= Spectral resolution of a quantum propagator
{title2=$K_U(x,t;x',t')=\sum_\alpha\varphi_\alpha(x)\varphi_\alpha^*(x')e^{-iE_\alpha(t-t')/\hbar}$}
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.
Back to article page
