Harmonic oscillator | Quantum mechanics

The wave functions of the harmonic oscillator are given by the equation (\ref{ec17}), where N is the normalization constant, which we can calculate with the following equation: \begin{equation}\label{ec-43} \int_ {-\infty}^{+\infty}\Psi_{v}^{\ast}(x)\Psi_{v}(x)dx=1 \end{equation} The normalization of the wave function for a state general $v$ gives us the following result: \begin{equation} N_v =\left(2^vv!\right)^{-1/2}\left(\frac{\beta}{\pi}\right) ^{1/4} \end{equation}