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The Matrix Perturbation Method in Quantum Mechanics

  • Book
  • © 2023

Overview

  • Offers a simplified technique for perturbative solutions to problems with a Schrödinger-like structure
  • Demonstrates how this new analysis applies to time evolution of mixed states in a Lindblad master equation framework
  • Accessible to both students and researchers

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Table of contents (7 chapters)

Keywords

About this book

This book provides an alternative approach to time-independent perturbation theory in non-relativistic quantum mechanics. It allows easy application to any initial condition because it is based on an approximation to the evolution operator and may also be used on unitary evolution operators for the unperturbed Hamiltonian in the case where the eigenvalues cannot be found. This flexibility sets it apart from conventional perturbation theory. The matrix perturbation method also gives new theoretical insights; for example, it provides corrections to the energy and wave function in one operation.  Another notable highlight is the facility to readily derive a general expression for the normalization constant at m-th order, a significant difference between the approach within and those already in the literature. Another unique aspect of the matrix perturbation method is that it can be extended directly to the Lindblad master equation. The first and second-order corrections are obtained for this equation and the method is generalized for higher orders. An alternative form of the Dyson series, in matrix form instead of integral form, is also obtained. Throughout the book, several benchmark examples and practical applications underscore the potential, accuracy and good performance of this novel approach. Moreover, the method's applicability extends to some specific time-dependent Hamiltonians. This book represents a valuable addition to the literature on perturbation theory in quantum mechanics and is accessible to students and researchers alike.

Authors and Affiliations

  • Instituto Nacional de Astrofísica, Óptica y Electrónica, (INAOE), San Andrés Cholula, Mexico

    Francisco Soto-Eguibar, Braulio Misael Villegas-Martínez, Héctor Manuel Moya-Cessa

About the authors

Francisco Soto-Eguibar did his doctoral thesis on the foundations of quantum mechanics. He is currently a researcher in the quantum optics group of the National Institute of Astrophysics, Optics and Electronics (INAOE). He has published around 80 articles in international peer reviewed journals and 3 books. He teaches courses on quantum mechanics and electromagnetic theory.


Hector M. Moya-Cessa obtained his PhD at Imperial College in 1993 and since then he is a researcher/lecturer at Instituto Nacional de Astrofísica, Óptica y Electrónica in Puebla, Mexico where he works on Quantum Optics. He has published over 200 articles in international peer reviewed journals. He is a fellow of the Alexander von Humboldt Foundation.


Braulio Misael Villegas-Martínez is a researcher at the Center for Research in Engineering and Applied Science (CIICAp-UAEM), in Cuernavaca, Mexico. His research interests include quantum optics, PT symmetry, and waveguides. Villegas-Martinez received his PhD in Optical Sciences from the National Institute of Astrophysics, Optics, and Electrónics (INAOE).



Bibliographic Information

  • Book Title: The Matrix Perturbation Method in Quantum Mechanics

  • Authors: Francisco Soto-Eguibar, Braulio Misael Villegas-Martínez, Héctor Manuel Moya-Cessa

  • DOI: https://doi.org/10.1007/978-3-031-48546-6

  • Publisher: Springer Cham

  • eBook Packages: Physics and Astronomy, Physics and Astronomy (R0)

  • Copyright Information: The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023

  • Hardcover ISBN: 978-3-031-48545-9Published: 20 January 2024

  • Softcover ISBN: 978-3-031-48548-0Due: 20 February 2024

  • eBook ISBN: 978-3-031-48546-6Published: 19 January 2024

  • Edition Number: 1

  • Number of Pages: XV, 190

  • Number of Illustrations: 5 b/w illustrations, 34 illustrations in colour

  • Topics: Quantum Physics, Mathematical Methods in Physics, Numerical Analysis, Analysis, Quantum Optics

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