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European Journal of Physics

The one-dimensional hydrogen atom in momentum representation

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

Published Jul 1, 1987• 41 citations
Publisher

Abstract

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.

DOI
10.1088/0143-0807/8/3/008

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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.
Abstract In this chapter, we go to three dimensions in space and look at the solution of the time independent Schrödinger equation for the hydrogen atom. The Hamiltonian is then the kinetic energy plus the potential energy due to the Coulomb coupling between the positively charged nucleus and the electron. We construct the angular momentum operator and find that the partial differential equation for the angular momentum eigenfunctions of the spherical coordinates θ,ϕ is the same as the angular part of the ∇2 operator in spherical coordinates. The angular momentum eigenfunctions are the spherical harmonics, with two quantum numbers, l and m, and the solution to the radial part of the Hamiltonian including the Coulomb potential are Laguerre polynomials with one quantum number, called the principle quantum number, n. The hydrogen wave function is the product of a Laguerre polynomial and a spherical harmonic with three quantum numbers. Since these are two- and three-dimensional functions for angular momentum and hydrogen respectively, they are best understood in a series of plots. The chapter concludes by giving the historical letter names to specific orbitals, since they continue to be used today.
The hydrogen atom is a fundamental exactly soluble system for which the Wigner function, being a quantum analogue of the joint probability distribution of position and momentum, is unknown. In this paper, we present an effective method of calculating the Wigner function, for all bound states of the nonrelativistic hydrogen atom. The formal similarity between the eigenfunctions of the nonrelativistic hydrogen atom in the momentum representation and the Klein–Gordon propagator has allowed the calculation of the Wigner function for an arbitrary bound state of the hydrogen atom, using a simple atomic integral as a generator. These Wigner functions for some low-lying states are depicted and discussed.