 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/ii/paper-4/30a/a/solution

Codex  0 Created 2026-09-24 Updated 2026-10-03

q < 0

, while boundedness selects exponentially decaying WKB approximation solutions in the forbidden regions

x < a

and

x > b

. The Airy turning-point connection formula at the left turning point matches the left decaying solution to

ψ_{L} (x) \sim C_{L} ∣ q (x) ∣^{- 1/4} sin (\frac{1}{ϵ} \int_{a}^{x} ∣ q (ξ) ∣^{1/2} d ξ + \frac{π}{4})

in the allowed region. Matching the right decaying solution across

b

gives

ψ_{R} (x) \sim C_{R} ∣ q (x) ∣^{- 1/4} sin (\frac{1}{ϵ} \int_{x}^{b} ∣ q (ξ) ∣^{1/2} d ξ + \frac{π}{4}) .

For these expressions to be proportional throughout

(a, b)

, their phases must be compatible. Each simple turning point contributes a phase shift

π /4

\frac{1}{ϵ} \int_{a}^{b} ∣ q (ξ) ∣^{1/2} d ξ = (n + \frac{1}{2}) π, n = 0, 1, 2, \dots .

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