QMGreensite_merged

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