Past exam of the mathematics course of the University of Cambridge / 2026 / ib / Paper 1 / 13B (Quantum Mechanics)

The nonrelativistic Schrödinger Hamiltonian uses $p^2/(2m)$ and therefore has no massless limit; massless particles require a relativistic wave equation or quantum field theory.

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 $⟨{\psi }_n,{\psi }_m⟩={\delta }_{nm}$.

Using the quadratic form of the Hamiltonian,
$E_{av}=({\hslash }^2/2m)\int |\psi \prime {|}^{2}dx=2{\pi }^2{\hslash }^2/(m{a}^2)$.

Only odd stationary states contribute, and for $j=2n+1$ their squared coefficients are $128/[{\pi }^2(j^2-16{)}^2]$. Since $E_{av}/E_1=16$, substitution givesso $B=8$.