 Solution

ID: past-exam-of-the-chemistry-course-of-the-year-4-of-the-natural-sciences-course-of-the-university-of-cambridge/2022/paper-3/section-f/46/a/iii/solution

Write

ψ = C e^{i αx} + D e^{- i αx}

inside the barrier. Matching the wavefunction and derivative gives the four equations

A + B = C + D, k (A - B) = α (C - D),

C e^{i α a} + D e^{- i α a} = F e^{ika}, α (C e^{i α a} - D e^{- i α a}) = k F e^{ika} .

For sub-barrier energy the same equations hold by analytic continuation to imaginary

α

. No coefficient solution is needed for these matching conditions.

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