The one-dimensional stationary Schrödinger equation, coupled with the transparent boundary conditions and self-consistently linked to the Poisson equation, is a well-established model for describing quantum effects. In this paper, we introduce a general framework for constructing arbitrarily high-order finite difference schemes on arbitrary grids, whether they are uniform or nonuniform, inspired by the analytic discrete transparent boundary conditions [M. Guo et al., arXiv:2411.13175]. To enhance the accuracy of approximations while keeping computational costs low, we develop an optimal mesh refinement strategy that balances the need to resolve intervals with large gradient and high curvature of the potential function. We further propose an adaptive mesh refinement algorithm to solve the 1D Schr¨odinger-Poisson problem, incorporating a third-order compact finite difference discretization of the Schrödinger-Poisson system based on nonuniform grids, and a given mesh refinement strategy. Numerical experiments on a resonant tunneling diode are conducted to validate the algorithm’s high efficiency and to study the I-V characteristic curve of the device.