Schrödinger equation from a short-time path integral ($i{\partial }_t\psi =(-\frac{1}{2}{\partial }_q^2+V)\psi$)

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.