 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2006/ib/paper-4/6b/b/solution

Codex  0 Created 2026-09-24 Updated 2026-10-14

A

⟨ A ⟩_{ψ} = \int_{R^{3}} ψ^{*} (x) (A ψ) (x) d^{3} x,

Δ_{ψ} A = [\int_{R^{3}} ψ^{*} (A - ⟨ A ⟩_{ψ})^{2} ψ d^{3} x]^{1/2} .

Equivalently the squared uncertainty is

∥ (A - ⟨ A ⟩_{ψ}) ψ ∥^{2}

, which is visibly nonnegative. Since the expectation value of a self-adjoint operator is real, expansion of the square and

\int ∣ ψ ∣^{2} = 1

give

Δ_{ψ} A = ⟨ A^{2} ⟩_{ψ} - ⟨ A ⟩_{ψ}^{2} .

For a state not normalized to one, divide each defining expectation integral by

\int ∣ ψ ∣^{2}

before using this formula.

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