ANALYTICAL SOLUTION THE
RADIAL SCHRÖDINGER
EQUATION FOR THE QUARKANTIQUARK SYSTEM
S. M. Kuchin
The I.G. Petrovsky Bryansk State University
N. V. Maksimenko
The F. Skorina Gomel State University,
INTRODUCTION
The solution of the spectral problem for the Schrodinger
equation with spherically symmetric potentials is of major
concern in describing the spectra of quarkonia. Potential models
offer a rather good description of the mass spectra of systems
such as a quarkonium, a charmonium, etc. In simulating the
interaction potentials for these systems, confining-type
potentials are generally used. The holding potentials can be of
any form. For instance, a variety of this type of potential is the
so-called Cornel potential with two terms one of which is
responsible for the Coulomb interaction of quarks and the other
corresponds to a confining potential. Though this potential,
proposed to describe quarkonia with heavy quarks, has been
used for a long time, nevertheless the problem of finding the
interquark potential still remains incompletely solved. To solve
this problem is necessary both for finding the mass spectrum of
bound states and for describing the electromagnetic
characteristics of mesons.
Using the Nikiforov–Uvarov method, widely used to
solve Schrodinger equations, we obtain asymptotic
expressions for the eigenfunctions and eigenvalues of the
Schrodinger equation with the potential under
consideration and, using the expressions obtained, we
calculate the mass spectrum of quarkonia and 𝐵𝑐
mesons. The method used in this work allows one to
obtain approximate analytical formulas for energy
levels, which can be useful to analyze qualitatively the
spectrum of a model system.
THE NIKIFOROV–UVAROV METHOD
THE SCHRÖDINGER EQUATION
SUBSTITUTING THIS INTO THE RADIAL
SCHRÖDINGER EQUATION, WE OBTAIN
IN THIS CASE
THE SOLUTION OF THE
SCHRÖDINGER EQUATION
THE CORRESPONDING 𝑅𝑛𝑙 𝑟 WAVE FUNCTIONS ARE
THEN FOUND TO BE
DEPENDENCE
THE BINDING ENERGY OF THE QUARK MASS
DEPENDENCE
THE BINDING ENERGY OF THE QUARK MASS
CONCLUSIONS
Thus, in this work, the analytical expressions for the
wave functions and energy eigenvalues of the Schrödinger
equation with the Cornell potential. All calculations are
carried out in good agreement with the available
experimental data, and the process of determining the
energy spectrum and eigenfunctions of the Schrödinger
equation in this approach is much simpler than using the
standard perturbation theory or other methods. The
analytical solutions can be used not only to describe the
mass spectrum of the quark-antiquark systems, but also
other characteristics.
Thank you for
attention