Motion of heavy particles coupled to fermionic and bosonic environments in one dimension

Making use of a simple unitary transformation we change the Hamiltonian of a particle coupled to a one-dimensional gas of bosons or fermions to a new form from which the many-body degrees of freedom can be easily eliminated by taking the trace. The effective dynamics of the particle allows us to compute its damping constant in the semiclassical approximation in terms of the reflection coefficient of the interaction potential and the occupation number of the environmental particles. We apply our results to a delta-function repulsive potential.