Using the wavefunctions

\[u_{p}(\boldsymbol{r})=\frac{1}{\sqrt{ } V} e^{i(\boldsymbol{p} \cdot \boldsymbol{r}) \hbar}\]

to describe the motion of a free particle, write down the symmetrized wavefunctions for a pair of noninteracting bosons/fermions, and show that

\[\left\langle 1^{\prime}, 2^{\prime}\left|\hat{U}_{2}^{S / A}\right| 1,2\rightangle= \pm\left\langle 2^{\prime}\left|\hat{W}_{1}\right| 1\rightangle\left\langle 1^{\prime}\left|\hat{W}_{1}\right| 2\rightangle\]