 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2020/ib/paper-1/4a/solution

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

An operator

Q

is Hermitian when

⟨ ϕ, Q ψ ⟩ = ⟨ Qϕ, ψ ⟩

on its domain. Hermitian operators represent quantum observables because their expectation values and eigenvalues are real. In a normalized state,

(Δ Q)_{ψ} = ⟨ (Q - ⟨ Q ⟩_{ψ})^{2} ⟩_{ψ}^{1/2} .

Positivity of the norm of

(Q + iλ P) ψ

for every real

λ

gives

0 \leq ⟨ Q^{2} ⟩_{ψ} + λ ⟨ i [Q, P] ⟩_{ψ} + λ^{2} ⟨ P^{2} ⟩_{ψ} .

The middle expectation is real because

i [Q, P]

is Hermitian. The discriminant of this quadratic is nonpositive, so

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

Apply this to the centered operators

Q - ⟨ Q ⟩_{ψ}

and

P - ⟨ P ⟩_{ψ}

, whose commutator is still

[Q, P]

(Δ Q)_{ψ} (Δ P)_{ψ} \geq \frac{1}{2} ∣ ⟨ i [Q, P] ⟩_{ψ} ∣ .

Noncommuting observables therefore cannot both have arbitrarily sharply concentrated measurement distributions in the same state.

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