 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/ii/paper-1/35d/a/solution

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

For incidence from the left, normalize the asymptotic wavefunction by

ψ (x) \sim e^{ik x} + r e^{- ik x} (x \to - \infty), ψ (x) \sim t e^{ik x} (x \to + \infty) .

For incidence from the right, define

r^{'}, t^{'}

analogously with the directions reversed. Conservation of the probability current gives

∣ r ∣^{2} + ∣ t ∣^{2} = 1, ∣ r^{'} ∣^{2} + ∣ t^{'} ∣^{2} = 1.

Applying the same conserved sesquilinear current to the two scattering solutions gives

t^{' *} r + r^{' *} t = 0.

These are exactly the column orthonormality relations

S^{†} S = I

S

is unitary.

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