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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/ib/paper-3/6d/b/solution

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Past exam of the mathematics course of the University of Cambridge / 2023 / ib / Paper 3 / 6D / b / Solution by

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

∣ x ∣ \to \infty

, the equation approaches

χ^{''} + \frac{2 m}{ℏ ^{2}} (E - U_{0}) χ = 0.

(1)

E > U_{0}

, the asymptotic solutions are oscillatory and do not decay, so there are no bound states. If

0 < E < U_{0}

, the asymptotic solutions are exponential; one decaying branch can be selected at each end. Bound states are therefore possible, although only energies satisfying the global matching conditions actually occur.

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