Prove the following theorem due to Peierls:
``If H^ is the hermitian Hamiltonian operator of a given physical system and {ϕn} an arbitrary orthonormal set of wave functions satisfying the symmetry requirements and the boundary conditions of the problem, then the partition function of the system satisfies the following inequality:
Q(β)≥n∑exp{−β⟨ϕn∣H^∣ϕn⟩};
the equality holds when {ϕn} is a complete orthonormal set of eigenfunctions of the Hamiltonian itself.''