Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 1 / 35D / a / Solution

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

for the stated One-dimensional S-matrix, so

S

is unitary.

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