Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2014 / Paper 1 / 5 / ii / Solution

The absolutely convergent matrix exponential series givesfor Hermitian $H$. The exponents $iH$ and $-iH$ commute, so their exponentials multiply to the exponential of their sum:Hence $U$ is unitary. Alternatively, an orthonormal eigenbasis of $H$ turns the exponential eigenvalues into $e^{i{\lambda }_j}$, all on the unit circle.