Ullersma's model consistently describes the quantum-mechanical damped harmonic oscillator. Proper limiting processes in the initially reversible solutions lead to irreversible behavior. It is shown that the irreversible solutions can directly be derived from a field version of the discrete Ullersma Hamiltonian H (H\ensuremath{\rightarrow}
HF). In the ground state of the system the phonon occupation number of the oscillator does not vanish. Therefore, these ``bare'' phonons cannot be interpreted as ``physical'' quanta or particles. By suitable canonical transformations, the bare-field description is transformed into a physical description (
HF\ensuremath{\rightarrow}
Hph), which allows the definition of physical particles and which is characterized by a renormalized frequency, a screened interaction between oscillator and bath, and a symmetry of position and momentum operators. The physical description is closely related to the rotating-wave approximation of the original model. The relaxation behavior and the equilibrium properties of the different descriptions are discussed and compared.
The primary objective of the present work is an exact theoretical analysis of a vibrational dephasing process. The author has investigated a microscopic counterpart of the well known stochastic model of random harmonic oscillator frequency modulation. The author has found an exact expression for the oscillator coordinate relaxation. Several principal features of the solution appear to be quite different from the results based on the random frequency modulation model. (i) The envelope function of the oscillator coordinate time dependence is a function of temperature and it reveals definite types of coherent beats. (ii) The 'frequency' of the coordinate oscillations is time dependent and exhibits definite shift relative to the bare oscillator frequency when time goes from zero to infinity. (iii) Both the envelope function and the 'frequency' of the coordinate oscillations depend on the initial oscillator energy. This demonstrates the interaction-introduced non-linearity of the resulting dynamics.