Quantum propagator ($K_U(x,t;x^{\prime },t^{\prime })=⟨x|U(t,t^{\prime })|x^{\prime }⟩$)

The position-space kernel of the time-evolution operator propagates a wave function by $\psi (x,t)=\int K_U(x,t;x^{\prime },t^{\prime })\psi (x^{\prime },t^{\prime })d{x}^{\prime }$. It tends to $\delta (x-x^{\prime })$ as $t\to t^{\prime }$. A retarded convention multiplies it by the Heaviside step function: $K_R=\Theta (t-t^{\prime })K_U$. This convention obeys $(i\hslash {\partial }_t-H)K_R=i\hslash \delta (t-t^{\prime })\delta (x-x^{\prime })$; dividing by $i\hslash$ gives the Green function with a unit delta source. The time phase for an energy eigenstate is always $e^{-iE(t-t^{\prime })/\hslash }$.

Insert a complete resolution of identity in energy eigenstates on either side of the time-evolution operator. Each state evolves by its scalar energy phase, yielding the displayed quantum propagator. Its spectral sum includes an integral over any continuous spectrum, with the measure fixed by the continuum normalization. Omitting the scattering states when bound states are also present destroys completeness and the delta-function short-time limit.