© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

International Journal of Quantum Chemistry

Connections between perturbation theory and the Van Vleck transformation: Illustrative calculations on the perturbed harmonic oscillator

Paul A. Westhaus

Published Mar 1, 1982• 3 citations

Abstract

Abstract We continue to examine the connection between perturbation theory and the Van Vleck unitary transformation. Here we illustrate the formalism derived earlier by applying it to compute the stationary states of the perturbed harmonic oscillator. We find that each solution of the traditional Brillouin‐Wigner perturbation theory equations gives rise to a different unitary transformation which, when operating on the unperturbed ground state, produces one or the other of the perturbed eigenstates. With any of the perturbed states able to be reached by a unitary transformation on the unperturbed ground state, we advise caution in using approximate solutions of the perturbation equations in general cases, lest an unexpected stationary state be obtained.

DOI: 10.1002/qua.560210309

Related papers

Operator equations in perturbation theory

A. M. Arthurs, Peter D. Robinson

Abstract Quantum mechanical perturbation theory is discussed in terms of solutions W of commutator equations of the type [h, W] = A. The operators W lead to perturbation corrections for any eigenstate of the unperturbed system. The theory is illustrated by considering the linear harmonic oscillator perturbed by the Nth power of the displacement.

Proceedings of the Royal Society of London A Mathematical and Physical Sciences•Nov 7, 1967•9 citations

Stationary State Perturbation Theory

Gordon Baym

The bound states of a weak attractive potential never come out of doing perturbation theory on free particle states. This is a quite general rule: if the perturbation commutes with a constant of the motion, A, of the unperturbed Hamiltonian, then matrix elements of the perturbation between eigenstates of A with different eigenvalues must vanish. This rule, an example of a selection rule, often simplifies the problem of selecting the correct linear combination of degenerate states with which to do the perturbation expansion. The Rayleigh-Schrodinger perturbation theory – yields rather complicated expressions for the change in the states in second order and beyond. There is a somewhat different form of the perturbation theory, due to Brillouin and Wigner that enables to see more clearly the structure of the perturbed states.

CRC Press eBooks•Mar 5, 2018•0 citations

Perturbation theory

Giampiero V. M. Esposito, Giuseppe Marmo, George Sudarshan

The subject of perturbation theory in non-relativistic quantum mechanics is introduced. First, perturbation theory for stationary states in the absence of degeneracy is studied. The case of nearby levels, perturbations of a one-dimensional harmonic oscillator, the occurrence of degeneracy, Stark and Zeeman effects are then described in detail. After a brief outline of the variational method, the Dyson series for time-dependent perturbation theory is derived, with application to harmonic perturbations. The Fermi golden rule is also presented. The chapter ends with an assessment of four branches of the subject: regular perturbation theory, asymptotic perturbation theory, and summability methods, spectral concentration and singular perturbations. Approximate methods for stationary states It is frequently the case, in physical problems, that the full Hamiltonian operator H consists of an operator H 0 whose eigenvalues and eigenfunctions are known exactly, and a second term V resulting from a variety of sources (e.g. the interaction with an electric field, or a magnetic field, or the relativistic corrections to the kinetic energy, or the effects of spin). The problems and aims of perturbation theory for stationary states can be therefore summarized as follows. (i) Once the domain of (essential) self-adjointness of the ‘unperturbed’ Hamiltonian H 0 is determined, find on which domain the ‘sum’ H = H 0 + V represents an (essentially) self-adjoint operator (the reader should remember from appendix 4. A that in general, once some operators A and B are given, the intersection of their domains, on which their sum is defined, might be the empty set). […]

Cambridge University Press eBooks•Mar 11, 2004•0 citations

RESUMEN OSCILADOR ARMONICO PERTURBADO

Diógenes Campos

An harmonic oscillator is subject to a perturbation f{A)= VJA) I p = O) (p = 01 , where I p = O) is an eigenket of the momentum operator. The Brillouin-Wigner perturbation theory is applied to solve the eigenvalue equation of the perturbed Harniltonian.

•Jan 1, 1993•0 citations

Teoría de perturbaciones de Brillouin-Wigner: Oscilador armónico perturbado

Diógenes Campos

An harmonic oscillator is subject to a perturbation [Physical Formula] where [Physical Formula] is an eigenket of the momentum operator. The Brillouin-Wigner perturbation theory is applied to solve the eigenvalue equation of the perturbed Hamiltonian.

MOMENTO•Jan 1, 1993•0 citations

Hamiltonian point of view of quantum perturbation theory

A. D. Bermúdez Manjarres

Abstract We explore the relation of Van Vleck-Primas perturbation theory of quantum mechanics with the Lie-series-based perturbation theory of Hamiltonian systems in classical mechanics. In contrast to previous works on the relation of quantum and classical perturbation theories, our approach is not based on the conceptual similarities between the two methods. Instead, we show that for quantum systems with a finite-dimensional Hilbert space, the Van Vleck-Primas procedure can be recast exactly into a classical perturbation problem. As a non-obvious consequence, this approach gives a new way of calculating the geometric phase of quantum systems using tools from the theory of classical canonical transformations.

arXiv:2107.07050•Physica Scripta•Aug 28, 2025•0 citations

Abstract We continue to examine the connection between perturbation theory and the Van Vleck unitary transformation. Here we illustrate the formalism derived earlier by applying it to compute the stationary states of the perturbed harmonic oscillator. We find that each solution of the traditional Brillouin‐Wigner perturbation theory equations gives rise to a different unitary transformation which, when operating on the unperturbed ground state, produces one or the other of the perturbed eigenstates. With any of the perturbed states able to be reached by a unitary transformation on the unperturbed ground state, we advise caution in using approximate solutions of the perturbation equations in general cases, lest an unexpected stationary state be obtained.

The bound states of a weak attractive potential never come out of doing perturbation theory on free particle states. This is a quite general rule: if the perturbation commutes with a constant of the motion, A, of the unperturbed Hamiltonian, then matrix elements of the perturbation between eigenstates of A with different eigenvalues must vanish. This rule, an example of a selection rule, often simplifies the problem of selecting the correct linear combination of degenerate states with which to do the perturbation expansion. The Rayleigh-Schrodinger perturbation theory – yields rather complicated expressions for the change in the states in second order and beyond. There is a somewhat different form of the perturbation theory, due to Brillouin and Wigner that enables to see more clearly the structure of the perturbed states.

The subject of perturbation theory in non-relativistic quantum mechanics is introduced. First, perturbation theory for stationary states in the absence of degeneracy is studied. The case of nearby levels, perturbations of a one-dimensional harmonic oscillator, the occurrence of degeneracy, Stark and Zeeman effects are then described in detail. After a brief outline of the variational method, the Dyson series for time-dependent perturbation theory is derived, with application to harmonic perturbations. The Fermi golden rule is also presented. The chapter ends with an assessment of four branches of the subject: regular perturbation theory, asymptotic perturbation theory, and summability methods, spectral concentration and singular perturbations. Approximate methods for stationary states It is frequently the case, in physical problems, that the full Hamiltonian operator H consists of an operator H 0 whose eigenvalues and eigenfunctions are known exactly, and a second term V resulting from a variety of sources (e.g. the interaction with an electric field, or a magnetic field, or the relativistic corrections to the kinetic energy, or the effects of spin). The problems and aims of perturbation theory for stationary states can be therefore summarized as follows. (i) Once the domain of (essential) self-adjointness of the ‘unperturbed’ Hamiltonian H 0 is determined, find on which domain the ‘sum’ H = H 0 + V represents an (essentially) self-adjoint operator (the reader should remember from appendix 4. A that in general, once some operators A and B are given, the intersection of their domains, on which their sum is defined, might be the empty set). […]

Abstract We explore the relation of Van Vleck-Primas perturbation theory of quantum mechanics with the Lie-series-based perturbation theory of Hamiltonian systems in classical mechanics. In contrast to previous works on the relation of quantum and classical perturbation theories, our approach is not based on the conceptual similarities between the two methods. Instead, we show that for quantum systems with a finite-dimensional Hilbert space, the Van Vleck-Primas procedure can be recast exactly into a classical perturbation problem. As a non-obvious consequence, this approach gives a new way of calculating the geometric phase of quantum systems using tools from the theory of classical canonical transformations.