Source: Problem: The translation operator for a finite (spatial) displacement is given by , where p is the momentum operator. T ( l ) = e x p ( − i p . l ℏ ) {\displaystyle T(\mathbf {l} )=exp(-{\frac {i\mathbf {p} .\mathbf {l} }{\hbar }})}
a. Evaluate [ x i , T ( l ) ) ] {\displaystyle [x_{i},T(\mathbf {l} ))]} b. Using (a) (or otherwise), demonstrate how the expectation value < x > {\displaystyle <\mathbf {x} >} changes under translation.
Solution: a) [ x i , T ( l ) ) ] = i ℏ ∂ T ( l ) ∂ p i = i ℏ ( − i l i ℏ ) e x p ( − i p . l ℏ ) {\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 }})}
⇒= [ x i , T ( l ) ) ] = l i T ( l ) {\displaystyle \Rightarrow =[x_{i},T(\mathbf {l} ))]=l_{i}T(\mathbf {l} )}
b) < x i >=< α ∣ x i ∣ α > {\displaystyle <x_{i}>=<\alpha \mid x_{i}\mid \alpha >}
< α ∣ T + ( l ) [ x i , T ( l ) ) ] ∣ α >=< α ∣ T + ( l ) l i T ( l ) ∣ α >= l i {\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}}
![{\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.
![{\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)
![{\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)