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.