2. Spherical coordinates

2.1. Spherical coordinate system

\[\begin{split}x &= r \sin \theta \cos \phi \\ y &= r \sin \theta \sin \phi \\ z &= r \cos \theta\end{split}\]

where \(r \in [0,\infty)\), \(\theta \in [0,\pi]\) and \(\phi \in [0,2\pi)\).

Furthermore, the volume element is given by

\[dV = r^2 \sin \theta dr d\theta d\phi\]

and the Laplacian is given by

\[\begin{split}\nabla^2 &= \frac{1}{r^2} \frac{\partial}{\partial r}\left( r^2 \frac{\partial}{\partial r} \right) + \frac{1}{r^2} \left[\frac{1}{\sin(\theta)}\frac{\partial}{\partial \theta}\left(\sin(\theta) \frac{\partial}{\partial \theta}\right) +\frac{1}{\sin^2(\theta)}\frac{\partial^2}{\partial \phi^2}\right] \\ &= \frac{1}{r} \frac{\partial^2}{\partial r^2} r - \frac{\hat{L}^2}{r^2}\end{split}\]

Furthermore, the cartesian derivative operators are given by

\[\begin{split}\frac{\partial}{\partial x} &= \cos{\phi} \sin{\theta}\frac{\partial}{\partial r} - \frac{\sin{\phi}}{r\sin{\theta}}\frac{\partial}{\partial \phi} - + \frac{\cos{\phi}\cos{\theta}}{r}\frac{\partial}{\partial \theta} \\ \frac{\partial}{\partial y} &= \sin{\phi} \sin{\theta}\frac{\partial}{\partial r} + \frac{\cos{\phi}}{r\sin{\theta}}\frac{\partial}{\partial \phi} + + \frac{\sin{\phi}\cos{\theta}}{r}\frac{\partial}{\partial \theta} \\ \frac{\partial}{\partial z} &= \cos{\theta}\frac{\partial}{\partial r} - \frac{\sin{\theta}}{r}\frac{\partial}{\partial \theta}\end{split}\]

2.2. Wavefunction paramtetrization

In spherical coordinates, we parametrize the wavefunction as

\[\Psi(\mathbf{r}) = \sum_{l=0}^{l_{max}} \sum_{m=-l}^{l} r^{-1} u_{l,m}(r) Y_{l,m}(\theta, \phi),\]

where \(Y_{l,m}(\theta, \phi)\) are the spherical harmonics and \(l_{max}\) the maximum angular momentum.

\[\nabla^2 \Psi(\mathbf{r}) = \sum_{l,m} \frac{1}{r} \left(\frac{d^2 u_{l,m}(r)}{d r^2} - \frac{l(l+1)}{r^2} u_{l,m}(r) \right) Y_{l,m}(\theta, \phi)\]