 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/ib/paper-4/15b/b/ii/solution

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

With

X = x + g t^{2} /2

and

T = t

, the chain rule gives

Φ_{t} = Θ_{T} + g t Θ_{X}

and

Φ_{x} = Θ_{X}

. The drift terms cancel, so

i ℏ Θ_{T} = - \frac{ℏ ^{2}}{2 m} Θ_{XX} .

Applying Ehrenfest's theorem with

[X, P] = i ℏ

and

[P, P^{2}] = 0

gives

\frac{d}{d T} ⟨ X ⟩ = \frac{1}{m} ⟨ P ⟩, \frac{d}{d T} ⟨ P ⟩ = 0 .

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