 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/ib/paper-3/6b/a/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

H = - \frac{ℏ ^{2}}{2 m} \frac{d ^{2}}{d x ^{2}} .

Since

χ^{''} = - k^{2} χ

Hχ = \frac{ℏ ^{2} k ^{2}}{2 m} χ .

Thus

E = \frac{ℏ ^{2} k ^{2}}{2 m} \geq 0, p = ℏ k,

where the momentum follows because

- i ℏ d χ / d x = ℏ k χ

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