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

arXiv:quant-ph/0012023 · Journal of Physics A Mathematical and General

Non-perturbative solution of nonlinear Heisenberg equations

Ladislav Mišta, Radim Filip

Published Jun 27, 2001• 5 citations
arXivHTMLPDF

Abstract

A new non-perturbative method of solution of the nonlinear Heisenberg equations in a finite-dimensional subspace is illustrated. The method, being a counterpart of the traditional Schrödinger picture method, is based on a finite operator expansion into the elementary processes. It provides us with insight into the nonlinear quantal interaction from a different point of view. Thus, one can investigate the nonlinear system in both pictures of quantum mechanics.

DOI
10.1088/0305-4470/34/27/310

Related papers

Non-perturbative solution of nonlinear Heisenberg equations

Radim Filip

A new non-perturbative method of solution of the nonlinear Heisenberg equations in finite-dimensional subspace is illustrated. The method, being a counterpart of the traditional Schrödinger picture method, is based on a finite operator expansion into the elementary processes. It provides us with the insight into the nonlinear quantal interaction from the different point of view. Thus one can investigate the nonlinear system in both pictures of quantum mechanics. 1

•Jan 1, 2000•5 citations

Non-perturbative Treatment of Nonlinear Optical Systems

Ladislav Mišta, Radim Filip

A new non-perturbative method enabling to treat the nonlinear quantum optical systems in a finite-dimensional subspace is illustrated on the example of second-harmonic generation process. The method, being a counterpart of the traditional Schrödinger picture method, is based on a finite operator expansion into the elementary processes. In this way one can calculate the time dependence of the mean value of any operator in the subspace.

Fortschritte der Physik•Oct 1, 2001•1 citation

Heisenberg-picture operator perturbation theory

Jay R. Ackerhalt, Kazimierz Rza̧żewski

We develop a systematic method of operator perturbation theory in the Heisenberg picture, which is the formal analog of time-independent stationary-state perturbation theory in the Schr\"odinger picture. However, we use the eigenoperators and operator eigenfrequencies to calculate the time evolution of the dynamical variables of interest. The spectrum of the unperturbed Hamiltonian is assumed to be discrete. The method is especially well-suited for treating systems where the unperturbed Hamiltonian represents noninteracting fermions and/or bosons, and therefore, we illustrate the method using the exactly solvable model of Jaynes and Cummings: a single two-level atom interacting with a single quantized field mode. The time evolution of these operator solutions is shown to be unitary order by order in powers of the interaction strength. We find that the time evolution of exact operator solutions is well approximated for significantly longer times by the time evolution of this method's operator solutions than by the time evolution of the same operator solutions calculated using the Dyson expansion. We also demonstrate that time-dependent operator solutions are very convenient for computing quantities such as correlation functions.

Physical Review A•Dec 1, 1975•86 citations

A topical review on time-independent perturbation theory in one-dimensional quantum systems

Leonardo Lavagna, Sandra Carillo, Massimo Panella

Abstract This review presents time-independent perturbative methods for solving the one-dimensional Schrödinger equation, highlighting representative cases that reveal key aspects of the theory. The focus is on their relevance to quantum computing applications, particularly in systems with finite-dimensional state spaces. The main link between these methods and quantum processing emerges when we consider systems with only a finite number of well-defined energy levels, e.g., two levels corresponding to the ‘zero’ and ‘one’ states of a quantum bit (qubit), for which the Schrödinger equation describes a quantum algorithm corresponding to the unitary evolution of the qubits. Perturbative techniques can then be applied to this evolution to gain deeper insight into the algorithm or to simplify it in a controlled manner, as described in the last sections of this review. We initially discuss representative applications using benchmark cases that exemplify the essential features of perturbation theory while avoiding unnecessary technical complications. The methods presented involve direct approaches to solutions for quantum systems with small potential perturbations, with consideration of indirect and approximation techniques. Key cases examined include the particle in a potential well, the harmonic oscillator, and systems with polynomial or Dirac delta distribution perturbations, each chosen to highlight unique aspects of perturbative analysis. Starting with an overview of the unperturbed Schrödinger equation, we derive canonical solutions and then use perturbative techniques to approximate energy shifts and wavefunction corrections in both non-degenerate and degenerate settings. By combining theoretical discussion with case studies, this review underscores how perturbative methods are applied across a range of elementary quantum mechanical problems especially for applications in the qubit model, which, in many cases of interest, can be reduced to the explicit calculations shown here.

Physica Scripta•Sep 23, 2025•0 citations

On the link between Schrodinger and Heisenberg pictures of quantum mechanics

Peter Prešnajder, V. Pažma

The authors take into account boundary conditions in solutions of the Heisenberg equations of motion. They express the kernel of the evolution operator in the x-representation through the retarded propagator which solves the Schrodinger equation and satisfies the boundary conditions in question. As an illustration, they solve the Heisenberg equation for a free particle bound to the half-line (0, infinity ) with various boundary conditions at the point x=0.

European Journal of Physics•Jan 1, 1989•0 citations

Divergence-Free WKB Method

Tadanori Hyouguchi, Satoshi Adachi, Masahito Ueda

A new semiclassical approach to the linear and nonlinear one-dimensional Schrödinger equation is presented. For both equations our zeroth-order solutions include nonperturbative quantum corrections to the WKB solution and the Thomas-Fermi solution, thereby allowing us to make uniformly converging perturbative expansions of the wave functions. Our method leads to a new quantization condition that yields exact eigenenergies for the harmonic-oscillator and Morse potentials.

arXiv:cond-mat/0011374•Physical Review Letters•Apr 11, 2002•16 citations
A new non-perturbative method of solution of the nonlinear Heisenberg equations in a finite-dimensional subspace is illustrated. The method, being a counterpart of the traditional Schrödinger picture method, is based on a finite operator expansion into the elementary processes. It provides us with insight into the nonlinear quantal interaction from a different point of view. Thus, one can investigate the nonlinear system in both pictures of quantum mechanics.
We develop a systematic method of operator perturbation theory in the Heisenberg picture, which is the formal analog of time-independent stationary-state perturbation theory in the Schr\"odinger picture. However, we use the eigenoperators and operator eigenfrequencies to calculate the time evolution of the dynamical variables of interest. The spectrum of the unperturbed Hamiltonian is assumed to be discrete. The method is especially well-suited for treating systems where the unperturbed Hamiltonian represents noninteracting fermions and/or bosons, and therefore, we illustrate the method using the exactly solvable model of Jaynes and Cummings: a single two-level atom interacting with a single quantized field mode. The time evolution of these operator solutions is shown to be unitary order by order in powers of the interaction strength. We find that the time evolution of exact operator solutions is well approximated for significantly longer times by the time evolution of this method's operator solutions than by the time evolution of the same operator solutions calculated using the Dyson expansion. We also demonstrate that time-dependent operator solutions are very convenient for computing quantities such as correlation functions.
Abstract This review presents time-independent perturbative methods for solving the one-dimensional Schrödinger equation, highlighting representative cases that reveal key aspects of the theory. The focus is on their relevance to quantum computing applications, particularly in systems with finite-dimensional state spaces. The main link between these methods and quantum processing emerges when we consider systems with only a finite number of well-defined energy levels, e.g., two levels corresponding to the ‘zero’ and ‘one’ states of a quantum bit (qubit), for which the Schrödinger equation describes a quantum algorithm corresponding to the unitary evolution of the qubits. Perturbative techniques can then be applied to this evolution to gain deeper insight into the algorithm or to simplify it in a controlled manner, as described in the last sections of this review. We initially discuss representative applications using benchmark cases that exemplify the essential features of perturbation theory while avoiding unnecessary technical complications. The methods presented involve direct approaches to solutions for quantum systems with small potential perturbations, with consideration of indirect and approximation techniques. Key cases examined include the particle in a potential well, the harmonic oscillator, and systems with polynomial or Dirac delta distribution perturbations, each chosen to highlight unique aspects of perturbative analysis. Starting with an overview of the unperturbed Schrödinger equation, we derive canonical solutions and then use perturbative techniques to approximate energy shifts and wavefunction corrections in both non-degenerate and degenerate settings. By combining theoretical discussion with case studies, this review underscores how perturbative methods are applied across a range of elementary quantum mechanical problems especially for applications in the qubit model, which, in many cases of interest, can be reduced to the explicit calculations shown here.