Schrödinger equation from a short-time path integral
= Schrödinger equation from a short-time path integral
{c}
{title2=$i\partial_t\psi=(-\tfrac12\partial_q^2+V)\psi$}
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>.