Understanding the analysis of wave function

TL;DR

This paper analyzes wave functions with power-law potentials, deriving their order and type, and proposes an ansatz for their form involving polynomial functions, expanding understanding of their mathematical properties.

Contribution

It introduces a method to determine the order and type of wave functions with power-law potentials and proposes a polynomial-based ansatz for their form.

Findings

Order and type are compatible with $|\sigma| ho =\sqrt{v_m}$, except for $m=-2$.

Wave function form $\psi(r)=f(r)\exp[g(r)]$ with polynomial $f(r),g(r)$.

The polynomial degree of $g(r)$ does not exceed the order $ ho$.

Abstract

We obtained the order $\rho$ and the type $\sigma$ of wave function with power-law potentials, and found that the order and the type are compatible with the condition $|\sigma|\rho =\sqrt{v_m}$ except to $m=-2$. At the same time, we ansatz that the wave function satisfaction relation $\psi(r)=f(r)\exp[g(r)]$, where $f(r),g(r)$ are polynomials, and the power of $g(r)$ is no more than the order $\rho$.