In this paper, we handle the Hamiltonian of the particles that are affected by the linear-damping and time-dependent external force given by two different canonical transformations by using the theory of generalized quantum linear transformation. We give the rigorous solution of evolution operator, and the expectation values of coordinate and momentum of the particles quantum fluctuations. Results show that 1) the two regular translations are equivalent; 2) linear damping has a squeezing effect on the momentum of particle, and the deviation of the momentum attenuates with time t according to the rule of negative exponent, and the bigger the damping coefficient, the faster the attenuation is; 3) the expectation values of coordinate and momentum of the particle are equal to their classical values correspondingly.