Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 58 / 4 / b / i / Solution

For a unitary operator $W$, $W{W}^{†}=I$. Thus for every nonnegative integer $j$,Insert this into the convergent matrix exponential series in the finite-dimensional qubit setting:Consequently unitary conjugation commutes with the exponential:For an unbounded self-adjoint Hamiltonian, the same identity follows from the spectral theorem for normal operators, with the domain transported by $W$; no unbounded power-series manipulation is needed.