 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/d/solution

A stationary evanescent wave does not assign a classical trajectory or a real local speed inside the forbidden region. The dwell time in a semi-infinite potential step measures probability stored in

x > 0

divided by incident probability current. For incident amplitude one,

τ_{D} = \frac{\int _{0}^{\infty} ∣ c ∣ ^{2} e ^{- 2 κ x} d x}{ℏ k / m} = \frac{2 mk}{ℏ κ ( k ^{2} + κ ^{2} )} .

This gives a time in terms of

m, E, V_{0}

through the definitions of

k, κ

. A simpler dimensional estimate is the penetration length divided by the incident speed,

Δ x / v_{in} = m / (ℏ kκ) = ℏ/ [2 E (V_{0} - E)]

. The dwell time equals

(2 E / V_{0})

times that estimate. Different operational timing definitions need not agree; treating the imaginary under-barrier momentum as a real particle velocity would not justify a unique traversal time.

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