We calculate the mobility and conductivity of the proton transfer in hydrogen-bonded systems by quantum mechanics. We first give the equation of change of velocity of the kink-antikink pair resulting from proton transfer, from the Hamiltonian in our model. Utilizing the values of the physical parameters we finally find out the mobility of kink-antikink pairs and the electrical conductivity due to the proton transfer in an externally applied electric-field in the systems. These are about (6.5-6.9) x 10(-6) m(2)/V s and (7.6-8.1) x 10(-3)/(Omega m) for ice, respectively, which are basically consistent with experimental data. (C) 2003 Elsevier Science B.V. All rights reserved.