Spherical coordinates
#####################

Spherical coordinate system
===========================

.. math::

    x &= r \sin \theta \cos \phi \\
    y &= r \sin \theta \sin \phi \\
    z &= r \cos \theta

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

Furthermore, the volume element is given by 

.. math:: 
    
    dV = r^2 \sin \theta  dr d\theta d\phi 

and the Laplacian is given by 

.. math::

    \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}

Furthermore, the cartesian derivative operators are given by 

.. math::
    
    \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}



Wavefunction paramtetrization
=============================

In spherical coordinates, we parametrize the wavefunction as

.. math::

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

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

.. math::

    \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)