A remarkable mathematical property -- somehow hidden and recently rediscovered -- allows obtaining the eigenvectors of a Hermitian matrix directly from their eigenvalues. That opens the possibility to get the wavefunctions from the spectrum, an elusive goal of many fields in physics. Here, the formula is assessed for simple potentials, recovering the theoretical wavefunctions within machine accuracy. A striking feature of this eigenvalue--eigenvector relation is that it does not require knowing any of the entries of the working matrix. However, it requires the knowledge of the eigenvalues of the minor matrices (in which a row and a column have been deleted from the original matrix). We found a pattern in these sub-matrices spectra, allowing to get the eigenvectors analytically. The physical information hidden behind this pattern is analyzed.

中文翻译：

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来自能量的波函数：在简单势能中的应用

一个显着的数学属性——不知何故隐藏起来，最近又被重新发现——允许直接从厄米矩阵的特征值中获得它们的特征向量。这开启了从光谱中获取波函数的可能性，这是物理学中许多领域难以实现的目标。在这里，公式针对简单势进行评估，在机器精度范围内恢复理论波函数。这种特征值-特征向量关系的一个显着特点是它不需要知道工作矩阵的任何条目。但是，它需要知道次要矩阵的特征值（其中从原始矩阵中删除了一行和一列）。我们在这些子矩阵光谱中发现了一个模式，允许分析地获得特征向量。分析隐藏在该模式背后的物理信息。