Normalized short-time Schrödinger kernel ($K_{\Delta t}(q,q^{\prime })=(2\pi i\Delta t{)}^{-1/2}{e}^{i(q-q^{\prime }{)}^2/(2\Delta t)-i\Delta tV(q^{\prime })}$)

For unit mass and $\hslash =1$, this prepoint kernel tends to the identity and generates the Time-dependent Schrodinger equation. The square-root branch follows the damped Fresnel integral. Its normalized moments are $1$, $0$, and $i\Delta t$ at orders zero, one and two; these fix both the normalization and the kinetic sign.

Expand a smooth wavefunction and potential in the normalized short-time Schrödinger kernel. Its odd moment vanishes and its second moment is $i\Delta t$, so one step changes the wavefunction by $i\Delta t{\psi }^{\prime \prime }/2-i\Delta tV\psi +O(\Delta t^2)$. Division by the time step gives the Time-dependent Schrodinger equation.