Nachfolgend eine Aufstellung der normierten vollständigen Eigenfunktionen eines Elektrons im Coulombpotential $V(r) = -Ze^2/(4\pi\epsilon_0 r)$.
| n | l | m | $\boldsymbol{\psi_{nlm}(r,\vartheta,\varphi)}$ |
|---|---|---|---|
| 1 | 0 | 0 | $\frac{1}{\sqrt{\pi}}\left(\frac{Z}{a_0}\right)^{3/2}\exp\left\{-\frac{Zr}{a_0}\right\}$ |
| 2 | 0 | 0 | $\frac{1}{4\sqrt{2\pi}}\left(\frac{Z}{a_0}\right)^{3/2} \left(2-\frac{Zr}{a_0}\right)\exp\left\{-\frac{Zr}{2a_0}\right\}$ |
| 2 | 1 | 0 | $\frac{1}{4\sqrt{2\pi}}\left(\frac{Z}{a_0}\right)^{3/2}\frac{Zr}{a_0}\exp\left\{-\frac{Zr}{2a_0}\right\}\cos\vartheta$ |
| 2 | 1 | ±1 | $\frac{1}{8\sqrt{\pi}} \left(\frac{Z}{a_0}\right)^{3/2}\frac{Zr}{a_0}\exp\left\{-\frac{Zr}{2a_0}\right\}\sin\vartheta\exp\left\{\pm i\varphi\right\}$ |
| 3 | 0 | 0 | $\frac{1}{81\sqrt{3\pi}}\left(\frac{Z}{a_0}\right)^{3/2}\left(27-18\frac{Zr}{a_0}+2\frac{Z^2 r^2}{a_0^2}\right)\exp\left\{-\frac{Zr}{3a_0}\right\}$ |
| 3 | 1 | 0 | $\frac{\sqrt{2}}{81\sqrt{\pi}}\left(\frac{Z}{a_0}\right)^{3/2}\left(6-\frac{Zr}{a_0}\right)\frac{Zr}{a_0}\exp\left\{-\frac{Zr}{3a_0}\right\}\cos\vartheta$ |
| 3 | 1 | ±1 | $\frac{1}{81\sqrt{\pi}}\left(\frac{Z}{a_0}\right)^{3/2}\left(6-\frac{Zr}{a_0}\right)\frac{Zr}{a_0}\exp\left\{-\frac{Zr}{3a_0}\right\}\sin\vartheta \exp\left\{\pm 2 i\varphi\right\}$ |
| 3 | 2 | 0 | $\frac{1}{81\sqrt{6\pi}}\left(\frac{Z}{a_0}\right)^{3/2}\frac{Z^2 r^2}{a_0^2} \exp\left\{-\frac{Zr}{3a_0}\right\}\left(3\cos^2\vartheta - 1\right)$ |
| 3 | 2 | ±1 | $\frac{1}{81\sqrt{\pi}}\left(\frac{Z}{a_0}\right)^{3/2}\frac{Z^2 r^2}{a_0^2} \exp\left\{-\frac{Zr}{3a_0}\right\}\sin\vartheta\cos\vartheta\exp\left\{\pm 2 i\varphi\right\}$ |
| 3 | 2 | ±2 | $\frac{1}{162\sqrt{\pi}}\left(\frac{Z}{a_0}\right)^{3/2}\frac{Z^2 r^2}{a_0^2} \exp\left\{-\frac{Zr}{3a_0}\right\} \sin^2\vartheta\exp\left\{\pm 2 i\varphi\right\}$ |