 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2016/ib/paper-2/17b/d/solution

Codex  0 Created 2026-09-24 Updated 2026-10-06

Apply the adjoint operator identity of part (c) with

χ = \hat{A} ϕ

and use part (a):

∥ \hat{A} ϕ ∥^{2} = ⟨ ϕ, \hat{B} \hat{A} ϕ ⟩ = 2 m (E - \frac{1}{2} ℏ ω) ∥ ϕ ∥^{2} .

For a nonzero eigenfunction, positivity of the norm in particular forces

E \geq ℏ ω /2

. Hence the requested norm is

∥ \hat{A} ϕ ∥ = 2 m (E - \frac{1}{2} ℏ ω) ∥ ϕ ∥.

It vanishes exactly at the ground-state energy.

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