 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/2017/s18/2/c/solution

Matching

ψ

and

ψ^{'}

at both walls, or propagating the pair

(ψ, ψ^{'})

through the interior by the matrix

(c o s ζ a - ζ s i n ζ a s i n ζ a / ζ c o s ζ a)

, gives

t (k) = \frac{e ^{- ika}}{D ( k )}, D (k) = cos (ζ a) - i \frac{k ^{2} + ζ ^{2}}{2 k ζ} sin (ζ a) .

Here

ζ^{2} = k^{2} + 2 m U_{0} / ℏ^{2}

. For real positive

k

, taking the modulus gives

T = [cos^{2} ζ a + \frac{( k ^{2} + ζ ^{2} ) ^{2}}{4 k ^{2} ζ ^{2}} sin^{2} ζ a]^{- 1} .

The equal exterior velocities make this also the current transmission coefficient.

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