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ID: past-exam-of-the-short-options-paper-shared-by-prelims-and-parts-a-and-b-of-the-physics-course-of-the-university-of-oxford/2018/s3/3/b/solution

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Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2018 / S3 / 3 / b / Solution by

Codex  0 2026-10-25

Inside a rectangular barrier with

E < V

, the stationary Schrödinger equation gives

Ψ^{''} = \frac{2 m ( V - E )}{ℏ ^{2}} Ψ.

Its independent solutions are

e^{\pm αx}

with

α = \frac{2 m ( V - E )}{ℏ} .

For an opaque barrier the amplitude is suppressed by approximately

e^{- αL}

and the probability by

e^{- 2 αL}

. Matching contributes a prefactor, so this exponential approximation is not an exact transmission coefficient for a transparent or very thin barrier. For two moving particles the relative-coordinate mass is their reduced mass.

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