 Solution

ID: past-exam-of-the-chemistry-course-of-the-year-3-of-the-natural-sciences-course-of-the-university-of-cambridge/2022/paper-4/59/b/i/solution

ε (x - a /2)

, so inside the box

H = - \frac{ℏ ^{2}}{2 m} \frac{d ^{2}}{d x ^{2}} + qε (x - a /2), H^{(0)} = - \frac{ℏ ^{2}}{2 m} \frac{d ^{2}}{d x ^{2}}, H^{(1)} = x - a /2, λ = qε .

The infinite walls impose zero wavefunction at

0, a

. The normalized particle in a box states are

ψ_{n} (x) = 2/ a sin (nπ x / a)

and

E_{n}^{(0)} = n^{2} h^{2} / (8 m a^{2})

n

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