Читать книгу Quantum Mechanics for Nuclear Structure, Volume 2 - Professor Kris Heyde - Страница 19

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)

and from equations (1.229) and (1.231),

Pl(cosθ)=∑m4π2l+1Ylm(θ2,ϕ2)Ylm*(θ1,ϕ1).(1.237)

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