The effective $β$ value in a Simple Harmonic Oscillator wave function
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 $β$ value. We obtain the analytical expression of $β_{eff}$ ($β_{effective}$) 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 $β_{eff}$. The results show that $β_ {eff,\mathbf{r}}$ in coordinate space and $β_ {eff,\mathbf{p}}$ in momentum space are significantly different in the ground state, however, similar in the highly excited states with Cornell potential.
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