演示环境（数据为演示用途，非真实数据）

本文通过开发参数化维度映射策略，利用COMSOL Multiphysics软件实现了对一维无限深势阱中多粒子量子系统的数值模拟研究，通过建立粒子坐标与软件内置空间维度的非定域对应关系，克服了高维量子系统的计算复杂性。通过对单电子、双电子和三电子系统的分析，探讨了库仑相互作用对波函数分布和能级结构的影响。结果表明，在双电子系统中，库仑相互作用使波函数概率密度集中在电子距离较远的区域，并显著提高了系统总能量。对于不同电子数目，能量-宽度依赖关系(即基态能量随势阱宽度的变化规律)可以分为动能和势能两部分的贡献，前者与势阱宽度的二次方成反比，后者与势阱宽度的一次方成反比。通过引入有效幂次分析，发现随着电子数增加，能量与势阱宽度的关系趋于线性反比。该方法学创新显著降低了多体量子系统的教学演示难度，研究结果为深入理解低维量子系统中电子间相互作用的影响提供了新的视角。

This paper presents a numerical simulation study of a multi-particle quantum system in a one-dimensional infinite potential well using COMSOL Multiphysics software by developing a parameterized dimension mapping strategy. By establishing a non-separable correspondence between particle coordinates and the built-in spatial dimensions of the software, the computational complexity of high-dimensional quantum systems is overcome. The analysis of single-electron, two-electron, and three-electron systems reveals the impact of Coulomb interactions on the wave function distribution and energy level structure. It is found that in the two-electron system, Coulomb interaction concentrates the probability density of the wave function in regions where the electrons are farther apart and significantly increases the total energy of the system. For different numbers of electrons, the energy-width dependence relationship(the rule of how the ground state energy changes with the width of the potential well) can be divided into contributions from kinetic and potential energy. The former is inversely proportional to the square of the potential well width, while the latter is inversely proportional to the first power of the potential well width. By introducing effective power analysis, it is discovered that as the number of electrons increases, the relationship between energy and potential well width tends to be inversely proportional in a linear manner. This methodological innovation significantly reduces the difficulty of teaching demonstrations of multi-body quantum systems, and the research findings provide a new perspective for a deeper understanding of the effects of electron interactions in low-dimensional quantum systems.