Eigenfunctions of orbital angular momentum

In Cartesian coordinates, the three components of orbital angular momentum can be written

$$L_x = -{\rm i} \hbar\left(y \frac{\partial}{\partial z} - z \frac{\partial} {\partial y}\right),$$ (370)

$$L_y = -{\rm i} \hbar\left(z \frac{\partial}{\partial x} - x \frac{\partial} {\partial z}\right),$$ (371)

$$L_z = -{\rm i} \hbar\left(x \frac{\partial}{\partial y} - y \frac{\partial} {\partial x}\right),$$ (372)

using the Schrödinger representation. Transforming to standard spherical polar coordinates,

$$x = r \sin\theta \cos\varphi,$$ (373)

$$y = r \sin\theta \sin\varphi,$$ (374)

$$z = r \cos\theta,$$ (375)

we obtain

$$L_x = {\rm i} \hbar \left(\sin\varphi \frac{\partial}{\partial \theta} + \cot\theta \cos\varphi \frac{\partial}{\partial \varphi}\right)$$ (376)

$$L_y = -{\rm i} \hbar \left(\cos\varphi \frac{\partial}{\partial\theta} -\cot\theta \sin\varphi \frac{\partial}{\partial \varphi}\right)$$ (377)

$$L_z = -{\rm i} \hbar \frac{\partial}{\partial\varphi}.$$ (378)

Note that Eq. (378) accords with Eq. (353). The shift operators $L^\pm = L_x \pm {\rm i} L_y$ become

$$L^\pm = \pm \hbar \exp(\pm{\rm i} \varphi)\left(\frac{\par... ...{\rm i} \cot\theta \frac{\partial}{\partial\varphi}\right).$$ (379)

Now,

$$L^2 = L_x^{ 2}+L_y^{ 2}+L_z^{ 2} = L_z^{ 2} + (L^+ L^- + L^- L^+) /2,$$ (380)

so

$$L^2 = - \hbar^2\left( \frac{1}{\sin\theta}\frac{\partial}{\p... ...{1}{\sin^2\theta}\frac{\partial^2} {\partial\varphi^2}\right).$$ (381)

The eigenvalue problem for $L^2$ takes the form

$$L^2 \psi = \lambda \hbar^2 \psi,$$ (382)

where $\psi(r, \theta, \varphi)$ is the wave-function, and $\lambda$ is a number. Let us write

$$\psi(r, \theta, \varphi) = R(r) Y(\theta, \varphi).$$ (383)

Equation (382) reduces to

$$\left( \frac{1}{\sin\theta}\frac{\partial}{\partial\theta} \... ...rac{\partial^2} {\partial\varphi^2}\right)Y + \lambda Y = 0,$$ (384)

where use has been made of Eq. (381). As is well-known, square integrable solutions to this equation only exist when $\lambda$ takes the values $l (l+1)$, where $l$ is an integer. These solutions are known as spherical harmonics, and can be written

$$Y_l^m(\theta, \varphi) = \sqrt{ \frac{2 l+1}{4\pi} \frac{(l... ...-1)^m {\rm e}^{ {\rm i} m \varphi} P_l^m(\cos\varphi),$$ (385)

where $m$ is a positive integer lying in the range $0\leq m\leq l$. Here, $P_l^m(\xi)$ is an associated Legendre function satisfying the equation

$$\frac{d}{d\xi}\! \left[ (1-\xi^2)\frac{dP_l^m}{d\xi}\right] - \frac{m^2}{1-\xi^2} P_l^m + l (l+1) P_l^m = 0.$$ (386)

We define

$$Y_l^{-m} = (-1)^m (Y^{m}_l)^\ast,$$ (387)

which allows $m$ to take the negative values $-l\leq m< 0$. The spherical harmonics are orthogonal functions, and are properly normalized with respect to integration over the entire solid angle:

$$\int_0^\pi \int_0^{2\pi} Y_l^{m\ast} (\theta,\varphi) Y_{l... ...\sin\theta d\theta d\varphi = \delta_{l l'} \delta_{m m'}.$$ (388)

The spherical harmonics also form a complete set for representing general functions of $\theta$ and $\varphi$.

By definition,

$$L^2 Y_l^m = l (l+1) \hbar^2 Y_l^m,$$ (389)

where $l$ is an integer. It follows from Eqs. (378) and (385) that

$$L_z Y^m_l = m \hbar Y_l^m,$$ (390)

where $m$ is an integer lying in the range $-l\leq m \leq l$. Thus, the wave-function $\psi(r, \theta, \varphi) = R(r) Y_l^m(\theta, \phi)$, where $R$ is a general function, has all of the expected features of the wave-function of a simultaneous eigenstate of $L^2$ and $L_z$ belonging to the quantum numbers $l$ and $m$. The well-known formula

$$\frac{d P_l^m}{d\xi} = \frac{1}{\sqrt{1-\xi^2}}P_l^{m+1} - \frac{m \xi}{1-\xi^2} P_l^m$$

$$= - \frac{(l+m)(l-m+1)}{\sqrt{1-\xi^2}}P_l^{m-1} + \frac{m \xi} {1-\xi^2} P_l^m$$ (391)

can be combined with Eqs. (379) and (385) to give

$$L^+ Y_l^m = \sqrt{l (l+1)- m (m+1)} \hbar Y_l^{m+1},$$ (392)

$$L^- Y_l^m = \sqrt{l (l+1) - m (m-1)} \hbar Y_l^{m-1}.$$ (393)

These equations are equivalent to Eqs. (351)-(352). Note that a spherical harmonic wave-function is symmetric about the $z$-axis (i.e., independent of $\varphi$) whenever $m=0$, and is spherically symmetric whenever $l=0$ (since $Y_0^0 = 1/\sqrt{4\pi}$).

In summary, by solving directly for the eigenfunctions of $L^2$ and $L_z$ in Schrödinger's representation, we have been able to reproduce all of the results of Sect. 5.2. Nevertheless, the results of Sect. 5.2 are more general than those obtained in this section, because they still apply when the quantum number $l$ takes on half-integer values.