Hydrogen Atom Wave Functions, and Probability Densities

Table 1: Wave functions and their components
n	$\ell$	m		orbital shapes
1	0	0	${1\over \sqrt{\pi}}\left({1\over a_0}\right)^{3/2}e^{-r/a_0}$	1s
2	0	0	${1\over 4\sqrt{2\pi}}\left({1\over a_0}\right)^{3/2}\left(2-{r\over a_0}\right)e^{-r/2a_0}$	2s
2	1	0	${1\over 4\sqrt{2\pi}}\left({1\over a_0}\right)^{3/2} {r\over a_0}e^{-r/2a_0}\cos{\theta}$
2	1	$\pm 1$	${1\over 8} \sqrt{1\over \pi} \left({1\over a_0}\right)^{3/2}{r\over a_0}e^{-r/2a_0}\sin{\theta}e^{\pm i\phi}$
3	0	0	${1\over 81\sqrt{3\pi}}\left({1\over a_0}\right)^{3/2}\left(27 - 18{r\over a_0}+2(r/a_0)^2\right)e^{-r/3a_0}$	3s
3	1	0	${1\over 81}\sqrt{2\over \pi}\left({1\over a_0}\right)^{3/2}\left(6 - {r\over a_0}\right){r\over a_0}e^{-r/3a_0}\cos{\theta}$
3	1	$\pm 1$	${1\over 8\sqrt{\pi}}\left(1\over a_0\right)^{3/2}\left(6-{r\over a_0}\right){r\over a_0}e^{-r/3a_0}\sin{\theta}e^{\pm i\phi}$
3	2	0	${1\over 81\sqrt{6\pi}}\left({1\over a_0}\right)^{3/2}{r^2\over a_0^2}e^{-r/3a_0}\left(3\cos^2{\theta}-1\right)$
3	2	$\pm 1$	${1\over 81\sqrt{\pi}}\left({1\over a_0}\right)^{3/2} \left({r\over a_0}\right)^2e^{-r/3a_0}\sin{\theta}\cos{\theta}e^{\pm i\phi}$
3	2	$\pm 2$	${1\over 162\sqrt{\pi}} \left({1\over a_0}\right)^{3/2} \left({r\over a_0}\right)^2e^{-r/3a_0}\sin^2{\theta}e^{\pm 2i\phi}$