update 2002/ 4/12
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The spherical harmonics in the polar coordinate Y[lm](theta,phi) is given by: ![]() where P[lm] is the associated Legendre function. The value l is integer, l=0,1,2,3... For each l value, m takes the value of m=-l,-l+1,...l-1,l so that there are 2l+1 different m values. A square of this function is normalized to unity, < Y[lm] | Y[l'm'] > = delta(ll') delta(m,m'). The theta runs from 0 to pi, while phi runs from 0 to 2pi. ![]() In the gnuplot parametric representation, the angle u is the same as phi, while the definition of theta is different from v. From the left figure, theta = pi/2 - v, we get: cos(theta)=sin(v) sin(theta)=cos(v) |
The simplest spherical harmonics can be obtained by setting l=0 and m=0. This yields a constant value of 1/sqrt(4pi). A square of this, |Y[00]|^2=1/4pi, is of course constant, and this is a sphere with the radius of 1/4pi. It is easy to draw this function with gnuplot. The procedure is the same as the previous page.
![]() The left graph is a square of Y[00], which is a sphere whose center is the origin and the radius of 1/4pi. Since m=0, the exp(-im phi) term disappears and the function is independent of phi. The spherical harmonics becomes real when m is even. If m is an odd number the function has an imaginary term. |
Next, we consider the case of l=1 and m=0. This function becomes Y(theta)=sqrt(3/4pi)cos(theta). As shown above, cos(theta) is given by sin(v). The function can be plotted when the x,y,z coordinates are multiplied by sin(v). In the following example we show |Y[10]|^2=3/4pi cos^2(theta).
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