Template:!

Group work of Team 1

Questions

(a) Show that 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 \phi} \equiv \exp \left( \frac{i \Delta \phi \hat{L}_z}{\hbar} \right)}

when acting on the function Failed to parse (unknown function "\math"): {\displaystyle f(\pi) \!<\math> changes <math> f \! } by a rotation of coordinates about the $z\!$ axis so that the radius through $\phi \!$ is rotated to the radius through $\phi +\Delta \phi \!$. That is, show that

${\hat {R}}_{\Delta \phi }f(\phi )=f\left(\phi +\Delta \phi \right).(b)Showthattheoperator:<math>{\hat {R}}_{\Delta \phi }=\exp \left({\frac {i\Delta \mathbf {\phi } \cdot \mathbf {\hat {L}} }{\hbar }}\right)whenactionon<math>f(\mathbf {r} )\!$ changes $f\!$ by rotating $\mathbf {r} \!$ to a new value on the surface of the sphere of radius $r\!$, but rotated away from $\mathbf {r} \!$ through the azimuth $\Delta \phi \!$, so that $\mathbf {r} \left(\theta ,\phi \right)\rightarrow \mathbf {r} '=\mathbf {r} \left(\theta ,\phi +\Delta \phi \right)\!$. For infinitesimal displacement $\delta \mathbf {\phi } \!$, we may write

${\hat {R}}_{\delta \mathbf {\phi } }f(\mathbf {r} )=f\left(\mathbf {r} +\delta \mathbf {r} \right)$

$\delta \mathbf {r} =\delta \mathbf {\phi } \times \mathbf {r} .$

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Solutions

(a)

${\hat {R}}_{\Delta \phi }f=\left[\exp \left(\Delta \phi {\frac {\partial }{\partial \phi }}\right)\right]f$

$=f(\phi )+\Delta \phi {\frac {\partial f}{\partial \phi }}+{\frac {\left(\Delta \phi \right)^{2}}{2}}{\frac {\partial ^{2}f}{\partial \phi ^{2}}}+\cdots =f\left(\phi +\Delta \phi \right).$

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

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

so that

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

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

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

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

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

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

For a finite rotational displacement through the angle 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{\Delta} \mathbf{\phi} = n \delta \mathbf{\phi} \! } , we apply 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 \mathbf{\phi}} \! } , 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 n \! } times:

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

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

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

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\mathbf{\phi}} \! } rotates 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} \! } to 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\mathbf{\phi}\times\mathbf{r} \! } with respect to a fixed coordinate frame. If, on the other hand, the coordinate frame is rotated 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 \mathbf{\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 \mathbf{\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 \mathbf{\phi} \! } is generated by the operator <math> \hat{R}_{-\delta \mathbf{\phi}}.

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