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| (a)
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| :<math> \hat{R}_{\Delta \phi} f = \left[ \exp \left( \Delta \phi \frac{\partial}{\partial \phi} \right) \right] f </math> | | :<math> \hat{R}_{\Delta \phi} f = \left[ \exp \left( \Delta \phi \frac{\partial}{\partial \phi} \right) \right] f </math> |
| :<math> = f(\phi) + \Delta \phi \frac{\partial f}{\partial \phi} + \frac{\left(\Delta\phi\right)^2}{2} \frac{\partial^2 f}{\partial \phi^2} + \cdots = f \left( \phi + \Delta \phi \right).</math> | | :<math> = f(\phi) + \Delta \phi \frac{\partial f}{\partial \phi} + \frac{\left(\Delta\phi\right)^2}{2} \frac{\partial^2 f}{\partial \phi^2} + \cdots = f \left( \phi + \Delta \phi \right).</math> |
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| (b) Let <math> \delta \phi \!</math> be an infinitesimal angle so that <math> \Delta \phi = n \delta \phi \!</math> in the limit that <math> n >> 1 \!</math>. For the infinitesimal rotation
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| :<math> \mathbf{r}' = \mathbf{r} + \delta \mathbf{r} = \mathbf{r} + \delta \vec{\phi} \times \mathbf{r} </math>
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| so that
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| :<math> f \left( \mathbf {r} + \delta \mathbf{r} \right) = f(\mathbf{r})+ \delta \vec{\phi} \times \mathbf{r} \cdot \mathbf{\nabla} f(\mathbf{r}) </math>
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| :<math> = f(\mathbf{r})+ \delta \vec{\phi} \cdot \mathbf{r} \times \mathbf{\nabla} f(\mathbf{r}) </math>
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| :<math> = f(\mathbf{r})+ \frac{i}{\hbar} \delta \vec{\phi} \cdot \mathbf{r} \cdot \mathbf{\hat{p}} f(\mathbf{r}) </math>
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| :<math> = f(\mathbf{r})+ \frac{i}{\hbar}\delta \vec{\phi} \cdot \mathbf{\hat{L}} f(\mathbf{r}) </math>.
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| In the Taylor series expansion of <math> f\left( \mathbf{r}+\delta\mathbf{r} \right) \!</math> above we have only kept terms of <math> O \left(\delta \phi \right) \!</math>. [The expression <math> \delta \mathbf{r} = \delta \vec{\phi} \times \mathbf{r} \!</math> is valid only to terms of <math> O \left(\delta \phi \right) \!</math>.] In this manner we obtain
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| :<math> f\left( \mathbf{r} + \delta \mathbf{r} \right) = \left( \hat{I} + \frac{i}{\hbar} \delta \vec{\phi} \cdot \mathbf{\hat{L}} \right) f(\mathbf{r}) = \hat{R}_{\delta\vec{\phi}} f(\mathbf{r}) </math>
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| For a finite rotational displacement through the angle <math> \mathbf{\Delta} \vec{\phi} = n \delta \vec{\phi} \!</math>, we apply the operator <math> \hat{R}_{\delta \vec{\phi}} \!</math>, <math> n \!</math> times:
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| :<math> \hat{R}_{n\delta\vec{\phi}} = \left( \hat{R}_{\delta\vec{\phi}} \right)^n = \left( \hat{I} + \frac{i}{\hbar} \delta \vec{\phi} \cdot \mathbf{\hat{L}} \right)^n </math>
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| and pss to the limit <math> n \rightarrow \infty \!</math> or, equivalently, <math> \Delta \phi / \delta \phi \rightarrow \infty \!</math>.
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| :<math> \hat{R}_{\Delta \phi} = \lim_{\Delta \phi / \delta \phi \rightarrow \infty} \left( \hat{I} + \frac{i}{\hbar} \delta \vec{\phi} \cdot \mathbf{\hat{\mathbf{L}}} \right)^{\Delta \phi / \delta \phi} = e^{i \Delta \vec{\phi} \cdot \mathbf{\hat{\mathbf{L}}} \hbar} </math>.
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| The operator <math> \hat{R}_{\delta\vec{\phi}} \!</math> rotates <math> \mathbf{r} \!</math> to <math> \mathbf{r} + \delta\vec{\phi}\times\mathbf{r} \!</math> with respect to a fixed coordinate frame. If, on the other hand, the coordinate frame is rotated through <math> \delta \vec{\phi} \!</math> with <math> \mathbf{r} \!</math> fixed in space, then in the new coordinate frame this vector has the value <math> \mathbf{r} - \delta \vec{\phi} \times \mathbf{r} \!</math>. Thus, rotation of coordinates through <math> \delta \vec{\phi} \!</math> is generated by the operator <math> \hat{R}_{-\delta \vec{\phi}}.</math>
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| Back to [[Angular Momentum as a Generator of Rotations in 3D]] | | Back to [[Angular Momentum as a Generator of Rotations in 3D]] |
![{\displaystyle {\hat {R}}_{\Delta \phi }f=\left[\exp \left(\Delta \phi {\frac {\partial }{\partial \phi }}\right)\right]f}](https://wikimedia.org/api/rest_v1/media/math/render/svg/edac78cf5c44b4748da35831b77af2d34f35e198)
