Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

MDPI eBooks

Harmonic Oscillators and Two-by‑two Matrices in Symmetry Problems in Physics

Young Kim

Published Sep 25, 2017• 0 citations
Publisher

Abstract

This book consists of the articles published in the special issues of this Symmetry journal based on two-by-two matrices and harmonic oscillators. The book also contains additional articles published by the guest editor in this Symmetry journal. They are of course based on harmonic oscillators and/or two-by-two matrices. The subject of symmetry is based on exactly soluble problems in physics, and the physical theory is not soluble unless it is based on oscillators and/or two-by-two matrices. The authors of those two special issues were aware of this environment when they submitted their articles. This book could therefore serve as an example to illustrate this important aspect of symmetry problems in physics.

DOI
10.3390/books978-3-03842-501-4

Related papers

Coupled oscillators, entangled oscillators, and Lorentz-covariant harmonic oscillators

Y S Kim, Marilyn E. Noz

Other than scattering problems where perturbation theory is applicable, there are basically two ways to solve problems in physics. One is to reduce the problem to harmonic oscillators, and the other is to formulate the problem in terms of two-by-two matrices. If two oscillators are coupled, the problem combines both two-by-two matrices and harmonic oscillators. This method then becomes a powerful research tool to cover many different branches of physics. Indeed, the concept and methodology in one branch of physics can be translated into another through the common mathematical formalism. Coupled oscillators provide clear illustrative examples for some of the current issues in physics, including entanglement and Feynman's rest of the universe. In addition, it is noted that the present form of quantum mechanics is largely a physics of harmonic oscillators. Special relativity is the physics of the Lorentz group which can be represented by the group of two-by-two matrices commonly called $SL(2,c)$. Thus the coupled harmonic oscillators can play the role of combining quantum mechanics with special relativity. It is therefore possible to relate the current issues of physics to the Lorentz-covariant formulation of quantum mechanics.

arXiv:quant-ph/0502096•Journal of Optics B Quantum and Semiclassical Optics•Nov 4, 2005•22 citations

Harmonic Oscillators as Bridges between Theories: Einstein, Dirac, and Feynman

Y. S. Kim, Marilyn E. Noz

Other than scattering problems where perturbation theory is applicable, there are basically two ways to solve problems in physics. One is to reduce the problem to harmonic oscillators, and the other is to formulate the problem in terms of two-by-two matrices. If two oscillators are coupled, the problem combines both two-by-two matrices and harmonic oscillators. This method then becomes a powerful research tool to cover many different branches of physics. Indeed, the concept and methodology in one branch of physics can be translated into another through the common mathematical formalism. Coupled oscillators provide clear illustrative examples for some of the current issues in physics, including entanglement, decoherence, and Feynman's rest of the universe. In addition, it is noted that the present form of quantum mechanics is largely a physics of harmonic oscillators. Special relativity is the physics of the Lorentz group which can be represented by the group of by two-by-two matrices commonly called $SL(2,c)$. Thus the coupled harmonic oscillators can therefore play the role of combining quantum mechanics with special relativity. Both Paul A. M. Dirac and Richard P. Feynman were fond of harmonic oscillators, while they used different approaches to physical problems. Both were also keenly interested in making quantum mechanics compatible with special relativity. It is shown that the coupled harmonic oscillators can bridge these two different approaches to physics.

arXiv:quant-ph/0411017•Nov 2, 2004•0 citations

Introduction to Classical and Quantum Harmonic Oscillators

Heimo G. Latal

As the title aptly states, this book deals with harmonic oscillators of various kinds, from classical mechanical and electrical oscillations up to quantum oscillations. It is written in a lively language, and occasional interspersed anecdotes make the reading of an otherwise mathematically oriented text quite a pleasure. Although the author claims to have written an `elementary introduction', it is certainly necessary to have a good deal of previous knowledge in physics (mechanics, electrodynamics, quantum theory), electrical engineering and, of course, mathematics in order to follow the general line of his arguments. The book begins with a thorough treatment of classical oscillators (free, damped, forced) that is followed by an elaboration on Fourier analysis. Lagrange and Hamilton formalisms are then introduced before the problem of coupled oscillations is attacked. A chapter on statistical perspectives leads over to the final discussion of quantum oscillations. With the book comes a diskette containing a number of worksheets (Microsoft Excel) that can be used by the reader for instant visualization to get a better qualitative and quantitative understanding of the material. To the reviewer it seems difficult to pinpoint exactly the range of prospective readership of the book. It can certainly not be intended as a textbook for students, but rather as a reference book for teachers of physics or researchers, who want to look up one or other aspect of harmonic oscillations, for which purpose the diskette represents a very valuable tool.

European Journal of Physics•Nov 1, 1997•2 citations

Harmonic Oscillators as Bridges between Theories

Y. S. Kim

Other than scattering problems where perturbation theory is applicable, there are basically two ways to solve problems in physics. One is to reduce the problem to harmonic oscillators, and the other is to formulate the problem in terms of two‐by‐two matrices. If two oscillators are coupled, the problem combines both two‐by‐two matrices and harmonic oscillators. This method then becomes a powerful research tool to cover many different branches of physics. Indeed, the concept and methodology in one branch of physics can be translated into another through the common mathematical formalism. It is noted that the present form of quantum mechanics is largely a physics of harmonic oscillators. Special relativity is the physics of the Lorentz group which can be represented by the group of by two‐by‐two matrices commonly called SL(2, c). Thus the coupled harmonic oscillators can therefore play the role of combining quantum mechanics with special relativity. Both Paul A. M. Dirac and Richard P. Feynman were fond of harmonic oscillators, while they used different approaches to physical problems. Both were also keenly interested in making quantum mechanics compatible with special relativity. It is shown that the coupled harmonic oscillators can bridge these two different approaches to physics.

AIP conference proceedings•Jan 1, 2005•2 citations

Symmetries in Quantum Mechanics and Statistical Physics

Georg Junker

This book collects contributions to the Special Issue entitled "Symmetries in Quantum Mechanics and Statistical Physics" of the journal Symmetry. These contributions focus on recent advancements in the study of PT–invariance of non-Hermitian Hamiltonians, the supersymmetric quantum mechanics of relativistic and non-relativisitc systems, duality transformations for power–law potentials and conformal transformations. New aspects on the spreading of wave packets are also discussed.

MDPI eBooks•Dec 13, 2021•0 citations

Harmonic Oscillator

Olivier Pujol, R. Carles

An impressively comprehensive textbook adopting a phenomenological approach to quantum physics. The chapters cover everything from basic definitions of key concepts to detailed discussions of the underlying theoretical framework, walking students step-by-step through the necessary mathematics and drawing clear connections between the theory and the most important modern research applications including quantum optics, fluids, nanophysics, entanglement, information, and relativity. With this book, students and researchers will have access to hundreds of real-world examples, exercises, and illustrations to support and expand their understanding. Instructors can tailor the content to suit the length and level of their course and will have access to an online solutions manual with fully worked solutions to all 300+ exercises in the book. Other online resources include Python simulations, additional exercises, and detailed appendices.

Cambridge University Press eBooks•Mar 19, 2026•0 citations
This book consists of the articles published in the special issues of this Symmetry journal based on two-by-two matrices and harmonic oscillators. The book also contains additional articles published by the guest editor in this Symmetry journal. They are of course based on harmonic oscillators and/or two-by-two matrices. The subject of symmetry is based on exactly soluble problems in physics, and the physical theory is not soluble unless it is based on oscillators and/or two-by-two matrices. The authors of those two special issues were aware of this environment when they submitted their articles. This book could therefore serve as an example to illustrate this important aspect of symmetry problems in physics.
Other than scattering problems where perturbation theory is applicable, there are basically two ways to solve problems in physics. One is to reduce the problem to harmonic oscillators, and the other is to formulate the problem in terms of two-by-two matrices. If two oscillators are coupled, the problem combines both two-by-two matrices and harmonic oscillators. This method then becomes a powerful research tool to cover many different branches of physics. Indeed, the concept and methodology in one branch of physics can be translated into another through the common mathematical formalism. Coupled oscillators provide clear illustrative examples for some of the current issues in physics, including entanglement and Feynman's rest of the universe. In addition, it is noted that the present form of quantum mechanics is largely a physics of harmonic oscillators. Special relativity is the physics of the Lorentz group which can be represented by the group of two-by-two matrices commonly called SL(2,c)SL(2,c)SL(2,c). Thus the coupled harmonic oscillators can play the role of combining quantum mechanics with special relativity. It is therefore possible to relate the current issues of physics to the Lorentz-covariant formulation of quantum mechanics.
Other than scattering problems where perturbation theory is applicable, there are basically two ways to solve problems in physics. One is to reduce the problem to harmonic oscillators, and the other is to formulate the problem in terms of two-by-two matrices. If two oscillators are coupled, the problem combines both two-by-two matrices and harmonic oscillators. This method then becomes a powerful research tool to cover many different branches of physics. Indeed, the concept and methodology in one branch of physics can be translated into another through the common mathematical formalism. Coupled oscillators provide clear illustrative examples for some of the current issues in physics, including entanglement, decoherence, and Feynman's rest of the universe. In addition, it is noted that the present form of quantum mechanics is largely a physics of harmonic oscillators. Special relativity is the physics of the Lorentz group which can be represented by the group of by two-by-two matrices commonly called SL(2,c)SL(2,c)SL(2,c). Thus the coupled harmonic oscillators can therefore play the role of combining quantum mechanics with special relativity. Both Paul A. M. Dirac and Richard P. Feynman were fond of harmonic oscillators, while they used different approaches to physical problems. Both were also keenly interested in making quantum mechanics compatible with special relativity. It is shown that the coupled harmonic oscillators can bridge these two different approaches to physics.
As the title aptly states, this book deals with harmonic oscillators of various kinds, from classical mechanical and electrical oscillations up to quantum oscillations. It is written in a lively language, and occasional interspersed anecdotes make the reading of an otherwise mathematically oriented text quite a pleasure. Although the author claims to have written an `elementary introduction', it is certainly necessary to have a good deal of previous knowledge in physics (mechanics, electrodynamics, quantum theory), electrical engineering and, of course, mathematics in order to follow the general line of his arguments. The book begins with a thorough treatment of classical oscillators (free, damped, forced) that is followed by an elaboration on Fourier analysis. Lagrange and Hamilton formalisms are then introduced before the problem of coupled oscillations is attacked. A chapter on statistical perspectives leads over to the final discussion of quantum oscillations. With the book comes a diskette containing a number of worksheets (Microsoft Excel) that can be used by the reader for instant visualization to get a better qualitative and quantitative understanding of the material. To the reviewer it seems difficult to pinpoint exactly the range of prospective readership of the book. It can certainly not be intended as a textbook for students, but rather as a reference book for teachers of physics or researchers, who want to look up one or other aspect of harmonic oscillations, for which purpose the diskette represents a very valuable tool.
Other than scattering problems where perturbation theory is applicable, there are basically two ways to solve problems in physics. One is to reduce the problem to harmonic oscillators, and the other is to formulate the problem in terms of two‐by‐two matrices. If two oscillators are coupled, the problem combines both two‐by‐two matrices and harmonic oscillators. This method then becomes a powerful research tool to cover many different branches of physics. Indeed, the concept and methodology in one branch of physics can be translated into another through the common mathematical formalism. It is noted that the present form of quantum mechanics is largely a physics of harmonic oscillators. Special relativity is the physics of the Lorentz group which can be represented by the group of by two‐by‐two matrices commonly called SL(2, c). Thus the coupled harmonic oscillators can therefore play the role of combining quantum mechanics with special relativity. Both Paul A. M. Dirac and Richard P. Feynman were fond of harmonic oscillators, while they used different approaches to physical problems. Both were also keenly interested in making quantum mechanics compatible with special relativity. It is shown that the coupled harmonic oscillators can bridge these two different approaches to physics.
An impressively comprehensive textbook adopting a phenomenological approach to quantum physics. The chapters cover everything from basic definitions of key concepts to detailed discussions of the underlying theoretical framework, walking students step-by-step through the necessary mathematics and drawing clear connections between the theory and the most important modern research applications including quantum optics, fluids, nanophysics, entanglement, information, and relativity. With this book, students and researchers will have access to hundreds of real-world examples, exercises, and illustrations to support and expand their understanding. Instructors can tailor the content to suit the length and level of their course and will have access to an online solutions manual with fully worked solutions to all 300+ exercises in the book. Other online resources include Python simulations, additional exercises, and detailed appendices.