Energy equilibration between two weakly coupled quantum systems

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, ϱA(0) and ϱB(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.