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arXiv:quant-ph/0607013

Wronskian perturbation theoryWronskian perturbation theory

Paolo Amore, Francisco Marcelo Fernández

Published Jul 3, 2006• 0 citations
arXivHTMLPDF

Abstract

We develop a perturbation method that generalizes an approach proposed recently to treat velocity--dependent quantum--mechanical models. In order to test present approach we apply it to some simple trivial and nontrivial examples.
DOI
10.48550/arxiv.quant-ph/0607013

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We generalize a recently developed perturbation approach [F. M. Fern\'andez and E. A. Castro, Phys. Rev. A 46, 7288 (1992)], extending its applicability to a wider variety of separable quantum-mechanical problems. As an illustrative example, we consider a particle in a one-dimensional box with a finite potential wall perturbed by a linear potential and obtain the perturbation corrections to the wave function and energy of an arbitrary state in terms of the corresponding unperturbed quantities.
The standard textbook formulas for the perturbation corrections are somewhat cumbersome for a systematic calculation of sufficiently high order. In this book we show several alternative strategies that are easily programmable for numerical or algebraic calculation. We are mainly concerned with the derivation of exact perturbation corrections, and therefore concentrate on sufficiently simple nontrivial models having physical application. However, some of the algorithms discussed in this book are also suitable for numerical calculation.
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James' effective Hamiltonian method has been extensively adopted to investigate largely detuned interacting quantum systems. This method only corresponds to the second-order perturbation theory and cannot be exploited to treat problems which should be solved by using the third- or higher-order perturbation theory. In this paper, we generalize James' effective Hamiltonian method to the higher-order case. Using the method developed here, we reexamine two recently published examples [L. Garziano et al., Phys. Rev. Lett. 117, 043601 (2016); Ken K. W. Ma and C. K. Law, Phys. Rev. A 92, 023842 (2015)]; our results turn out to be the same as the original ones derived from the third-order perturbation theory and adiabatic elimination method, respectively. For some specific problems, this method can simplify the calculating procedure and the resultant effective Hamiltonian is more general.
In the course of preparing lectures on elementary quantum theory the author developed a way of presenting time-independent perturbation theory that seems to have some advantages over more familiar presentations. There is nothing fundamentally new in the method, and if any claim of greater power can be made for the new version, it is at best for situations in which high-order perturbations are of interest. Quantum-field theorists will no doubt recognize the relation of this presentation to much that has been done in their subject recently.