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

Hamilton-Jacobi Theory

Peter Mann

Published Jun 5, 2018• 1 citation
Publisher

Abstract

Abstract This chapter focuses on Liouville’s theorem and classical statistical mechanics, deriving the classical propagator. The terms ‘phase space volume element’ and ‘Liouville operator’ are defined and an n-particle phase space probability density function is constructed to derive the Liouville equation. This is deconstructed into the BBGKY hierarchy, and radial distribution functions are used to develop n-body correlation functions. Koopman–von Neumann theory is investigated as a classical wavefunction approach. The chapter develops an operatorial mechanics based on classical Hilbert space, and discusses the de Broglie–Bohm formulation of quantum mechanics. Partition functions, ensemble averages and the virial theorem of Clausius are defined and Poincaré’s recurrence theorem, the Gibbs H-theorem and the Gibbs paradox are discussed. The chapter also discusses commuting observables, phase–amplitude decoupling, microcanonical ensembles, canonical ensembles, grand canonical ensembles, the Boltzmann factor, Mayer–Montroll cluster expansion and the equipartition theorem and investigates symplectic integrators, focusing on molecular dynamics.

DOI
10.1093/oso/9780198822370.003.0019

Related papers

From classical to quantum statistical mechanics

Giampiero V. M. Esposito, Giuseppe Marmo, George Sudarshan

Classical statistical mechanics tries to derive the macroscopic properties of matter starting from the mechanical laws that rule the behaviour of single particles. To describe equilibrium states, only observables accounting for correlations among states are considered, and only systems consisting of a large number of particles are taken into account. The observables are described by continuous functions on phase space, and the states are represented by linear assignments of a number to each observable. Within the framework of the canonical ensemble, one deals with mechanical systems in thermal equilibrium with a thermal reservoir, and the equilibrium state is reached as a result of the interaction with the external world. The external world may be really external, or equally well, the unrecognized internal degrees of freedom, like in the calculation of viscosity or thermal conductivity. If really only the external world is the cause we may expect some surface dependence while an internal unrecognized degree of freedom would have a volume effect, unless the interactions are long range. The microcanonical ensemble is instead introduced to study isolated mechanical systems, and the equilibrium is viewed as a temporal average, rather than as a limit. Attention is then focused on partition functions, the theorem of equipartition of energy and an elementary theory of specific heats. In the second part, the Planck derivation of the law of black-body radiation is analysed, presenting in chronological order the Kirchhoff theorem, the Stefan law, the Wien displacement law, the Rayleigh–Jeans formula and the Planck hypothesis. Further topics discussed are the Einstein and Debye quantum models for specific heats of solids. These topics prepare the ground for the introduction of quantum statistical mechanics. The third part is, in fact, devoted to the analysis of identical particles in quantum mechanics.

Cambridge University Press eBooks•Mar 11, 2004•0 citations

Hamiltonian theory of wave and particle in quantum mechanics I. Liouville's theorem and the interpretation of the de Broglie-Bohm theory

Peter Holland

This paper and a following one develop a Hamiltonian formulation of the de Broglie-Bohm (deBB) interpretation of quantum mechanics. Drawing upon a unified Hamiltonian treatment of a classical particle and its associated ensemble, it is shown how the interaction of a particle and a spin-0 quantum wave can be consistently expressed in a canonical framework. A formulation is found in which the particle is acted upon by a force determined by the quantum potential but the particle does not react on the Schrodinger field. This requires the introduction of an additional field which the particle does act upon, and allows the deBB theory to be expressed in Hamiltonian terms. An ensemble theory for the wave-particle system based on Liouville's equation is developed, and it is shown that for a pure quantum state the particle obeys its own Liouville equation. The general particle phase space distribution that is compatible with ‖ψ‖ 2 is derived. A special case of this corresponds to the relation p = ∂S/∂q (and thus the guidance law of the deBB theory) but more general distributions, and hence non-deBB motions, are possible. This relation is interpreted as a constraint on the phase space of the composite system. Using the theory of Hamiltonian constraints it is shown that the first-order guidance law and Hamilton's second-order particle equations are dynamically equivalent.

Il Nuovo Cimento B•Sep 1, 2001•12 citations

Hamiltonian theory of wave and particle in quantum mechanics II: Hamilton-Jacobi theory and particle back-reaction

Peter Holland

Pursuing the Hamiltonian formulation of the de Broglie-Bohm (deBB) theory presented in the preceding paper, the Hamilton-Jacobi (HJ) theory of the wave-particle system is developed. It is shown how to derive a HJ equationfor the particle, which enables trajectories to be computed algebraically using Jacobi's method. Using Liouville's equation in the HJ representation we find the restriction on the Jacobi solutions which implies the quantal distribution. This gives a first method for interpreting the deBB theory in HJ terms. A second method proceeds via an explicit solution of the field+particle HJ equation. Both methods imply that the quantum phase may be interpreted as an incomplete integral. Using these results and those of the first paper it is shown how Schrodinger's equation can be represented in Liouvillian terms, and vice versa. The general theory of canonical transformations that represent quantum unitary transformations is given, and it is shown in principle how the trajectory theory may be expressed in other quantum representations. Using the solution found for the total HJ equation, an explicit solution for the additional field containing a term representing the particle back-reaction is found. The conservation of energy and momentum in the model is established, and a weak form of the action-reaction principle is shown to hold. Alternative forms for the Hamiltonian are explored and it is shown that, within this theoretical context, the deBB theory is not unique. The theory potentially provides an alternative way of obtaining the classical limit.

Il Nuovo Cimento B•Oct 1, 2001•41 citations

Phase space formulation of the quantum many-body problem

Paul H. Levine

By means of a quantum mechanical phase space distribution function introduced by von Roos, the Schroedinger equation for a non-relativistic system of N identical particles with scalar interactions is transformed into a quantum mechanical generalization of the Liouville equation, thereby formulating the problem in terms of a generalized density in phase space, a quantity of primary interest in most treatments of the corresponding classical system (or plasma). This transformation permits a parallel development of the theories of classical and quantum plasmas and thus allows the quantum many-body problem to be discussed virtually completely in classical terms. In particular, a kinetic theory of quantum plasmas is obtained by deriving the quantum analogue of the BBGKY hierarchy, and applying thereto approximation techniques similar to those of Rostocker and Rosenbluth, and Bogoliubov. The point of departure from similar previous studies based on the Wigner distribution function is that the proper exchange symmetry can be tractably introduced into the formalism. Attention is first focused on the Hartree and Hartree-Fock approximations, in which case the quantum BBGKY system reduces to a simple quantum generalization of the Vlasov equation. This equation is used to study the response of spatially homogeneous systems to weak external forces, and the associated problems of plasmon and spin-wave excitations. It is also used to derive the quantum and exchange corrected equations of inviscid hydrodynamical transport which are then applied to the problem of sound propagation in the degenerate electron gas. The second part of the study is concerned with the theory of the many-electron atom in the Hartree and Hartree-Fock approximations. The relevant quantum Vlasov equations lead naturally to a statistical theory of the atom which reduces to the Thomas-Fermi-Amaldi and Thomas Fermi models (respectively) as ħ → 0. For ħ ≠ 0, the quantum and exchange corrections to these models are simultaneously generated. The quantum hydrodynamical theory developed earlier is used to determine the influence of these corrections on the boundary conditions of the model, and a theory of the compressed atom is consequently obtained. Considered in somewhat less detail are the effects of non-zero temperature, net orbital angular momentum, relativity and correlations, as well as time dependent processes. The final part deals with the problem of the degenerate electron gas with a uniform neutralizing background. Going beyond the Hartree-Fock approximation, the pair functions for particles with parallel and anti-parallel spin are obtained by neglecting three particle correlations. From these functions, a quantum-mechanical collision integral is derived which differs from that obtained by Silin and Guernsey and conjectured by Wyld and Pines in that dynamical exchange effects are included. Also obtained from the pair function is an expression for the correlation which reduces in the high density limit to the result of Gell-Mann and Brueckner. At intermediate densities an additional term appears in the energy due to the screening of the exchange interaction by the dielectric properties of the medium. It is evaluated in the high density limit and found to be -0.151 r_s ln r_s Ryd/electron in marked disagreement with the corresponding value obtained by DuBois.

•Jan 1, 1963•0 citations

Statistics and Nos\\'e formalism for Ehrenfest dynamics

José L. Alonso, Castro, Alberto, Jesús Clemente-Gallardo +3 more

Quantum dynamics (i.e. the Schrödinger equation) and classical dynamics (i.e. Hamilton equations) can both be formulated in equal geometric terms: a Poisson bracket defined on a manifold. In this paper, we first show that the hybrid quantum-classical dynamics prescribed by the Ehrenfest equations can also be formulated within this general framework, what has been used in the literature to construct propagation schemes for Ehrenfest dynamics. Then, the existence of a well-defined Poisson bracket allows us to arrive to a Liouville equation for a statistical ensemble of Ehrenfest systems. The study of a generic toy model shows that the evolution produced by Ehrenfest dynamics is ergodic and therefore the only constants of motion are functions of the Hamiltonian. The emergence of the canonical ensemble characterized by the Boltzmann distribution follows after an appropriate application of the principle of equal a priori probabilities to this case. Once we know the canonical distribution of an Ehrenfest system, it is straightforward to extend the formalism of Nosé (invented to do constant temperature molecular dynamics by a non-stochastic method) to our Ehrenfest formalism. This work also provides the basis for extending stochastic methods to Ehrenfest dynamics.

•May 8, 2019•30 citations

Wave Mechanics in Classical Phase Space, Brownian Motion, and Quantum Theory

Giacomo Della Riccia, Norbert Wiener

A wave dynamics of fields φ(p, q; t) ∈ L2(Γ) over the phase space Γ(p, q) of a classical system 𝒮 is derived from the Liouville theorem. We define the energy contained in a given field φ(p, q; t). We show that for a special class of fields, selected on physical grounds, the energy spectrum is given by a time-independent Schrödinger equation. This allows us to associate with 𝒮 an ordinary quantum system Q such that the values of the quantized energy coincide for the fields in the phase space of 𝒮 and for Q. Then we make use of Wiener's stochastic integral based on the theory of Brownian motion to derive probabilities which are the same as those one would obtain through Born's statistical postulate of quantum theory. From this it follows that we can regard normalized fields φ(p, q; t) as ``probability amplitudes'' leading to a probability density function ρ(p, q; t) = φφ* in the sense of Gibbs' statistical mechanics. Our work therefore appears as a bridge between a statistical theory (in the sense of Gibbs) of a mechanical system 𝒮 and the usual quantum theory of the related quantum system Q.

Journal of Mathematical Physics•Aug 1, 1966•40 citations
Abstract This chapter focuses on Liouville’s theorem and classical statistical mechanics, deriving the classical propagator. The terms ‘phase space volume element’ and ‘Liouville operator’ are defined and an n-particle phase space probability density function is constructed to derive the Liouville equation. This is deconstructed into the BBGKY hierarchy, and radial distribution functions are used to develop n-body correlation functions. Koopman–von Neumann theory is investigated as a classical wavefunction approach. The chapter develops an operatorial mechanics based on classical Hilbert space, and discusses the de Broglie–Bohm formulation of quantum mechanics. Partition functions, ensemble averages and the virial theorem of Clausius are defined and Poincaré’s recurrence theorem, the Gibbs H-theorem and the Gibbs paradox are discussed. The chapter also discusses commuting observables, phase–amplitude decoupling, microcanonical ensembles, canonical ensembles, grand canonical ensembles, the Boltzmann factor, Mayer–Montroll cluster expansion and the equipartition theorem and investigates symplectic integrators, focusing on molecular dynamics.
Classical statistical mechanics tries to derive the macroscopic properties of matter starting from the mechanical laws that rule the behaviour of single particles. To describe equilibrium states, only observables accounting for correlations among states are considered, and only systems consisting of a large number of particles are taken into account. The observables are described by continuous functions on phase space, and the states are represented by linear assignments of a number to each observable. Within the framework of the canonical ensemble, one deals with mechanical systems in thermal equilibrium with a thermal reservoir, and the equilibrium state is reached as a result of the interaction with the external world. The external world may be really external, or equally well, the unrecognized internal degrees of freedom, like in the calculation of viscosity or thermal conductivity. If really only the external world is the cause we may expect some surface dependence while an internal unrecognized degree of freedom would have a volume effect, unless the interactions are long range. The microcanonical ensemble is instead introduced to study isolated mechanical systems, and the equilibrium is viewed as a temporal average, rather than as a limit. Attention is then focused on partition functions, the theorem of equipartition of energy and an elementary theory of specific heats. In the second part, the Planck derivation of the law of black-body radiation is analysed, presenting in chronological order the Kirchhoff theorem, the Stefan law, the Wien displacement law, the Rayleigh–Jeans formula and the Planck hypothesis. Further topics discussed are the Einstein and Debye quantum models for specific heats of solids. These topics prepare the ground for the introduction of quantum statistical mechanics. The third part is, in fact, devoted to the analysis of identical particles in quantum mechanics.
This paper and a following one develop a Hamiltonian formulation of the de Broglie-Bohm (deBB) interpretation of quantum mechanics. Drawing upon a unified Hamiltonian treatment of a classical particle and its associated ensemble, it is shown how the interaction of a particle and a spin-0 quantum wave can be consistently expressed in a canonical framework. A formulation is found in which the particle is acted upon by a force determined by the quantum potential but the particle does not react on the Schrodinger field. This requires the introduction of an additional field which the particle does act upon, and allows the deBB theory to be expressed in Hamiltonian terms. An ensemble theory for the wave-particle system based on Liouville's equation is developed, and it is shown that for a pure quantum state the particle obeys its own Liouville equation. The general particle phase space distribution that is compatible with ‖ψ‖ 2 is derived. A special case of this corresponds to the relation p = ∂S/∂q (and thus the guidance law of the deBB theory) but more general distributions, and hence non-deBB motions, are possible. This relation is interpreted as a constraint on the phase space of the composite system. Using the theory of Hamiltonian constraints it is shown that the first-order guidance law and Hamilton's second-order particle equations are dynamically equivalent.
Pursuing the Hamiltonian formulation of the de Broglie-Bohm (deBB) theory presented in the preceding paper, the Hamilton-Jacobi (HJ) theory of the wave-particle system is developed. It is shown how to derive a HJ equationfor the particle, which enables trajectories to be computed algebraically using Jacobi's method. Using Liouville's equation in the HJ representation we find the restriction on the Jacobi solutions which implies the quantal distribution. This gives a first method for interpreting the deBB theory in HJ terms. A second method proceeds via an explicit solution of the field+particle HJ equation. Both methods imply that the quantum phase may be interpreted as an incomplete integral. Using these results and those of the first paper it is shown how Schrodinger's equation can be represented in Liouvillian terms, and vice versa. The general theory of canonical transformations that represent quantum unitary transformations is given, and it is shown in principle how the trajectory theory may be expressed in other quantum representations. Using the solution found for the total HJ equation, an explicit solution for the additional field containing a term representing the particle back-reaction is found. The conservation of energy and momentum in the model is established, and a weak form of the action-reaction principle is shown to hold. Alternative forms for the Hamiltonian are explored and it is shown that, within this theoretical context, the deBB theory is not unique. The theory potentially provides an alternative way of obtaining the classical limit.
By means of a quantum mechanical phase space distribution function introduced by von Roos, the Schroedinger equation for a non-relativistic system of N identical particles with scalar interactions is transformed into a quantum mechanical generalization of the Liouville equation, thereby formulating the problem in terms of a generalized density in phase space, a quantity of primary interest in most treatments of the corresponding classical system (or plasma). This transformation permits a parallel development of the theories of classical and quantum plasmas and thus allows the quantum many-body problem to be discussed virtually completely in classical terms. In particular, a kinetic theory of quantum plasmas is obtained by deriving the quantum analogue of the BBGKY hierarchy, and applying thereto approximation techniques similar to those of Rostocker and Rosenbluth, and Bogoliubov. The point of departure from similar previous studies based on the Wigner distribution function is that the proper exchange symmetry can be tractably introduced into the formalism. Attention is first focused on the Hartree and Hartree-Fock approximations, in which case the quantum BBGKY system reduces to a simple quantum generalization of the Vlasov equation. This equation is used to study the response of spatially homogeneous systems to weak external forces, and the associated problems of plasmon and spin-wave excitations. It is also used to derive the quantum and exchange corrected equations of inviscid hydrodynamical transport which are then applied to the problem of sound propagation in the degenerate electron gas. The second part of the study is concerned with the theory of the many-electron atom in the Hartree and Hartree-Fock approximations. The relevant quantum Vlasov equations lead naturally to a statistical theory of the atom which reduces to the Thomas-Fermi-Amaldi and Thomas Fermi models (respectively) as ħ → 0. For ħ ≠ 0, the quantum and exchange corrections to these models are simultaneously generated. The quantum hydrodynamical theory developed earlier is used to determine the influence of these corrections on the boundary conditions of the model, and a theory of the compressed atom is consequently obtained. Considered in somewhat less detail are the effects of non-zero temperature, net orbital angular momentum, relativity and correlations, as well as time dependent processes. The final part deals with the problem of the degenerate electron gas with a uniform neutralizing background. Going beyond the Hartree-Fock approximation, the pair functions for particles with parallel and anti-parallel spin are obtained by neglecting three particle correlations. From these functions, a quantum-mechanical collision integral is derived which differs from that obtained by Silin and Guernsey and conjectured by Wyld and Pines in that dynamical exchange effects are included. Also obtained from the pair function is an expression for the correlation which reduces in the high density limit to the result of Gell-Mann and Brueckner. At intermediate densities an additional term appears in the energy due to the screening of the exchange interaction by the dielectric properties of the medium. It is evaluated in the high density limit and found to be -0.151 r_s ln r_s Ryd/electron in marked disagreement with the corresponding value obtained by DuBois.
Quantum dynamics (i.e. the Schrödinger equation) and classical dynamics (i.e. Hamilton equations) can both be formulated in equal geometric terms: a Poisson bracket defined on a manifold. In this paper, we first show that the hybrid quantum-classical dynamics prescribed by the Ehrenfest equations can also be formulated within this general framework, what has been used in the literature to construct propagation schemes for Ehrenfest dynamics. Then, the existence of a well-defined Poisson bracket allows us to arrive to a Liouville equation for a statistical ensemble of Ehrenfest systems. The study of a generic toy model shows that the evolution produced by Ehrenfest dynamics is ergodic and therefore the only constants of motion are functions of the Hamiltonian. The emergence of the canonical ensemble characterized by the Boltzmann distribution follows after an appropriate application of the principle of equal a priori probabilities to this case. Once we know the canonical distribution of an Ehrenfest system, it is straightforward to extend the formalism of Nosé (invented to do constant temperature molecular dynamics by a non-stochastic method) to our Ehrenfest formalism. This work also provides the basis for extending stochastic methods to Ehrenfest dynamics.
A wave dynamics of fields φ(p, q; t) ∈ L2(Γ) over the phase space Γ(p, q) of a classical system 𝒮 is derived from the Liouville theorem. We define the energy contained in a given field φ(p, q; t). We show that for a special class of fields, selected on physical grounds, the energy spectrum is given by a time-independent Schrödinger equation. This allows us to associate with 𝒮 an ordinary quantum system Q such that the values of the quantized energy coincide for the fields in the phase space of 𝒮 and for Q. Then we make use of Wiener's stochastic integral based on the theory of Brownian motion to derive probabilities which are the same as those one would obtain through Born's statistical postulate of quantum theory. From this it follows that we can regard normalized fields φ(p, q; t) as ``probability amplitudes'' leading to a probability density function ρ(p, q; t) = φφ* in the sense of Gibbs' statistical mechanics. Our work therefore appears as a bridge between a statistical theory (in the sense of Gibbs) of a mechanical system 𝒮 and the usual quantum theory of the related quantum system Q.