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

Modern Physics Letters B

Novel Technique for Quantum-Mechanical Eigenstate and Eigenvalue Calculations based on Seke's Self-Consistent Projection-Operator Method

JOSIP SEKE, Andrey Vladimirovich Soldatov, Nikolay N. Bogolubov

Published Mar 10, 1997• 1 citation
Publisher

Abstract

Seke's self-consistent projection-operator method has been developed for deriving non-Markovian equations of motion for probability amplitudes of a relevant set of state vectors. This method, in a Born-like approximation, leads automatically to an Hamiltonian restricted to a subspace and thus enables the construction of effective Hamiltonians. In the present paper, in order to explain the efficiency of Seke's method in particular applications, its algebraic operator structure is analyzed and a new successive approximation technique for the calculation of eigenstates and eigenvalues of an arbitrary quantum-mechanical system is developed. Unlike most perturbative techniques, in the present case each order of the approximation determines its own effective (approximating) Hamiltonian ensuring self-consistency and formal exactness of all results in the corresponding approximation order.

DOI
10.1142/s0217984997000311

Related papers

A New Expansion of the Heisenberg Equation of Motion with Projection Operator

Tomoi Koide C, M. Maruyama

We derive a new expansion of the Heisenberg equation of motion based on the projection operator method proposed by Shibata, Hashitsume and Shingū. In their projection operator method, a certain restriction is imposed on the initial state. As a result, one cannot prepare arbitrary initial states, for example a coherent state, to calculate the time development of quantum systems. In this paper, we generalize the projection operator method by relaxing this restriction. We explain our method in the case of a Hamiltonian both with and without explicit time dependence. Furthermore, we apply it to an exactly solvable model called the damped harmonic oscillator model and confirm the validity of our method.

arXiv:hep-th/0008182•Progress of Theoretical Physics•Sep 1, 2000•14 citations

Algebraic Aspects of the Dynamics of Quantum Multilevel Systems in the Projection Operator Technique

N. N. Bogolyubov, Andrey Vladimirovich Soldatov

Using the projection operator method, we obtain approximate time-local and time-nonlocal master equations for the reduced statistical operator of a multilevel quantum system with a finite number N of quantum eigenstates coupled simultaneously to arbitrary classical fields and a dissipative environment. We show that the structure of the obtained equations is significantly simplified if the free Hamiltonian dynamics of the multilevel system under the action of external fields and also its Markovian and non-Markovian evolutions due to coupling to the environment are described via the representation of the multilevel system in terms of the SU(N) algebra, which allows realizing effective numerical algorithms for solving the obtained equations when studying real problems in various fields of theoretical and applied physics.

Theoretical and Mathematical Physics•Feb 1, 2018•0 citations

Self-consistent projection operator theory in nonlinear quantum optical systems: A case study on degenerate optical parametric oscillators

Peter Degenfeld-Schonburg, Carlos Navarrete–Benlloch, Michael J. Hartmann

Nonlinear quantum optical systems are of paramount relevance for modern quantum technologies, as well as for the study of dissipative phase transitions. Their nonlinear nature makes their theoretical study very challenging and hence they have always served as great motivation to develop new techniques for the analysis of open quantum systems. We apply the recently developed self-consistent projection operator theory to the degenerate optical parametric oscillator to exemplify its general applicability to quantum optical systems. We show that this theory provides an efficient method to calculate the full quantum state of each mode with a high degree of accuracy, even at the critical point. It is equally successful in describing both the stationary limit and the dynamics, including regions of the parameter space where the numerical integration of the full problem is significantly less efficient. We further develop a Gaussian approach consistent with our theory, which yields sensibly better results than the previous Gaussian methods developed for this system, most notably standard linearization techniques.

arXiv:1503.03466•Physical Review A•May 26, 2015•18 citations

Projection Operators in Quantum-Mechanical Perturbation Theory

W. R. Salzman, M. E. Baur

The construction of operator functionals of the full Hamiltonian of a quantum-mechanical system which act as projection operators for a specified exact eigenstate is studied. In general these operator functionals are not equivalent to the formal operator |ψ〉〈ψ|, where |ψ〉 is the state of interest, and are not always idempotent. A formal method by which standard perturbation theoretic expansions for the exact ground-state vector and ground-state energy of the system can be recovered by expansion of a certain operator functional of this type is presented. Related work is briefly discussed.

The Journal of Chemical Physics•May 1, 1965•3 citations

On Construction of Projection Operators

Artur F. Izmaylov

The problem of construction of projection operators on eigensubspaces of symmetry operators is considered. This problem arises in many approximate methods for solving time-independent and time-dependent quantum problems, and its solution ensures proper physical symmetries in development of approximate methods. The projector form is sought as a function of symmetry operators and their eigenvalues characterizing the eigensubspace of interest. This form is obtained in two steps: (1) identification of algebraic structures within a set of symmetry operators (e.g., groups and Lie algebras) and (2) construction of the projection operators for individual symmetry operators. The first step is crucial for efficient projection operator construction because it allows for use of information on irreducible representations of the present algebraic structure. The discussed approaches have potential to stimulate further developments of variational approaches for electronic structure of strongly correlated systems and in quantum computing.

arXiv:1902.07865•The Journal of Physical Chemistry A•Mar 27, 2019•10 citations

Self-consistent projection-operator method for describing the non-Markovian time-evolution of subsystems

JOSIP SEKE

It is pointed out that the usual projection-operator method yields non-positive-definite reduced density operators for subsystems. In order to obtain non-negative probabilities, a new self-consistent projection-operator method for treating non-Markovian behaviour is developed. Finally, the applicability of the method is shown by an example.

Journal of Physics A Mathematical and General•Jan 21, 1990•28 citations
Seke's self-consistent projection-operator method has been developed for deriving non-Markovian equations of motion for probability amplitudes of a relevant set of state vectors. This method, in a Born-like approximation, leads automatically to an Hamiltonian restricted to a subspace and thus enables the construction of effective Hamiltonians. In the present paper, in order to explain the efficiency of Seke's method in particular applications, its algebraic operator structure is analyzed and a new successive approximation technique for the calculation of eigenstates and eigenvalues of an arbitrary quantum-mechanical system is developed. Unlike most perturbative techniques, in the present case each order of the approximation determines its own effective (approximating) Hamiltonian ensuring self-consistency and formal exactness of all results in the corresponding approximation order.
We derive a new expansion of the Heisenberg equation of motion based on the projection operator method proposed by Shibata, Hashitsume and Shingū. In their projection operator method, a certain restriction is imposed on the initial state. As a result, one cannot prepare arbitrary initial states, for example a coherent state, to calculate the time development of quantum systems. In this paper, we generalize the projection operator method by relaxing this restriction. We explain our method in the case of a Hamiltonian both with and without explicit time dependence. Furthermore, we apply it to an exactly solvable model called the damped harmonic oscillator model and confirm the validity of our method.
Using the projection operator method, we obtain approximate time-local and time-nonlocal master equations for the reduced statistical operator of a multilevel quantum system with a finite number N of quantum eigenstates coupled simultaneously to arbitrary classical fields and a dissipative environment. We show that the structure of the obtained equations is significantly simplified if the free Hamiltonian dynamics of the multilevel system under the action of external fields and also its Markovian and non-Markovian evolutions due to coupling to the environment are described via the representation of the multilevel system in terms of the SU(N) algebra, which allows realizing effective numerical algorithms for solving the obtained equations when studying real problems in various fields of theoretical and applied physics.
Nonlinear quantum optical systems are of paramount relevance for modern quantum technologies, as well as for the study of dissipative phase transitions. Their nonlinear nature makes their theoretical study very challenging and hence they have always served as great motivation to develop new techniques for the analysis of open quantum systems. We apply the recently developed self-consistent projection operator theory to the degenerate optical parametric oscillator to exemplify its general applicability to quantum optical systems. We show that this theory provides an efficient method to calculate the full quantum state of each mode with a high degree of accuracy, even at the critical point. It is equally successful in describing both the stationary limit and the dynamics, including regions of the parameter space where the numerical integration of the full problem is significantly less efficient. We further develop a Gaussian approach consistent with our theory, which yields sensibly better results than the previous Gaussian methods developed for this system, most notably standard linearization techniques.
The construction of operator functionals of the full Hamiltonian of a quantum-mechanical system which act as projection operators for a specified exact eigenstate is studied. In general these operator functionals are not equivalent to the formal operator |ψ〉〈ψ|, where |ψ〉 is the state of interest, and are not always idempotent. A formal method by which standard perturbation theoretic expansions for the exact ground-state vector and ground-state energy of the system can be recovered by expansion of a certain operator functional of this type is presented. Related work is briefly discussed.
The problem of construction of projection operators on eigensubspaces of symmetry operators is considered. This problem arises in many approximate methods for solving time-independent and time-dependent quantum problems, and its solution ensures proper physical symmetries in development of approximate methods. The projector form is sought as a function of symmetry operators and their eigenvalues characterizing the eigensubspace of interest. This form is obtained in two steps: (1) identification of algebraic structures within a set of symmetry operators (e.g., groups and Lie algebras) and (2) construction of the projection operators for individual symmetry operators. The first step is crucial for efficient projection operator construction because it allows for use of information on irreducible representations of the present algebraic structure. The discussed approaches have potential to stimulate further developments of variational approaches for electronic structure of strongly correlated systems and in quantum computing.