The Hydrogen Atom:
Wave Functions,
Probability Density "pictures"

Table 1: Wave functions and their components
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}$ ${1\over 4}\sqrt{15\over 2\pi}\sin^2{\theta}e^{\pm 2i\phi}$ ${1\over 162\sqrt{\pi}} \left({1\over a_0}\right)^{3/2} \left({r\over a_0}\right)^2e^{-r/3a_0}\sin^2{\theta}e^{\pm 2i\phi}$

Figure 1: The coordinate system used to define the above variables.
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Figure 2:   Graphs of the radial wave functions for the first three principal quantum numbers of hydrogen.

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Figure 3: Probability distributions (per unit volume). A slice through the phi = 0 plane.
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Figure 4: The radial probability function r2R2 which gives the relative probability of finding the electron at a given distance r/a0.
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Figure 5: The energy ladder for hydrogen corresponding to its eigenstates.
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