The one-dimensional (1D) hydrogen atom with potential energy V ( q ) = − e 2 /| q |, with e the electron charge and q its position coordinate, has been a source of discussion and controversy for more than 55 years. A number of incorrect claims have been made about its spectrum; for example, that its ground state has infinite binding energy, that bound states associated with a continuum of negative energy values exist, or that anomalous non-Balmer energy levels are present in the system. Given such claims and the ongoing controversy, we have re-analysed the 1D hydrogen atom, first from a classical and then from a quantum perspective both in the coordinate and in the momentum representations. This work exhibits that certain classical properties of the system may serve to clarify the properties of the quantum problem. Using the Dirichlet boundary condition, we show that the singularity of the potential prevents any relation between the ...

中文翻译：

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经典和量子空间分裂：一维氢原子

势能为V（q）=-e 2 / |的一维（1D）氢原子 在过去的55年中，带有电子电荷和q的位置坐标的q |一直是讨论和争议的源头。关于它的频谱，有许多错误的说法；例如，其基态具有无限的结合能，存在与连续的负能量值关联的结合态，或者系统中存在异常的非巴尔默能级。鉴于此类主张和持续存在的争议，我们从坐标和动量表示两方面，首先从经典角度，然后从量子角度，重新分析了一维氢原子。这项工作表明，系统的某些经典性质可以用来阐明量子问题的性质。