Consider the isospin subgroup chain, \(\mathrm{U}(2) \supset \mathrm{U}(1)_{B} \times \mathrm{SU}(2) \supset \mathrm{U}(1)_{B} \times \mathrm{U}(1)_{T_{3}}\), where subscripts distinguish the \(\mathrm{U}(1)\) group generated by baryon number \(B\) from the U(1) group generated by the third component of isospin, \(T_{3}\). Write a Hamiltonian consisting of a linear combination of the Casimir invariants for this chain and show that the resulting spectrum can be obtained analytically as

\[E\left(N_{B}, T, Q\right)=\left(a-\frac{c}{2}\right) N_{\mathrm{B}}+b T(T+1)+c Q\]

where a, b, and c are constants, and Q = T3+ B.