Spinless Hydrogen: Energy Eigenfunctions

Brian P. Anderson

Yu-Xian, WeiZhiyong

Hans Albrecht Bethe

H. N. Núñez‐Yépez, Carlos Alejandro Vargas, A. L. Salas Brito

A momentum representation treatment of the one-dimensional hydrogen atom with Coulomb interaction is presented. The authors obtain the energy levels and bound-state eigenfunctions for the problem and discuss the general properties of the solution. The energy spectrum is discrete and given by the Balmer formula. They show that, despite the symmetry of the potential, it is impossible to find eigenfunctions with definite parity for the system.

Brian P. Anderson

Yi-Xuan 艺轩 Yin 殷, Tiantian Wang, Biao Wu

Abstract We show that a highly-excited energy eigenfunction ψ nlm ( r ) of hydrogen atom can be approximated as an equal-weight superposition of classical elliptic orbits with energy E n and angular momentum L = l ( l + 1 ) ħ , and z component of angular momentum L z = mħ . This correspondence is established by comparing the quantum probability distribution | ψ nlm ( r )| 2 and the classical probability distribution p c ( r ) of an ensemble of such orbits. This finding illustrates a general principle: in the semi-classical limit, an energy eigenstate of a quantum system is in general reduced to a collection of classical orbits, rather than a single classical orbit.