130_notes.dvi

∫

dΩYℓm∗Yℓ′m′=δℓℓ′δmm′

We will use theactual functionin some problems.

Y 00 =

1

√

4 π

Y 11 = −

√

3

8 π

eiφsinθ

Y 10 =

√

3

4 π

cosθ

Y 22 =

√

15

32 π

e^2 iφsin^2 θ

Y 21 = −

√

15

8 π

eiφsinθcosθ

Y 20 =

√

5

16 π

(3 cos^2 θ−1)

The spherical harmonics with negativemcan be easily compute from those with positivem.

Yℓ(−m)= (−1)mYℓm∗

Any function ofθandφcan beexpanded in the spherical harmonics.

f(θ,φ) =

∑∞

ℓ=0

∑ℓ

m=−ℓ

CℓmYℓm(θ,φ)

The spherical harmonics form acomplete set.

∑∞

ℓ=0

∑ℓ

m=−ℓ

|Yℓm〉 〈Yℓm|=

∑∞

ℓ=0

∑ℓ

m=−ℓ

|ℓm〉 〈ℓm|= 1

When using bra-ket notation,|ℓm〉is sufficient to identify the state.

The spherical harmonics arerelated to the Legendre polynomialswhich are functions ofθ.

Yℓ 0 (θ,φ) =

(

2 ℓ+ 1

4 π

)^12

Pℓ(cosθ)

Yℓm = (−1)m

[

2 ℓ+ 1

4 π

(ℓ−m)!

(ℓ+m)!

]^12

Pℓm(cosθ)eimφ