In this paper we investigated the single particle dynamics in a colloidal crystal by means of video microscopy and digital imaging methods. To obtain the self term of the van Hove function G(s)(x, tau) with a reasonable good statistical accuracy we analyzed up to two million particle displacements per time channel tau of G(s)(x, tau), which corresponds to the analysis of 3000 video frames, each with approximately 1000 particles. It was found that the Gaussian approximation almost perfectly accounts for the experimental results, so that the time evolution of G(s)(x, tau) can solely be described by the mean square displacement [x(2)(tau)]. The measured mean square displacements were compared with the model of a Brownian particle bound in a harmonic potential [Physica A 157, 705 (1989)]. The experimental curves approach the plateau value [x(2)(tau-->(infinity))] Of the mean square displacement much slower than is predicted by the model. From the plateau value of the mean square displacement the Lindemann parameter was determined for various samples.