Wavefunctions of the Hydrogen Atom

 

n

$\ell$

m

$R_{n\ell}$

$Y_{\ell m}$

1

0

0

$2\left({1\over a_0}\right)^{3/2}e^{-r/a_0}$

${1\over 2\sqrt{\pi}}$

${1\over \sqrt{\pi}}\left({1\over a_0}\right)^{3/2}e^{-r/a_0}$

2

0

0

$\left({1\over 2a_0}\right)^{3/2}\left(2-{r\over a_0}\right)e^{-r/2a_0}$

${1\over 2\sqrt{\pi}}$

${1\over 4\sqrt{2\pi}}\left({1\over a_0}\right)^{3/2}\left(2-{r\over a_0}\right)e^{-r/2a_0}$

2

1

0

$\left({1\over 2a_0}\right)^{3/2}{1\over \sqrt{3}}{r\over a_0}e^{-r/2a_0}$

${1\over 2}\sqrt{3\over\pi}\cos{\theta}$

$ {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

$\pm 1$

$\left({1\over 2a_0}\right)^{3/2}{1\over \sqrt{3}}{r\over a_0}e^{-r/2a_0}$

$\pm {1\over 2} \sqrt{3\over 2\pi}\sin{\theta}e^{\pm i\phi}$

${1\over 8} \sqrt{1\over \pi} \left({1\over a_0}\right)^{3/2}{r\over a_0}e^{-r/2a_0}\sin{\theta}e^{\pm i\phi}$

3

0

0

$2\left({1\over 3a_0}\right)^{3/2}\left(1 - {2\over 3}{r\over
a_0}+{2\over 27}(r/a_0)^2\right)e^{-r/3a_0}$

${1\over 2\sqrt{\pi}}$

${1\over 81\sqrt{3\pi}}\left({1\over a_0}\right)^{3/2}\left(27 - 18{r\over
a_0}+2(r/a_0)^2\right)e^{-r/3a_0}$

3

1

0

$\left({1\over 3a_0}\right)^{3/2}{4\sqrt{2}\over 3}\left(1 -
{1\over 6}{r\over a_0}\right){r\over
a_0}e^{-r/3a_0}$

${1\over 2}\sqrt{3\over\pi}\cos{\theta}$

${1\over
81}\sqrt{2\over \pi}\left({1\over a_0}\right)^{3/2}\left(6 -
{r\over a_0}\right){r\over a_0}e^{-r/3a_0}\cos{\theta}$

3

1

$\pm 1$

$\left({1\over 3a_0}\right)^{3/2}{4\sqrt{2}\over 3}\left(1 -
{1\over 6}{r\over a_0}\right){r\over
a_0}e^{-r/3a_0}$

$\pm {1\over 2} \sqrt{3\over 2\pi}\sin{\theta}e^{\pm i\phi}$

$ {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

$ \left({1\over 3a_0}\right)^{3/2} {2\sqrt{2}\over 27\sqrt{5}}\left({r\over
a_0}\right)^2e^{-r/3a_0}$

${1\over 4} \sqrt{5\over\pi}\left( 3\cos^2{\theta}-1\right) $

${1\over
81\sqrt{6\pi}}\left({1\over a_0}\right)^{3/2}{r^2\over a_0^2}e^{-r/3a_0}\left(3\cos^2{\theta}-1\right)$

3

2

$\pm 1$

$ \left({1\over 3a_0}\right)^{3/2} {2\sqrt{2}\over 27\sqrt{5}}\left({r\over
a_0}\right)^2e^{-r/3a_0}$

$\pm {1\over 2}\sqrt{15\over 2\pi}\sin{\theta}\cos{\theta}e^{\pm i\phi}$

${1\over 81\sqrt{\pi}}\left({1\over a_0}\right)^{3/2} \left({r\over a_0}\right)^2e^{-r/3a_0}\sin{\theta}\cos{\theta}e^{\pm i\phi}$

3

2

$\pm 2$

$ \left({1\over 3a_0}\right)^{3/2} {2\sqrt{2}\over 27\sqrt{5}}\left({r\over
a_0}\right)^2e^{-r/3a_0}$

${1\over 4}\sqrt{15\over 2\pi}\sin^2{\theta}e^{\pm 2i\phi}$

${1\over
162\sqrt{\pi}} \left({1\over a_0}\right)^{3/2} \left({r\over a_0}\right)^2e^{-r/3a_0}\sin^2{\theta}e^{\pm 2i\phi}$