The ensemble average of the expectation value of the Hamiltonian of a free particle (chosen, for the sake of simplicity in the subsequent discussion), contained in a cubical box, is evaluated in both coordinate representation and momentum representation explicitly (step by step) in the density matrix formalism. We see that these results are independent of the choice of the representation, provided we are using the correct representation of the wave function and the operator and evaluate some new calculations in addition. We then discuss the applications of the density matrix theory in diverse areas of physics and beyond in a concise way and conclude.