 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/ii/paper-1/39d/c/i/solution

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

x

from the axis, the gap is

h (x) = h_{0} + a - a^{2} - x^{2} .

For

∣ x ∣ ≪ a

a^{2} - x^{2} = a - \frac{x ^{2}}{2 a} + O (x^{4} / a^{3}),

h (x) ≃ h_{0} + \frac{x ^{2}}{2 a} = h_{0} (1 + \frac{x ^{2}}{2 a h _{0}}) .

The narrow-gap length scale is therefore

(a h_{0})^{1/2}

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