Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 4 / 33B / c / Solution

A $j=0$ state exists only when $j_1=j_2=:j$. Since its total magnetic quantum number is zero, every uncoupled basis term must have $m_1+m_2=0$, soApply the total raising operator $J_{+}=J_{1+}+J_{2+}$. The coefficient of
$|j,m+1{⟩}_1|j,-m{⟩}_2$ isThe angular momentum singlet state is annihilated by $J_{+}$, henceAll coefficients consequently have the same modulus and alternating signs. Normalization fixes that modulus to $(2j+1{)}^{-1/2}$, and an arbitrary overall phase may be chosen so thatThus