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Physics LibreTexts

7.2: Representation of Angular Momentum

  • Page ID
    15765
  • 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

    • Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)

      \( \newcommand {\ltapp} {\stackrel {_{\normalsize<}}{_{\normalsize \sim}}}\) \(\newcommand {\gtapp} {\stackrel {_{\normalsize>}}{_{\normalsize \sim}}}\) \(\newcommand {\btau}{\mbox{\boldmath$\tau$}}\) \(\newcommand {\bmu}{\mbox{\boldmath$\mu$}}\) \(\newcommand {\bsigma}{\mbox{\boldmath$\sigma$}}\) \(\newcommand {\bOmega}{\mbox{\boldmath$\Omega$}}\) \(\newcommand {\bomega}{\mbox{\boldmath$\omega$}}\) \(\newcommand {\bepsilon}{\mbox{\boldmath$\epsilon$}}\)