Past exam of the chemistry A course of the University of Cambridge / 2024 / Paper 1 / A1 / a / iv

For distinct positive integers $n,m$, use the product-to-sum identity in the inner product:Every endpoint sine vanishes. For $m=n$, the normalization integral is one; the expression with a denominator $m-n$ should not be substituted directly. Thus the normalized functions form an orthonormal set, $⟨{\Psi }_m|{\Psi }_n⟩={\delta }_{mn}$.