Time-Dependent Perturbation Theory

Symmetry Integrability and Geometry-methods and Applications, 2005

Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2003

Dr. Mustafa Alsayed, 2023

So far we have concentrated on systems for which we could find exactly the eigenvalues and eigenfunctions of the Hamiltonian, like e.g. the harmonic oscillator, the quantum rotator, or the hydrogen atom. HoweverthevastmajorityofsystemsinNaturecannotbesolvedexactly,andweneed to develop appropriate tools to deal with them. Perturbation theory is extremely successful in dealing with those cases that can be mod- elled as a “small deformation” of a system that we can solve exactly. Letustranslatetheabovestatementintoaprecisemathematicalframework.Weare going to consider systems that have an Hamiltonian: Hˆ=Hˆ0+Vˆ, (17.1) whereHˆ0istheHamiltonianoftheunperturbedsystem,isasmallparameter,andVˆ isthepotentialdescribingtheperturbation.WeshallassumethattheperturbationVis independent of time. Letusalsoassumethatwecansolvethetime-independentSchr¨odingerequationforHˆ0, i.e.thatweknowitseigenvaluesandeigenfunctions: Hˆ0ψ(n)(x)=E(n)ψ(n)(x).

Physical review letters, 2021

We present a systematic treatment of scattering processes for quantum systems whose time evolution is discrete. We define and show some general properties of the scattering operator, in particular the conservation of quasienergy which is defined only modulo 2π. Then we develop two perturbative techniques for the power series expansion of the scattering operator, the first one analogous to the iterative solution of the Lippmann-Schwinger equation, the second one to the Dyson series of perturbative quantum field theory. We use this formalism to compare the scattering amplitudes of a continuous-time model and of the corresponding discretized one. We give a rigorous assessment of the comparison for the case of bounded free Hamiltonian, as in a lattice theory with a bounded number of particles. Our framework can be applied to a wide class of quantum simulators, like quantum walks and quantum cellular automata. As a case study, we analyze the scattering properties of a one-dimensional cel...

American Journal of Physics, 1967

A formalism is presented in which both the degenerate and nondegenerate cases can be treated in the same fashion. Nonlinear integral equations are written for the perturbed wavefunctions, which lead to the usual formulas when solved by iteration. The method provides a simple way to set —up to any desired order— secular equations for the energy shifts and zeroth-order perturbed wavefunctions.

2009

The first order estimate of energy and eigenstate of a perturbed system are found by summation over its basis.