水素原子におけるシュレーディンガー方程式の解
具体的な値

水素原子の波動関数のℓ  = 0~3における角因子は以下のようになる。ここでΘ(θ)、Φ(φ)はそれぞれ、動径方向の関数
の右辺の積の第一成分と第二成分を規格化したものである。なお、Φ(φ)の指数関数の虚数部分はオイラーの公式により一対のΦ(φ)関数の一次結合で書き換えられる。
| ℓ | m | Φ(φ) | Θ(θ) |  | Φ(φ)Θ(θ)（極座標） | Φ(φ)Θ(θ)（直交座標） | 記号 |
| --- | --- | --- | --- | --- | --- | --- | --- |
| 0 | 0 |  {\frac {1}{\sqrt {2\pi }}} |  {\frac {1}{\sqrt {2}}} |  |  {\frac {1}{2{\sqrt {\pi }}}} |  {\frac {1}{2{\sqrt {\pi }}}} |  {\mbox{s}}\, |
| 1 | 0 |  {\frac {1}{\sqrt {2\pi }}} |  {\sqrt {\frac {3}{2}}}\cos \theta  |  |  {\frac {1}{2}}{\sqrt {\frac {3}{\pi }}}\cos \theta  |  {\frac {1}{2}}{\sqrt {\frac {3}{\pi }}}{\frac {z}{r}} |  {\mbox{p}}_{z}\, |
| 1 | +1 |  {\frac {1}{\sqrt {2\pi }}}\exp(i\phi ) |  {\frac {\sqrt {3}}{2}}\sin \theta  |  {\Bigg \{} |  {\frac {1}{2}}{\sqrt {\frac {3}{\pi }}}\sin \theta \cos \phi  |  {\frac {1}{2}}{\sqrt {\frac {3}{\pi }}}{\frac {x}{r}} |  {\mbox{p}}_{x}\, |
| 1 | -1 |  {\frac {1}{\sqrt {2\pi }}}\exp(-i\phi ) |  {\frac {\sqrt {3}}{2}}\sin \theta  |  {\Bigg \{} |  {\frac {1}{2}}{\sqrt {\frac {3}{\pi }}}\sin \theta \sin \phi  |  {\frac {1}{2}}{\sqrt {\frac {3}{\pi }}}{\frac {y}{r}} |  {\mbox{p}}_{y}\, |
| 2 | 0 |  {\frac {1}{\sqrt {2\pi }}} |  {\frac {1}{2}}{\sqrt {\frac {5}{2}}}(3\cos ^{2}\theta -1) |  |  {\frac {1}{4}}{\sqrt {\frac {5}{\pi }}}(3\cos ^{2}\theta -1) |  {\frac {1}{4}}{\sqrt {\frac {5}{\pi }}}{\frac {2z^{2}-x^{2}-y^{2}}{r^{2}}} |  {\mbox{d}}_{3z^{2}-r^{2}} |
| 2 | +1 |  {\frac {1}{\sqrt {2\pi }}}\exp(i\phi ) |  {\frac {\sqrt {15}}{2}}\sin \theta \cos \theta  |  {\Bigg \{} |  {\frac {1}{2}}{\sqrt {\frac {15}{\pi }}}\sin \theta \cos \theta \cos \phi  |  {\frac {1}{2}}{\sqrt {\frac {15}{\pi }}}{\frac {zx}{r^{2}}} |  {\mbox{d}}_{zx}\, |
| 2 | -1 |  {\frac {1}{\sqrt {2\pi }}}\exp(-i\phi ) |  {\frac {\sqrt {15}}{2}}\sin \theta \cos \theta  |  {\Bigg \{} |  {\frac {1}{2}}{\sqrt {\frac {15}{\pi }}}\sin \theta \cos \theta \sin \phi  |  {\frac {1}{2}}{\sqrt {\frac {15}{\pi }}}{\frac {yz}{r^{2}}} |  {\mbox{d}}_{yz}\, |
| 2 | +2 |  {\frac {1}{\sqrt {2\pi }}}\exp(2i\phi ) |  {\frac {\sqrt {15}}{4}}\sin ^{2}\theta  |  {\Bigg \{} |  {\frac {1}{4}}{\sqrt {\frac {15}{\pi }}}\sin ^{2}\theta \cos 2\phi  |  {\frac {1}{4}}{\sqrt {\frac {15}{\pi }}}{\frac {x^{2}-y^{2}}{r^{2}}} |  {\mbox{d}}_{x^{2}-y^{2}} |
| 2 | -2 |  {\frac {1}{\sqrt {2\pi }}}\exp(-2i\phi ) |  {\frac {\sqrt {15}}{4}}\sin ^{2}\theta  |  {\Bigg \{} |  {\frac {1}{4}}{\sqrt {\frac {15}{\pi }}}\sin ^{2}\theta \sin 2\phi  |  {\frac {1}{2}}{\sqrt {\frac {15}{\pi }}}{\frac {xy}{r^{2}}} |  {\mbox{d}}_{xy}\, |
| 3 | 0 |  {\frac {1}{\sqrt {2\pi }}} |  {\frac {1}{2}}{\sqrt {\frac {7}{2}}}(5\cos ^{3}\theta -3\cos \theta ) |  |  {\frac {1}{4}}{\sqrt {\frac {7}{\pi }}}(5\cos ^{3}\theta -3\cos \theta ) |  {\frac {1}{4}}{\sqrt {\frac {7}{\pi }}}{\frac {z(2z^{2}-3x^{2}-3y^{2})}{r^{3}}} |  {\mbox{f}}_{z(5z^{2}-3r^{2})} |
| 3 | +1 |  {\frac {1}{\sqrt {2\pi }}}\exp(i\phi ) |  {\frac {1}{4}}{\sqrt {\frac {21}{2}}}(5\cos ^{2}\theta -1)\sin \theta  |  {\Bigg \{} |  {\frac {1}{4}}{\sqrt {\frac {21}{2\pi }}}(5\cos ^{2}\theta -1)\sin \theta \cos \phi  |  {\frac {1}{4}}{\sqrt {\frac {21}{2\pi }}}{\frac {x(5z^{2}-r^{2})}{r^{3}}} |  {\mbox{f}}_{x(5z^{2}-r^{2})} |
| 3 | -1 |  {\frac {1}{\sqrt {2\pi }}}\exp(-i\phi ) |  {\frac {1}{4}}{\sqrt {\frac {21}{2}}}(5\cos ^{2}\theta -1)\sin \theta  |  {\Bigg \{} |  {\frac {1}{4}}{\sqrt {\frac {21}{2\pi }}}(5\cos ^{2}\theta -1)\sin \theta \sin \phi  |  {\frac {1}{4}}{\sqrt {\frac {21}{2\pi }}}{\frac {y(5z^{2}-r^{2})}{r^{3}}} |  {\mbox{f}}_{y(5z^{2}-r^{2})} |
| 3 | +2 |  {\frac {1}{\sqrt {2\pi }}}\exp(2i\phi ) |  {\frac {\sqrt {105}}{4}}\cos \theta \sin ^{2}\theta  |  {\Bigg \{} |  {\frac {1}{4}}{\sqrt {\frac {105}{\pi }}}\cos \theta \sin ^{2}\theta \cos 2\phi  |  {\frac {1}{4}}{\sqrt {\frac {105}{\pi }}}{\frac {z(x^{2}-y^{2})}{r^{3}}} |  {\mbox{f}}_{z(x^{2}-y^{2})} |
| 3 | -2 |  {\frac {1}{\sqrt {2\pi }}}\exp(-2i\phi ) |  {\frac {\sqrt {105}}{4}}\cos \theta \sin ^{2}\theta  |  {\Bigg \{} |  {\frac {1}{4}}{\sqrt {\frac {105}{\pi }}}\cos \theta \sin ^{2}\theta \sin 2\phi  |  {\frac {1}{2}}{\sqrt {\frac {105}{\pi }}}{\frac {xyz}{r^{3}}} |  {\mbox{f}}_{xyz}\, |
| 3 | +3 |  {\frac {1}{\sqrt {2\pi }}}\exp(3i\phi ) |  {\frac {1}{4}}{\sqrt {\frac {35}{2}}}\sin ^{3}\theta  |  {\Bigg \{} |  {\frac {1}{4}}{\sqrt {\frac {35}{2\pi }}}\sin ^{3}\theta \cos 3\phi  |  {\frac {1}{4}}{\sqrt {\frac {35}{2\pi }}}{\frac {x(x^{2}-3y^{2})}{r^{3}}} |  {\mbox{f}}_{x(x^{2}-3y^{2})} |
| 3 | -3 |  {\frac {1}{\sqrt {2\pi }}}\exp(-3i\phi ) |  {\frac {1}{4}}{\sqrt {\frac {35}{2}}}\sin ^{3}\theta  |  {\Bigg \{} |  {\frac {1}{4}}{\sqrt {\frac {35}{2\pi }}}\sin ^{3}\theta \sin 3\phi  |  {\frac {1}{4}}{\sqrt {\frac {35}{2\pi }}}{\frac {y(3x^{2}-y^{2})}{r^{3}}} |  {\mbox{f}}_{y(3x^{2}-y^{2})} |
原子番号Zの水素様原子の動径関数は以下のようになる。
| 0 | 1 | 2 | 3 |
| --- | --- | --- | --- |

1s軌道の動径関数: 

| 2s軌道の動径関数 | 2p軌道の動径関数 |  |  |

| 3s軌道の動径関数 | 3p軌道の動径関数 | 3d軌道の動径関数 |  |

| 4s軌道の動径関数 | 4p軌道の動径関数 | 4d軌道の動径関数 | 4f軌道の動径関数 |
動径関数を2乗しr2を掛けた動径分布r2R(r)2は、核の中心からのある距離における電子の存在確率に相当する。
| 0 | 1 | 2 | 3 |
| --- | --- | --- | --- |

1s軌道の動径分布: 

| 2s軌道の動径分布 | 2p軌道の動径分布 |  |  |

| 3s軌道の動径分布 | 3p軌道の動径分布 | 3d軌道の動径分布 |  |

| 4s軌道の動径分布 | 4p軌道の動径分布 | 4d軌道の動径分布 | 4f軌道の動径分布 |
詳しくは電子配置の項を参照のこと。