 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-341/section-a/3/a/solution

Codex  0 Created 2026-09-28 Updated 2026-10-14

Write

H = \partial_{x}^{2} - V (x)

. The real potential and periodic boundary condition make

H

u_{t} = - i H u

\frac{d}{d t} ∥ u (t) ∥_{L^{2}}^{2} = 2 Re ⟨ u, u_{t} ⟩ = 2 Re (- i ⟨ u, H u ⟩) = 0,

⟨ u, H u ⟩

is real. Thus the Schrödinger equation preserves the

L^{2}

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