Hydrogen atom in one dimension

Chemistry in one dimension

“…44 Despite its simplicity, this model has been useful for studying the behavior of many physical systems, such as Rydberg atoms in external fields 45,46 or the dynamics of surface-state electrons in liquid helium 47,48 and its potential application to quantum computing. 49,50 Most work since Loudon has focused on one-electron ions 44,[51][52][53][54][55][56][57] and, to the best of our knowledge, no calculation has been reported for larger chemical systems. In part, this can be attributed to the ongoing controversy concerning the mathematical structure of the eigenfunctions.…”

Dimensionalities of weak solutions in hydrogenic systems

“…Spector and Lee [17] presented a relativistic treatment that removed the problem of infinite binding energy of the ground state. Several other works [18,19,20,21,22,5,23,24] (see also references therein) have discussed this (apparent) simple problem.…”

On Solvable Potentials, Supersymmetry, and the One-Dimensional Hydrogen Atom

“…The results (8) to (13) establish that the one dimensional potential V 1DH can be regarded as stemming from the superpotential W(x) in Equation (8). In [4,29], we have discussed a complete supersymmetric extension of the one-dimensional hydrogen atom problem, with Hamiltonian 2 2 2 2…”

“…For example, it has been claimed that the potntial energy term in Hamiltonian (4) is its own supersymmetric partner [8], or, as in [1], that the Hamiltonian itself cannot really be written since its potential energy function does not exist. On the other hand, it has been proven that it violateswhich the nondegeneracy theorem for onedimensional quantum problems [5], and it has been shown that a superselection rule, analogous to the one preventing the so-called paradox of optical isomers of quantum chemistry, operates in the system [6,[9][10][11][12][13]; see also [14,15] for other similar points of view. The Hamiltonian (4) is not in general self-adjoint (in conventional physics parlance, is not Hermitian).…”

“…This is one of the manifestations of the superselection rule which, among other things, prohibits any superposition of the right ψ 0 + with the left ψ 0 -eigenstates. The energy eigenstates of the problem are given by a Balmer-like formula [4,13,25,26] E n = -1/2 n 2 , n=1,2,3, …, so the ground state energy is E 1 = -1/2.…”