Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 2 / 36E (Applications of Quantum Mechanics)

For any normalized trial state $\psi$, the spectral expansion gives $⟨\psi |H|\psi ⟩\ge E_0$. Minimize this Rayleigh quotient over a trial family to obtain the best upper bound.

Write the normalized trial state as $\sqrt{1-{ϵ}^2}|0⟩+ϵ|\eta ⟩$ with $\eta \perp 0$. The cross term vanishes because $H|0⟩=E_0|0⟩$, so the energy error is ${ϵ}^2(⟨\eta |H|\eta ⟩-E_0)=O({ϵ}^2)$.

The unperturbed ground state is even about $a/2$, while $V_{\mathcal{E}}=q\mathcal{E}(x-a/2)$ is odd. Its first-order expectation therefore vanishes.

The correction is odd, obeys the well boundary conditions, and is linear in the field, exactly the parity and order of the first-order wavefunction correction.

Put $y=x-a/2$ and $ϵ=q\mathcal{E}$. To quadratic order the Rayleigh quotient contains ${ϵ}^2(A{\alpha }^2+2B\alpha )$, where $A={\hslash }^2/(2m)$ and $B=⟨y^2⟩=a^2({\pi }^2-6)/(12{\pi }^2)$. Hence