 Solution

Let $\Delta ={\theta }_1-{\theta }_0>0$. For one observation from the unit-variance normal location family,The sample log-likelihood ratio is therefore $n{T}_n$. By the Neyman-Pearson lemma, the level-$\alpha$ most powerful test rejects for a sufficiently large likelihood ratio, equivalently when $T_n>C_{\alpha }$, with $C_{\alpha }$ chosen to give null rejection probability $\alpha$.