7.2: Representation of Angular Momentum
- Page ID
- 15765
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Now, we saw earlier, in Section [s7.2], that the operators, \(p_i\), which represent the Cartesian components of linear momentum in quantum mechanics, can be represented as the spatial differential operators \(-{\rm i}\,\hbar\,\partial/\partial x_i\). Let us now investigate whether angular momentum operators can similarly be represented as spatial differential operators.
It is most convenient to perform our investigation using conventional spherical polar coordinates: that is, \(r\), \(\theta\), and \(\phi\). These are defined with respect to our usual Cartesian coordinates as follows:
\[\begin{align} \label{e8.21} x &= r\,\sin\theta\,\cos\phi,\\[4pt] y&= r\,\sin\theta\,\sin\phi,\\[4pt] z&= r\,\cos\theta.\label{e8.23}\end{align} \]
We deduce, after some tedious analysis, that
\[\begin{align} \frac{\partial}{\partial x} &= \sin\theta\,\cos\phi\,\frac{\partial}{\partial r} + \frac{\cos\theta\,\cos\phi}{r}\,\frac{\partial}{\partial\theta} - \frac{\sin\phi}{r\,\sin\theta}\,\frac{\partial}{\partial\phi},\label{e8xx}\\[4pt] \frac{\partial}{\partial y} &= \sin\theta\,\sin\phi\,\frac{\partial}{\partial r} + \frac{\cos\theta\,\sin\phi}{r}\,\frac{\partial}{\partial\theta} + \frac{\cos\phi}{r\,\sin\theta}\,\frac{\partial}{\partial\phi},\label{e8yy}\\[4pt] \frac{\partial}{\partial z} &= \cos\theta\,\frac{\partial}{\partial r} -\frac{\sin\theta}{r}\,\frac{\partial}{\partial \theta}.\label{e8zz}\end{align} \]
Making use of the definitions \ref{e8.1}–\ref{e8.3}, \ref{e8.9}, and \ref{e8.13}, the fundamental representation \ref{e6.12}–\ref{e6.14} of the \(p_i\) operators as spatial differential operators, Equations \ref{e8.21}–\ref{e8zz}, and a great deal of tedious analysis, we finally obtain
\[\begin{align} L_x &= - {\rm i}\,\hbar\left(-\sin\phi\,\frac{\partial}{\partial\theta} -\cos\phi\,\cot\theta\,\frac{\partial}{\partial\phi}\right),\\[4pt] L_y &= - {\rm i}\,\hbar\left(\cos\phi\,\frac{\partial}{\partial\theta} -\sin\phi\,\cot\theta\,\frac{\partial}{\partial\phi}\right),\\[4pt] L_z &= -{\rm i}\,\hbar\,\frac{\partial}{\partial\phi},\label{e8.26}\end{align} \]
as well as
\[L^2 = -\hbar^2\left[\frac{1}{\sin\theta}\frac{\partial}{\partial\theta}\left( \sin\theta\,\frac{\partial}{\partial\theta}\right) + \frac{1}{\sin^2\theta}\frac{\partial^2}{\partial\phi^2}\right], \nonumber \]
and
\[ L_\pm = \hbar\,\rm e^{\pm{\rm i}\,\phi}\left(\pm\frac{\partial}{\partial\theta} +{\rm i}\,\cot\theta\,\frac{\partial}{\partial\phi}\right). \label{e8.28} \]
We, thus, conclude that all of our angular momentum operators can be represented as differential operators involving the angular spherical coordinates, \(\theta\) and \(\phi\), but not involving the radial coordinate, \(r\).