l=1,m=-1,+1
In the quantum mechanics, l and m values of the
spherical harmonics Y[lm] are the angular momentum and magnetic
momentum numbers. Here we show several spherical harmonics which
have relatively small l values. The method is the same as
the previous section --- express x,y,z by parameters
u and v.
When m is odd, the spherical harmonics contains an
imaginary part which comes from exp(-im phi) term. For m=-1 and 1,
the function becomes as follows:
Y[1,-1](t,p) = sqrt(3/8pi) sin(t) exp(-i p)
Y[1, 1](t,p) = -sqrt(3/8pi) sin(t) exp( i p)
|Y|^2 can be calculated easily by multiplying Y and its complex
conjugate, and both above become the same function, Y(t)=3/8 pi
sin^2(t). The complex conjugate function of the spherical harmonics
can be given by the relation, (Y[l,m])^* = (-1)^m Y[l,-m].
gnuplot> set parametric
dummy variable is t for curves, u/v for surfaces
gnuplot> set angle degree
gnuplot> set urange [0:360]
gnuplot> set vrange [0:360]
gnuplot> set isosample 18,18
gnuplot> set ticslevel 0
gnuplot> set size 0.65,1.0
gnuplot> a=3.0/(8*pi)
gnuplot> fx(u,v)=cos(u)*cos(v)
gnuplot> fy(u,v)=sin(u)*cos(v)
gnuplot> fz(v)=sin(v)
gnuplot> g(v)=cos(v)*cos(v)
gnuplot> splot a*g(v)*fx(u,v),a*g(v)*fy(u,v),a*g(v)*fz(v)
|
The left torus-shape figure is |Y[11]|^2. Since this function
contains sin(t)=cos(v), we defined g(v)=cos(v)*cos(v)
.