 Solution

ID: past-exam-of-the-chemistry-a-course-of-the-university-of-cambridge/2021/paper-1/a2/b/ii/solution

\hat{H} sin \frac{nπ x}{a} = \frac{n ^{2} π ^{2} ℏ ^{2}}{2 m a ^{2}} sin \frac{nπ x}{a} .

n

n = 0

is not an eigenfunction, and negative

n

merely changes an overall sign. Therefore

n = 1, 2, \dots, E_{n} = \frac{n ^{2} π ^{2} ℏ ^{2}}{2 m a ^{2}} .

The corresponding normalized eigenfunction is

2/ a sin (nπ x / a)

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