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

by same on x component and y component:

and by

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

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