186 CHAPTER11. ANGULARMOMENTUM
Asanexampleoftheprocedure,letuscomputethel= 1 multipletofspherical
harmonics,i.e.Y 11 , Y 10 , Y 1 ,− 1 .Webeginwith
Y 11 = (−1)
[
Γ(^52 )
2 π^3 /^2
] 1 / 2
sinθeiφ
= −
√
3
8 π
sinθeiφ (11.98)
then
L−Y 11 = −
√
3
8 π
̄he−iφ
(
icotθ
∂
∂φ
−
∂
∂θ
)
sinθeiφ
̄h
√
2 Y 10 =
√
3
8 π
h ̄(cotθsinθ+cosθ)
Y 10 =
√
3
4 π
cosθ (11.99)
Andapplyingtheloweringoperatoragain
L ̃−Y 10 =
√
3
4 π
̄he−iφ
(
icotθ
∂
∂φ
−
∂
∂θ
)
cosθ
h ̄
√
2 Y 1 ,− 1 =
√
3
4 π
̄he−iφsinθ
Y 1 ,− 1 =
√
3
8 π
e−iφsinθ (11.100)
Itiseasytocheckthatapplyingtheloweringoperatoronemoretimeannihilatesthe
state
L ̃−Y 1 ,− 1 = 0 (11.101)
asitshould. Infact,inconstructingthesphericalharmonics,wecouldstartwiththe
solutionofthedifferentialequation
L ̃−Yl,−l= 0 (11.102)
andobtaintherestoftheYlmbyusingtheraisingoperatorL ̃+.
Problem: Obtainthe l= 1 multipletbysolving(11.102),andthenapplyingthe
raisingoperator.
Ofcourse,onerarelyhastogotothetroubleofcomputingthesphericalharmonics,
sincetheyarelistedinmanybooks. Thefirstfew,includingthel= 1 multipletwe