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

Journal of Mathematical Physics

On the time evolution operator for time-dependent quadratic Hamiltonians

Francisco Marcelo Fernández

Published Jul 1, 1989• 7 citations
Publisher

Abstract

The Schrödinger equation with a time-dependent quadratic Hamiltonian is investigated. The time-evolution operator is written as a product of exponential operators determined by the Heisenberg equations of motion. This product operator is shown to be global in the occupation number representation when the Hamiltonian is Hermitian. The success of some physical applications of the product-form representation is explained.

DOI
10.1063/1.528284

Related papers

Time-evolution operator and Lie algebras

Francisco Marcelo Fernández

We discuss the Schr\"odinger equation with a time-dependent Hamiltonian that can be written as a linear combination of operators which span a finite-dimensional Lie algebra. The equations of motion for the operators in the Heisenberg representation are shown to be useful in calculating matrix elements and transition probabilities, as well as in obtaining the time-evolution operator. A general time-dependent one-dimensional bilinear Hamiltonian is considered as an illustrative example, and the product form for the time-evolution operator is shown to be global.

Physical Review A•Jul 1, 1989•38 citations

Time-evolution operator and propagator for quadratic Hamiltonians

Francisco Marcelo Fernández

The time-evolution operator and the propagator for a general one-dimensional quadratic Hamiltonian are obtained. The method is based on the equations of motion for the coordinate and momentum in the Heisenberg representation and the problem is reduced to solving the classical equations of motion.

Journal of Physics A Mathematical and General•Mar 21, 1988•3 citations

Exponential time-evolution operator for the time-dependent harmonic oscillator

Francisco Marcelo Fernández

The time-evolution operator for the time-dependent harmonic oscillator H= (1)/(2) {α(t)p2 +β(t)q2} is exactly obtained as the exponential of an anti-Hermitian operator. The method is based on the equations of motion for the coordinate and momentum operators in the Heisenberg representation. The problem is reduced to solving the classical equations of motion.

Journal of Mathematical Physics•Dec 1, 1987•16 citations

Generalized adiabatic product expansion: A nonperturbative method of solving the time-dependent Schrödinger equation

Ali Mostafazadeh

We outline a method based on successive canonical transformations which yields a product expansion for the evolution operator of a general (possibly non-Hermitian) Hamiltonian. For a class of such Hamiltonians this expansion involves a finite number of terms, and our method gives the exact solution of the corresponding time-dependent Schrödinger equation. We apply this method to study the dynamics of a general nondegenerate two-level quantum system, a time-dependent classical harmonic oscillator, and a degenerate system consisting of a spin 1 particle interacting with a time-dependent electric field E⃗(t) through the Stark Hamiltonian H=λ(J⃗⋅E⃗)2.

arXiv:quant-ph/9904026•Journal of Mathematical Physics•Jul 1, 1999•12 citations

Quantum Dynamics

Uri Peskin

Explicitly time-dependent Hamiltonians are ubiquitous in applications of quantum theory. It is therefore necessary to solve the time-dependent Schrödinger equation directly. The system’s dynamics is associated with a unitary time-evolution operator (a propagator), formally given as an infinite Dyson series. Time-dependent observables are invariant under unitary time-dependent transformations, where it is sometimes useful to transform the time-evolution from the states into the corresponding operators. This is carried out in part (in full) by transforming to the interaction (Heisenberg) picture. The corresponding equations of motion for the time-dependent operators are introduced. For quadratic potential energy functions, the time evolution of quantum expectation values coincides with the corresponding classical dynamics. This is demonstrated and analyzed in detail for Gaussian wave packets and a coherent state. Finally, we derive exact and approximate expressions for time-dependent transition probabilities and transition rates between quantum states. The validity of time-dependent perturbation theory is analyzed by comparison to exact dynamics.

Cambridge University Press eBooks•May 11, 2023•0 citations

The Schrödinger and Heisenberg Representations: Time-Evolution Operator

Giancarlo Calvanese Strinati

This chapter introduces the Hamiltonian operator in the language of second quantization, which is associated with the many-particle system to be considered throughout, including its time-dependent part. An expression is derived for the corresponding time-evolution operator, which depends only on the Hamiltonian and not on the initial preparation of the system before the time-dependent part begins to act. The connection between the Schrödinger and Heisenberg representations is discussed.

Cambridge University Press eBooks•Feb 6, 2026•0 citations
The Schrödinger equation with a time-dependent quadratic Hamiltonian is investigated. The time-evolution operator is written as a product of exponential operators determined by the Heisenberg equations of motion. This product operator is shown to be global in the occupation number representation when the Hamiltonian is Hermitian. The success of some physical applications of the product-form representation is explained.
We discuss the Schr\"odinger equation with a time-dependent Hamiltonian that can be written as a linear combination of operators which span a finite-dimensional Lie algebra. The equations of motion for the operators in the Heisenberg representation are shown to be useful in calculating matrix elements and transition probabilities, as well as in obtaining the time-evolution operator. A general time-dependent one-dimensional bilinear Hamiltonian is considered as an illustrative example, and the product form for the time-evolution operator is shown to be global.
We outline a method based on successive canonical transformations which yields a product expansion for the evolution operator of a general (possibly non-Hermitian) Hamiltonian. For a class of such Hamiltonians this expansion involves a finite number of terms, and our method gives the exact solution of the corresponding time-dependent Schrödinger equation. We apply this method to study the dynamics of a general nondegenerate two-level quantum system, a time-dependent classical harmonic oscillator, and a degenerate system consisting of a spin 1 particle interacting with a time-dependent electric field E⃗(t) through the Stark Hamiltonian H=λ(J⃗⋅E⃗)2.
Explicitly time-dependent Hamiltonians are ubiquitous in applications of quantum theory. It is therefore necessary to solve the time-dependent Schrödinger equation directly. The system’s dynamics is associated with a unitary time-evolution operator (a propagator), formally given as an infinite Dyson series. Time-dependent observables are invariant under unitary time-dependent transformations, where it is sometimes useful to transform the time-evolution from the states into the corresponding operators. This is carried out in part (in full) by transforming to the interaction (Heisenberg) picture. The corresponding equations of motion for the time-dependent operators are introduced. For quadratic potential energy functions, the time evolution of quantum expectation values coincides with the corresponding classical dynamics. This is demonstrated and analyzed in detail for Gaussian wave packets and a coherent state. Finally, we derive exact and approximate expressions for time-dependent transition probabilities and transition rates between quantum states. The validity of time-dependent perturbation theory is analyzed by comparison to exact dynamics.