Here is the problem. The parity of the negative pion was deduced from a reaction of pion with deuteron which results in two neutrons. The deuteron has spin equal to one, pion has zero spin, both had zero orbital angular momentum. The resulting protons can have spin either $0$ or $1$, from the requirement of antisymetric wavefunction, spin 0 corresponds to orbital momentums of 0,2,4,... and spin $1$ corresponds to angular momentums of 1,3,...

The total angular momentum of deuteron and pion is $J=1$, which has to be conserved. Now they supposedly deduced, that the orbital angular momentum of the two neutrons must be $L=1$ and spin $S=1$.

My question is, how can this be, since $J=L+S=1+1=2$ which is not $1$, so the total angular momentum should not be conserved and yet wikipedia and my textbook says it is?!

  • 1
    $\begingroup$ have you realized that angular momentum is a vector? $\endgroup$
    – anna v
    Commented Oct 25, 2016 at 18:44
  • $\begingroup$ WP entry. $\endgroup$ Commented May 17, 2017 at 18:49

2 Answers 2


$J$ can take values $|L-S|$ to $|L+S|$, so \begin{equation} \begin{aligned} |1-1|&=0\\ |1-1|+1&=1\\ |1-1|+2&=2 \end{aligned} \end{equation}


final j ranges from S + L to |S-L| , since we require j = 1 for total angular momentum conservation , the final state is contained in it.


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