where \(H_n(x)\) are the Hermite polynomials. Here, we stress the fact that (\ref{n-functions}) are eigenfunctions of the operator \(I_Q\), but they are also solutions of the Schrödinger equation \(\left(i\hbar \partial_t -H\right)\psi=0\). These states are the same found in \cite{Dodonov_1979}, where coherent states of the general one-dimensional oscillator are discussed.
Writing the canonical variables in the form