Читать книгу Quantum Mechanics for Nuclear Structure, Volume 2 - Professor Kris Heyde - Страница 18

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.