 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/s7/1/e/solution

Expand

A (r) = A (r_{0}) + A^{'} (r_{0}) s + O (s^{2})

about

q A (r_{0}) = p_{z}

. Then

p_{z} - q A ≃ - q A^{'} (r_{0}) s

\overset{s}{˙}^{2} + Ω^{2} s^{2} = \frac{2 H}{m} = Ω^{2} ρ^{2}, ρ = \frac{2 H / m}{∣Ω∣}, s (t) = ρ cos (∣Ω∣ t + ϕ) .

Here

Ω = q A^{'} (r_{0}) / m

can be signed, but the physical oscillation frequency and amplitude are nonnegative. The small-energy condition ensures that this local expansion samples only a narrow range about

r_{0}

, provided the field varies smoothly there.

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