 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2018/ib/paper-3/16b/solution

Codex  0 Created 2026-09-24 Updated 2026-10-03

The expectation value is the mean outcome over many identically prepared measurements of the observable. For real

λ

, positivity of the norm in the Hilbert space gives

0 \leq ∥ (Q + iλ P) ψ ∥^{2} = ⟨ Q^{2} ⟩ + λ^{2} ⟨ P^{2} ⟩ + iλ ⟨[Q, P]⟩ .

This real quadratic has nonpositive quadratic discriminant, so

⟨ Q^{2} ⟩ ⟨ P^{2} ⟩ \geq \frac{1}{4} ∣ ⟨[Q, P]⟩ ∣^{2} .

Apply this to the centered observables

Q - ⟨ Q ⟩

and

P - ⟨ P ⟩

Δ Q Δ P \geq \frac{1}{2} ∣ ⟨[Q, P]⟩ ∣.

[x, p] = i ℏ

and

E = \frac{⟨ p ^{2} ⟩}{2 m} + \frac{m ω ^{2} ⟨ x ^{2} ⟩}{2} \geq \frac{( Δ p ) ^{2}}{2 m} + \frac{m ω ^{2} ( Δ x ) ^{2}}{2} .

Δ x Δ p \geq ℏ/2

now give

E \geq ω Δ x Δ p \geq \frac{1}{2} ℏ ω .

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