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

Proceedings of the Royal Society of London Series A Containing Papers of a Mathematical and Physical Character

Perturbation problems in quantum mechanics

John Edward Lennard-Jones

Published Nov 3, 1930• 138 citations
Publisher

Abstract

Abstract One of the great achievements of the Schrödinger wave-mechanics is the elegance of its perturbation theory, which has brought many problems, formerly considered intractable, within the range of a highly-developed mathematical technique. It is not necessary at this stage to review the numerous applications which have been made of this perturbation theory or to dwell upon its many advantages. The important advance towards an understanding of chemical forces which it has made possible is in itself a considerable achievement. There are, however, certain disadvantages in the perturbation theory in its present form, which limit the extent of its applications to complex problems of atomic and molecular structure. If the interaction of atoms, for instance, is to be calculated, as it most desirable, improved methods will have to be found.

DOI
10.1098/rspa.1930.0177

Related papers

Perturbation theory in quantum mechanics

Alan Herries Wilson

Abstract The theory of perturbations of electronic systems has been considered by various writers, and the method is the basis of the solution of most quantum problems. It therefore seems desirable that the theory should be placed on a firm footing, and this necessitates the discussion of the convergence of the series of perturbations, which up to the present has been avoided. This investigation is all the more necessary since the wave equation of the Stark Effect has no solution of the form assumed by Schrödinger in his original discussion of the perturbation theory. Dirac’s theory leads to a system of linear differential equations in an infinite number of variables, and 2 is devoted to the discussion of the existence theorem for such a system by a matrix method. In 3 the application is made to the perturbation theory in the case of perturbations satisfying the conditions assumed in 2. In 4 the theory is extended so as to include more general types of disturbances. It is shown that, though the series of perturbations does not in general converge, yet it usually possesses the same asymptotic character as in the classical theory, and its use can therefore be justified.

Proceedings of the Royal Society of London Series A Containing Papers of a Mathematical and Physical Character•Feb 1, 1929•8 citations

- Approximation Methods I: Time-Independent Perturbation Theory

S. Rajasekar, R. Velusamy

The starting point in a quantum mechanical problem is the construction of a Schrödinger equation. We try to solve it exactly to obtain wave function. A Schrödinger equation is exactly solvable only for a certain number of simple potentials. In most of the cases the Schrödinger equation cannot be solved exactly. The solution of the Schrödinger equation for the real systems is usually complicated and often exact solutions are very difficult to find or solutions not exist. Examples include as anharmonic oscillator, hydrogen atom in an electric field, a plane rotator in the presence of an applied electric field and so on. Therefore, we need to rely on approximation methods. The simple and most interesting and powerful approximate method is the perturbation theory developed by Schrödinger in 1926 [1]. Earlier Lord Rayleigh [2] analysed harmonic vibrations of a string perturbed by small inhomogeneities. The work of Rayleigh was pointed out by Schrödinger. Consequently, the method is also called Rayleigh-Schrödinger perturbation theory. This method is powerful for determining the changes in the discrete energy values and the associated eigenfunctions of a system due to a weak perturbation, provided the eigenpairs of the undisturbed system are available. The approximations obtained from the perturbation theory are used to calculate energy level diagrams and to understand the splitting and assignments of atomic and molecular spectra [35]. The perturbation theory has been proven to be useful in several fields of theoretical physics and also in chemistry. It provides considerable insight into various phenomena. The main disadvantage of this method is that in certain problems the perturbation series converges slowly or diverges.

CRC Press eBooks•Dec 11, 2014•0 citations

Approximation Methods I: Time-Independent Perturbation Theory

S. Rajasekar, R. Velusamy

The starting point in a quantum mechanical problem is the construction of a Schrödinger equation. A Schrödinger equation is exactly solvable only for a certain number of simple potentials. In most of the cases the Schrödinger equation can be solved exactly. The solution of the Schrödinger equation for the real systems is usually complicated and often exact solutions. The perturbation theory has been proven to be useful in several fields of theoretical physics and also in chemistry. It provides considerable insight into various phenomena. The main disadvantage of this method is that in certain problems the perturbation series converges slowly or diverges. The splitting of an energy level due to the interaction of the nuclear magnetic dipole and the magnetic field Be produced by the electron orbit current is called the fine structure. The term hyperfine structure is used to describe extremely small splitting of spectral lines.

CRC Press eBooks•Sep 29, 2022•0 citations

Quantum-mechanical perturbation theory

John P. Killingbeck

Modern computers have made possible the evaluation of higher order terms of perturbation series. Apart from the empirical work involving numerical calculations, much formal mathematical work has been done on the convergence properties of perturbation series and of Pade approximants which represent them. Some of the modern work in perturbation theory is reviewed in the context of traditional results from the theory of real and complex variables. The two major versions of time-independent perturbation theory, the Rayleigh-Schrodinger (RS) and Brillouin-Wigner (BW) theories, are compared and contrasted, and the alternative techniques for evaluating the terms in the energy series are discussed. The sum-over-states method, the differential equation method and the variational principle method are treated, with emphasis on their inter-relations.

Reports on Progress in Physics•Sep 1, 1977•163 citations

Mathematical Physics and the Study of Complex Quantum Systems

Ana Uzla BatuBara, Zainab Ali, Arnes Yuli Vandika

The study of complex quantum systems is a fundamental aspect of modern physics, providing insights into the behavior of matter at microscopic scales. Mathematical physics plays a crucial role in developing the theoretical frameworks necessary for understanding these systems, yet challenges remain in applying these concepts to real-world scenarios. This research aims to investigate the application of mathematical techniques in analyzing complex quantum systems. The focus is on identifying effective mathematical models and methods that can enhance our understanding of quantum phenomena. A comprehensive literature review was conducted, analyzing various mathematical approaches utilized in quantum mechanics, including perturbation theory, group theory, and numerical simulations. Case studies were examined to illustrate the successful application of these methods in real-world quantum systems. Findings indicate that advanced mathematical techniques significantly improve the modeling and analysis of complex quantum systems. The application of perturbation theory and numerical simulations provided deeper insights into system behaviors, while group theory facilitated a better understanding of symmetry properties. This research highlights the indispensable role of mathematical physics in the study of complex quantum systems. By emphasizing the integration of mathematical techniques, the study contributes to the advancement of theoretical physics and offers pathways for future research in quantum mechanics.

Research of Scientia Naturalis•Dec 27, 2024•0 citations

Introduction

Rajat Kumar Chaudhuri, Sudip Kumar Chattopadhyay

Perturbation theory, in the arena of quantum mechanics, refers to a set of physically motivated and intuitive approximations engineered by mathematical perturbation to describe and explore the properties of a complicated system. The philosophy of perturbation theory is to begin with a simpler system, whose mathematical solution is known to the best of one’s knowledge, and then “perturb” the system by means of an externally tunable influence. For a disturbance that is not too strong, the physical quantities associated with the perturbed system can be expressed in terms of the “corrections” of the corresponding quantities of the simpler (unperturbed) system. For small perturbation, the magnitudes of these correction terms are really small in light of the magnitude of the quantities themselves and are therefore subject to treatments by approximations like the asymptotic series. The accuracy of the evaluated physical quantity, among other factors, depends on the extent to which such correction terms are incorporated. Therefore, perturbation theory is a useful avenue to explore the properties of a complicated system, where the exact (analytic) solution of the system is either difficult or impossible to achieve.

CRC Press eBooks•Feb 17, 2017•0 citations
Abstract One of the great achievements of the Schrödinger wave-mechanics is the elegance of its perturbation theory, which has brought many problems, formerly considered intractable, within the range of a highly-developed mathematical technique. It is not necessary at this stage to review the numerous applications which have been made of this perturbation theory or to dwell upon its many advantages. The important advance towards an understanding of chemical forces which it has made possible is in itself a considerable achievement. There are, however, certain disadvantages in the perturbation theory in its present form, which limit the extent of its applications to complex problems of atomic and molecular structure. If the interaction of atoms, for instance, is to be calculated, as it most desirable, improved methods will have to be found.
Abstract The theory of perturbations of electronic systems has been considered by various writers, and the method is the basis of the solution of most quantum problems. It therefore seems desirable that the theory should be placed on a firm footing, and this necessitates the discussion of the convergence of the series of perturbations, which up to the present has been avoided. This investigation is all the more necessary since the wave equation of the Stark Effect has no solution of the form assumed by Schrödinger in his original discussion of the perturbation theory. Dirac’s theory leads to a system of linear differential equations in an infinite number of variables, and 2 is devoted to the discussion of the existence theorem for such a system by a matrix method. In 3 the application is made to the perturbation theory in the case of perturbations satisfying the conditions assumed in 2. In 4 the theory is extended so as to include more general types of disturbances. It is shown that, though the series of perturbations does not in general converge, yet it usually possesses the same asymptotic character as in the classical theory, and its use can therefore be justified.
The starting point in a quantum mechanical problem is the construction of a Schrödinger equation. We try to solve it exactly to obtain wave function. A Schrödinger equation is exactly solvable only for a certain number of simple potentials. In most of the cases the Schrödinger equation cannot be solved exactly. The solution of the Schrödinger equation for the real systems is usually complicated and often exact solutions are very difficult to find or solutions not exist. Examples include as anharmonic oscillator, hydrogen atom in an electric field, a plane rotator in the presence of an applied electric field and so on. Therefore, we need to rely on approximation methods. The simple and most interesting and powerful approximate method is the perturbation theory developed by Schrödinger in 1926 [1]. Earlier Lord Rayleigh [2] analysed harmonic vibrations of a string perturbed by small inhomogeneities. The work of Rayleigh was pointed out by Schrödinger. Consequently, the method is also called Rayleigh-Schrödinger perturbation theory. This method is powerful for determining the changes in the discrete energy values and the associated eigenfunctions of a system due to a weak perturbation, provided the eigenpairs of the undisturbed system are available. The approximations obtained from the perturbation theory are used to calculate energy level diagrams and to understand the splitting and assignments of atomic and molecular spectra [35]. The perturbation theory has been proven to be useful in several fields of theoretical physics and also in chemistry. It provides considerable insight into various phenomena. The main disadvantage of this method is that in certain problems the perturbation series converges slowly or diverges.
The starting point in a quantum mechanical problem is the construction of a Schrödinger equation. A Schrödinger equation is exactly solvable only for a certain number of simple potentials. In most of the cases the Schrödinger equation can be solved exactly. The solution of the Schrödinger equation for the real systems is usually complicated and often exact solutions. The perturbation theory has been proven to be useful in several fields of theoretical physics and also in chemistry. It provides considerable insight into various phenomena. The main disadvantage of this method is that in certain problems the perturbation series converges slowly or diverges. The splitting of an energy level due to the interaction of the nuclear magnetic dipole and the magnetic field Be produced by the electron orbit current is called the fine structure. The term hyperfine structure is used to describe extremely small splitting of spectral lines.
Modern computers have made possible the evaluation of higher order terms of perturbation series. Apart from the empirical work involving numerical calculations, much formal mathematical work has been done on the convergence properties of perturbation series and of Pade approximants which represent them. Some of the modern work in perturbation theory is reviewed in the context of traditional results from the theory of real and complex variables. The two major versions of time-independent perturbation theory, the Rayleigh-Schrodinger (RS) and Brillouin-Wigner (BW) theories, are compared and contrasted, and the alternative techniques for evaluating the terms in the energy series are discussed. The sum-over-states method, the differential equation method and the variational principle method are treated, with emphasis on their inter-relations.
The study of complex quantum systems is a fundamental aspect of modern physics, providing insights into the behavior of matter at microscopic scales. Mathematical physics plays a crucial role in developing the theoretical frameworks necessary for understanding these systems, yet challenges remain in applying these concepts to real-world scenarios. This research aims to investigate the application of mathematical techniques in analyzing complex quantum systems. The focus is on identifying effective mathematical models and methods that can enhance our understanding of quantum phenomena. A comprehensive literature review was conducted, analyzing various mathematical approaches utilized in quantum mechanics, including perturbation theory, group theory, and numerical simulations. Case studies were examined to illustrate the successful application of these methods in real-world quantum systems. Findings indicate that advanced mathematical techniques significantly improve the modeling and analysis of complex quantum systems. The application of perturbation theory and numerical simulations provided deeper insights into system behaviors, while group theory facilitated a better understanding of symmetry properties. This research highlights the indispensable role of mathematical physics in the study of complex quantum systems. By emphasizing the integration of mathematical techniques, the study contributes to the advancement of theoretical physics and offers pathways for future research in quantum mechanics.
Perturbation theory, in the arena of quantum mechanics, refers to a set of physically motivated and intuitive approximations engineered by mathematical perturbation to describe and explore the properties of a complicated system. The philosophy of perturbation theory is to begin with a simpler system, whose mathematical solution is known to the best of one’s knowledge, and then “perturb” the system by means of an externally tunable influence. For a disturbance that is not too strong, the physical quantities associated with the perturbed system can be expressed in terms of the “corrections” of the corresponding quantities of the simpler (unperturbed) system. For small perturbation, the magnitudes of these correction terms are really small in light of the magnitude of the quantities themselves and are therefore subject to treatments by approximations like the asymptotic series. The accuracy of the evaluated physical quantity, among other factors, depends on the extent to which such correction terms are incorporated. Therefore, perturbation theory is a useful avenue to explore the properties of a complicated system, where the exact (analytic) solution of the system is either difficult or impossible to achieve.