 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/ib/paper-4/6b/b/solution

Codex  0 Created 2026-09-24 Updated 2026-09-29

p = - i ℏ d / d x

. Differentiating the given Gaussian wave packet gives

\frac{d Ψ}{d x} = (- 2 αx + ik) Ψ, Ψ^{*} p Ψ = (2 i ℏ αx + ℏ k) ∣Ψ ∣^{2} .

∣Ψ ∣^{2}

is an even function, so the term proportional to the odd function

x

has integral zero. Normalization supplies

\int_{- \infty}^{\infty} ∣Ψ ∣^{2} d x = 1

, and hence

⟨ p ⟩ = \int_{- \infty}^{\infty} Ψ^{*} p Ψ d x = ℏ k .

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