A proton and two neutrons are combined into a p n n system. Each particle has zero orbital angular momentum, and the individual spin quantum numbers are s1 = 1/2, m1 = +1/2 for the proton and s2 = 1/2, m2 = +1/2; s3 = 1/2, m3 = −1/2 for the two neutrons. What are the probabilities of finding the pnn system in states characterized by the following total angular momentum and z-component of total angular momentum quantum numbers?
a) j = 3/2 and m = − 1/2
b) j = 3/2 and m = +1/2
c) j = 1/2 and m = − 1/2
d) j = 1/2 and m = +1