Numerical Solution of the Schrödinger Equation for Quarkonium Bound States

Abstract

Quarkonium, which is a bound state of a heavy quark and its corresponding antiquark, is similar to the hydrogen atom in particle physics, and its spectrum can provide information about the interquark potential of quantum chromodynamics. Numerical solution of the radial Schrödinger equation for charmonium uses the Cornell potential model, which combines the short-range Coulomb term and the linearly rising confining term. Fourth-order Numerov integration is used in combination with inward and outward shooting; they are connected at the classical turning point, and a node labelled energy scan is added to link each root to an excited state. First, this way is validated against the analytical hydrogen spectrum; it has been reproduced to better than one millielectronvolt. Then, based on the experimental measurement of the three lowest S-wave vector states of charmonium, the mass of the charm quark, the strong coupling constant and the confinement coefficient are determined by fitting, and a coarse random search followed by Nelder-Mead simplex refinement is employed for the fit. The converged fit is within 0.01 MeV of the experimental masses, and its root mean square deviation is 0.005 MeV; however, the fitted string tension is lower than its independent value and the fitted coupling is higher than its independent value, which may be due to the omission of spin-spin interaction, relativistic correction and open charm coupled-channel effects. The calculated wave functions have a node structure and their spatial extent grows slowly; it is characteristic of confinement.