On optimum Hamiltonians for state transformations

For a prescribed pair of quantum states |ψ I ⟩ and |ψ F ⟩ we establish an elementary derivation of the optimum Hamiltonian, under constraints on its eigenvalues, that generates the unitary transformation |ψ I ⟩ → |ψ F ⟩ in the shortest duration. The derivation is geometric in character and does not rely on variational calculus.