The motion of quantum mechanical systems in physical sciences is described by partial differential equations, usually of second order with respect to spatial coordinates. The required solutions of the time-dependent type of equations are, in general, functions of the temporal variable t and the state vectors determining the positions of the system’s particles at the time t. Only a small number, however of those differential equations can however be solved analytically, while the majority of them must be solved numerically by applying specific very advanced integration techniques. Among these equations, the fundamental Schrödinger equation offers great insight towards numerical solving. In this work we present an effective method that solves numerically the time-independent Schrodinger equation on the basis of neural networks techniques. Analytical and numerical results for the radial part of this equation with the corresponding energies are then compared so as to estimate the performance of our method.

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在 Python 中使用神经网络数值求解薛定谔方程

物理科学中量子力学系统的运动是通过偏微分方程描述的，通常是相对于空间坐标的二阶方程。一般来说，时间相关类型方程的所需解是时间变量的函数t状态向量决定系统粒子在时间 t 的位置。然而，这些微分方程中只有一小部分可以解析求解，而大多数微分方程必须通过应用特定的非常先进的积分技术来数值求解。在这些方程中，基本的薛定谔方程为数值求解提供了深刻的见解。在这项工作中，我们提出了一种基于神经网络技术数值求解与时间无关的薛定谔方程的有效方法。然后将该方程的径向部分的分析结果和数值结果与相应的能量进行比较，以估计我们方法的性能。