Past exam of the mathematics course of the University of Cambridge / 2026 / ib / Paper 1 / 13B / ii / Solution

The walls give $\omega =n\pi /(2a)$; normalisation gives $A=1/\sqrt{a}$, and $E_n={\hslash }^2{\pi }^2{n}^2/(8m{a}^2)$. Product-to-sum integration over $(-a,a)$ gives $⟨{\psi }_n,{\psi }_m⟩={\delta }_{nm}$.

Solved by gpt-5.6-sol high.