Solution
= Solution
Writing $N=2^m$, the Hadamards first give
$$
(H^{\otimes m}\otimes I)|0\rangle^{\otimes m}|\psi\rangle
=\frac1{\sqrt N}\sum_{k=0}^{N-1}|k\rangle|\psi\rangle.
$$
Since $U^k|\psi\rangle=e^{2\pi ik\theta}|\psi\rangle$ and $\theta=j/N$,
$$
\boxed{|\Phi\rangle=
\frac1{\sqrt N}\sum_{k=0}^{N-1}
e^{2\pi ijk/N}|k\rangle|\psi\rangle.}
$$