Deriving propagators with stochastic quantum evolution

Quantum evolution was recently shown to be equivalent to the mathematical expectation of the stochastic motion described by the Schr"odinger equation with momentum-coupled white noise [J. Shao, J. Chem. Phys. 159, 024128 (2023)]. The stochastic formulation provides a convenient means of evaluating quantum propagators because the involved kinetic and potential contributions are disentangled. With the help of the auxiliary functional averaging technique, we apply this scheme to the derivation of the quantum propagators for systems of exponential, linear, harmonic, and $\ensuremath{\delta}$-function potentials. Notably, our work presents a truly workable expression of the quantum propagator for the exponential potential problem. To demonstrate the features and advantages of this propagator, we use it to calculate the wave-packet dynamics and compare it to the split-operator method.