We consider the Dirac-Frenkel variational principle in Wigner phase-space and apply it to the Wigner-Liouville equation for both imaginary and real time dynamical problems. The variational principle allows us to deduce the optimal time-evolution of the parameter-dependent Wigner distribution. It is shown that the variational principle can be formulated alternatively as a "principle of least action." Several low-dimensional problems are considered. In imaginary time, high-temperature classical distributions are "cooled" to arrive at low-temperature quantum Wigner distributions whereas in real time, the coherent dynamics of a particle in a double well is considered. Especially appealing is the relative ease at which Feynman's path integral centroid variable can be incorporated as a variational parameter. This is done by splitting the high-temperature Boltzmann distribution into exact local centroid constrained distributions, which are thereafter cooled using the variational principle. The local distributions are sampled by Metropolis Monte Carlo by performing a random walk in the centroid variable. The combination of a Monte Carlo and a variational procedure enables the study of quantum effects in low-temperature many-body systems, via a method that can be systematically improved.
We investigate variational principles for the approximation of quantum dynamics that apply for approximation manifolds that do not have complex linear tangent spaces. The first one, dating back to McLachlan [Mol. Phys. 8, 39–44 (1964)], minimizes the residuum of the time-dependent Schrödinger equation, while the second one, originating from the lecture notes of Kramer and Saraceno [Geometry of the Time-Dependent Variational Principle in Quantum Mechanics, Lecture Notes in Physics Vol. 140 (Springer, Berlin, 1981)], imposes the stationarity of an action functional. We characterize both principles in terms of metric and symplectic orthogonality conditions, consider their conservation properties, and derive an elementary a posteriori error estimate. As an application, we revisit the time-dependent Hartree approximation and frozen Gaussian wave packets.
In this paper we give a partial answer to the problem: When does an initially non-negative Wigner function remain non-negative under the effect of the time evolution? We show that, for pure states, this is the case for linear systems only; to prove this we use the fact that the Wigner function is non-negative if and only if the wavefunction is Gaussian. We also prove that the Green’s solution of the evolution equation of the Wigner function, which in the framework of probability theory corresponds to the conditional probability density, takes on negative values. We utilize a theorem, about moments, borrowed from Pawula. We conclude that the Wigner phase-space formulation of quantum mechanics cannot receive a genuine probabilistic interpretation.
Variational Gaussian wave-packet dynamics is usually obtained by restricting quantum evolution to Gaussian wave packets. We instead construct finite-
N dynamics for a labeled bundle of classical trajectories under a constraint fixing the symplectic-area scale of its covariance. At finite
N, the bundle itself is the dynamical state: it may be non-Gaussian, and the potential force on each trajectory is evaluated from the full potential without a local harmonic approximation. For sequences whose initial empirical distributions approach a smooth Gaussian density as
N→∞, the limiting relative motion becomes linear and preserves Gaussianity. With the area scale set to
ℏ/2, the limiting centroid and width equations coincide with those of the Gaussian time-dependent variational principle, and the limiting phase-space density coincides with the corresponding Gaussian Wigner density for matched initial data. Here
N counts trajectories, not orders in a Moyal or
ℏ expansion. This phase-space correspondence reveals two complementary constructions of the same Gaussian dynamics: a variational reduction of quantum evolution and the Gaussian large-
N limit of a constrained classical trajectory bundle.
Starting with the exact variational formulation of quantum mechanics (or quantum field theory) in the Schroedinger picture, and using Gaussian trail wave functions, an exact functional Schroedinger equation is obtained. The time-dependent Hartree-Fock approximation is obtained and extended to many oscillators and to field theory. The problem of quantum roll, in which an initial Gaussian wave packet starting at an unstable classical equilibrium point evolves in time, is treated using this formalism. Quantum tunnelling is also studied. (LEW)
Abstract. Previously, an explicit solution for the time evolution of the Wigner function was presented in terms of auxiliary phase space coordinates which obey simple equations that are analogous with, but not identical to, the classical equations of motion. They can be solved easily and their solutions can be utilized to construct the time evolution of the Wigner function. The usefulness of this explicit solution is demonstrated by solving a numerical example in which the Wigner function has strong spatial and temporal variations as well as regions with negative values. It is found that the explicit solution gives a correct description of the time evolution of the Wigner function. We examine next the pseudoparticle approximation which uses classical trajectories to evolve the Wigner function. We find that the pseudoparticle approximation reproduces the general features of the time evolution, but there are deviations. We show how these deviations can be systematically reduced by including higher-order correction terms in powers of ¯h 2.