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Angular Momentum Operators

It is well known from quantum mechanics that Equation (1417) can be written in the form

$\displaystyle L^2\, Y_{lm} = l\,(l+1)\,Y_{lm}.$ (1434)

Here, the differential operator $ L^2$ is given by

$\displaystyle L^2 = L_x^{\,2} + L_y^{\,2} + L_z^{\,2},$ (1435)

where

$\displaystyle {\bf L} = -{\rm i}\,{\bf r}\times \nabla$ (1436)

is $ 1/\hbar$ times the orbital angular momentum operator of wave mechanics.

The components of $ {\bf L}$ are conveniently written in the combinations

$\displaystyle L_+$ $\displaystyle =L_x + {\rm i}\,L_y = {\rm e}^{\,{\rm i}\,\varphi} \left(\frac{\p...
...\partial\theta} +{\rm i}\,\cot\theta\, \frac{\partial}{\partial\varphi}\right),$ (1437)
$\displaystyle L_-$ $\displaystyle = L_x - {\rm i}\,L_y={\rm e}^{-{\rm i}\,\varphi} \left(-\frac{\pa...
...\partial\theta} +{\rm i}\,\cot\theta\, \frac{\partial}{\partial\varphi}\right),$ (1438)
$\displaystyle L_z$ $\displaystyle = -{\rm i}\, \frac{\partial}{\partial\varphi}.$ (1439)

Note that $ {\bf L}$ only operates on angular variables, and is independent of $ r$ . It is evident from the definition (1438) that

$\displaystyle {\bf r}\cdot{\bf L} = 0.$ (1440)

It is easily demonstrated from Equations (1439)-(1441) that

$\displaystyle L^2 = - \frac{1}{\sin\theta}\frac{\partial}{\partial\theta}\, \si...
...ial\theta}-\frac{1}{\sin^2\theta} \frac{\partial^{\,2}}{\partial\varphi^{\,2}}.$ (1441)

The following results are well known in quantum mechanics:

$\displaystyle L_+ \,Y_{lm}$ $\displaystyle = \sqrt{(l-m)\,(l+m+1)}\,Y_{l,m+1},$ (1442)
$\displaystyle L_-\,Y_{lm}$ $\displaystyle =\sqrt{(l+m)\,(l-m+1)}\,Y_{l,m-1},$ (1443)
$\displaystyle L_z\,Y_{lm}$ $\displaystyle = m\,Y_{lm}.$ (1444)

In addition,

$\displaystyle L^2\,{\bf L}$ $\displaystyle = {\bf L} \,L^2,$ (1445)
$\displaystyle {\bf L}\times{\bf L}$ $\displaystyle = {\rm i}\,{\bf L},$ (1446)
$\displaystyle L_j \,\nabla^{\,2}$ $\displaystyle = \nabla^{\,2} L_j,$ (1447)

where

$\displaystyle \nabla^{\,2} = \frac{1}{r^{\,2}}\frac{\partial}{\partial r} \,r^{\,2}\frac{\partial}{\partial r} - \frac{L^2}{r^{\,2}}.$ (1448)


next up previous
Next: Multipole Expansion of Vector Up: Multipole Expansion Previous: Multipole Expansion of Scalar
Richard Fitzpatrick 2014-06-27