Translation operator problem

Source: Problem: The translation operator for a finite (spatial) displacement is given by , where p is the momentum operator. $T(\mathbf {l} )=exp(-{\frac {i\mathbf {p} .\mathbf {l} }{\hbar }})$

a. Evaluate $[x_{i},T(\mathbf {l} ))]$ b. Using (a) (or otherwise), demonstrate how the expectation value $<\mathbf {x} >$ changes under translation.

Solution: a) $[x_{i},T(\mathbf {l} ))]=i\hbar {\frac {\partial T(\mathbf {l} )}{\partial p_{i}}}=i\hbar (-i{\frac {l_{i}}{\hbar }})exp(-{\frac {i\mathbf {p} .\mathbf {l} }{\hbar }})$

$\Rightarrow =[x_{i},T(\mathbf {l} ))]=l_{i}T(\mathbf {l} )$

b) $<x_{i}>=<\alpha \mid x_{i}\mid \alpha >$ ,$\mid \alpha >$ is a general ket

$<\alpha \mid \ T^{+}(\mathbf {l} )[x_{i},T(\mathbf {l} ))]\mid \alpha >=<\alpha \mid T^{+}(\mathbf {l} )l_{i}T(\mathbf {l} )\mid \alpha >=l_{i}$

$<\alpha \mid \ T^{+}(\mathbf {l} )[x_{i},T(\mathbf {l} ))]\mid \alpha >=<\alpha \mid T^{+}x_{i}T\mid \alpha \ >-<\alpha \mid T^{+}Tx_{i}\mid \alpha >$

$\Rightarrow <x_{i}>_{translated}=<x_{i}>+l_{i}\Rightarrow <\mathbf {x} >_{translated}=<\mathbf {x} >+\mathbf {l}$