 Solution

ID: past-exam-of-the-chemistry-course-of-the-year-3-of-the-natural-sciences-course-of-the-university-of-cambridge/2022/paper-2b/section-b/35/b/i/solution

d Ω = sin θ d θ d ϕ

, not the flat measure

d θ d ϕ

. Since

∣ e^{- i ϕ} ∣ = 1

\int ∣ ψ ∣^{2} d Ω = \int_{0}^{2 π} d ϕ \int_{0}^{π} sin^{3} θ d θ = 2 π \int_{- 1}^{1} (1 - u^{2}) d u = \frac{8 π}{3},

where

u = cos θ

. Therefore

Ψ (θ, ϕ) = \frac{3}{8 π} sin θ e^{- i ϕ} .

With the usual phase convention this is the spherical harmonic

Y_{1, - 1}

. An arbitrary global unit-modulus factor would not change normalization or any observable.

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