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Spherical Harmonics

Polar diagrams of the shperical harmonics: In each direction $(\vartheta,\varphi)$ we plot a point at distance $r=|Y_{l,m}(\vartheta,\varphi)|$ from the origin. We color this point with the phase of $Y_{l,m}(\vartheta,\varphi)$. Connecting these colored points gives the surfaces that are shown. Observe that all spherical harmonics are rotationally symmetric about the $z$ axis, with the phase given by $e^{im\varphi}$.
When you click on a plot you can rotate and zoom the spherical harmonic.
l=0
m=0
{ "orbtIndex":0, "bkgdColor":[1.0,1.0,1.0], "gridColor":[0.9,0.9,0.9], "panelOn":false, "cWidth":1.0, "cHeight":1.0 }
l=1
m=-1m=0m=1
{ "orbtIndex":1, "bkgdColor":[1.0,1.0,1.0], "gridColor":[0.9,0.9,0.9], "panelOn":false, "cWidth":1.0, "cHeight":1.0 }
{ "orbtIndex":2, "bkgdColor":[1.0,1.0,1.0], "gridColor":[0.9,0.9,0.9], "panelOn":false, "cWidth":1.0, "cHeight":1.0 }
{ "orbtIndex":3, "bkgdColor":[1.0,1.0,1.0], "gridColor":[0.9,0.9,0.9], "panelOn":false, "cWidth":1.0, "cHeight":1.0 }
l=2
m=-2m=-1m=0m=1m=2
{ "orbtIndex":4, "bkgdColor":[1.0,1.0,1.0], "gridColor":[0.9,0.9,0.9], "panelOn":false, "cWidth":1.0, "cHeight":1.0 }
{ "orbtIndex":5, "bkgdColor":[1.0,1.0,1.0], "gridColor":[0.9,0.9,0.9], "panelOn":false, "cWidth":1.0, "cHeight":1.0 }
{ "orbtIndex":6, "bkgdColor":[1.0,1.0,1.0], "gridColor":[0.9,0.9,0.9], "panelOn":false, "cWidth":1.0, "cHeight":1.0 }
{ "orbtIndex":7, "bkgdColor":[1.0,1.0,1.0], "gridColor":[0.9,0.9,0.9], "panelOn":false, "cWidth":1.0, "cHeight":1.0 }
{ "orbtIndex":8, "bkgdColor":[1.0,1.0,1.0], "gridColor":[0.9,0.9,0.9], "panelOn":false, "cWidth":1.0, "cHeight":1.0 }
l=3
m=-3m=-2m=-1m=0m=1m=2m=3
{ "orbtIndex":9, "bkgdColor":[1.0,1.0,1.0], "gridColor":[0.9,0.9,0.9], "panelOn":false, "cWidth":1.0, "cHeight":1.0 }
{ "orbtIndex":10, "bkgdColor":[1.0,1.0,1.0], "gridColor":[0.9,0.9,0.9], "panelOn":false, "cWidth":1.0, "cHeight":1.0 }
{ "orbtIndex":11, "bkgdColor":[1.0,1.0,1.0], "gridColor":[0.9,0.9,0.9], "panelOn":false, "cWidth":1.0, "cHeight":1.0 }
{ "orbtIndex":12, "bkgdColor":[1.0,1.0,1.0], "gridColor":[0.9,0.9,0.9], "panelOn":false, "cWidth":1.0, "cHeight":1.0 }
{ "orbtIndex":13, "bkgdColor":[1.0,1.0,1.0], "gridColor":[0.9,0.9,0.9], "panelOn":false, "cWidth":1.0, "cHeight":1.0 }
{ "orbtIndex":14, "bkgdColor":[1.0,1.0,1.0], "gridColor":[0.9,0.9,0.9], "panelOn":false, "cWidth":1.0, "cHeight":1.0 }
{ "orbtIndex":15, "bkgdColor":[1.0,1.0,1.0], "gridColor":[0.9,0.9,0.9], "panelOn":false, "cWidth":1.0, "cHeight":1.0 }

Cubic Harmonics

The cubic harmonics are linear combinations of the spherical harmonics $Y_{l,m}$ and $Y_{l,-m}$ resulting in real functions. They are adapted to cubic symmetry.
l=0
$s$: $Y_{0,0}$
{ "orbtIndex":16, "bkgdColor":[1.0,1.0,1.0], "gridColor":[0.9,0.9,0.9], "panelOn":false, "cWidth":1.0, "cHeight":1.0 }
l=1
$p_x$: $(Y_{1,-1}-Y_{1,1})/\sqrt{2}$ $p_z$: $Y_{1,0}$ $p_y$: $i(Y_{1,-1}+Y_{1,1})/\sqrt{2}$
{ "orbtIndex":18, "bkgdColor":[1.0,1.0,1.0], "gridColor":[0.9,0.9,0.9], "panelOn":false, "cWidth":1.0, "cHeight":1.0 }
{ "orbtIndex":17, "bkgdColor":[1.0,1.0,1.0], "gridColor":[0.9,0.9,0.9], "panelOn":false, "cWidth":1.0, "cHeight":1.0 }
{ "orbtIndex":19, "bkgdColor":[1.0,1.0,1.0], "gridColor":[0.9,0.9,0.9], "panelOn":false, "cWidth":1.0, "cHeight":1.0 }
l=2
$d_{x^2-y^2}$: $(Y_{2,-2}+Y_{2,2})/\sqrt{2}$ $d_{xz}$: $(Y_{2,-1}-Y_{2,1})/\sqrt{2}$ $d_{3z^2-1}$: $Y_{2,0}$ $d_{yz}$: $i(Y_{2,-1}+Y_{2,1})/\sqrt{2}$ $d_{xy}$: $i(Y_{2,-2}-Y_{2,2})/\sqrt{2}$
{ "orbtIndex":23, "bkgdColor":[1.0,1.0,1.0], "gridColor":[0.9,0.9,0.9], "panelOn":false, "cWidth":1.0, "cHeight":1.0 }
{ "orbtIndex":21, "bkgdColor":[1.0,1.0,1.0], "gridColor":[0.9,0.9,0.9], "panelOn":false, "cWidth":1.0, "cHeight":1.0 }
{ "orbtIndex":20, "bkgdColor":[1.0,1.0,1.0], "gridColor":[0.9,0.9,0.9], "panelOn":false, "cWidth":1.0, "cHeight":1.0 }
{ "orbtIndex":22, "bkgdColor":[1.0,1.0,1.0], "gridColor":[0.9,0.9,0.9], "panelOn":false, "cWidth":1.0, "cHeight":1.0 }
{ "orbtIndex":24, "bkgdColor":[1.0,1.0,1.0], "gridColor":[0.9,0.9,0.9], "panelOn":false, "cWidth":1.0, "cHeight":1.0 }
l=3
$f_{x(x^2-3y^2)}$: $(Y_{3,-3}-Y_{3,3})/\sqrt{2}$ $f_{z(x^2-y^2)}$: $(Y_{3,-2}+Y_{3,2})/\sqrt{2}$ $f_{x(5z^2-1)}$: $(Y_{3,-1}-Y_{3,1})/\sqrt{2}$ $f_{z(5z^2-3)}$: $Y_{3,0}$ $f_{y(5z^2-1)}$: $i(Y_{3,-1}+Y_{3,1})/\sqrt{2}$ $f_{xyz}$: $i(Y_{3,-2}-Y_{3,2})/\sqrt{2}$ $f_{y(3x^2-y^2)}$: $i(Y_{3,-3}+Y_{3,3})/\sqrt{2}$
{ "orbtIndex":30, "bkgdColor":[1.0,1.0,1.0], "gridColor":[0.9,0.9,0.9], "panelOn":false, "cWidth":1.0, "cHeight":1.0 }
{ "orbtIndex":28, "bkgdColor":[1.0,1.0,1.0], "gridColor":[0.9,0.9,0.9], "panelOn":false, "cWidth":1.0, "cHeight":1.0 }
{ "orbtIndex":26, "bkgdColor":[1.0,1.0,1.0], "gridColor":[0.9,0.9,0.9], "panelOn":false, "cWidth":1.0, "cHeight":1.0 }
{ "orbtIndex":25, "bkgdColor":[1.0,1.0,1.0], "gridColor":[0.9,0.9,0.9], "panelOn":false, "cWidth":1.0, "cHeight":1.0 }
{ "orbtIndex":27, "bkgdColor":[1.0,1.0,1.0], "gridColor":[0.9,0.9,0.9], "panelOn":false, "cWidth":1.0, "cHeight":1.0 }
{ "orbtIndex":29, "bkgdColor":[1.0,1.0,1.0], "gridColor":[0.9,0.9,0.9], "panelOn":false, "cWidth":1.0, "cHeight":1.0 }
{ "orbtIndex":31, "bkgdColor":[1.0,1.0,1.0], "gridColor":[0.9,0.9,0.9], "panelOn":false, "cWidth":1.0, "cHeight":1.0 }

Explore

Here you can explore the harmonic functions in more detail. Either choose from the drop-down menue, or define your own linear combination.
{ "orbtIndex":6, "bkgdColor":[1.0,1.0,1.0], "gridColor":[0.9,0.9,0.9], "panelOn":true, "cWidth":0.80, "cHeight":0.50 }

Instructions

Credits

The WebGL routines were programmed by Qian Zhang. An OpenGL version with other visualization programs was developed by Khaldoon Ghanem during the JSC Guest Student Programme.