Relation between a relaxation time and the friction coefficient used in the theory of damped motion

The paper deals with the Caldirola–Kanai Hamiltonian in which the potential energy is defined as a linear function of the position variable. The exact solution of the Schrodinger equation using this Hamiltonian is presented. It is shown that this solution can be utilized in the theory of electric conduction of metals and semiconductors. The main idea of this paper is that if the Caldirola–Kanai Hamiltonian is applied to conduction electrons , it is possible to calculate successfully the drift velocity of the electrons in a constant electric field. The alternative calculation of the drift velocity in the framework of the density-matrix theory which is also presented leads to the same result. A useful lesson is drawn: if the friction coefficient , the parameter defining the Caldirola–Kanai Hamiltonian, concerns the conduction electrons, it is precisely equal to the reciprocal value of the relaxation time used in the kinetic theory of the electron transport in solids.