Consider a 1-dimensional harmonic oscillator with mass \(m\) moving along the \(x\)-axis. Its angular frequency is \(\omega=\sqrt{\frac{k}{m}}\), where \(k\) is the spring constant. The wave function of its first excited state \(n=1\) is \(\left(\beta=\frac{h}{m \omega}ight)\).

\[\psi(x)=\frac{1}{(\pi \beta)^{\frac{1}{4}}} \sqrt{\frac{2}{\beta}} x e^{-\frac{x^{2}}{2 \beta}}\]

Use Schrödinger's wave equation to calculate the energy of this state.