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}.

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 | jIf 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:

| j_{1}, m_{1}; j_{2}, m_{2} > = | 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_{+} | j_{1}=1/2, m_{1}=-1/2; j_{2}=1/2, m_{2}=-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 >

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

Copyright © Robert Harr 2005