Quantum motion of h-bond protons on a thermally fluctuating lattice

The quantum mechanical motion of protons on hydrogen bonds in order-disorder type ferroelectrics was investigated. The protons move in double well potentials. All protons but the central one are represented by spins flipping in a way corresponding to barrier height and attack frequency of the potential, to the momentary spin configuration in their neighbourhood, and to the temperature. Their time evolution is treated by an MC-simulation. For the central proton the time dependent Schrödmger Langevin equation, is solved numerically. Its potential contains two time dependent terms - one which describes the momentary interactions with the neighbouring spins and another one, varying stochastically, which simulates the interaction with the rest of the lattice - and a (q.m.) damping term which makes the Schrödinger equation nonlinear. An expression is derived for the dependence of its magnitude on the fluctuating terms and on temperature. The time evolution of the system was determined for different potentials and temperatures. The time dependence of the expectation value of the position of the central proton was analysed. Statistical characteristics of the proton motion are compared with corresponding properties following from standard spin flip calculations.