There is a discussion of this in appendix C of Perkins.

Recall that in quantum mechanics, to the measurable angular momentum is associated an operator, just like any other measurable quantity.
In fact, angular momentum is more complex because there is a group of operators associated with it.
Conceptually, it is easiest to begin with the operators for the angular momentum about each of the x, y, and z axes; I'll call these three operators J_{x}, J_{y}, and J_{z}.

Earlier I wrote out J_{z} in terms of the coordinates:

J_{z} = -i hbar(x d/dy - y d/dx)

The three angular momentum operators don't commute:
[J_{x}, J_{y}] = i J_{z}

[J_{y}, J_{z}] = i J_{x}

[J_{z}, J_{x}] = i J_{y}

Notice the cyclic nature of this relation.
[J

[J

The fact that the operators don't commute means that a simultaneous measurement of more than one is impossible. Another operator that does commute with any of the three is the square of the total angular momentum:

J^{2} = J_{x}^{2} + J_{y}^{2} + J_{z}^{2}

The usual choice for simultaneous measurement is JAnother pair of useful operators is

J_{±} = J_{x} ± i J_{y}

These are known as raising (+) and lowering (-) operators of angular momentum because when acting on a state with angular momentum quantum numbers j and m:
J_{+}|j, m> = C_{+}^{jm} |j, m+1> = sqrt{j(j+1)-m(m+1)} |j, m+1>

J_{-}|j, m> = C_{-}^{jm} |j, m-1> = sqrt{j(j+1)-m(m-1)} |j, m-1>

The CJ

Now suppose we have a system with two (or more) angular momenta given by the states |j_{1}, m_{1}> and |j_{2}, m_{2}>, where the j's and m's are the corresponding quantum numbers (m's measured for a common axis), and we want to determine the angular momentum of the whole system, |J, M>.
Classically we would simply add the vectors to get **J** = **J**_{1} + **J**_{2}.
But quantum mechanically, we can't simultaneously know all the components of angular momentum, so we can't perform the vector addition.
In fact, the rules of quantum mechanics require that the x and y components of angular momenta take on all possible allowed values.

Therefore, the addition of |j_{1}, m_{1}> and |j_{2}, m_{2}> can result in a state with |j_{1} - j_{2}| < J < j_{1} + j_{2}.
The z components are known though, and their addition yields the z component of the total angular momentum, M = m_{1} + m_{2}.
The result is that the whole system is a superposition of states with the same M and different J's:

|j_{1}, m_{1}; j_{2}, m_{2}> = Σ_{J=|j1-j2|}^{j1+j2} |J, M><J, M|j_{1} j_{2} m_{1} m_{2}>

where the quantities <J, M|jCopyright © Robert Harr 2003