Quantum-mechanical irreversible motion of an infinite chain

Abstract

The rate of the energy transfer between the particles in an infinite uniform chain provides an interesting example of a system exhibiting a nonexponential approach to equilibrium. For this system, the quasi-particle description is not convenient, hence, for the purpose of quantization, the total Hamiltonian is decomposed into a sum of single particle Hamiltonians, each containing a time-dependent effective potential energy. The resulting Schrödinger equation for a single particle has a wave-packet-like solution in which the center of the packet moves according to the classical law of motion. The quantal problem shows the same kind of oscillatory decay law as the classical one, and there is no characteristic relaxation time for the system. From the Schrödinger equation, expressions for the total energy, the momentum, and the probability of excitation for each subsystem are obtained. It is shown that the ground state wave function satisfies a Fokker–Planck type equation, and that the equation of motion of each particle remains invariant under the time-reversal transformation. However, the direction of the flow of energy is the same for the actual and the time-reversed motions.