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: \[\begin{aligned} \label{e8.21} x &= r\,\sin\theta\,\cos\phi,\\[0.5ex] y&= r\,\sin\theta\,\sin\phi,\\[0.5ex] z&= r\,\cos\theta.\label{e8.23}\end{aligned}\] We deduce, after some tedious analysis, that \[\begin{aligned} \frac{\partial}{\partial x} &= \sin\theta\,\cos\phi\,\frac{\partial}{\partial r} + \frac{\cos\theta\,\cos\phi}{r}\,\frac{\partial}{\partial\theta} - \frac{\sin\phi}{r\,\sin\theta}\,\frac{\partial}{\partial\phi},\label{e8xx}\\[0.5ex] \frac{\partial}{\partial y} &= \sin\theta\,\sin\phi\,\frac{\partial}{\partial r} + \frac{\cos\theta\,\sin\phi}{r}\,\frac{\partial}{\partial\theta} + \frac{\cos\phi}{r\,\sin\theta}\,\frac{\partial}{\partial\phi},\label{e8yy}\\[0.5ex] \frac{\partial}{\partial z} &= \cos\theta\,\frac{\partial}{\partial r} -\frac{\sin\theta}{r}\,\frac{\partial}{\partial \theta}.\label{e8zz}\end{aligned}\] Making use of the definitions ([e8.1])–([e8.3]), ([e8.9]), and ([e8.13]), the fundamental representation ([e6.12])–([e6.14]) of the \(p_i\) operators as spatial differential operators, Equations ([e8.21])–([e8zz]), and a great deal of tedious analysis, we finally obtain \[\begin{aligned} L_x &= - {\rm i}\,\hbar\left(-\sin\phi\,\frac{\partial}{\partial\theta} -\cos\phi\,\cot\theta\,\frac{\partial}{\partial\phi}\right),\\[0.5ex] L_y &= - {\rm i}\,\hbar\left(\cos\phi\,\frac{\partial}{\partial\theta} -\sin\phi\,\cot\theta\,\frac{\partial}{\partial\phi}\right),\\[0.5ex] L_z &= -{\rm i}\,\hbar\,\frac{\partial}{\partial\phi},\label{e8.26}\end{aligned}\] as well as \[L^2 = -\hbar^{\,2}\left[\frac{1}{\sin\theta}\frac{\partial}{\partial\theta}\left( \sin\theta\,\frac{\partial}{\partial\theta}\right) + \frac{1}{\sin^2\theta}\frac{\partial^{\,2}}{\partial\phi^{\,2}}\right],\] and \[\label{e8.28} L_\pm = \hbar\,{\rm e}^{\pm{\rm i}\,\phi}\left(\pm\frac{\partial}{\partial\theta} +{\rm i}\,\cot\theta\,\frac{\partial}{\partial\phi}\right).\] 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, \(r\).
Contributors and Attributions
Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)
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