In molecular dynamics applications there is a growing interest in mixed quantum-classical models. The quantum-classical Liouville equation (QCL) describes most atoms of the molecular system under consideration by means of a classical phase space density but an important, small portion of the system by means of quantum mechanics. The QCL is derived from the full quantum dynamical (QD) description by applying the Wigner transform to the \\classical part" of the system only. We discuss the conditions under which the QCL model approximates the full QD evolution of the system. First, analysis of the asymptotic properties of the Wigner transform shows that solving the QCL yields a rst order approximation of full quantum dynamics. Second, we discuss the adiabatic limit of the QCL. This discussion shows that the QCL solutions may be interpretated as classical phase space densities, at least near the adiabatic limit. Third, it is demonstrated that the QCL yields good approximations of non-adia...In molecular dynamics applications there is a growing interest in mixed quantum-classical models. The quantum-classical Liouville equation (QCL) describes most atoms of the molecular system under consideration by means of a classical phase space density but an important, small portion of the system by means of quantum mechanics. The QCL is derived from the full quantum dynamical (QD) description by applying the Wigner transform to the \\classical part" of the system only. We discuss the conditions under which the QCL model approximates the full QD evolution of the system. First, analysis of the asymptotic properties of the Wigner transform shows that solving the QCL yields a rst order approximation of full quantum dynamics. Second, we discuss the adiabatic limit of the QCL. This discussion shows that the QCL solutions may be interpretated as classical phase space densities, at least near the adiabatic limit. Third, it is demonstrated that the QCL yields good approximations of non-adia...

Apr 12, 19992 citations

Quantum Liouville Equation

Slobodan Prvanović

International Journal of Theoretical PhysicsApr 11, 20122 citations

Classical stochastic approach to quantum mechanics and quantum thermodynamics

Mario J. de Olliveira

We derive the equations of quantum mechanics and quantum thermodynamics from the assumption that a quantum system can be described by an underlying classical system of particles. Each component $ϕ_j$ of the wave vector is understood as a stochastic complex variable whose real and imaginary parts are proportional to the coordinate and momentum associated to a degree of freedom of the underlying classical system. From the classical stochastic equations of motion, we derive a general equation for the covariance matrix of the wave vector which turns out to be of the Lindblad type. When the noise changes only the phase of $ϕ_j$ , the Schrödinger and the quantum Liouville equation are obtained. The component $ψ_j$ of the wave vector obeying the Schrödinger equation is related to stochastic wave vector by $|ψ_j|^2=\langle|ϕ_j|^2\rangle$ .We derive the equations of quantum mechanics and quantum thermodynamics from the assumption that a quantum system can be described by an underlying classical system of particles. Each component $ϕ_j$ of the wave vector is understood as a stochastic complex variable whose real and imaginary parts are proportional to the coordinate and momentum associated to a degree of freedom of the underlying classical system. From the classical stochastic equations of motion, we derive a general equation for the covariance matrix of the wave vector which turns out to be of the Lindblad type. When the noise changes only the phase of $ϕ_j$, the Schrödinger and the quantum Liouville equation are obtained. The component $ψ_j$ of the wave vector obeying the Schrödinger equation is related to stochastic wave vector by $|ψ_j|^2=\langle|ϕ_j|^2\rangle$.

arXiv:2309.01851Sep 4, 20230 citations

Classical stochastic approach to quantum mechanics and quantum thermodynamics

Mário J. de Oliveira

Abstract We derive the equations of quantum mechanics and quantum thermodynamics from the assumption that a quantum system can be described by an underlying classical system of particles. Each component φ j of the wave vector is understood as a stochastic complex variable whose real and imaginary parts are proportional to the coordinate and momentum associated with a degree of freedom of the underlying classical system. From the classical stochastic equations of motion, we derive a general equation for the covariance matrix of the wave vector, which turns out to be of the Lindblad type. When the noise changes only the phase of φ j , the Schrödinger and the quantum Liouville equations are obtained. The component ψ j of the wave vector obeying the Schrödinger equation is related to the stochastic wave vector by | ψ j | 2 = ⟨ | ϕ j | 2 ⟩ .Abstract We derive the equations of quantum mechanics and quantum thermodynamics from the assumption that a quantum system can be described by an underlying classical system of particles. Each component φ j of the wave vector is understood as a stochastic complex variable whose real and imaginary parts are proportional to the coordinate and momentum associated with a degree of freedom of the underlying classical system. From the classical stochastic equations of motion, we derive a general equation for the covariance matrix of the wave vector, which turns out to be of the Lindblad type. When the noise changes only the phase of φ j , the Schrödinger and the quantum Liouville equations are obtained. The component ψ j of the wave vector obeying the Schrödinger equation is related to the stochastic wave vector by | ψ j | 2 = ⟨ | ϕ j | 2 ⟩ .

Journal of Statistical Mechanics Theory and ExperimentMar 22, 20243 citations

We derive the equations of quantum mechanics and quantum thermodynamics from the assumption that a quantum system can be described by an underlying classical system of particles. Each component

ϕ_j

of the wave vector is understood as a stochastic complex variable whose real and imaginary parts are proportional to the coordinate and momentum associated to a degree of freedom of the underlying classical system. From the classical stochastic equations of motion, we derive a general equation for the covariance matrix of the wave vector which turns out to be of the Lindblad type. When the noise changes only the phase of

ϕ_j

, the Schrödinger and the quantum Liouville equation are obtained. The component

ψ_j

of the wave vector obeying the Schrödinger equation is related to stochastic wave vector by

|ψ_j|^2=\langle|ϕ_j|^2\rangle