Alternative analysis to perturbation theory in quantum mechanics

Abstract

We develop an alternative approach to time independent perturbation theory in non-relativistic quantum mechanics. The method developed has the advantage to provide in one operation the correction to the energy and to the wave function, additionally we can analyze the time evolution of the system. To verify our results, we apply our method to the harmonic oscillator perturbed by a quadratic potential. An alternative form of the Dyson series, in matrix form instead of integral form, is also obtained.

The su(1,1)-like intensity-dependent Rabi model: A perturbative analysis of weak and strong-coupling regimes

“…15 In particular, it would be interesting to evaluate signi¯cant physical quantities such as the atomic population inversion hÉðtÞjS 3 jÉðtÞi and the mean photon number hÉðtÞjnjÉðtÞi to evidence possible e®ects related to the intensitydependent coupling for di®erent regimes. Moreover, the regime g !…”

Normalization corrections to perturbation theory based on a matrix method

“…Thus, the Matrix Method allows to transform the Taylor series of the formal solution of the time-dependent Schrödinger equation in a power series of the matrix M , which can be handled easily. Likewise, this process allows us to find any kth-order correction in a simple and straightforward way through the following relation [16,17]…”

“…This has prompted researchers to develop and implement new techniques to get better approximate solutions for the time-dependent Schrödinger equation. An alternative perturbative approach, that we will focus along this work, is the Matrix Method [16,17]. This new scheme, based on the implementation of triangular matrices, allows to solve approximately the time-dependent Schrödinger equation in an elegant and simple manner.…”

Application of Perturbation Theory to a Master Equation

“…We simplify the above expression using the matrix method [23]; we define a triangular 2 × 2 array of superoperators, where the diagonal elements are given by the nonperturbed system and the superior triangle contains the perturbation:…”