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

The European Physical Journal Plus

A simple iterative method to write the terms of any order of perturbation theory in quantum mechanics

L. Bracci, Luigi E. Picasso

Published Oct 1, 2012• 8 citations
Publisher
DOI
10.1140/epjp/i2012-12119-6

Related papers

Perturbation Theory in Quantum Mechanics

CRC Press eBooks•Sep 19, 2000•2 citations

An alternative to the magnus expansion in time-dependent perturbation theory

W. R. Salzman

An alternative to the Magnus expansion in time-dependent quantum mechanical perturbation theory is presented. An exponential form of the time-development operator, given as the exponential of a sum of products of integrals well known in time-dependent quantum mechanics, is derived from the standard perturbation expansion. The derivation is simple and the form of the terms in the exponential is so straightforward that the Nth order term can be written by inspection. The standard Magnus terms can be obtained by rearrangement of the terms of the new expansion probably easier than they can be obtained by iteration of the original equations. A proof by induction is presented to establish the form of the Nth term and it is shown how the new expansion terms relate to the Magnus terms. Some apparent differences between Magnus terms in the literature are resolved.

The Journal of Chemical Physics•Jan 15, 1985•55 citations

The eigenvalue spectrum in quantum mechanics and the nonlinearization procedure

Alexander V. Turbiner

A new approach to the eigenvalue problem in quantum mechanics is proposed. This approach is based on three propositions: 1) a perturbation theory which does not require knowledge of the entire eigenvalue spectrum of the unperturbed problem and which uses a nonlinearization procedure (leaving some latitude in the choice of a zeroth order approximation); 2) a relationship between the perturbation theory and a variational principle, namely that any variational calculation is none other than the first two terms of some nontrivial perturbation theory which, when developed further, can reveal the accuracy of the variational calculations and can refine them by an iterative procedure; 3) Dyson's argument, which serves as a criterion for the reasonableness of the choice of a zeroth order approximation (the unperturbed problem). The realization of this perturbation theory in a k-dimensional space is equivalent to the solution of a k-dimensional electrostatic problem with a variable dielectric permittivity. In the one-dimensional case and in cases which reduce to the one-dimensional case, all the corrections are written in quadratures. It is shown that the construction of an ordinary perturbation theory (in which the zeroth order approximation is an exactly solvable problem) within the framework of this perturbation theory is a purely algebraic procedure, which reduces to the solution of some simple recurrence relations. An approximation analogous to the leading logarithmic approximation of quantum field theory is constructed. Some standard problems of quantum mechanics—the anharmonic oscillator, the Zeeman effect, and the Stark effect—are treated as examples. It is shown that this new approach makes it possible to develop systematically a theory for strong coupling and large perturbations.

Soviet Physics Uspekhi•Sep 30, 1984•70 citations

Perturbation Theory in Quantum Mechanics

Luigi E. Picasso, L. Bracci, Emilio d’Emilio

Springer eBooks•Oct 19, 2011•1 citation

CALCULATION OF PERTURBATIONS IN QUANTUM MECHANICS

Márcel Mayot

Compt. rend.•Feb 25, 1952•0 citations

Teaching Perturbation Theory through Spin Systems

L. E. Radford, J. S. Willis, P. W. Forbes

The spin system provides a relatively simple and compact vehicle for discussion of non-degenerate perturbation theory. In fact in many such systems the terms required to write the electron's energy correct to second order may be read directly from the spin-energy matrix. In this work a hypothetical spin system and three different physical spin systems are treated in order to demonstrate the conciseness of the spin-energy-matrix approach to some perturbation problems. The combination of a perturbation problem which is calculable and which may be related to experimental data provides an approach to perturbation theory which is particularly satisfying to the student.

American Journal of Physics•May 1, 1970•0 citations
An alternative to the Magnus expansion in time-dependent quantum mechanical perturbation theory is presented. An exponential form of the time-development operator, given as the exponential of a sum of products of integrals well known in time-dependent quantum mechanics, is derived from the standard perturbation expansion. The derivation is simple and the form of the terms in the exponential is so straightforward that the Nth order term can be written by inspection. The standard Magnus terms can be obtained by rearrangement of the terms of the new expansion probably easier than they can be obtained by iteration of the original equations. A proof by induction is presented to establish the form of the Nth term and it is shown how the new expansion terms relate to the Magnus terms. Some apparent differences between Magnus terms in the literature are resolved.
A new approach to the eigenvalue problem in quantum mechanics is proposed. This approach is based on three propositions: 1) a perturbation theory which does not require knowledge of the entire eigenvalue spectrum of the unperturbed problem and which uses a nonlinearization procedure (leaving some latitude in the choice of a zeroth order approximation); 2) a relationship between the perturbation theory and a variational principle, namely that any variational calculation is none other than the first two terms of some nontrivial perturbation theory which, when developed further, can reveal the accuracy of the variational calculations and can refine them by an iterative procedure; 3) Dyson's argument, which serves as a criterion for the reasonableness of the choice of a zeroth order approximation (the unperturbed problem). The realization of this perturbation theory in a k-dimensional space is equivalent to the solution of a k-dimensional electrostatic problem with a variable dielectric permittivity. In the one-dimensional case and in cases which reduce to the one-dimensional case, all the corrections are written in quadratures. It is shown that the construction of an ordinary perturbation theory (in which the zeroth order approximation is an exactly solvable problem) within the framework of this perturbation theory is a purely algebraic procedure, which reduces to the solution of some simple recurrence relations. An approximation analogous to the leading logarithmic approximation of quantum field theory is constructed. Some standard problems of quantum mechanics—the anharmonic oscillator, the Zeeman effect, and the Stark effect—are treated as examples. It is shown that this new approach makes it possible to develop systematically a theory for strong coupling and large perturbations.
The spin system provides a relatively simple and compact vehicle for discussion of non-degenerate perturbation theory. In fact in many such systems the terms required to write the electron's energy correct to second order may be read directly from the spin-energy matrix. In this work a hypothetical spin system and three different physical spin systems are treated in order to demonstrate the conciseness of the spin-energy-matrix approach to some perturbation problems. The combination of a perturbation problem which is calculable and which may be related to experimental data provides an approach to perturbation theory which is particularly satisfying to the student.