Angular Momentum Operators

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

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

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

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

where

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

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

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

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

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

$$L_z = -{\rm i}\, \frac{\partial}{\partial\varphi}.$$ (1439)

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

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

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

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

The following results are well known in quantum mechanics:

$$L_+ \,Y_{lm} = \sqrt{(l-m)\,(l+m+1)}\,Y_{l,m+1},$$ (1442)

$$L_-\,Y_{lm} =\sqrt{(l+m)\,(l-m+1)}\,Y_{l,m-1},$$ (1443)

$$L_z\,Y_{lm} = m\,Y_{lm}.$$ (1444)

In addition,

$$L^2\,{\bf L} = {\bf L} \,L^2,$$ (1445)

$${\bf L}\times{\bf L} = {\rm i}\,{\bf L},$$ (1446)

$$L_j \,\nabla^{\,2} = \nabla^{\,2} L_j,$$ (1447)

where

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