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| Source: "Modern Quantum Mechanics",Sakurai,problem1.30
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| Problem: The translation operator for a finite (spatial) displacement is given by ,
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| where '''p''' is the momentum operator.
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| <math>T(\mathbf{l})=exp(-\frac{i\mathbf{p}.\mathbf{l}}{\hbar})</math>
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| a. Evaluate <math>[x_{i},T(\mathbf{l}))]</math>
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| b. Using (a) (or otherwise), demonstrate how the expectation value <math><\mathbf{x}></math>
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| changes under translation.
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| Solution:
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| a) <math>[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})</math> | | a) <math>[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})</math> |
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Revision as of 16:00, 8 July 2013
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)
