# Radial and angular wavefunctions
The few first wavefunctions are listed below,

\begin{align}
R_{10} & = 2a^{-3/2}\ \exp(-r/a) \\
R_{20} & = \frac{1}{\sqrt{2}} a^{-3/2}\ (1-\frac{1}{2}\frac{r}{a}) \exp(-r/2a) \\
R_{21} & = \frac{1}{\sqrt{24}} a^{-3/2}\ \frac{r}{a} \exp(-r/2a) \\
R_{30} & = \frac{2}{\sqrt{27}} a^{-3/2}\ \left[
1-\frac{2}{3}\frac{r}{a} + \frac{2}{27}(\frac{r}{a})^2 \right]
\exp(-r/3a) \\
R_{31} & = \frac{8}{27\sqrt{6}} a^{-3/2}\ ( 1-\frac{1}{6}\frac{r}{a} ) (\frac{r}{a}) \exp(-r/3a) \\
R_{32} & = \frac{4}{81\sqrt{30}} a^{-3/2}\ (\frac{r}{a})^2 \exp(-r/3a) \\
\\
Y^0_0 & = (\frac{1}{4\pi})^{1/2} \\
Y^0_1 & = (\frac{3}{4\pi})^{1/2} \cos{\theta} \\
Y^{\pm 1}_1 & = \mp (\frac{3}{8\pi})^{1/2} \sin{\theta} e^{\pm i\phi} \\
Y^{0}_2 & = (\frac{5}{16\pi})^{1/2} (3\cos^2{\theta} - 1) \\
Y^{\pm 1}_2 & = \mp (\frac{15}{8\pi})^{1/2} \sin{\theta} \cos{\theta} e^{\pm i\phi} \\
Y^{\pm 2}_2 & = (\frac{15}{32\pi})^{1/2} \sin^2{\theta} e^{\pm 2i\phi}.
\end{align}

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For a hydrogem atom, the quantized energy level and the Bohr radius of are

\begin{equation}
E_n = - \left[
\frac{m}{2\hbar^2}(\frac{e^2}{4\pi \epsilon_0})^2 \right] \frac{1}{n^2}
= -\frac{\hbar^2}{2ma^2}\ \frac{1}{n^2}
= \frac{E_1}{n^2}
=\frac{-13.6 \text{eV}}{n^2}
\end{equation}

\begin{equation}
a \equiv \frac{4\pi\epsilon_0\hbar^2}{me^2} = 0.529 \text{Å}
\end{equation}