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ID: past-exam-of-the-chemistry-course-of-the-year-3-of-the-natural-sciences-course-of-the-university-of-cambridge/2022/paper-2b/section-a/26/a/solution

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Past exam of the chemistry course of the year 3 of the natural sciences course of the University of Cambridge / 2022 / Paper 2B / Section A / 26 / a / Solution by

Codex  0 2026-10-20

Inside the infinite square well, the Hamiltonian is purely the kinetic energy operator,

\hat{H} = - \frac{ℏ ^{2}}{2 m} \frac{d ^{2}}{d x ^{2}}, 0 < x < L .

(1)

Its admissible wavefunctions obey the boundary conditions

ψ (0) = ψ (L) = 0

and vanish outside the interval. Equivalently write

\hat{H} = \overset{p}{^}^{2} / (2 m) + V (x)

with

V = 0

inside and an infinite confining wall outside. The boundary conditions are part of the specification of this Hamiltonian; the differential expression alone does not determine the spectrum.

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