,...

m = −l,−l+ 1 ,−l+ 2 ,......,l− 3 ,l− 2 ,l− 1 ,l (11.51)

Note that,as predicted, the eigenvalues of L ̃^2 aredegenerate: to eacheigenvalue
L^2 =l(l+1) ̄h^2 thereare 2 l+ 1 linearlyindependenteigenstatesφlm,forvaluesof
mintherange−l ≤ m≤ l, asin(11.51). Inotherwords,eachL^2 eigenvalueis
2 l+1-folddegenerate.

11.3 The Angular MomentumCones

Itistimetopauseandinterprettheremarkableresult,eq.(11.51),ofthelastsection.
Whatwehavefoundisthatangularmomentumis”quantized”,inthesensethata
measurementofthemagnitudeofangularmomentumwillonlyfindoneofthediscrete
setofvalues

|L|=

√

l(l+1) ̄h, l= 0 ,

1

2

, 1 ,

3

2

, 2 ,

5

2

, 3 ,... (11.52)

andameasurementofthecomponentofangularmomentumalongacertainaxis,e.g.
thez-axis,wouldonlyfindoneofthepossiblevalues

Lz=m ̄h m∈{−l,−l+ 1 ,−l+ 2 ,...l− 2 ,l− 1 ,l} (11.53)

Theseeigenvalueshavebeendeduced withoutsolvingtheSchrodingerequation,or
anyotherdifferentialequation. Theywereobtainedfromthecommutationrelations
(11.14)andabitofcleveralgebra,nothingmore.