Abstract: The Brownian motion is a basis subject in non-equilibrium statistical physics. However，the current textbooks have inappropriate problems for explaining this subject，namely，by using the particle's coordinate to multiply the Langevin equation and then averaging over the two sides of the equation，thus the time-dependent mean squared displacement of the force-free particle is calculated. We think that the expression proposed by the moment method of this kind is not rigorous and the explanation for the mechanism of ballistic diffusion motion is also incorrect. Based on the accurate solution of the linear Langevin equation，this paper proposes the expressions of variances of velocity and coordinate of the particle. We also modify the moment method used in the present textbooks and make it give the correct result of the mean squared displacement of the force-free particle.

Key words: Brownian motion, Langevin equation, transition solution, diffusion law