Schrödinger equation for the two-dimensional Coulomb potential

Abstract

General properties of the Sturm-Liouville equation and numerical methods are used to determine the eigenquantities of the Schrödinger equation of a (+ -) pair interacting via the two-dimensional Coulomb potential $q^2$ lnr. The spectrum is shown to be purely discrete and semibounded in its lower part with a density of levels of $\frac{dN\left(\lambda \right)}{d\lambda }\sim e^{\frac{2\lambda }{q^2}}$, while the wave functions behave like those of a harmonic oscillator. The spectrum is studied as a function of the coupling parameter $q$. Wave functions $\psi$ and radial densities $r{\left|\psi \right|}^2$ are plotted against $r$ and some matrix elements $〈\psi \left|r^{\nu }\right|\psi 〉$ are given. The $q=1\mathrm{}$ spectrum is used to evaluate the corresponding sum-over-states which allows us to give a clear meaning to the canonical thermodynamical functions of the two-dimensional Coulomb gas below the transition temperature $T_c=\frac{q^2}{2k_B}$.