Quantum Mechanics of One‐Dimensional Two‐Particle Models. Electrons Interacting in an Infinite Square Well

Articles you may be interested in Pairing interaction and reaction mechanism for one-and two-particle transfer reactions: A simple model in one dimension AIP Conf. Two charged particles in a one-dimensional well Am. Density functional theory of one-dimensional two-particle systems Am. One‐dimensional hydrogen atom in an infinite square well Am. Solutions of SchrOdinger's equation for the system of two parti~les bound in a one-dime~sion.al infinite square well and repelling each other with a Coulomb force are obtaIned by the m~thod of. fimte d~fferences. For the case of a 4.0-a.u. well, the energy levels are shifted above those of the nonmteractmg-partlcle m~del by as much as a factor of 4 although the exc~tation energies are on.ly about ~O% greater. The analytIcal form of the solutions is also obtained and it IS shown that every eigenstate IS doubly degenerate d~e to the "pathological" nature of the one-dimensional Coulomb potential. This degeneracy i~ verified numencally by the finite-difference method. The properties of the model system are compared With ~hose of t~e free-electron and hard-sphere models; perturbation and variational treatments are a~so c~rned out. usmg the hard-sphere Hamiltonian as a zero!h-order approximation: The lowest several fin!~e-dl~e~ence elge~v~lues converge from below with decreasmg mesh size to ener?Ies ?elow those o~ th~ best lm~ar vanatlO~al function consisting of hard-sphere eigenfunctions. The fimte-dlfference.solu~lOns m general gIve expectatIOn values and matrix elements more accurately than do the other approXImatIOns.