The computation of allowed energy levels for a particle bounded to a finite square well potential is ubiquitous in modern physics and introductory quantum mechanics textbooks. For stationary bound states, the matching conditions for the wave functions lead to a pair of transcendental equations whose roots correspond to the energy eigenvalues. However, the graphical solutions available do not make clear the dependence of the energies on the well potential parameters. In this note, I present a simpler graphical solution involving only one dimensionless parameter that determines a straight-line crossing identical sinusoidal curves. I then reduce this solution to a single cosine curve, and from a three-point interpolation, I derive an approximate formula for all energy levels valid for any square quantum well and that demands only a pocket calculator.

中文翻译：

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有限正方形量子阱中能量特征值的更简单图形解和近似公式

计算限定于有限平方势阱的粒子的允许能级在现代物理学和量子力学入门教科书中无处不在。对于平稳边界状态，波函数的匹配条件导致一对超越方程，其根对应于能量特征值。然而，可用的图形解决方案并没有明确能量对阱势参数的依赖性。在这篇笔记中，我提出了一个更简单的图形解决方案，它只涉及一个无量纲参数，该参数确定与相同正弦曲线相交的直线。然后我将这个解简化为一条余弦曲线，并从三点插值中推导出一个适用于任何方形量子阱的所有能级的近似公式，并且只需要一个袖珍计算器。