where \(H_n(x)\) are the Hermite polynomials. Here, we must stress the fact that (\ref{n-functions}) are eigenfunctions of the operator \(I_Q\) and, at the same time, solutions of the Schrödinger equation \(\left(i\hbar \frac{\partial}{\partial t}-H\right)\psi=0\).
The simple harmonic oscillator
Let us consider the case of the simple harmonic oscillator (SHO), characterized by a constant \(\omega\), and \(g=F=0\). Equations for \(\beta\) and \(\beta^*\) are given by