 a situation in which there is no thermal disorder in the quantum degrees of freedom so that $\iota=1$: *viz.*, the density matrix becomes that of a pure state $\rho_{\alpha\alpha^{\prime}}(X,t)\to C_{\alpha}(X,t)C_{\alpha^{\prime}}^{*}(X,t)$. Then, equations (\[eq:c\]) and (\[eq:cstar\]) remain unaltered and one has just to remove the index $\iota$ from the coefficients. Therefore, it can be realized that no classical trajectory propagation, and no state switching are involved by Eqs. (\[eq:c\]) and (\[eq:cstar\]). Instead, in order to calculate averages according to Eq. (\[eq:qc-ave-ad\]), one has to sample phase space points and integrate the matrix equations.

In the next section, an equilibrium approximation of Eqs. (\[eq:c\]) and (\[eq:cstar\]), along the lines followed by the method of *Wigner trajectories* [@wignertraj], is given and applied, with good numerical results, to the adiabatic and nonadiabatic dynamics of the spin-boson model.

Wave dynamics of the spin-boson model {#sec:sb}
=====================================

The theory developed in the previous sections can be applied to simulate the relaxation dynamics of the spin-boson system [@sb]. This system has already been studied within the framework of quantum-classical dynamics of operators in Ref. [@qc-sb] and “exact” numerical results were obtained at short-time by means of an iterative path integral procedure developed by Nancy Makri and co-worker [@makri]. The short-time results of Ref. [@sb] numerically coincide with those obtained by the path integral calculation of Ref. [@makri]. However, as it is shown later by Fig. \[fig:fig2\], the quantum-classical results of Ref. [@sb] have some limitations concerning the numerical stability of the algorithm beyong a certain time length. Using the dimensionless variables of Ref. [@qc-sb], the quantum-classical Hamiltonian operator of the spin-boson system reads $$\begin{aligned}
\hat{H}(X)&=&-\Omega\hat{\sigma}_x
+\sum_{j=1}^N\left(\frac{P_j^2}{2}+
\frac{1}{2}\omega_j^2R_j^2-c_j\ha