Calculate the matrix elements $$ \langle \alpha_{1}\alpha_{2}|\hat{F}|\alpha_{1}\alpha_{2}\rangle $$ and $$ \langle \alpha_{1}\alpha_{2}|\hat{G}|\alpha_{1}\alpha_{2}\rangle $$ with $$ |\alpha_{1}\alpha_{2}\rangle=a_{\alpha_{1}}^{\dagger}a_{\alpha_{2}}^{\dagger}|0\rangle , $$ $$ \hat{F}=\sum_{\alpha\beta}\langle \alpha|\hat{f}|\beta\rangle a_{\alpha}^{\dagger}a_{\beta} , $$ $$ \langle \alpha|\hat{f}|\beta\rangle=\int \psi_{\alpha}^{*}(x)f(x)\psi_{\beta}(x)dx , $$ $$ \hat{G} = \frac{1}{2}\sum_{\alpha\beta\gamma\delta} \langle \alpha\beta |\hat{g}|\gamma\delta\rangle a_{\alpha}^{\dagger}a_{\beta}^{\dagger}a_{\delta}a_{\gamma} , $$ and $$ \langle \alpha\beta |\hat{g}|\gamma\delta\rangle= \int\int \psi_{\alpha}^{*}(x_{1})\psi_{\beta}^{*}(x_{2})g(x_{1}, x_{2})\psi_{\gamma}(x_{1})\psi_{\delta}(x_{2})dx_{1}dx_{2} $$ Compare these results with those from exercise 3c).