We analyze the energy exchange between two quantum (sub)systems with identical non-degenerate energy spectra weakly coupled to each other. Initially both the subsystems are assumed to be in pure states, ϱ A (0) and ϱ B (0), characterized by populations of the energy levels, pjA and pjB, respectively. Then the subsystems are brought into a thermal contact (no particle exchange between the subsystems is allowed). We prove that the composite system evolves toward a mixed state, where the reduced density matrices of the subsystems are characterized by the same diagonal elements, pjA+pjB/2. This result is shown also to be valid for arbitrary, diagonal initial subsystem density matrices. We numerically illustrate that by the exact diagonalization of a randomly “synthesized” model Hamiltonian.