Phy5645/Angular Momentum Problem 1

(a)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{R}_{\Delta \phi} f = \left[ \exp \left( \Delta \phi \frac{\partial}{\partial \phi} \right) \right] f }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = 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).}

(b) Let $\delta \phi \!$ be an infinitesimal angle so that $\Delta \phi =n\delta \phi \!$ in the limit that $n>>1\!$. For the infinitesimal rotation

$\mathbf {r} '=\mathbf {r} +\delta \mathbf {r} =\mathbf {r} +\delta {\vec {\phi }}\times \mathbf {r}$

so that

$f\left(\mathbf {r} +\delta \mathbf {r} \right)=f(\mathbf {r} )+\delta {\vec {\phi }}\times \mathbf {r} \cdot \mathbf {\nabla } f(\mathbf {r} )$

$=f(\mathbf {r} )+\delta {\vec {\phi }}\cdot \mathbf {r} \times \mathbf {\nabla } f(\mathbf {r} )$

$=f(\mathbf {r} )+{\frac {i}{\hbar }}\delta {\vec {\phi }}\cdot \mathbf {r} \cdot \mathbf {\hat {p}} f(\mathbf {r} )$

$=f(\mathbf {r} )+{\frac {i}{\hbar }}\delta {\vec {\phi }}\cdot \mathbf {\hat {L}} f(\mathbf {r} )$.

In the Taylor series expansion of $f\left(\mathbf {r} +\delta \mathbf {r} \right)\!$ above we have only kept terms of $O\left(\delta \phi \right)\!$. [The expression $\delta \mathbf {r} =\delta {\vec {\phi }}\times \mathbf {r} \!$ is valid only to terms of $O\left(\delta \phi \right)\!$.] In this manner we obtain

$f\left(\mathbf {r} +\delta \mathbf {r} \right)=\left({\hat {I}}+{\frac {i}{\hbar }}\delta {\vec {\phi }}\cdot \mathbf {\hat {L}} \right)f(\mathbf {r} )={\hat {R}}_{\delta {\vec {\phi }}}f(\mathbf {r} )$

For a finite rotational displacement through the angle $\mathbf {\Delta } {\vec {\phi }}=n\delta {\vec {\phi }}\!$, we apply the operator ${\hat {R}}_{\delta {\vec {\phi }}}\!$, $n\!$ times:

${\hat {R}}_{n\delta {\vec {\phi }}}=\left({\hat {R}}_{\delta {\vec {\phi }}}\right)^{n}=\left({\hat {I}}+{\frac {i}{\hbar }}\delta {\vec {\phi }}\cdot \mathbf {\hat {L}} \right)^{n}$

and pss to the limit $n\rightarrow \infty \!$ or, equivalently, $\Delta \phi /\delta \phi \rightarrow \infty \!$.

${\hat {R}}_{\Delta \phi }=\lim _{\Delta \phi /\delta \phi \rightarrow \infty }\left({\hat {I}}+{\frac {i}{\hbar }}\delta {\vec {\phi }}\cdot \mathbf {\hat {\mathbf {L} }} \right)^{\Delta \phi /\delta \phi }=e^{i\Delta {\vec {\phi }}\cdot \mathbf {\hat {\mathbf {L} }} \hbar }$.

The operator ${\hat {R}}_{\delta {\vec {\phi }}}\!$ rotates $\mathbf {r} \!$ to $\mathbf {r} +\delta {\vec {\phi }}\times \mathbf {r} \!$ with respect to a fixed coordinate frame. If, on the other hand, the coordinate frame is rotated through $\delta {\vec {\phi }}\!$ with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{r} \!} fixed in space, then in the new coordinate frame this vector has the value Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{r} - \delta \vec{\phi} \times \mathbf{r} \!} . Thus, rotation of coordinates through Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta \vec{\phi} \!} is generated by the operator Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{R}_{-\delta \vec{\phi}}.}

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(Note: This problem is excerpted from Introductory Quantum Mechanics, 2nd edition, p377-p379, which is written by Richard L. Liboff.)