 Solution

ID: past-exam-of-the-physics-course-of-the-year-4-of-the-natural-sciences-course-of-the-university-of-cambridge/2016/paper-1/section-a/a8/solution

The wave numbers before and after the drop are

k = 2 m E /ℏ

and

q = 2 m (3 E) /ℏ = 3 k

. Continuity of

ψ

and

ψ^{'}

at the finite step gives

1 + r = t, k (1 - r) = qt, r = \frac{k - q}{k + q} .

Thus the reflected probability flux fraction is

R = ∣ r ∣^{2} = (\frac{1 - 3}{1 + 3})^{2} = 7 - 43 ≃ 0.0718.

(q / k) ∣ t ∣^{2} = 1 - R

, rather than simply

∣ t ∣^{2}

. Reflection is possible at a downward potential step even though both sides are classically allowed.

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