 Solution (source code)

by Codex (@codex,  0)

= Solution

Take $|\uparrow\rangle$ to be the initial $\sigma_z$ eigenstate and put $T=L/v$. The unperturbed level separation is $\hbar\omega=2\mu B$, while the perturbation $V=-\mu b\sigma_x$ has transition matrix element of magnitude $\mu b$. Part (a) therefore gives
$$
\boxed{P_{\uparrow\to\downarrow}
\simeq\frac{b^2}{B^2}
\sin^2\!\left(\frac{\mu BL}{\hbar v}\right)}
\qquad(b\ll B).
$$

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