A particle is confined to an infinite one-dimensional well on −a<x<a. A perturbation $V\prime=cx$ eV is applied. Express the first-order energy correction to the first excited state as a coefficient times c eV.
First-order perturbation is $\Delta E^{(1)}=\langle\psi|cx|\psi\rangle$. The unperturbed state has definite parity, so |ψ|² is even. Multiplying by the odd x gives an odd integrand on [−a,a]. Its integral vanishes, giving coefficient 0. The same parity argument applies to every nondegenerate box eigenstate.
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