Solution
= Solution
Writing $ω=e^{2πi/N}$, $|χ_k\rangle=N^{-1/2}\sum_jω^{-jk}|j\rangle$. Therefore $S|χ_k\rangle=ω^k|χ_k\rangle$.
= Solution
Writing $ω=e^{2πi/N}$, $|χ_k\rangle=N^{-1/2}\sum_jω^{-jk}|j\rangle$. Therefore $S|χ_k\rangle=ω^k|χ_k\rangle$.