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4.4: Eigenfunctions of Orbital Angular Momentum

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

\[L_x = -{\rm i}\,\hbar\left(y\,\dfrac{\partial}{\partial z} - z\,\dfrac{\partial} {\partial y}\right) \tag{363}\]

\[ L_y = -{\rm i}\,\hbar\left(z\,\dfrac{\partial}{\partial x} - x\,\dfrac{\partial} {\partial z}\right) \tag{364}\]

\[ L_z = -{\rm i}\,\hbar\left(x\,\dfrac{\partial}{\partial y} - y\,\dfrac{\partial} {\partial x}\right) \tag{365}\]

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

\[x =  r \,\sin\theta\, \cos\varphi \tag{366}\]

\[ y = r\, \sin\theta\, \sin\varphi \tag{367}\]

\[ z =r \cos \theta \tag{ 368}\]

we obtain

\[ L_x = {\rm i}\,\hbar\,\left(\sin\varphi\, \dfrac{\partial}{\partial \theta} + \cot\theta \cos\varphi\,\dfrac{\partial}{\partial \varphi}\right) \tag{369}\]

\[ L_y =  -{\rm i} \,\hbar\,\left(\cos\varphi\, \dfrac{\partial}{\partial\theta} -\cot\theta \sin\varphi \,\dfrac{\partial}{\partial \varphi}\right),\tag{370}\]

\[ L_z = -{\rm i}\,\hbar\,\dfrac{\partial}{\partial\varphi} \tag{371}\]

Note that Equation 371 accords with Equation 346. The shift operators \(L^\pm = L_x \pm {\rm i} \,L_y\) become

\[L^\pm = \pm \hbar\,\exp(\pm{\rm i}\,\varphi)\left(\dfrac{\partial}{\partial\theta} \pm{\rm i} \,\cot\theta\,\dfrac{\partial}{\partial\varphi}\right) \tag{372}\]

Now,

\[L^2 = L_x^{\,2}+L_y^{\,2}+L_z^{\,2} = L_z^{\,2} + (L^+\, L^- + L^- \,L^+) /2 \tag{373}\]

so

\[L^2 = - \hbar^2\left( \dfrac{1}{\sin\theta}\dfrac{\partial}{\partial \theta} \sin \theta \dfrac{\partial}{\partial \theta} + \dfrac{1}{\sin^2\theta}\dfrac{\partial^2} {\partial\varphi^2}\right) \tag{374}\]

The eigenvalue problem for \(L^2\) takes the form

\[L^2 \,\psi = \lambda \,\hbar^2\, \psi \tag{375}\]

where \(\psi(r, \theta, \varphi)\) is the wavefunction, and \(\lambda\) is a number. Let us write

\[ \psi(r, \theta, \varphi) = R(r) \,Y(\theta, \varphi) \tag{376}\]

Equation 375 reduces to

\[\left( \dfrac{1}{\sin\theta}\dfrac{\partial}{\partial\theta}\, \sin \theta \dfrac{\partial}{\partial \theta} + \dfrac{1}{\sin^2\theta}\dfrac{\partial^2} {\partial\varphi^2}\right)Y + \lambda \,Y = 0 \tag{377}\]

where use has been made of Equation 374. 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

\[Y_{l\,m}(\theta, \varphi) = \sqrt{ \dfrac{2\,l+1}{4\pi} \dfrac{(l-m)!}{(l+m)!}} \,(-1)^m\, {\rm e}^{\,{\rm i} \,m\,\varphi}\, P_{l\,m}(\cos\varphi) \tag{378}\]

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

\[ \dfrac{d}{d\xi}\! \left[ (1-\xi^{\,2})\,\dfrac{dP_{l\,m}}{d\xi}\right] - \dfrac{m^2}{1-\xi^{\,2}}\, P_{l\,m} + l\,(l+1)\,P_{l\,m} = 0 \tag{379}\]

We define

\[ Y_{l\,\,-m} = (-1)^m\, Y_{l\,m}^{\,\ast} \tag{380}\]

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:

\[ \int_0^\pi \int_0^{2\pi}d\theta\,d\varphi \, \sin\theta\,Y_{l\,m} (\theta,\varphi)\, Y_{l'\,m'}(\theta, \varphi) = \delta_{l \,l'} \,\delta_{m \,m'} \tag{381}\]

The spherical harmonics also form a complete set for representing general functions of \(\theta) and \(\phi\).

By definition,

\[L^2 \,Y_{l\,m} = l\,(l+1)\,\hbar^2\,Y_{l\,m} \tag{382}\]

where \(l\) is an integer. It follows from Equations 371 and 378 that

\[L_z \,Y_{l\,m} = m\,\hbar\,Y_{l\,m} \tag{383}\]

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

$\displaystyle \frac{d P_{l\,m}}{d\xi} = -\frac{1}{\sqrt{1-\xi^{\,2}}}\,P_{l\,\,...
...+1)}{\sqrt{1-\xi^{\,2}}}\,P_{l\,\,m-1} + \frac{m\,\xi} {1-\xi^{\,2}}\, P_{l\,m}$ (384)

 

can be combined with Equations (372) and (378) to give

 

$\displaystyle L^+ \,Y_{l\,m}$ $\displaystyle = [l\,(l+1)- m\,(m+1)]^{1/2}\,\hbar\,Y_{l\,\,m+1},$ (385)
$\displaystyle L^- \,Y_{l\,m}$ $\displaystyle = [l\,(l+1) - m \,(m-1)]^{1/2} \,\hbar \,Y_{l\,\,m-1}.$ (386)

 

These equations are equivalent to Equations (344)-(345). Note that a spherical harmonic wavefunction 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 the Schrödinger representation, we have been able to reproduce all of the results of Section 4.2. Nevertheless, the results of Section 4.2 are more general than those obtained in this section, because they still apply when the quantum number $ l$ takes on half-integer values.

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