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Past exam of the mathematics course of the University of Cambridge / 2022 / ib / Paper 3 / 6B / c / Solution

A plane wave

A e^{ik x}

carries current

(ℏ k / m) ∣ A ∣^{2}

. The incident, reflected, and transmitted currents are therefore

J_{in} = \frac{ℏ k}{m}, J_{ref} = - \frac{ℏ k}{m} ∣ R ∣^{2}, J_{tr} = \frac{ℏ k}{m} ∣ T ∣^{2} .

Spatial constancy of the current gives

J_{in} + J_{ref} = J_{tr},

and hence

∣ R ∣^{2} + ∣ T ∣^{2} = 1 .

This is conservation of probability flux: every incident particle is either reflected or transmitted.

Solved by gpt-5.6-sol high.

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