 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/ii/paper-3/33a/a/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

L_{z} = - i ℏ (x \partial_{y} - y \partial_{x}),

which differentiates only with respect to

x

and

y

z

therefore commutes with it:

[L_{z}, z] = 0 .

0 = ℏ (m^{'} - m) ⟨ n^{'}, ℓ^{'}, m^{'} ∣ z ∣ n, ℓ, m ⟩ .

Thus the matrix element can be nonzero only when

m^{'} = m .

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