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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/s18/1/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 / 2018 / S18 / 1 / a / Solution by

Codex  0 2026-10-25

Assume a real, time-independent potential with the same zero asymptote on both sides, so

k = 2 m E /ℏ

is the same incident and transmitted wave number. For unit incoming amplitudes,

ψ_{L} \sim {e^{ik x} + r_{L} e^{- ik x}, t_{L} e^{ik x}, x \to - \infty, x \to + \infty, ψ_{R} \sim {t_{R} e^{- ik x}, e^{- ik x} + r_{R} e^{ik x}, x \to - \infty, x \to + \infty.

Multiplying the two Schrödinger equations crosswise and subtracting shows that the Wronskian

W = ψ_{L} ψ_{R}^{'} - ψ_{L}^{'} ψ_{R}

is constant. Its two asymptotic values are

- 2 ik t_{R}

and

- 2 ik t_{L}

, hence

t_{L} = t_{R}

.

For a real potential the probability current

j = (ℏ/ m) Im (ψ^{*} ψ^{'})

is conserved, giving

∣ r_{L} ∣^{2} + ∣ t_{L} ∣^{2} = ∣ r_{R} ∣^{2} + ∣ t_{R} ∣^{2} = 1

. Therefore

T_{L} = T_{R} = ∣ t ∣^{2}, R_{L} = R_{R} = 1 - T .

The reflection amplitudes can have different phases. Reality or self-adjointness is essential for the reflection-probability conclusion: a complex absorbing potential need not have equal left and right reflectances. Equal asymptotic speeds are also used in identifying transmission with the squared amplitude.

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