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:1402.1604

Balance equations-based properties of the Rabi Hamiltonian

ANTONINO MESSINA, Anna Napoli, Margarita A. Man’ko, Vladimir I. Man’ko

Published Feb 7, 2014• 0 citations
arXivHTMLPDF

Abstract

A stationary physical system satisfies peculiar balance conditions involving mean values of appropriate observables. In this paper we show how to deduce such quantitative links, named balance equations, demonstrating as well their usefulness in bringing to light physical properties of the system without solving the Schrodinger equation. The knowledge of such properties in the case of Rabi Hamiltonian is exploit to provide arguments to make easier the variational engineering of the ground state of this model.

DOI
10.48550/arxiv.1402.1604

Related papers

Eigenvalues and Eigenstates of Quantum Rabi Model

Andrei V. Ivanov

The scientific interest in the analytical solution of the quantum Rabi model is due to the widespread use of this simple model in quantum optics, quantum computing, cavity QED, and nanoelectromechanical systems. This interest is related to the need for the theoretical description of the interaction of a two-level system with a quantum oscillator in the case when the rotating wave approximation fails. In this Letter, we present an approach to the exact diagonalization of the quantum Rabi Hamiltonian. This approach is based on the properties of the Pauli operators and allows us to readily solve the stationary Schrodinger equation for a two-level system. First, we demonstrate the applicability of the approach to the Jaynes-Cummings Hamiltonian to get the well-known solution. Then, we obtain the eigenvalues and eigenstates for the quantum Rabi Hamiltonian using the proposed approach. It is shown that the obtained eigenstates can be represented in the basis of the eigenstates of the Jaynes-Cummings Hamiltonian.

arXiv:2104.12771•Apr 26, 2021•0 citations

Risultati rigorosi riguardanti lo stato fondamentale dell'interazione spin-fonone

Elisabetta Paladino, ANTONINO MESSINA

We present some characteristic properties of the ground state of a two-level unit linearly coupled to a bosonic field, exactly deduced without previously solving the eigenvalue problem for the physical system. Such properties are valid in the whole range of the characteristic parameters appearing in the Hamiltonian modelling this system. Our treatment is based on operatorial methods which combine symmetry considerations with some general peculiarities of the lowest energy state. In particular we present rigorous conclusions concerning the expectation value of the two-level unit energy and the mean value of the interaction energy for arbitrary strenght of the coupling constant. The relevance of these rigorous results in connection with an approximate variational determination of the ground state of the system is evidenced.

•Jan 1, 1994•0 citations

Simple applications

Peter Holland

Stationary states General properties We shall consider first some general characteristics of the ‘stationary states’. These are eigenfunctions of the Hamiltonian operator: The requirement that ψ satisfies the Schrödinger equation fixes its time dependence: Notice that (4.1.2) can only be a solution if the external potential V is independent of the time: V = V ( x ). The initial function ψ 0 is not arbitrary but must satisfy the time-independent Schrödinger equation (4.1.1). Writing ψ = R e iS /ħ one thus has The following deductions may be made from (4.1.3): ( a ) The probability density is independent of the time: |ψ| 2 = R 2 0 ( X ). ( b ) The quantum potential Q = (- ħ2/2 mR 0 )∇ 2 R 0 is time-independent. Therefore so is the total effective potential: ( c ) The velocity field v = (l/ m )∇ S 0 is independent of time and hence the trajectories, found by solving m = v ( x )| x = x ( t ) , are solutions of Moreover, ψ 0 ( x ) is often a real function for many bound state problems of interest (although not for scattering states) which implies that the velocity is zero. The particle is at rest where one would classically expect it to move since the quantum force (- ∇ Q ) cancels the classical force (- ∇ K ). ( d ) The energy of all particles in the ensemble, -∂ S /∂ t , is a constant of the motion and equal to the energy eigenvalue: with (4.1.4), we have an example of the conservation of energy (cf. §3.9.3). ( e ) The total field energy is conserved: 〈Ĥ〉 = E . […]

Cambridge University Press eBooks•May 13, 1993•0 citations

Quantization and molecular energy levels

John M. Brown

This chapter discusses the physical properties of atoms and molecules, which are quantized. This means that they can only take certain discrete values rather than a full range of continuously varying values. The chapter examines the quantized energy levels of a molecule. These are called stationary states, which describe an energy that is independent of time. The chapter also outlines the procedure for constructing a quantum mechanical Hamiltonian, which includes writing down the complete and classical expression for the energy of the system. The chapter reviews the general approach to the determination of energy levels of a physical system. This involves two stages: First, the appropriate Hamiltonian is defined and, in a second step, the eigenfunctions and eigenvalues of the Hamiltonian are determined. The chapter also discusses the Schrödinger equation, which can only be solved for one-body systems.

Oxford University Press eBooks•Jul 23, 1998•0 citations

A Physical Axiomatic Approach to Schrodinger's Equation

Rajesh R. Parwani

The Schrodinger equation for non-relativistic quantum systems is derived from some classical physics axioms within an ensemble hamiltonian framework. Such an approach enables one to understand the structure of the equation, in particular its linearity, in intuitive terms. Furthermore it allows for a physically motivated and systematic investigation of potential generalisations which are briefly discussed.

•Aug 17, 2005•8 citations

Approximating Ground State Energies and Wave Functions of Physical Systems with Neural Networks

Lema, Cesar, Anna Choromanska

Quantum theory has been remarkably successful in providing an understanding of physical systems at foundational scales. Solving the Schrödinger equation provides full knowledge of all dynamical quantities of the physical system. However closed form solutions to this equation are only available for a few systems and approximation methods are typically used to find solutions. In this paper we address the problem of solving the time independent Schrödinger equation for the ground state solution of physical systems. We propose using end-to-end deep learning approach in a variational optimization scheme for approximating the ground state energies and wave functions of these systems. A neural network realizes a universal trial wave function and is trained in an unsupervised learning framework by optimizing the expectation value of the Hamiltonian of a physical system. The proposed approach is evaluated on physical systems consisting of a particle in a box with and without a perturbation. We demonstrate that our approach obtains approximations of ground state energies and wave functions that are highly accurate, which makes it a potentially plausible candidate for solving more complex physical systems for which analytical solutions are beyond reach.

arXiv:2011.10694•Nov 21, 2020•1 citation
A stationary physical system satisfies peculiar balance conditions involving mean values of appropriate observables. In this paper we show how to deduce such quantitative links, named balance equations, demonstrating as well their usefulness in bringing to light physical properties of the system without solving the Schrodinger equation. The knowledge of such properties in the case of Rabi Hamiltonian is exploit to provide arguments to make easier the variational engineering of the ground state of this model.
The scientific interest in the analytical solution of the quantum Rabi model is due to the widespread use of this simple model in quantum optics, quantum computing, cavity QED, and nanoelectromechanical systems. This interest is related to the need for the theoretical description of the interaction of a two-level system with a quantum oscillator in the case when the rotating wave approximation fails. In this Letter, we present an approach to the exact diagonalization of the quantum Rabi Hamiltonian. This approach is based on the properties of the Pauli operators and allows us to readily solve the stationary Schrodinger equation for a two-level system. First, we demonstrate the applicability of the approach to the Jaynes-Cummings Hamiltonian to get the well-known solution. Then, we obtain the eigenvalues and eigenstates for the quantum Rabi Hamiltonian using the proposed approach. It is shown that the obtained eigenstates can be represented in the basis of the eigenstates of the Jaynes-Cummings Hamiltonian.
We present some characteristic properties of the ground state of a two-level unit linearly coupled to a bosonic field, exactly deduced without previously solving the eigenvalue problem for the physical system. Such properties are valid in the whole range of the characteristic parameters appearing in the Hamiltonian modelling this system. Our treatment is based on operatorial methods which combine symmetry considerations with some general peculiarities of the lowest energy state. In particular we present rigorous conclusions concerning the expectation value of the two-level unit energy and the mean value of the interaction energy for arbitrary strenght of the coupling constant. The relevance of these rigorous results in connection with an approximate variational determination of the ground state of the system is evidenced.
Stationary states General properties We shall consider first some general characteristics of the ‘stationary states’. These are eigenfunctions of the Hamiltonian operator: The requirement that ψ satisfies the Schrödinger equation fixes its time dependence: Notice that (4.1.2) can only be a solution if the external potential V is independent of the time: V = V ( x ). The initial function ψ 0 is not arbitrary but must satisfy the time-independent Schrödinger equation (4.1.1). Writing ψ = R e iS /ħ one thus has The following deductions may be made from (4.1.3): ( a ) The probability density is independent of the time: |ψ| 2 = R 2 0 ( X ). ( b ) The quantum potential Q = (- ħ2/2 mR 0 )∇ 2 R 0 is time-independent. Therefore so is the total effective potential: ( c ) The velocity field v = (l/ m )∇ S 0 is independent of time and hence the trajectories, found by solving m = v ( x )| x = x ( t ) , are solutions of Moreover, ψ 0 ( x ) is often a real function for many bound state problems of interest (although not for scattering states) which implies that the velocity is zero. The particle is at rest where one would classically expect it to move since the quantum force (- ∇ Q ) cancels the classical force (- ∇ K ). ( d ) The energy of all particles in the ensemble, -∂ S /∂ t , is a constant of the motion and equal to the energy eigenvalue: with (4.1.4), we have an example of the conservation of energy (cf. §3.9.3). ( e ) The total field energy is conserved: 〈Ĥ〉 = E . […]
This chapter discusses the physical properties of atoms and molecules, which are quantized. This means that they can only take certain discrete values rather than a full range of continuously varying values. The chapter examines the quantized energy levels of a molecule. These are called stationary states, which describe an energy that is independent of time. The chapter also outlines the procedure for constructing a quantum mechanical Hamiltonian, which includes writing down the complete and classical expression for the energy of the system. The chapter reviews the general approach to the determination of energy levels of a physical system. This involves two stages: First, the appropriate Hamiltonian is defined and, in a second step, the eigenfunctions and eigenvalues of the Hamiltonian are determined. The chapter also discusses the Schrödinger equation, which can only be solved for one-body systems.
Quantum theory has been remarkably successful in providing an understanding of physical systems at foundational scales. Solving the Schrödinger equation provides full knowledge of all dynamical quantities of the physical system. However closed form solutions to this equation are only available for a few systems and approximation methods are typically used to find solutions. In this paper we address the problem of solving the time independent Schrödinger equation for the ground state solution of physical systems. We propose using end-to-end deep learning approach in a variational optimization scheme for approximating the ground state energies and wave functions of these systems. A neural network realizes a universal trial wave function and is trained in an unsupervised learning framework by optimizing the expectation value of the Hamiltonian of a physical system. The proposed approach is evaluated on physical systems consisting of a particle in a box with and without a perturbation. We demonstrate that our approach obtains approximations of ground state energies and wave functions that are highly accurate, which makes it a potentially plausible candidate for solving more complex physical systems for which analytical solutions are beyond reach.