Invariance Principles and Conservation Laws

Recall from last lecture:

Parity and C-parity are multiplicative quantities.

Parity and C-parity are conserved in strong and electromagnetic interactions.

The parity of a state of angular momentum L is (-1)^L.

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.

Example: Combining q and qbar to form a meson

If we combine a quark and an anti-quark to form a meson, what are the possible states of total angular momentum (spin of the meson) that can be formed? Begin by assuming that the q and qbar have orbital angular momentum L=0, so that the problem is simplified to combining to spin 1/2 particles. The combination of two spin 1/2 particles can yield J = 0 or 1. Begin with both particles aligned with spins down. This combination has M = -1, and can only be in the J = 1 state:

| j₁, m₁; j₂, m₂ > = | 1/2, -1/2; 1/2, -1/2 > = | J, M > = | 1, -1 >

The Clebsch-Gordon coefficient for this combinatino is 1.

Now apply the raising operator to this relation:

J₊ | j₁=1/2, m₁=-1/2; j₂=1/2, m₂=-1/2 > = | 1/2, 1/2; 1/2, -1/2 > + | 1/2, -1/2; 1/2, 1/2 > = J₊ | J=1, M=-1 > = sqrt(2) | 1, 0 >

Applying the raising operator a second time yields:

J₊ { | 1/2, 1/2; 1/2, -1/2 > + | 1/2, -1/2; 1/2, 1/2 > } = | 1/2, 1/2; 1/2, 1/2 > + | 1/2, 1/2; 1/2, 1/2 > = 2 | 1/2, 1/2; 1/2, 1/2 > = sqrt(2) J₊ | J=1, M=0 > = 2 | 1, 1 >

Finally, the J=0, M=0 state is obtained by noting that there is a combination of momenta that is orthogonal to the J=1, M=0 state, namely:

1/sqrt(2) | 1/2, 1/2; 1/2, -1/2 > -1/sqrt(2) | 1/2, -1/2; 1/2, 1/2 > = | 0, 0 >

Example: Decay of a pseudoscalar to a pseudoscalar and a vector

Let's determine the angular momentum states possible in the decay of a pseudoscalar meson to a pseudoscalr meson and a vector meson (abbreviated as P to PV). A pseudoscalar meson has J^P = 0^- and a vector meson has J^P = 1^-. The initial state is