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 > {\displaystyle \mid \alpha >} is a general ket
< 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}}
< α ∣ T + ( l ) [ x i , T ( l ) ) ] ∣ α >=< α ∣ T + x i T ∣ α > − < α ∣ T + T x i ∣ α > {\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 >}
⇒< x i > t r a n s l a t e d =< x i > + l i ⇒< x > t r a n s l a t e d =< x > + l {\displaystyle \Rightarrow <x_{i}>_{translated}=<x_{i}>+l_{i}\Rightarrow <\mathbf {x} >_{translated}=<\mathbf {x} >+\mathbf {l} }
![{\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)
,
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)
