A note for the effective $β$ value in a Simple Harmonic Oscillator wave function

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 β value. We obtain the analytical expression of β e f f (β e f f ective ) of the SHO wave function in coordinate space and momentum space. The expression is applied to the light meson system (uū, u s) to compare the behavior of β e f f . The results show that β e f f (r) in coordinate space and β e f f (p) in momentum space are significantly different in the ground state, however, similar in the highly excited states.