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.

since

which can be simplify

then, we have

However, even we have the functional form, it is still not good. we need the ladder operator

notice that

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.