📂Quantum Mechanics

The eigenfunctions of the angular momentum operator are spherical harmonics.

Summary

The angular momentum operator $L^{2}$ and $L_{z}$ have simultaneous eigenfunctions determined by constants $l$, $m$ $\ket{\ell, m}$.

$$\begin{align*} L^{2}\ket{\ell, m} &= \hbar^{2}\ell(\ell+1)\ket{\ell, m} \\ L_{z}\ket{\ell, m} &= m\hbar\ket{\ell, m} \end{align*}$$

Here, the eigenfunction of the angular momentum operator $\ket{\ell, m}$ is actually spherical harmonics $Y_{l}^{m}$.

$$\ket{\ell, m} = Y_{l}^{m}$$

Proof

In spherical coordinates, the angular momentum operator $L_{z}$ is as follows:

$$L_{z} = -\i\hbar\frac{\partial}{\partial \phi}$$

Also, the ladder operator of angular momentum is as follows:

$$L_{+}L_{-} = -\hbar ^{2} \left( \frac{\partial^{2}}{\partial \theta^{2}} + \cot \theta \frac{\partial}{\partial \theta}+\cot ^{2}\theta \frac{\partial^{2}}{\partial \phi^{2}} +\i\frac{\partial}{\partial \phi}\right)$$

Then, since $L^{2} = L_{+}L_{-} + L_{z}^{2} - \hbar L_{z}$, we obtain the following equation.

$$\begin{align*} L^{2} &= -\hbar ^{2} \left( \frac{\partial^{2}}{\partial \theta^{2}} + \cot \theta \frac{\partial}{\partial \theta}+\cot ^{2}\theta \frac{\partial^{2}}{\partial \phi^{2}} +\i\frac{\partial}{\partial \phi}\right) + \left( -\i\hbar\frac{\partial}{\partial \phi} \right)^{2} -\hbar \left( -\i\hbar\frac{\partial}{\partial \phi} \right)\\ &= -\hbar ^{2} \left( \frac{\partial^{2}}{\partial \theta^{2}} + \cot \theta \frac{\partial}{\partial \theta}+\cot ^{2}\theta \frac{\partial^{2}}{\partial \phi^{2}} + \i\frac{\partial}{\partial \phi}\right) - \hbar ^{2}\frac{\partial^{2}}{\partial \phi^{2}} + \i \hbar ^{2}\frac{\partial}{\partial \phi} \\ &= -\hbar ^{2} \left( \frac{\partial^{2}}{\partial \theta^{2}} + \cot \theta \frac{\partial}{\partial \theta}+\cot ^{2}\theta \frac{\partial^{2}}{\partial \phi^{2}} + \frac{\partial^{2}}{\partial \phi^{2}}\right) \\ &= -\hbar ^{2} \left( \dfrac{1}{\sin\theta} \dfrac{\partial }{\partial \theta} \left( \sin\theta \frac{\partial}{\partial \theta} \right) + (\cot ^{2} + 1) \theta \frac{\partial^{2}}{\partial \phi^{2}} \right) \\ &= -\hbar ^{2} \left( \dfrac{1}{\sin\theta} \dfrac{\partial }{\partial \theta} \left( \sin\theta \frac{\partial}{\partial \theta} \right) + \dfrac{1}{\sin^{2}\theta}\frac{\partial^{2}}{\partial \phi^{2}} \right) \\ \end{align*}$$

Applying this to the eigenfunctions,

$$L^{2}\ket{\ell, m}=-\hbar ^{2}\left[ \frac{1}{\sin \theta}\frac{\partial}{\partial \theta}\left(\sin \theta \frac{\partial}{\partial \theta} \right)+\frac{1}{\sin ^{2} \theta}\frac{\partial^{2}}{ \partial \phi^{2} } \right]\ket{\ell, m} = \ell(\ell + 1)\hbar^{2}\ket{\ell, m}$$

Therefore, we obtain the following.

$$\begin{align*} && \left[ \frac{1}{\sin \theta}\frac{\partial}{\partial \theta}\left(\sin \theta \frac{\partial}{\partial \theta} \right)+\frac{1}{\sin ^{2} \theta}\frac{\partial^{2}}{ \partial \phi^{2} } \right]\ket{\ell, m} &= -\ell(\ell + 1)\ket{\ell, m} \\ \end{align*}$$

However, assuming that it can be separated by variables as $\ket{\ell, m} = \Theta(\theta)\Phi(\phi)$, we find that it is exactly the same as the differential equation that the spherical harmonics satisfy.

$$\begin{align*} && \left[ \frac{1}{\sin \theta}\frac{\partial}{\partial \theta}\left(\sin \theta \frac{\partial}{\partial \theta} \right)+\frac{1}{\sin ^{2} \theta}\frac{\partial^{2}}{ \partial \phi^{2} } \right]\Theta(\theta)\Phi(\phi) &= -\ell(\ell + 1)\Theta(\theta)\Phi(\phi) \\ \implies && \frac{\Phi}{\sin \theta}\frac{d}{d \theta}\left(\sin \theta \frac{d \Theta}{d \theta} \right) + \frac{\Theta}{\sin ^{2} \theta}\frac{d^{2} \Phi}{d \phi^{2} } &= -\ell(\ell + 1)\Theta\Phi \\ \implies && \frac{1}{\Theta \sin \theta}\frac{d}{d \theta}\left(\sin \theta \frac{d \Theta}{d \theta} \right) + \frac{1}{\Phi \sin ^{2} \theta}\frac{d^{2} \Phi}{d \phi^{2} } &= -\ell(\ell + 1) \\ \end{align*}$$

Hence, the eigenfunction $\ket{\ell, m}$ of the angular momentum operator is actually the spherical harmonics $Y_{l}^{m}$.

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