Translation operator problem

(a) $[{\hat {x}}_{i},{\hat {T}}(\mathbf {l} )]=i\hbar {\frac {\partial T(\mathbf {l} )}{\partial {\hat {p}}_{i}}}=i\hbar \left(-i{\frac {l_{i}}{\hbar }}\right)\exp \left(-{\frac {i{\hat {\mathbf {p} }}\cdot \mathbf {l} }{\hbar }}\right)=l_{i}{\hat {T}}(\mathbf {l} )$

(b) Given a general state $|\alpha \rangle ,$ the expectation value of ${\hat {x}}_{i}$ is $\langle {\hat {x}}_{i}\rangle =\langle \alpha |{\hat {x}}_{i}|\alpha \rangle .$

Let us now find the expectation value for the translated state ${\hat {T}}(\mathbf {l} )|\alpha \rangle .$

$\langle \alpha |{\hat {T}}^{\dagger }(\mathbf {l} ){\hat {x}}_{i}{\hat {T}}(\mathbf {l} )|\alpha \rangle =\langle \alpha |{\hat {x}}_{i}|\alpha \rangle +\langle \alpha |{\hat {T}}^{\dagger }(\mathbf {l} )[{\hat {x}}_{i},{\hat {T}}(\mathbf {l} )]|\alpha \rangle =\langle {\hat {x}}_{i}\rangle +\langle \alpha |{\hat {T}}^{\dagger }(\mathbf {l} )l_{i}{\hat {T}}(\mathbf {l} )|\alpha \rangle =\langle {\hat {x}}_{i}\rangle +l_{i}$

Therefore, the effect of the translation operator ${\hat {T}}(\mathbf {l} )$ is to shift the expectation value of the position operator ${\hat {\mathbf {x} }}$ by the vector $\mathbf {l} .$

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