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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/ib/paper-2/15a/vi/solution

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Past exam of the mathematics course of the University of Cambridge / 2024 / ib / Paper 2 / 15A / vi / Solution by

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

z = r e^{i ϕ}

and

L = - i ℏ \frac{\partial}{\partial ϕ} .

(1)

Consequently

L ψ = ℏ ψ, L ψ^{*} = - ℏ ψ^{*} .

(2)

The two states are orthogonal. They are degenerate energy eigenstates, so their coefficients acquire the same overall time-dependent phase; equivalently,

[L, H] = 0

makes

⟨ L ⟩

constant. For the normalized state

\frac{5}{3} ψ + \frac{2}{3} ψ^{*},

(3)

the angular momentum of a complex-coordinate Gaussian state is therefore

⟨ L ⟩_{t} = \frac{5}{9} ℏ - \frac{4}{9} ℏ = \frac{ℏ}{9}, t > 0.

(4)

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