= Solution
For collinear antiferromagnetic order in the $xy$ plane, take $\mathbf S_n=(-1)^nS\hat{\mathbf x}$. Nearest neighbors are antiparallel, giving $-S^2$, whereas second neighbors lie on the same sublattice and remain parallel, giving $+S^2$. Thus
$$
\boxed{E_{\rm AF}=J_1S^2-J_2S^2=S^2(J_1-J_2).}
$$
The result follows from substituting into the specified XY <Hamiltonian operator>, regardless of whether that trial state is the global minimum for the chosen exchanges. Comparing trial-state energies does not eliminate the possibility of a lower helical state.
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