AI 中文总结
该研究对比了分形结构（谢尔宾斯基地毯）与均匀晶格上单个量子粒子的动力学，发现分形晶格会使粒子被自相似地捕获在初始位置附近，而均匀晶格上粒子会弹道式到达对角。
AI 中文摘要
我们研究有限尺寸方形晶格上单个量子粒子的动力学，该晶格具有类似谢尔宾斯基地毯的分形结构，并将其与相同尺寸均匀晶格上的动力学进行比较。对于初始定域在晶格一角的粒子，我们采用区域积分概率监测其在初始角和对角附近被探测到的概率，从而实现跨分形阶的一致比较。在均匀晶格上，粒子以与晶格尺寸成正比的时间弹道式到达对角；而在谢尔宾斯基晶格上，粒子则越来越多地被限制在初始位置附近。我们表明，这种捕获现象在晶格的整个角区域层次中以自相似方式累积。
英文摘要
We study the dynamics of a single quantum particle on a finite-size square lattice with a fractal structure resembling the Sierpiński carpet, and compare it to the dynamics on a uniform lattice of the same size. For a particle initially localized at a corner of the lattice, we monitor the probability of finding it near the initial and opposite corners using zone-integrated probabilities, allowing a consistent comparison across fractal orders. While on the uniform lattice the particle reaches the opposite corner ballistically, in a time proportional to the lattice size, on the Sierpiński lattice it becomes increasingly confined to the vicinity of its initial position. We show that this trapping builds up self-similarly across the whole hierarchy of corner zones of the lattice.
