Using the identification up to the phase convention fixing the basis

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Linear algebra and vector spaces · Special functions and orthogonal polynomials

The Structure of Quantum Mechanics, Ch Angular momentum from the rotation group

Problem

Using the identification Ylm(θ,ϕ)=⟨n^∣l,m⟩Y_l^m(\theta,\phi) = \langle\hat n|l,m\rangle up to the phase convention fixing the basis, and the fact that {∣l,m⟩}\{|l,m\rangle\} are orthonormal simultaneous eigenstates of L^2\hat L^2 and L^z\hat L_z, evaluate ∫Y20∗(θ,ϕ) Y00(θ,ϕ) dΩ\int Y_2^{0*}(\theta,\phi)\,Y_0^0(\theta,\phi)\,d\Omega without carrying out any integration over θ\theta or ϕ\phi, using only orthogonality of the states ∣l,m⟩|l,m\rangle and the resolution of the identity ∫∣n^⟩⟨n^∣ dΩ=1\int |\hat n\rangle\langle\hat n|\,d\Omega = \mathbb1 on the space of functions on the sphere.

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