the Momentum operator in spherical coordinate
where L is the Reduced angular momentum operator. the minus sign is very important for giving a correct sign. the original angular momentum operator J is related by:
by compare the Laplacian in spherical coordinate, the L is
But this complicated form is rather useless, expect you are mathematic madman.
we can start from classical mechanic
with the change of coordinate
and the Jacobian Matrix , which is used for related the derivatives.
which can be simplify
then, we have
However, even we have the functional form, it is still not good. we need the ladder operator
so, just replacing .
when we looking for the Maximum state of the spherical Harmonic
use the separable variable assumption.
the solution is
an application on Hydrogen wave function is here.