Express wavefunction in spherical harmonics & show normalisation

Question

Ψ(Θ,φ)=√(27/40π)sinθcosθ e ^iφ -√(3/10π) sin ² θ e ^2iφ

1-write Ψ in terms of the spherical harmonics.

2-Prove that the wave function is normalized?

Answer 1

Note Double Check the Algebra

1.

Consider the spherical harmonics

Y¹₂ = -1/2 √15/2/π sinθ cosθ e^iφ

Y²₂ = 1/4 √15/2/π sin²θ e^2iφ

Notice that ψ

ψ = -3/5 Y¹₂ - 4/5 Y²₂

2.

Remember the orthogonality of the spherical harmonics.

∫∫ Y^m_l Y^m'_l'^* sinθ dθ dΦ = δ_mm'δ_ll'

Then

∫∫ ^{ψ ψ*} sinθ dθ dΦ

= ∫∫(-3/5 Y¹₂ - 4/5 Y²₂)(-3/5 Y¹₂^* - 4/5 Y²₂^*) sinθ dθ dΦ

= ∫∫ (9/25 Y¹₂ Y¹₂^* - 12/25 Y²₂ Y¹₂^* - 12/25 Y²₂^* Y¹₂ + 16/25 Y²₂ Y²₂^*) sinθ dθ dΦ

= 9/25 δ₁₁δ₂₂ - 12/25 δ₁₂δ₂₂ - 12/25 δ₂₁δ₂₂ + 16/25 δ₂₂δ₂₂

= 9/25 + 16/25 = 1