Compounded infinitesimal Hamiltonian perturbations: A model of Hamiltonian transitions by a continuous perturbation on a free Hamiltonian
Dynamical systems with explicit time-dependent Hamiltonians have attracted considerable attention, mainly because the only legitimate closed system to which the stationary Schrödinger's equation applies is the entire universe. In general, the temporal evolution of the Hamiltonian is modeled as a continuous variation of an initial Hamiltonian to a certain final Hamiltonian. One such model − which is of our concern in this paper − is the evolution of a given free Hamiltonian H0 by switching on a perturbing Hamiltonian H1, that is,
HE(t) = H0 + λ(t)H1,
where 0 ≤ λ(t) ≤ 1 describes how H1 is infused into the system. This Hamiltonian, for example, serves as the Hamiltonian in tunneling problems through a modulated potential barrier and in dynamical capture in quantum scattering by asymptotically switched on potentials.
In this work we postulate that the variation of the free Hamiltonian by continuously switching on a perturbing Hamiltonian is accomplished by compounding infinitesimal perturbations on H0. That is perturbing H0 by H1 by an infinitesimal strength is formally achieved by a non-bijective (we call) infinitesimal unitary transformation
H0 → H(δλ) = H0 + δλH1 → H(δλ) = Ω(δλ)H0 Ω−1(δλ),
where Ω(δλ) = 1−i(δλ)ω, in which ω is hereafter referred to as the generator of infinitesimal Hamiltonian perturbations (GIHP). We refer to the transformation as non-bijective because the initial and the final Hamiltonians have different spectra. This is so as the transformation is not intended as a canonical transformation on the conjugate operators q and p − the free and the perturbed Hamiltonians are expressed in terms of the same conjugate variables. The state ket then evolves not according to Ω(δλ)|Ψ0〉, but according to the nonstationary Schrödinger's equation.