Invariance Principles and Conservation Laws

Addition of Angular Momenta

Review of Rotation Operators

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.

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:

J² = J_x² + J_y² + J_z²

The usual choice for simultaneous measurement is J² with eigenvalue J(J+1), and J_z with eigenvalue M.

Another 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 C_±^jm are coefficients that are given by the square root expression. Notice that if m=j, then the coefficient C₊^jm = 0, and if m=-j, C_-^jm = 0. Thus the raising and lowering operators "obey the rules" for angular momentum quantum numbers and do not produce states with m>j or m<-j.

Example of Adding Angular Momenta States

Now suppose we have a system with two (or more) angular momenta given by the states |j₁, m₁> and |j₂, m₂>, 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 = J₁ + J₂. 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 |j₁, m₁> and |j₂, m₂> can result in a state with |j₁ - j₂| < J < j₁ + j₂. The z components are known though, and their addition yields the z component of the total angular momentum, M = m₁ + m₂. The result is that the whole system is a superposition of states with the same M and different J's:

|j₁, m₁; j₂, m₂> = Σ_{J=|j₁-j₂|}^j₁+j₂ |J, M><J, M|j₁ j₂ m₁ m₂>

where the quantities <J, M|j₁ j₂ m₁ m₂> are numbers called Clebsch-Gordon coefficients.