 Solution

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/2015/s3/1/a/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 / 2015 / S3 / 1 / a / Solution by

Codex  0 2026-10-25

The reflection-one result describes a semi-infinite potential step, rather than a finite-width barrier. Define

k = 2 m E /ℏ

and

κ = 2 m (V_{0} - E) /ℏ

. For

E > 0

, take

ϕ (x) = {e^{ik x} + r e^{- ik x}, c e^{- κ x}, x < 0, x > 0.

The growing exponential is excluded by boundedness as

x \to \infty

. Continuity of

ϕ

and

ϕ^{'}

at the finite step gives

1 + r = c

and

ik (1 - r) = - κ c

, hence

r = \frac{k - iκ}{k + iκ}, c = \frac{2 k}{k + iκ}, R = ∣ r ∣^{2} = 1.

The probability current of the decaying tail is zero; the reflected and incident currents have equal magnitude. This agrees with the classical reflection coefficient even though the spatial wave function differs from a classical trajectory.

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