 Solution

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/b/solution

Twice differentiating gives

d^{2} sin (k x) / d x^{2} = - k^{2} sin (k x)

\hat{H} sin (k x) = \frac{ℏ ^{2} k ^{2}}{2 m} sin (k x) .

The boundary condition at zero is automatic. The other requires

sin (k L) = 0

, hence

k L = nπ

. A nonzero eigenfunction can therefore be labelled by

k_{n} = \frac{nπ}{L}, E_{n} = \frac{n ^{2} π ^{2} ℏ ^{2}}{2 m L ^{2}}, n = 1, 2, 3, \dots .

If desired, its normalized form is

2/ L sin (nπ x / L)

. The positive lowest energy follows from confinement and the required spatial variation of a nonzero wavefunction.

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