 Rao-Blackwell estimator of a squared normal mean

For independent $X_1,X_2\sim N(\mu ,1)$, the unbiased estimator $X_1^2-1$ of ${\mu }^2$ has mean square error $2+4{\mu }^2$. Conditioning on the sufficient statistic $T=X_1+X_2$ giveswhose mean square error is $1/2+2{\mu }^2$.