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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/i/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 / i / Solution by

Codex  0 2026-10-20

Take

0 < E < V_{0}

and equal electron mass in all regions. Define

k = 2 m E /ℏ

and

κ = 2 m (V_{0} - E) /ℏ

. Then

I : II : III : - \frac{ℏ ^{2}}{2 m} ψ^{''} = E ψ, ψ_{I} = A e^{ik x} + B e^{- ik x}, - \frac{ℏ ^{2}}{2 m} ψ^{''} + V_{0} ψ = E ψ, ψ_{II} = C e^{κ x} + D e^{- κ x}, - \frac{ℏ ^{2}}{2 m} ψ^{''} = E ψ, ψ_{III} = F e^{ik x} .

(1)

There is no wave incident from the far right. Region I contains oscillatory incident and reflected waves; region II is evanescent, with real exponentials rather than propagating plane waves. Both exponential terms are allowed in a finite barrier; deleting the growing term before boundary matching is generally incorrect. A complex superposition in the barrier can carry a nonzero conserved tunnelling current.

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