Experimental results from literature show equidistant energy levels in thin Bi films on surfaces, suggesting a harmonic oscillator description. Yet this conclusion is by no means imperative, especially considering that any measurement only yields energy levels in a finite range and with a nonzero uncertainty. Within this study we review isospectral potentials from the literature and investigate the applicability of the harmonic oscillator hypothesis to recent measurements. First, we describe experimental results from literature by a harmonic oscillator model, obtaining a realistic size and depth of the resulting quantum well. Second, we use the shift-operator approach to calculate anharmonic non-polynomial potentials producing (partly) equidistant spectra. We discuss different potential types and interpret the possible modeling applications. Finally, by applying nth order perturbation theory we show that exactly equidistant eigenenergies cannot be achieved by polynomial potentials, except by the harmonic oscillator potential. In summary, we aim to give an overview over which conclusions may be drawn from the experimental determination of energy levels and which may not.

文献中的实验结果显示表面上的薄 Bi 膜中的等距能级，表明谐振子描述。然而，这个结论绝不是强制性的，特别是考虑到任何测量都只产生有限范围内的能量水平，并且具有非零不确定性。在这项研究中，我们回顾了文献中的等谱电位，并研究了谐振子假设对最近测量的适用性。首先，我们通过谐振子模型描述了文献中的实验结果，获得了所得量子阱的真实尺寸和深度。其次，我们使用移位运算符方法来计算产生（部分）等距谱的非谐非多项式势。我们讨论不同的潜在类型并解释可能的建模应用。n阶微扰理论表明 ，除谐振子势之外，多项式势不能实现精确等距的本征能。总之，我们旨在概述哪些结论可以从能级的实验确定中得出，哪些不能。