Prove the following theorem due to Peierls.

"If \(\hat{H}\) is the hermitian Hamiltonian operator of a given physical system and \(\left\{\phi_{n}ight\}\) an arbitrary orthonormal set of wavefunctions satisfying the symmetry requirements and the boundary conditions of the problem, then the partition function of the system satisfies the following inequality:

\[Q(\beta) \geq \sum_{n} \exp \left\{-\beta\left\langle\phi_{n}|\hat{H}| \phi_{n}ightangleight\}\]

the equality holds when \(\left\{\phi_{n}ight\}\) constitute a complete orthonormal set of eigenfunctions of the Hamiltonian itself."