for general angular momentum state, it is a “ugly”combination of the orbital state and the spin state:
the above combination only possible when:
OK, we start with basic result.
the above are basic properties shared by spin angular momentum operator.
thus, we have follow result:
and the total angular momentum operator is
and the state is
by same on x component and y component:
To find the possible , we can use the highest state withe and . Since the highest state are always symmetric. we have to find the others orthogonal states, which are anti-symmetric.
the result is