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The quantum-mechanical system which consists of a particle in one dimension subjected to a Coulomb attraction (the one-dimensional hydrogen atom) is shown to have a ground state of infinite binding energy, all the excited bound states of the system having a twofold degeneracy. The breakdown of the theorem that a one-dimensional system cannot have degeneracy is examined. The treatment illustrates a number of properties common to the quantum mechanics of one-dimensional systems.
R. Loudon (Tue,) studied this question.