= Quantum propagator
{title2=$K_U(x,t;x',t')=\langle x|U(t,t')|x'\rangle$}
The position-space kernel of the <time-evolution operator> propagates a <wave function> by $\psi(x,t)=\int K_U(x,t;x',t')\psi(x',t')\,dx'$. It tends to $\delta(x-x')$ as $t\to t'$. A retarded convention multiplies it by the <Heaviside step function>: $K_R=\Theta(t-t')K_U$. This convention obeys $(i\hbar\partial_t-H)K_R=i\hbar\delta(t-t')\delta(x-x')$; dividing by $i\hbar$ gives the <Green function> with a unit delta source. The time phase for an <energy eigenstate> is always $e^{-iE(t-t')/\hbar}$.
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