a$.$ Prove the following:

$($i$)$

$($:$|)$

$($ii$),($:$|x|$:$)={\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}d{p}^{\text{'}}{}_{}^{**}(p^{\text{'}})i\frac{\text{d}\text{e}\text{l}}{del{p}^{\text{'}}}{}_{}^{}(p^{\text{'}}),$

where $(a$:$|)$ and $($:$|)$ are momentum$-$space wave functions.

b$.$ What is the physical significance of

exp$(\frac{ix}{})$

where $x$ is the position operator and $$ is some number with the dimension of momentum? Justify your answer.