For an example of friction forces directly proportional to the velocity of motion, a new approach is proposed to the problem of quantization of systems with energy dissipation. An analogy with the radiative friction forces of Lorentz is widely used. The power of dissipation forces is expanded in harmonic components and, by using the correspondence principle, recalculated into the rate of change of the population probability of an excited state. Quantum dynamics is described by the temporal Green functions in the Feynman integral representation. In the case of a harmonic oscillator, analytic expressions for temporal behavior of the probability amplitudes are derived, and the entire spectrum of complex energies is obtained. Their imaginary part gives a quantitative description of the energy dissipation. In our scheme, the ground state is found to be stable.