3.15: Spherical Harmonics

The rigid rotor problem was solved using the Schrödinger equation

\[-\dfrac{\hbar^{2}}{2 \mu r^{2}}\left(\dfrac{1}{\sin \theta} \dfrac{\partial}{\partial \theta} \sin \theta \dfrac{\partial}{\partial \theta}+\dfrac{1}{\sin ^{2} \theta} \dfrac{\partial^{2}}{\partial \phi^{2}}\right) \psi(\theta, \phi)=E \psi(\theta, \phi)\nonumber\]

As it turns out, the solutions to this equation are very important in a number of areas in chemistry and physics. The eigenfunctions are known as the spherical harmonics \(\left(Y_{l}^{m_{l}}(\theta, \phi)\right)\) and they appear in every problem that has spherical symmetry. The Spherical Harmonics satisfy the relationship

\[\left(\dfrac{1}{\sin \theta} \dfrac{\partial}{\partial \theta} \sin \theta \dfrac{\partial}{\partial \theta}+\dfrac{1}{\sin ^{2} \theta} \dfrac{\partial^{2}}{\partial \phi^{2}}\right) Y_{l}^{m_{l}}(\theta, \phi)=\hbar^{2} l(l+1) Y_{l}^{m_{l}}(\theta, \phi)\nonumber\]

Each function \(Y_{l}^{m_{l}}(\theta, \phi)\) has three parts: 1) a normalization constant, 2) an associated Legendre polynomial in \(\cos (\theta)\), and 3) an imaginary (for \(m_{l} \neq 0\) ) exponential in \(\phi\).

\[Y_{l}^{m_{l}}(\theta, \phi)=\left[\dfrac{(2 l+1)\left(l-\left|m_{l}\right|\right)}{4 \pi\left(l+\left|m_{l}\right|\right)}\right]^{\dfrac{1}{2}} P_{l}^{\left|m_{l}\right|}(\cos \theta) e^{i m_{l} \phi}\nonumber\]

The first few Spherical harmonics are shown in the table below.

\(\boldsymbol{I}\)	\(\boldsymbol{m}_{\boldsymbol{I}}\)	\(Y_{l}^{m_{l}}(\theta, \phi)\)
0	0	\(\sqrt{\dfrac{1}{4 \pi}}\)
\[1\nonumber\]	0	\(\sqrt{\dfrac{3}{4 \pi}} \cos (\theta)\)
	\(\pm 1\)	\(\pm 1\)
\[2\nonumber\]	\(\pm 1\)	\(\sqrt{\dfrac{5}{16 \pi}}\left(3 \cos ^{2}(\theta)-1\right)\)
	0	\(\sin (\theta) \cos (\theta) e^{\pm i \phi}\)
	\(\pm 2\)	\(\sqrt{\dfrac{15}{32 \pi}} \sin ^{2}(\theta) e^{\pm 2 i \phi}\)

Notice the \((2 l+1)\) degeneracy in these functions, due to the \((2 l+1)\) values of \(m_{l}\) for each value of \(l\). Also, it is useful to not that these functions all have \(l\) angular nodes (values of \(\theta\) that cause the wavefunction to vanish.) For the \(l=1\) wavefunctions, these nodes occur at \(\theta=\pi / 2\) for \(m_{l}=0\) and at \(\theta=0\) for \(m_{l}=\pm 1\). The number of nodes in each wavefunction is a useful property to know when discussing how these functions related to the radial wavefunction in the Hydrogen atom.