The quantum mechanical bound states of the −α/x2 potential are truly anomalous. We revisit this problem by adopting a slightly modified version of this potential, one that adopts a cutoff in the potential arbitrarily close to the origin. The resulting solutions are completely well-defined and “normal,” and provide us with additional insight into the solutions of the “bare” −α/x2 potential. We present results here as a case study in undergraduate research—two independent methodologies are used: one analytical (albeit with very unfamiliar non-elementary functions) and one numerical (with a very straightforward methodology). These play complementary roles in arriving at solutions and achieving insights into this problem.

中文翻译：

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-α/x2 势能束缚态的数值分析研究

-α/x2 势的量子力学束缚态确实是反常的。我们通过采用这种势能的一个稍微修改的版本来重新审视这个问题，一种在任意接近原点的势能中采用截止。由此产生的解决方案是完全明确定义的和“正常的”，并为我们提供了对“裸”-α/x2 势的解决方案的额外洞察。我们在此将结果作为本科研究的案例研究——使用了两种独立的方法：一种是分析的（尽管使用非常陌生的非初等函数）和一种数值的（使用非常简单的方法）。这些在达成解决方案和深入了解这个问题方面发挥着互补作用。