球谐函数
前几阶球谐函数

条目：球谐函数表
|  l |  m |  \Phi (\varphi ) |  \Theta (\theta ) |  | 極坐標中的表達式 | 直角坐標中的表達式 | 量子力學中的記号 |
| --- | --- | --- | --- | --- | --- | --- | --- |
| 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\varphi ) |  {\frac {\sqrt {3}}{2}}\sin \theta  |  {\Bigg \{} |  {\frac {1}{2}}{\sqrt {\frac {3}{\pi }}}\sin \theta \cos \varphi  |  {\frac {1}{2}}{\sqrt {\frac {3}{\pi }}}{\frac {x}{r}} |  {\mbox{p}}_{x}\, |
| 1 | -1 |  {\frac {1}{\sqrt {2\pi }}}\exp(-i\varphi ) |  {\frac {\sqrt {3}}{2}}\sin \theta  |  {\Bigg \{} |  {\frac {1}{2}}{\sqrt {\frac {3}{\pi }}}\sin \theta \sin \varphi  |  {\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\varphi ) |  {\frac {\sqrt {15}}{2}}\sin \theta \cos \theta  |  {\Bigg \{} |  {\frac {1}{2}}{\sqrt {\frac {15}{\pi }}}\sin \theta \cos \theta \cos \varphi  |  {\frac {1}{2}}{\sqrt {\frac {15}{\pi }}}{\frac {zx}{r^{2}}} |  {\mbox{d}}_{zx}\, |
| 2 | -1 |  {\frac {1}{\sqrt {2\pi }}}\exp(-i\varphi ) |  {\frac {\sqrt {15}}{2}}\sin \theta \cos \theta  |  {\Bigg \{} |  {\frac {1}{2}}{\sqrt {\frac {15}{\pi }}}\sin \theta \cos \theta \sin \varphi  |  {\frac {1}{2}}{\sqrt {\frac {15}{\pi }}}{\frac {yz}{r^{2}}} |  {\mbox{d}}_{yz}\, |
| 2 | +2 |  {\frac {1}{\sqrt {2\pi }}}\exp(2i\varphi ) |  {\frac {\sqrt {15}}{4}}\sin ^{2}\theta  |  {\Bigg \{} |  {\frac {1}{4}}{\sqrt {\frac {15}{\pi }}}\sin ^{2}\theta \cos 2\varphi  |  {\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\varphi ) |  {\frac {\sqrt {15}}{4}}\sin ^{2}\theta  |  {\Bigg \{} |  {\frac {1}{4}}{\sqrt {\frac {15}{\pi }}}\sin ^{2}\theta \sin 2\varphi  |  {\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\varphi ) |  {\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 \varphi  |  {\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\varphi ) |  {\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 \varphi  |  {\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\varphi ) |  {\frac {\sqrt {105}}{4}}\cos \theta \sin ^{2}\theta  |  {\Bigg \{} |  {\frac {1}{4}}{\sqrt {\frac {105}{\pi }}}\cos \theta \sin ^{2}\theta \cos 2\varphi  |  {\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\varphi ) |  {\frac {\sqrt {105}}{4}}\cos \theta \sin ^{2}\theta  |  {\Bigg \{} |  {\frac {1}{4}}{\sqrt {\frac {105}{\pi }}}\cos \theta \sin ^{2}\theta \sin 2\varphi  |  {\frac {1}{2}}{\sqrt {\frac {105}{\pi }}}{\frac {xyz}{r^{3}}} |  {\mbox{f}}_{xyz}\, |
| 3 | +3 |  {\frac {1}{\sqrt {2\pi }}}\exp(3i\varphi ) |  {\frac {1}{4}}{\sqrt {\frac {35}{2}}}\sin ^{3}\theta  |  {\Bigg \{} |  {\frac {1}{4}}{\sqrt {\frac {35}{2\pi }}}\sin ^{3}\theta \cos 3\varphi  |  {\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\varphi ) |  {\frac {1}{4}}{\sqrt {\frac {35}{2}}}\sin ^{3}\theta  |  {\Bigg \{} |  {\frac {1}{4}}{\sqrt {\frac {35}{2\pi }}}\sin ^{3}\theta \sin 3\varphi  |  {\frac {1}{4}}{\sqrt {\frac {35}{2\pi }}}{\frac {y(3x^{2}-y^{2})}{r^{3}}} |  {\mbox{f}}_{y(3x^{2}-y^{2})} |
l=0
l=1
l=2
實值的球諧函數 Ylm，l = 0 到 4 （由上至下），m=0 到 4（由左至右）。負數階球諧函數 Yl,-m 可由正數階函數對 z-軸轉 90/m 度得到。