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 ground state \(n=0\) is \(\left(\beta=\frac{\hbar}{m \omega}ight)\).

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

(a) Show that this wave function is normalized.

(b) What is the expectation value \(\langle xangle\) of \(x\) ?