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.

翻译：我们研究了用于求解Schrödinger方程的蛙跳时域有限差分方法的概率与能量守恒性质。提出了区域内总数值概率与能量、概率流密度与功率通过边界通量的表达式，证明了在适当条件下这些表达式满足概率与能量的守恒性，并揭示了其与基于Courant-Friedrichs-Lewy稳定性条件的关联。论证表明，该研究成果可发展为基于时域有限差分法求解Schrödinger方程的先进算法中建立模块化稳定性分析框架的基础。