Semiclassical approximations to quantum dynamics are almost as old as quantum mechanics itself. In the approach pioneered by Wigner, the evolution of his quasiprobability density function on phase space is expressed as an asymptotic series in increasing powers of Planck's constant, with the classical Liouvillean evolution as leading term. Successive semiclassical approximations to quantum dynamics are defined by successive terms in the series. We consider a complementary approach, which explores the quantum-clssical interface from the other direction. Classical dynamics is formulated in Hilbert space, with the Groenewold quasidensity operator as the image of the Liouville density on phase space. The evolution of the Groenewold operator is then expressed as an asymptotic series in increasing powers of Planck's constant. Successive semiquantum approximations to classical dynamics are defined by successive terms in this series, with the familiar quantum evolution as leading term.
Semiclassical approximations to quantum dynamics are almost as old as quantum mechanics itself. In the approach pioneered by Wigner, the evolution of his quasiprobability density function on phase space is expressed as an asymptotic series in increasing powers of Planck's constant, with the classical Liouvillean evolution as leading term. Successive semiclassical approximations to quantum dynamics are defined by successive terms in the series. We consider a complementary approach, which explores the quantum-clssical interface from the other direction. Classical dynamics is formulated in Hilbert space, with the Groenewold quasidensity operator as the image of the Liouville density on phase space. The evolution of the Groenewold operator is then expressed as an asymptotic series in increasing powers of Planck's constant. Successive semiquantum approximations to classical dynamics are defined by successive terms in this series, with the familiar quantum evolution as leading term.
Quantum mechanics has been formulated in phase space, with the Wigner function as the representative of the quantum density operator, and classical mechanics has been formulated in Hilbert space, with the Groenewold operator as the representative of the classical Liouville density function. Semiclassical approximations to the quantum evolution of the Wigner function have been defined, enabling the quantum evolution to be approached from a classical starting point. Now analogous semiquantum approximations to the classical evolution of the Groenewold operator are defined, enabling the classical evolution to be approached from a quantum starting point. Simple nonlinear systems with one degree of freedom are considered, whose Hamiltonians are polynomials in the Hamiltonian of the simple harmonic oscillator. The behavior of expectation values of simple observables and of eigenvalues of the Groenewold operator are calculated numerically and compared for the various semiclassical and semiquantum approximations.
Classical mechanics is formulated in complex Hilbert space with the introduction of a commutative product of operators, an antisymmetric bracket and a quasidensity operator that is not positive definite. These are analogues of the star product, the Moyal bracket, and the Wigner function in the phase space formulation of quantum mechanics. Quantum mechanics is then viewed as a limiting form of classical mechanics, as Planck's constant approaches zero, rather than the other way around. The forms of semiquantum approximations to classical mechanics, analogous to semiclassical approximations to quantum mechanics, are indicated.
The Moyal description of quantum mechanics, based on the Wigner-Weyl isomorphism between operators and symbols, provides a comprehensive phase space representation of dynamics. The Weyl symbol image of the Heisenberg picture evolution operator is regular in ħ and so presents a preferred foundation for semiclassical analysis. Its semiclassical expansion "coefficients," acting on symbols that represent observables, are simple, globally defined (phase space) differential operators constructed in terms of the classical flow. The first of two presented methods introduces a cluster-graph expansion for the symbol of an exponentiated operator, which extends Groenewold′s formula for the Weyl product of two symbols and has ħ as its natural small parameter. This Poisson bracket based cluster expansion determines the Jacobi equations for the semiclassical expansion of "quantum trajectories." Their Green function solutions construct the regular ħ ⇓ 0 asymptotic series for the Heisenberg-Weyl evolution map. The second method directly substitutes such a series into the Moyal equation of motion and determines the ħ coefficients recursively. In contrast to the WKB approximation for propagators, the Heisenberg-Weyl description of evolution involves no essential singularity in ħ, no Hamilton-Jacobi equation to solve for the action, and no multiple trajectories, caustics, or Maslov indices.
We investigate the suitability of the Wigner method as a tool for semiclassical dynamics. In spite of appearances, the dynamical time evolution of Wigner phase space densities is found not to reduce to classical dynamics in most circumstances, even as h→0. In certain applications involving highly ’coherent’ density matrices, this precludes direct h-expansion treatment of quantum corrections. However, by selective resummation of terms in the Wigner–Moyal series for the quantum phase space propagation it is possible to arrive at a revised or renormalized classicallike dynamics which solves the difficulties of the direct approach. In this paper, we review the Wigner method, qualitatively introduce the difficulties encountered in certain semiclassical applications, and derive quantitative means of surmounting these difficulties. Possible practical applications are discussed.
Abstract. We establish the difference between the propagation of semiclassical Wigner functions and classical Liouville propagation. First we re-discuss the semiclassical limit for the propagator of Wigner functions, which on its own leads to their classical propagation. Then, via stationary phase evaluation of the full integral evolution equation, using the semiclassical expressions of Wigner functions, we provide the correct geometrical prescription for their semiclassical propagation. This is determined by the classical trajectories of the tips of the chords defined by the initial semiclassical Wigner function and centered on their arguments, in contrast to the Liouville propagation which is determined by the classical trajectories of the arguments themselves. 1. introduction The Wigner [11] function W is a representation of the quantum density operator ˆρ as a real function on phase space, for a non-relativistic dynamical system with classical-quantum correspondence and n degrees of freedom. Here, all the underlying geometry is euclidean and we can include the Wigner function within the general framework of Weyl representation of operators: Â → A ∈ Ck IR (IR2n), so W(x) = (2π�) −n ρ(x) , where x ≡ (p1,...pn, q1,...qn) denotes a point in phase space. In this context, the quantum evolution of the density operator ˆρt effected by an autonomous external hamiltonian
A semiclassical approximation is derived by using a family of wavepackets to map arbitrary wavefunctions into phase space. If the Hamiltonian can be approximated as linear over each individual wavepacket, as often done when presenting Ehrenfest's theorem, the resulting approximation is a linear first-order partial differential equation on phase space, which will be referred to as the Schrödinger-Ehrenfest or SE equation. This advectively transports wavefunctions along classical trajectories, so that as a trajectory is followed in time the amplitude remains constant while the phase changes by the action divided by ℏ. The wavefunction's squared-magnitude is a plausible phase space density and obeys Liouville's equation for the classical time evolution. This is a derivation of the Koopman-von~Neumann (KvN) formulation of classical mechanics, which previously was postulated but not derived. With the time-independent SE equation, continuity of the wavefunction requires the change of phase around any closed path in the torus covered by a classical trajectory to be an integer multiple of 2π, giving a standing wave picture of old quantum mechanics. While this applies to any system, for separable systems it gives Bohr-Sommerfeld quantization.