= Solution
Use $\hbar=1$ and $S_n^\pm=S_n^x\pm iS_n^y$. Operators on different sites commute, and expanding the two products gives $S_n^+S_{n+1}^-+S_n^-S_{n+1}^+=2(S_n^xS_{n+1}^x+S_n^yS_{n+1}^y)$. Therefore the <quantum XX chain> has
$$
\boxed{H=-\frac J2\sum_{n=1}^N(S_n^+S_{n+1}^-+S_n^-S_{n+1}^+)-h\sum_{n=1}^NS_n^z.}
$$
Each exchange moves a reversed <spin> between neighbours, conserving the total number of reversed <spins>. The factor of one half follows from defining $S^\pm$ without a normalizing square root.
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