Changes

</math>

</math>

This is one way to describe the final state, but it is not the only way.  Instead of specifying the final-state particles' spin state via their helicities, we can first couple their spins together independent of their momentum direction, to obtain total spin ''S'', then couple ''S'' to their relative orbital angular momentum ''L'' to obtain their total angular momentum ''J''.   

This is one way to describe the final state, but it is not the only way.  Instead of specifying the final-state particles' spin state via their helicities, we can first couple their spins together independent of their momentum direction, to obtain total spin ''S'', then couple ''S'' to their relative orbital angular momentum ''L'' to obtain their total angular momentum ''J''.  When we do this, we give up our knowledge of the particles' helicities, having replaced those two quantum numbers with the alternative pair ''L,S''.  These two bases, the helicity basis and the ''L,S'' basis, are each individually complete and orthonormal within themselves.  Following on from the above expression, let us insert a sum over the ''L,S'' basis.

:<math>

:<math>

\langle \Omega \lambda_1 \lambda_2 | U | J m \rangle

\langle \Omega \lambda_1 \lambda_2 | U | J m \rangle

  J m L S \rangle \langle J m L S |

  J m L S \rangle \langle J m L S |

U | J m \rangle

U | J m \rangle

</math></td>

</math>

:<math>

 

<td><math>

=\sum_{L,S}

=\sum_{L,S}

\left[ \sqrt{\frac{2J+1}{4\pi}} D_{m \lambda}^{J *}(\Omega,0) \right]

\left[ \sqrt{\frac{2J+1}{4\pi}} D_{m \lambda}^{J *}(\Omega,0) \right]