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

Physical Review A

Equation of Motion for the P Representation

B. Crosignani, U. Ganiel, S. Solimeno, Paolo Di Porto

Published Oct 1, 1971• 4 citations
Publisher

Abstract

A new equation of motion, which describes the time evolution of the $P$ representation in the Schr\"odinger picture, is derived. The equation applies to a very general class of Hamiltonians, and thus should be useful in many problems. As a demonstration of the usefulness of this equation, some known results are recovered by its direct application.

DOI
10.1103/physreva.4.1570

Related papers

Master Equation for the P Representation

B. Crosignani, S. Solimeno, Paolo Di Porto

A master equation for the P representation in the Schrödinger picture is derived for a general class of Hamiltonians.Received 15 March 1971DOI:https://doi.org/10.1103/PhysRevLett.26.1130©1971 American Physical Society

Physical Review Letters•May 3, 1971•4 citations

Equation of Motion for the Quantum Characteristic Functions

Itay Hen, Amir Kalev

In this paper, we derive equations of motion for the normal-order, the symmetric-order and the antinormal-order quantum characteristic functions, applicable for general Hamiltonian systems. We do this by utilizing the `characteristic form' of both quantum states and Hamiltonians. The equations of motion we derive here are rather simple in form and in essence, and as such have a number of attractive features. As we shall see, our approach enables the descriptions of quantum and classical time evolutions in one unified language. It allows for a direct comparison between quantum and classical dynamics, providing insight into the relations between quantum and classical behavior, while also revealing a smooth transition between quantum and classical time evolutions. In particular, the $\hbar\to 0$ limit of the quantum equations of motion instantly recovers their classical counterpart. We also argue that the derived equations may prove to be very useful in numerical simulations.

arXiv:0803.0108•Mar 2, 2008•0 citations

Brief Note on Derivation of Equation of Motion in Relativistic Quantum Mechanics

shoichiro otsuka

In this paper, I show the solution I gave in the previous paper is compatible with ordinary relativistic and non-relativistic Quantum Mechanics.

•Sep 6, 2026•0 citations

Bilinear oscillators

Francisco Marcelo Fernández, Eduardo A. Castro

The matrix U' contains the full dynamics of a bilinear oscillator, because any observable of the system is an analytical function of the operators that determine the matrix representation. One cannot solve the time evolution equation in the general case, but there are many exactly solvable problems that simulate and enable one to understand the dynamics of more elaborate and realistic quantum-mechanical models. In the case of the solvable model just discussed it is easy to derive analytical expressions for operators in the Heisenberg picture as well as for invariants or constants of the motion. The mathematical treatment of the driven oscillator just discussed is trivial because the Hamiltonian operator belongs to a solvable Lie algebra. This model is a convenient starting point to illustrate the use of the Lie algebraic method, but it is also an oversimplification of most oscillatory phenomena. Nonlinear Hamiltonians are useful in many physical applications such as the simulation of frictional forces and dissipation effects.

CRC Press eBooks•Jan 16, 2020•0 citations

Formulation of Schrödinger-Like Relativistic Wave Equation of Motion

Young‐Sea Huang

Foundations of Physics•Oct 1, 1998•2 citations

Invariants and Lie algebraic solutions of Schrödinger equation

Franco Salmistraro, Riccardo Rosso

A representation of invariants in quantum mechanics for a general time-dependent Hamiltonian is provided. The main properties of these operators and their relevance for solving the Schrödinger equation are discussed in light of this representation. In particular, the well-known Lewis and Riesenfeld’s method is shown to be recovered quite directly by the present approach. A detailed analysis is concerned with Hamiltonians which at each instant are elements of a finite dimensional Lie algebra. Exploiting the Wei and Norman’s theory for such Hamiltonians, it is found that the evolution of a quantum system is always determined, in part or fully, by an invariant which belongs to the same algebra.

Journal of Mathematical Physics•Sep 1, 1993•8 citations
A new equation of motion, which describes the time evolution of the PPP representation in the Schr"odinger picture, is derived. The equation applies to a very general class of Hamiltonians, and thus should be useful in many problems. As a demonstration of the usefulness of this equation, some known results are recovered by its direct application.
In this paper, we derive equations of motion for the normal-order, the symmetric-order and the antinormal-order quantum characteristic functions, applicable for general Hamiltonian systems. We do this by utilizing the `characteristic form' of both quantum states and Hamiltonians. The equations of motion we derive here are rather simple in form and in essence, and as such have a number of attractive features. As we shall see, our approach enables the descriptions of quantum and classical time evolutions in one unified language. It allows for a direct comparison between quantum and classical dynamics, providing insight into the relations between quantum and classical behavior, while also revealing a smooth transition between quantum and classical time evolutions. In particular, the ℏ→0\hbar\to 0ℏ→0 limit of the quantum equations of motion instantly recovers their classical counterpart. We also argue that the derived equations may prove to be very useful in numerical simulations.
The matrix U' contains the full dynamics of a bilinear oscillator, because any observable of the system is an analytical function of the operators that determine the matrix representation. One cannot solve the time evolution equation in the general case, but there are many exactly solvable problems that simulate and enable one to understand the dynamics of more elaborate and realistic quantum-mechanical models. In the case of the solvable model just discussed it is easy to derive analytical expressions for operators in the Heisenberg picture as well as for invariants or constants of the motion. The mathematical treatment of the driven oscillator just discussed is trivial because the Hamiltonian operator belongs to a solvable Lie algebra. This model is a convenient starting point to illustrate the use of the Lie algebraic method, but it is also an oversimplification of most oscillatory phenomena. Nonlinear Hamiltonians are useful in many physical applications such as the simulation of frictional forces and dissipation effects.
A representation of invariants in quantum mechanics for a general time-dependent Hamiltonian is provided. The main properties of these operators and their relevance for solving the Schrödinger equation are discussed in light of this representation. In particular, the well-known Lewis and Riesenfeld’s method is shown to be recovered quite directly by the present approach. A detailed analysis is concerned with Hamiltonians which at each instant are elements of a finite dimensional Lie algebra. Exploiting the Wei and Norman’s theory for such Hamiltonians, it is found that the evolution of a quantum system is always determined, in part or fully, by an invariant which belongs to the same algebra.