= Solution
Expand $A(r)=A(r_0)+A'(r_0)s+O(s^2)$ about $qA(r_0)=p_z$. Then $p_z-qA\simeq-qA'(r_0)s$ and <conservation of energy> gives
$$
\boxed{\dot s^2+\Omega^2s^2=\frac{2H}{m}=\Omega^2\rho^2,\quad \rho=\frac{\sqrt{2H/m}}{|\Omega|},\quad s(t)=\rho\cos(|\Omega|t+\phi).}
$$
Here $\Omega=qA'(r_0)/m$ 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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