Time-ordered exponential
= Time-ordered exponential
{title2=$U(t,t_0)=\mathcal T\exp[-i\int_{t_0}^t H(s)ds/\hbar]$}
The time-ordered exponential solves $i\hbar\partial_tU=H(t)U$ with initial condition $U(t_0,t_0)=I$. Its expansion is the <Dyson series>. For bounded continuous <Hamiltonian operators> the series converges, as in <differentiation of the Dyson time-ordered exponential>. If the generators commute at all times, time ordering is unnecessary and an ordinary operator exponential suffices.