latter case the scattering process is inelastic because the time dependent potential allow the transfer of energy between the two. We observe that the constant average energy also has steps as a function of the left Fermi level which become more pronounced with increasing coupling strength. Hence, the oscillatory part of the behavior of the mechanical oscillator is damped by coupling with the electrons on the dot but the constant part is not. The damping mechanism in the transient dynamics is due to transfer of energy from the nanomechanical oscillator to the electrons on the dot while when the oscillator is in any of the pure eigenstate then there is no mechanism for the transfer of energy between the two. This same physical phenomenon also applies to the net current flowing through the dot as well. This appear to be a specifically new quantum phenomena in the study of nanomechanical systems. Can we compare this quantum phenomena with the classical mechanical oscillator? Yes, the nanomechanical oscillator has to enter the classical regime in the limit of small ℏ . For this, we study the dynamics of the quantum oscillator in the classical limit, in which ℏ in the mechanical oscillator part of the Hamiltonian given in Eq. (1) goes to zero, where ℏ ω < Γ . To see this, we have plotted the average energy as a function of Y = ℏ ′ ℏ in the nanomechanical part of the system in fig. 7 for fixed values of tunneling Γ = 1, ϵ$_{F}$$_{R}$ = 0 , ϵ$_{F}$$_{L}$ = 1 and coupling strength η = 0 . 05. We found that the average energy of the quantum nanomechanical oscillator scales as ℏ $^{2}$. We set the average energy in the limit ℏ ′ → 0 to see what happen to the system for long time. It implies that in this limit, the energy transferred to the 16 FIG. 7. Average energy transferred to the oscillator as a function of ℏ ′ ℏ and for fixed values of ϵ$_{0}$ = 0 . 5 , t = 1000 , ϵ$_{FR}$ = 0 , ϵ$_{FL}$ = 1 , Γ = 1 and η = 0 . 02. Units: all the parameters have same dimension as of $^{ℏ}$ω . 0.10 1.00 0.50 0.20 0.30 0.15 0.70 1.00 0.50 0.20 0.10 0.05 0.02 Y GLYPH<Less> E Ph GLYPH<Greater>