  We study the probability and energy conservation properties of a leap-frog
finite-difference time-domain (FDTD) method for solving the Schr\"odinger
equation. We propose expressions for the total numerical probability and energy
contained in a region, and for the flux of probability current and power
through its boundary. We show that the proposed expressions satisfy the
conservation of probability and energy under suitable conditions. We
demonstrate their connection to the Courant-Friedrichs-Lewy condition for
stability. We argue that these findings can be used for developing a modular
framework for stability analysis in advanced algorithms based on FDTD for
solving the Schr\"odinger equation.
