 Solution

ID: past-exam-of-the-advanced-quantum-physics-course-of-the-university-of-cambridge/2014/paper-1/1/b/solution

An impulse integrated across its full support produces the impulsive quantum kick

K = e^{- i \hat{U} /ℏ}

. Thus the leading transition amplitude is

- i U_{21} /ℏ

and

P_{1 \to 2} = \frac{∣ U _{21} ∣ ^{2}}{ℏ ^{2}} + O (∣ U_{21} /ℏ ∣^{4}) .

Here

U_{21}

P = sin^{2} (∣ U_{21} ∣/ℏ)

for a Hermitian two-state impulse with zero diagonal. No energy denominator occurs: the impulse is instantaneous.

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