Quantum Mechanics for Nuclear Structure, Volume 2. Professor Kris Heyde

      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, P_l 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)