# Invariance Principles and Conservation Laws

### 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 Jx, Jy, and Jz.

Earlier I wrote out Jz in terms of the coordinates:

Jz = -i hbar(x d/dy - y d/dx)
The three angular momentum operators don't commute:
[Jx, Jy] = i Jz
[Jy, Jz] = i Jx
[Jz, Jx] = i Jy
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 = Jx2 + Jy2 + Jz2
The usual choice for simultaneous measurement is J2 with eigenvalue J(J+1), and Jz with eigenvalue M.

Another pair of useful operators is

J± = Jx ± i Jy
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> = ΣJ=|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.