Analytical results are provided for the mean-square displacement and velocity of the particles of a simple one-dimensional lattice with free ends in the T = 0°K and high-temperature limits. It is pointed out that care must be used in generalizing from one to three dimensions since the mean-square displacement of a one-dimensional lattice depends on the number of particles. Despite this number dependence the results for the mean-square displacement can be used to show that the effective Debye temperature (as it would be measured by the Mössbauer effect or diffraction experiments) for a central atom is twice that of an end atom, in rough qualitative agreement with detailed three-dimensional calculations. Other effective Debye temperatures are also obtained.