Compute \(\left(\frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2}i\right)^2\)
( sqrt (2) / 2 + sqrt (2) /2 i )^2 =
(sqrt (2) + sqrt (2) i) / 4 =
(sqrt (2) + sqrt (2) i) ( sqrt (2) + sqrt (2) i) / 4 =
(2 + 2i + 2i + 2i^2) / 4 =
( 2 + 4i - 2) / 4 =
4i / 4 =
i
