Source: "Modern Quantum Mechanics",Sakurai,problem1.30
Problem: The translation operator for a finite (spatial) displacement is given by ,
where p is the momentum operator.
a. Evaluate ![{\displaystyle [x_{i},T(\mathbf {l} ))]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c03d978439690500ab223cd3852c9c01a1712efa)
b. Using (a) (or otherwise), demonstrate how the expectation value 
changes under translation.
Solution:
a) ![{\displaystyle [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 }})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/82a0cc0ad8c3acf60a79efc25d88363446226d75)
![{\displaystyle \Rightarrow =[x_{i},T(\mathbf {l} ))]=l_{i}T(\mathbf {l} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6ccd5a7b5fe5a3ab6d5a1581e818392d473d73c2)
b) 
,
is a general ket
![{\displaystyle <\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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/37331cb577440e505cb56b46afa6411b89bdf47a)
![{\displaystyle <\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 >}](https://wikimedia.org/api/rest_v1/media/math/render/svg/efe8843a3faa5b33839bbabce526d1c6491d8e91)
