Consider a composite system \(A B\) composed of two noninteracting subsystems \(A\) and \(B\), with the principle of independence of noninteracting subsystems requiring that \(\hat{ho}_{A B}=\hat{ho}_{A} \hat{ho}_{B}\). Use this to show that the equilibrium density matrix must be of the form \(\hat{ho}=e^{-\beta \hat{H}} / \operatorname{Tr} e^{-\beta \hat{H}}\). Further show that if the spectrum of the Hamiltonian is unbounded above, then we must require that \(\beta>0\). On the other hand, if the spectrum of the Hamiltonian is bounded, \(\beta\) may be positive, negative, or zero, and can exhibit negative temperatures.