Classical limit of the probability density for a perturbed change in the Hamiltonian

Abstract

The stationary perturbation series is derived from the time-dependent perturbation series up to any order by using a formula derived from the theory of divergent series. The probability density of the perturbed state turns out as the evolution of the density operator of the pure unperturbed initial state as ħ → 0. This result, for ħ → 0, indicates that the particle moves in the classical trajectory with the HamiltonianH=H ₀+V immediately afterV is turned on, if it is in the trajectory ofH ₀ initially. This confirms the classical dynamics. Further, the adiabatic theorem is introduced in order to get the same conclusion for arbitrary finite potentialV.