Summary: For a prescribed pair of quantum states \(|\psi_{I}\rangle\) and \(|\psi_{F}\rangle\) we establish an elementary derivation of the optimum Hamiltonian, under constraints on its eigenvalues, that generates the unitary transformation \(|\psi_{I}\rangle \rightarrow |\psi_{F}\rangle\) in the shortest duration. The derivation is geometric in character and does not rely on variational calculus.