Fast Solution of Schrödinger Equation Based on Complex One-Dimensional Potential Wells

Abstract The Schrödinger equation (SE) is a fundamental equation in quantum mechanics that describes the behavior of quantum systems. Solving the SE is crucial for understanding quantum phenomena, but it is a challenging task, particularly for complex potential wells, which require significant computational resources to obtain high accuracy by using traditional numerical methods. Recently, deep neural networks, such as Fourier neural operator (FNO), have made significant breakthroughs in solving partial differential equations. This paper explores solving the one-dimensional SE with diversified but classified potential wells using FNO. Gaussian random fields are used to generate the potential function, and then FNO is used to learn the mapping relationship between different potential functions and the solutions to the SE. The results show that the trained FNO can accurately output the solution to the SE under the corresponding potential function in a very short time, and the output results are completely consistent with expectations. Moreover, using FNO to solve the SE is significantly faster than traditional solvers, especially when the grid size is large. Therefore, in the future, FNO will provide a new solving method in complex, high-dimensional cases, such as two-dimensional, three-dimensional, and time-dependent SEs.