GATE 2025 PH – Question 35
The $\Delta^+$ baryon with spin 3/2, at rest, decays to a proton and a pion ($\Delta^+\to p+\pi^0$). The $\Delta^+$ has positive intrinsic parity and $\pi^0$ has negative intrinsic parity. The orbital angular momentum of the proton–pion system (in integer) is _____
Practise this question in The GATE Grind →
Show answer and explanation
Correct answer: 1
Explanation
**Conservation laws in $\Delta^+\to p+\pi^0$.**
**Parity.** The initial parity is that of the $\Delta^+$ (positive). The final parity is the product of the intrinsic parities of the proton (+) and pion ($-$) and the orbital factor $(-1)^l$:
$$+1=(+1)(-1)(-1)^l\;\Rightarrow\;(-1)^l=-1\;\Rightarrow\;l\text{ is odd}.$$
**Angular momentum.** The pion has spin 0 and the proton has spin $\tfrac12$. The total angular momentum $J=\tfrac32$ is the vector sum of the orbital $l$ and the proton spin $\tfrac12$, so $l$ can only be 1 or 2 ($l+\tfrac12\geq\tfrac32$ and $|l-\tfrac12|\leq\tfrac32$).
**Combine:** $l$ is odd and equal to 1 or 2, so $l=\mathbf{1}$.