We are talking about the algebra of angular momentum operators, deriving commutation relations, and relations among the orbital angular momentum squared and the ladder operators. These relations are written out, derived, and discussed in the textbook, pages 150-155. The algebra turns out to be the same for spin angular momentum. All interesting things about angular momentum can be derived from the algebra. This algebra includes:
Commutation relations between components of the angular momentum;
The angular momentum squared commutes with components (these statements refer to the operators);
Definitions of ladder operators and derivation of commutation relations of the ladder operators with Lz;
Expressions for the square of the angular momentum in terms of ladder operators and Lz.
Then one can show that the azimuthal quantum number m of a state is changed to m+1 when the raising operator is applied, but that the total angular momentum quantum number l is not changed. Then one finds that for a given l, m varies from -l to +l in integral steps. That means for example that l=1/2, m= -1/2 or +1/2 is a possibility (spin).
Calculation of the effect of the raising or creation operator to a state |l m) can be shown to give:
L+|l m) = Sqrt[l(l+1)-m(m+1)] |l m+1)
L-|l m) = Sqrt[l(l+1)-m(m-1)] |l m-1)
In the above two equations, l is always positive or zero, but m can be negative.
We worked out problem 4.24 in class, based on the fact that eigenvectors of a Hermitian operator are orthogonal if the eigenvalues are unequal.
The above algebra is the same no matter what the value of l is. If l=1/2, then we describe this case in terms of "spin." The case l=1/2 is of particular intereste as this describes the intrinsic angular momentum of the electron, independent of and in addition to whatever orbital angular momentum it may have. In this case m=-1/2 or +1/2, so there are two states, called up and down, or + and -. Matrix representations of these states were derived in class. Ladder operators were defined and their matrix representations were obtained from the algebra.
The operator Sz is diagonal and has eigenvalues hbar/2 and -hbar/2. The operator Sx is not diagonal but still has eigenvalues hbar/2 and -hbar/2.
Matrix representations for Sx were obtained and eigenvectors were found.