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

The squeezed state approach of the semiclassical limit of the time-dependent Schrödinger equation

Monique Combescure

Published Nov 1, 1992• 31 citations
Publisher

Abstract

It is by now well recognized how the quantum time evolution for time-dependent Hamiltonians that are quadratic both in position and momentum variables generates ‘‘squeezing.’’ On the other hand, ‘‘squeezed states’’ and their ‘‘high-n generalizations’’ are known to provide a simple classical mechanical description of general quantum time evolution. In this paper, these ideas are combined to show the relevance of ‘‘generalized squeezed states’’ in the semiclassical limit of the evolution of time-dependent wave packets for general time-dependent Hamiltonians, including the multidimensional case.

DOI
10.1063/1.529884

Related papers

Squeezed states in the semiclassical limit

Pavel S. Alekseev, Fedor V. Moroseev

A symplectically covariant quantum-mechanical uncertainty relation more accurate than previously known ones is derived for multidimensional systems. It is shown that the quantum-mechanical description of a linear Hamiltonian system in terms of squeezed states is completely equivalent to its description in terms of a phase-space distribution function. A new approach to the semiclassical limit is proposed, based on the use of squeezed states. By analyzing explicit formulas for squeezed states, a semiclassical asymptotic form of the solution to the Cauchy problem for a multidimensional Schrödinger equation is found in the limit of ħ → 0. The behavior of the semiclassical solution in the neighborhood of a caustic is analyzed in the one-dimensional case, and the phase shift across the caustic is determined. General properties and examples of squeezed states are discussed that point to the fundamental importance of squeezed states for developing a nonrelativistic quantum-mechanical description of a system of charged particles in an electromagnetic field in the dipole approximation.

Journal of Experimental and Theoretical Physics•Apr 1, 2009•3 citations

Time-Dependent Variational Approach in Terms of Squeezed Coherent States

Y. Tsue, Yoshikazu Fujiwara

A general framework for a time-dependent variational approach in terms of squeezed coherent states is given with the aid of canonicity conditions developed in the time-dependent Hartree-Fock theory. By using a formula given by Balian and Brezin for canonical decompositions of generalized Bogoliubov transformations, we show that this approach is entirely equivalent to that given by Jackiw and Kerman in terms of the most general Gaussian functions, which is originally motivated for the variational definition of the effective action in the quantum field theory. The resultant equations of motion have structure of classical Hamiltonian systems consisting of displacement degree of freedom and squeezing degree of freedom in the configuration space. The initial conditions for solving these equations, particularly for the squeezing degree of freedom, are discussed in connection with the minimum-uncertainty relations of the squeezed coherent state. For a simple one-dimensional quantum-mechanical Hamiltonian with potentials of quadratic terms or less, the present formulation gives an exact result for the time-evolution of the Gaussian wave packet.

Progress of Theoretical Physics•Aug 1, 1991•36 citations

Time-evolution of squeezed coherent states of a generalized quantum parametric oscillator

Şirin A. Büyükaşık, Zehra Çayiç

Time evolution of squeezed coherent states for a quantum parametric oscillator with the most general self-adjoint quadratic Hamiltonian is found explicitly. For this, we use the unitary displacement and squeeze operators in coordinate representation and the evolution operator obtained by the Wei-Norman Lie algebraic approach. Then, we analyze squeezing properties of the wave packets according to the complex parameter of the squeeze operator and the time-variable parameters of the Hamiltonian. As an application, we construct all exactly solvable generalized quantum oscillator models classically corresponding to a driven simple harmonic oscillator. For each model, defined according to the frequency modification in position space, we describe explicitly the squeezing and displacement properties of the wave packets. This allows us to see the exact influence of all parameters and make a basic comparison between the different models.

Journal of Mathematical Physics•Jun 1, 2019•11 citations

The Time-Dependent Schrödinger Equation

Reinhold A. Bertlmann, Nicolai Friis

Abstract The second chapter introduces the notion of wave functions and their equation of motion proposed by Erwin Schrödinger. The interpretation and normalization of the wave function are discussed, leading to the continuity equation. The chapter also introduces basic mathematical operations that can be carried out for wave functions and operators, describing e.g., how to calculate scalar products and commutators, as well as expectation values and variances of observables. With these tools at hand, Heisenberg’s uncertainty principle is re-examined, focusing on the example of a Gaussian wave packet, which is studied both in position space and momentum space. We also discuss the time evolution of general states and of the exemplary wave packet. Finally, we will see how classical physics is recovered as a limiting case of the quantum-mechanical description

•Sep 7, 2023•0 citations

Quadratic Quantum Hamiltonians revisited

Monique Combescure, Robert, Didier

Time dependent quadratic Hamiltonians are well known as well in classical mechanics and in quantum mechanics. In particular for them the correspondance between classical and quantum mechanics is exact. But explicit formulas are non trivial (like the Mehler formula). Moreover, a good knowlege of quadratic Hamiltonians is very useful in the study of more general quantum Hamiltonians and associated Schrödinger equations in the semiclassical regime. Our goal here is to give our own presentation of this important subject. We put emphasis on computations with Gaussian coherent states. Our main motivation to do that is application concerning revivals and Loschmidt echo.

arXiv:math-ph/0509027•Cubo•Sep 13, 2005•17 citations

Classical equations for quantum squeezing and coherent pumping by the time-dependent quadratic Hamiltonian

A. F. R. de Toledo Piza

The time evolution, under the general time-dependent quadratic quantum Hamiltonian, of mean values of a class of observables that includes the field quadratures and their dispersions, is obtained exactly in classical Hamiltonian form. Quantum states are described in terms of density operators, so that arbitrary initial states (both pure states and statistical mixtures) are allowed.

Physical Review A•Feb 1, 1995•11 citations
It is by now well recognized how the quantum time evolution for time-dependent Hamiltonians that are quadratic both in position and momentum variables generates ‘‘squeezing.’’ On the other hand, ‘‘squeezed states’’ and their ‘‘high-n generalizations’’ are known to provide a simple classical mechanical description of general quantum time evolution. In this paper, these ideas are combined to show the relevance of ‘‘generalized squeezed states’’ in the semiclassical limit of the evolution of time-dependent wave packets for general time-dependent Hamiltonians, including the multidimensional case.
A symplectically covariant quantum-mechanical uncertainty relation more accurate than previously known ones is derived for multidimensional systems. It is shown that the quantum-mechanical description of a linear Hamiltonian system in terms of squeezed states is completely equivalent to its description in terms of a phase-space distribution function. A new approach to the semiclassical limit is proposed, based on the use of squeezed states. By analyzing explicit formulas for squeezed states, a semiclassical asymptotic form of the solution to the Cauchy problem for a multidimensional Schrödinger equation is found in the limit of ħ → 0. The behavior of the semiclassical solution in the neighborhood of a caustic is analyzed in the one-dimensional case, and the phase shift across the caustic is determined. General properties and examples of squeezed states are discussed that point to the fundamental importance of squeezed states for developing a nonrelativistic quantum-mechanical description of a system of charged particles in an electromagnetic field in the dipole approximation.
A general framework for a time-dependent variational approach in terms of squeezed coherent states is given with the aid of canonicity conditions developed in the time-dependent Hartree-Fock theory. By using a formula given by Balian and Brezin for canonical decompositions of generalized Bogoliubov transformations, we show that this approach is entirely equivalent to that given by Jackiw and Kerman in terms of the most general Gaussian functions, which is originally motivated for the variational definition of the effective action in the quantum field theory. The resultant equations of motion have structure of classical Hamiltonian systems consisting of displacement degree of freedom and squeezing degree of freedom in the configuration space. The initial conditions for solving these equations, particularly for the squeezing degree of freedom, are discussed in connection with the minimum-uncertainty relations of the squeezed coherent state. For a simple one-dimensional quantum-mechanical Hamiltonian with potentials of quadratic terms or less, the present formulation gives an exact result for the time-evolution of the Gaussian wave packet.
Time evolution of squeezed coherent states for a quantum parametric oscillator with the most general self-adjoint quadratic Hamiltonian is found explicitly. For this, we use the unitary displacement and squeeze operators in coordinate representation and the evolution operator obtained by the Wei-Norman Lie algebraic approach. Then, we analyze squeezing properties of the wave packets according to the complex parameter of the squeeze operator and the time-variable parameters of the Hamiltonian. As an application, we construct all exactly solvable generalized quantum oscillator models classically corresponding to a driven simple harmonic oscillator. For each model, defined according to the frequency modification in position space, we describe explicitly the squeezing and displacement properties of the wave packets. This allows us to see the exact influence of all parameters and make a basic comparison between the different models.
Abstract The second chapter introduces the notion of wave functions and their equation of motion proposed by Erwin Schrödinger. The interpretation and normalization of the wave function are discussed, leading to the continuity equation. The chapter also introduces basic mathematical operations that can be carried out for wave functions and operators, describing e.g., how to calculate scalar products and commutators, as well as expectation values and variances of observables. With these tools at hand, Heisenberg’s uncertainty principle is re-examined, focusing on the example of a Gaussian wave packet, which is studied both in position space and momentum space. We also discuss the time evolution of general states and of the exemplary wave packet. Finally, we will see how classical physics is recovered as a limiting case of the quantum-mechanical description
Time dependent quadratic Hamiltonians are well known as well in classical mechanics and in quantum mechanics. In particular for them the correspondance between classical and quantum mechanics is exact. But explicit formulas are non trivial (like the Mehler formula). Moreover, a good knowlege of quadratic Hamiltonians is very useful in the study of more general quantum Hamiltonians and associated Schrödinger equations in the semiclassical regime. Our goal here is to give our own presentation of this important subject. We put emphasis on computations with Gaussian coherent states. Our main motivation to do that is application concerning revivals and Loschmidt echo.