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Solution To Quantum Liouville Equation In Phase Space Via A Stochastic Process

Sarah John

Published Jan 1, 1986• 2 citations

Abstract

A numerical technique for the solution of the Quantum Liouville equation is discussed which is expected to be useful for mixed states describing highly excited many body systems.;The equation is represented in phase-space, where the density operator becomes the Wigner function. The Liouville operator which determines its time evolution, separates into a classical and quantum term. When the Wigner function varies slowly in momentum, the quasi-classical approximation is valid in which one keeps only the first order quantum term in the Liouville equation.;The Wigner function is represented by a sample set of gaussian smeared points. In an iterative time development scheme, the points are propagated first along a classical phase-space trajectory, which is then followed by a quantum development. The latter is represented by a stochastic process incorporated into the numerical procedure as a generalized Markovian process.;This method is numerically tested for an anharmonic quartic potential. The contours of the Wigner function at different time intervals are compared with the purely classical and exact results. It is seen that it provides a definite improvement over the purely classical approximation. Various averages are also compared and sources of error discussed.;In addition it was discovered that intermediate mixed states show a 'quantum focusing' effect, wherein the function peaks to values greater than the original maximum. This quantum-mechanical effect did not appear in the pure case, and disappears because of Liouville's theorem for highly mixed states which are essentially classical. There are other numerical schemes, which treat the points as moving in an effective potential, wherein the points retain their identity. These again by Liouville's theorem cannot show this focusing effect.

DOI: 10.21220/s2-rmv8-qj13

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A numerical technique for the solution of the Quantum Liouville equation is discussed which is expected to be useful for mixed states describing highly excited many body systems.;The equation is represented in phase-space, where the density operator becomes the Wigner function. The Liouville operator which determines its time evolution, separates into a classical and quantum term. When the Wigner function varies slowly in momentum, the quasi-classical approximation is valid in which one keeps only the first order quantum term in the Liouville equation.;The Wigner function is represented by a sample set of gaussian smeared points. In an iterative time development scheme, the points are propagated first along a classical phase-space trajectory, which is then followed by a quantum development. The latter is represented by a stochastic process incorporated into the numerical procedure as a generalized Markovian process.;This method is numerically tested for an anharmonic quartic potential. The contours of the Wigner function at different time intervals are compared with the purely classical and exact results. It is seen that it provides a definite improvement over the purely classical approximation. Various averages are also compared and sources of error discussed.;In addition it was discovered that intermediate mixed states show a 'quantum focusing' effect, wherein the function peaks to values greater than the original maximum. This quantum-mechanical effect did not appear in the pure case, and disappears because of Liouville's theorem for highly mixed states which are essentially classical. There are other numerical schemes, which treat the points as moving in an effective potential, wherein the points retain their identity. These again by Liouville's theorem cannot show this focusing effect.

The quantum Liouville equation in the Wigner representation is solved numerically by using Monte Carlo methods. For incremental time steps, the propagation is implemented as a classical evolution in phase space modified by a quantum correction. The correction, which is a momentum jump function, is simulated in the quasi-classical approximation via a stochastic process. In this paper the technique, which is developed and validated in two- and three-dimensional momentum space, extends an earlier one-dimensional work. Also, by developing a new algorithm, the application to bound state motion in an anharmonic quartic potential shows better agreement with exact solutions in two-dimensional phase space. I. Introduction Nuclear interaction theory is formulated in the language of quantum mechanics, and hence the development of general methods of solutions to quantum dynamics will provide useful tools for application to a large class of problems in nuclear many-body theory. Different approaches...

The quantum Liouville equation is solved in the Wigner representation using generalized Monte Carlo techniques. For small increments of time, the solution is represented as a sequential classical evolution in phase space followed by a quantum "jump" distribution in momentum space, with the latter simulated via a stochastic method. Extending the work initiated by John and Remler [Ann Phys. (N.Y.) 180, 152 (1987)] the technique is developed and validated for higher dimensions. Also, an alternative algorithm is developed and applied to study motion of a quantum system in an anharmonic quartic potential well, with significantly improved results.

A general approximation scheme for solving the quantum Liouville equation has been developed and is presented here specifically within the Hartree approximation. This method uses the connection between the density matrix and the Wigner phase-space representations. By use of the Wigner representation the off-diagonal elements of the density matrix are expanded in powers of deviations from the mean momentum value. The time evolution of these deviations is governed by a closed self-consistent set of equations.

We present a comprehensive study of semiclassical phase-space propagation in the Wigner representation, emphasizing numerical applications, in particular as an initial-value representation. Two semiclassical approximation schemes are discussed. The propagator of the Wigner function based on van Vleck's approximation replaces the Liouville propagator by a quantum spot with an oscillatory pattern reflecting the interference between pairs of classical trajectories. Employing phase-space path integration instead, caustics in the quantum spot are resolved in terms of Airy functions. We apply both to two benchmark models of nonlinear molecular potentials, the Morse oscillator and the quartic double well, to test them in standard tasks such as computing autocorrelation functions and propagating coherent states. The performance of semiclassical Wigner propagation is very good even in the presence of marked quantum effects, e.g., in coherent tunneling and in propagating Schrodinger cat states, and of classical chaos in four-dimensional phase space. We suggest options for an effective numerical implementation of our method and for integrating it in Monte-Carlo-Metropolis algorithms suitable for high-dimensional systems.

When a dynamical system has a perturbation which is considered as a stochastic process, the Liouville equation for the system in the phase space or the space of quantum-mechanical density operators is a sort of stochastic equation. The ensemble average of its formal integral defines the relaxation operator Φ(t) of the system. By the definition Φ(t) = exp K(t), the cumulant function K(t) may be introduced. Some general properties are first discussed for a simple example of an oscillator with random frequency modulation and then, concepts of slow and fast modulation are considered. These concepts can be generalized to more general types of stochastic Liouville equations. It is shown that by various possibilities of defining generalized exponential functions, this approach may be useful to understand some essential features of the problem from an unified point of view.

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