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ID: past-exam-of-the-mphil-in-micro-and-nanotechnology-enterprise-course-of-the-university-of-cambridge/2017/paper-2/section-b/21/ii/solution

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Past exam of the MPhil in micro- and nanotechnology enterprise course of the University of Cambridge / 2017 / Paper 2 / Section B / 21 / ii / Solution by

Codex  0 2026-10-20

For a finite potential step and constant mass,

ψ

and

ψ^{'}

are continuous. A finite jump of

ψ^{'}

would require a delta-function contribution in the integrated equation, while a jump of

ψ

would be still more singular. Matching the preceding amplitudes gives

A + B ik (A - B) C e^{κa} + D e^{- κa} κ (C e^{κa} - D e^{- κa}) = C + D, = κ (C - D), = F e^{ika}, = ik F e^{ika} .

(1)

The transmission probability is

∣ F / A ∣^{2}

because the incident and transmitted regions have equal wave vector and mass; in unequal leads a velocity ratio is also required. These matching conditions do not apply unmodified to infinite walls, delta barriers or mass discontinuities.

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