The effective $\beta$ value in a Simple Harmonic Oscillator wave function

TL;DR

This paper derives an analytical expression for the effective $eta$ parameter in SHO wave functions, compares its behavior in coordinate and momentum space for light mesons, and finds significant differences in ground states but similarities in excited states.

Contribution

It provides the first analytical expression for the effective $eta$ in SHO wave functions and applies it to light mesons to analyze its behavior across states.

Findings

$eta_{eff}$ differs significantly between coordinate and momentum space in ground states.

$eta_{eff}$ becomes similar in highly excited states with Cornell potential.

The analytical expression aids in better modeling of meson wave functions.

Abstract

When a Simple Harmonic Oscillator (SHO) wave function is used as an effective wave function, a very important parameter in the SHO wave function is the effective $\beta$ value. We obtain the analytical expression of $\beta_{eff}$ ($\beta_{effective}$) of the SHO wave function in coordinate space and momentum space. The expression is applied to the light meson system $(u\bar{u},~u\bar{s})$ to compare the behavior of $\beta_{eff}$. The results show that $\beta_ {eff,\mathbf{r}}$ in coordinate space and $\beta_ {eff,\mathbf{p}}$ in momentum space are significantly different in the ground state, however, similar in the highly excited states with Cornell potential.