where (2l)!!≔(2l)(2l−2)(2l−4)…2or1 and
∫02πdϕ∫0πsinθdθ∣Yl,±l(θ,ϕ)∣2=1.(1.210)
Thus,
ϕl,±l(r⃗)=rl4π(2l)!!(2l+1)!!Yl,±l(θ,ϕ).(1.211)
It then follows from
ϕlm(r⃗)=(l−m)!(2l)!(l+m)!(Lˆ+)l+m(x−iy)l,(1.212)
which is obtained by repeated application of equations (1.186)–(1.188), that a general spherical harmonic is given by
Ylm(θ,ϕ)=12ll!(2l+1)(l−m)!4π(l+m)!1rl(Lˆ+)l+m(x−iy)l.(1.213)
This leads to the general expression for spherical harmonics:
Ylm(θ,ϕ)=12ll!(2l+1)(l−m)!4π(l+m)!eimϕ(−sinθ)mdd(cosθ)l+m(cos2θ−1)l.(1.214)
The spherical harmonics are related to the Legendre polynomials, Pl by:
Pl(cosθ)=4π2l+1Yl,m=0(θ,ϕ).(1.215)
Table 1.1. The spherical harmonics, Ylm(θ,ϕ), m=l,l−1,l−2,…,1,0,−1,…,−l+1,−l, for l=0,1,2, and 3. They are normalized for 0⩽ϕ⩽2π, 0⩽θ⩽π.
| l | m | Ylm(θ,ϕ) |
|---|---|---|
| 0 | 0 | 14π |
| 1 | 0 | 34πcosθ |
| 1 | ±1 | ∓38πe±iϕsinθ |
| 2 | 0 | 516π(3cos2θ−1) |
| 2 | ± 1 | ∓158πe±iϕcosθsinθ |
| 2 | ±2 | 1532πe±2iϕsin2θ |
| 3 | 0 | 6316π53cos3θ−cosθ |
| 3 | ±1 | ∓2164πe±iϕ(5cos2θ−1)sinθ |
| 3 | ±2 | 10532πe±2iϕsin2θcosθ |
| 3 | ±3 | ∓3564πe±3iϕsin3θ |
1.10 Spherical harmonics and wave functions
Spherical harmonics naturally arise when using three-dimensional position wave functions in quantum mechanics. Thus, for the position eigenkets ∣r⃗〉:
∣α〉=∫dr⃗∣r⃗〉〈r⃗∣α〉,(1.216)
the position wave function Ψα(r⃗) is the amplitude 〈r⃗∣α〉 and Ψα(r⃗) is often expressed in spherical polar coordinates:
Ψα(r⃗)=Rα(r)Ωα(θ,ϕ).(1.217)
The functions Ωα(θ,ϕ) are then expanded in terms of spherical harmonics
Ωα(θ,ϕ)=∑lmcαlmYlm(θ,ϕ).(1.218)
Within the above framework, we can define direction eigenkets ∣nˆ〉, nˆ=r⃗r:
∣α〉=∫dnˆ∣nˆ〉〈nˆ∣α〉;(1.219)
and for
∣lm〉=∫dnˆ∣nˆ〉〈nˆ∣lm〉,(1.220)
〈nˆ∣lm〉=Ylm(θ,ϕ)=Ylm(nˆ),(1.221)
i.e. Ylm(θ,ϕ) is the amplitude for the state ∣lm〉 to be found in the direction nˆ specified by θ and ϕ.
1.11 Spherical harmonics and rotation matrices
Spherical harmonics can be related to (the elements of) rotation matrices because of their connection to direction eigenkets:
∣nˆ〉=∑lm∣lm〉〈lm∣nˆ〉=∑lmYlm*(θ,ϕ)∣lm〉.(1.222)
To see this, consider
∣nˆ〉=D(R)∣zˆ〉,(1.223)
i.e. ∣nˆ〉 is obtained by the rotation of ∣zˆ〉. Evidently,
D(R)=D(α=ϕ,β=θ,γ=0)(1.224)
will do the job. Then for equation (1.223), from the completeness relation:
∣nˆ〉=∑lmD(R)∣lm〉〈lm∣zˆ〉,(1.225)
∴〈l′m′∣nˆ〉=∑lm〈l′m′∣D(R)∣lm〉〈lm∣zˆ〉=Dm′m(l′)(α=0,β=θ,γ=0)〈l′m∣zˆ〉.(1.226)
But, 〈l′m∣zˆ〉 is just Yl′m*(θ=0,ϕ) and Yl′m(θ=0,ϕ)=0 for m≠0: this is seen by inspection of table 1.1. Thus,
〈l′m∣zˆ〉=Yl′m*(θ=0,ϕ)δm0=2l′+14πPl′(cosθ)∣θ=0δm0=2l′+14πδm0,(1.227)
where the Pl′(cosθ) are the Legendre polynomials given by equation (1.215). Hence, from equations (1.226), (1.223) and (1.227):
Yl′m′*(θ,ϕ)=Dm′0(l′)(α=ϕ,β=θ,γ=0)2l′+14π,(1.228)
or
Dm0(l)(α,β,γ=0)=4π2l+1Ylm*(θ,ϕ)∣θ=β,ϕ=α;(1.229)
and for m = 0
D00(l)(α,β,γ)=d00(l)(β),(1.230)
and
∴d00(l)(β)=Pl(cosθ)∣θ=β.(1.231)
Theorem 1.11.1. The addition theorem for spherical harmonics,
Pl(cosθ)=∑m4π2l+1Ylm(θ2,ϕ2)Ylm*(θ1,ϕ1),(1.232)
where θ is defined by
cosθ≔cosθ1cosθ2+sinθ1sinθ2cos(ϕ1−ϕ2).(1.233)
Proof. Consider
〈l0∣D(ϕ,θ,0)∣l0〉=〈l0∣D(ϕ2,θ2,0)D(ϕ1,θ1,0)∣l0〉,(1.234)
where the group properties of rotations in ket space have been used. Then, from the completeness relation
〈l0∣D(ϕ,θ,0)∣l0〉=∑m〈l0∣D(ϕ2,θ2,0)∣lm〉〈lm∣D(ϕ1,θ1,0)∣l0〉,(1.235)
∴D00(l)(ϕ,θ,0)=∑mD0m(l)(ϕ2,θ2,0)Dm0(l)(ϕ1,θ1,0),(1.236)