Total wavefunction of a hydrogenic atom

The total wavefunction $\psi$ of a hydrogenic atom characterises the probability of locating an electron around the nucleus by incorporating both radial and angular components.

This spatial wavefunction is mathematically defined as:

$\psi=R_{n,l}(r)Y_l^{\vert m_l\vert}(\theta,\phi)\;\;\;\;\;\;\;\;470$

where

$R_{n,l}(r)=N_{n,l}\frac{1}{r}\biggr$\frac{r}{na_0}\biggr$^{l+1}e^{-\frac{r}{na_0}}L_{n-l-1}^{2l+1}\biggr$\frac{2r}{na_0}\biggr$$ , the radial wavefunction, is the radial component of $\psi$ .
$Y_l^{\vert m_l\vert}(\theta,\phi)=N_l^{\vert m_l\vert}P_l^{\vert m_l\vert}(cos\theta)e^{im_l\phi}$ , the spherical harmonics, is the angular component of $\psi$ .
$L_{n-l-1}^{2l+1}=\sum_{k=0}^{n-l-1}(-1)^k\frac{(n+l)!}{k!(k+2l+1)!(n-l-1-k)!}\biggr$\frac{2r}{na_0}\biggr$^k$ are the associated Laguerre polynomials.
$P_l^{\vert m_l\vert}(cos\theta)=\frac{1}{2^ll!}(1-cos^2\theta)^{\frac{\vert m_l\vert}{2}}\frac{d^{l+\vert m_l\vert}}{dcos\theta^{l+\vert m_l\vert}}(cos^2\theta-1)^l$ are the associated Legendre polynomials.
$N_{n,l}=\sqrt{\frac{(n-l-1)!}{2n(n+l)!}\frac{2^{2l+3}}{na_0}}$ is the normalisation constant of the radial wavefunction.
$N_l^{\vert m_l\vert}=\frac{1}{\sqrt{2\pi}}\biggr\[\frac{2l+1}{2}\frac{(l-\vert m_l\vert)!}{(l+\vert m_l\vert)!}\biggr\]^{\frac{1}{2}}$ is the normalisation constant of the spherical harmonics.
$n$ is the principal quantum number, where $n=1,2,\cdots$
$l$ is the orbital angular momentum quantum number (also known as azimuthal quantum number), where $l=0,1,2,\cdots$ and $l\leq n-1$ .
$m_l$ is the magnetic quantum number, where $-l\leq m_l\leq l$ .
$r$ is the distance between an electron and the nucleus of a hydrogenic atom.
$\theta$ is the polar angle.
$\phi$ is the azimuthal angle.
$a_0=\frac{4\pi\varepsilon_0\hbar^2}{\mu Ze^2}$ is the Bohr radius.
$\mu$ is the reduced mass of the atom.
$\hbar$ is the ratio of the Planck constant and $2\pi$ .
$Z$ is the atomic number of the atom.
$\varepsilon_0$ is the vacuum permittivity.