Q$4.$

$(3$ marks$)$

Evaluate the average potential energy, $($:$E_{\text{p}\text{o}\text{t}\text{e}\text{n}\text{t}\text{i}\text{a}\text{l}}$:$),$ for the ground state $)=(0$ of the harmonic oscillator by carrying out the appropriate integration $($with respect to $x).($:$E_{\text{p}\text{o}\text{t}\text{e}\text{n}\text{t}\text{i}\text{a}\text{l}}$:$)=\frac{1}{2}k($:$x^2$:$)$

${}_0=(\frac{1}{{}^{\frac{1}{2}}}{)}^{\frac{1}{2}}{e}^{-(\frac{x^2}{2{}^2})}$