The dynamics of quantum systems can be approximated by the time propagation of Gaussian wave packets. Applying a time dependent variational principle, the time evolution of the parameters of the coupled Gaussian wave packets can be calculated from a set of ordinary differential equations. Unfortunately, the set of equations is ill behaved in most practical applications, depending on the number of propagated Gaussian wave packets, and methods for regularization are needed. We present a general method for regularization based on applying adequate nonholonomic inequality constraints to the evolution of the parameters, keeping the equations of motion well behaved. The power of the method is demonstrated for a nonintegrable system with two degrees of freedom.
The dynamics of quantum systems can be approximated by the time propagation of Gaussian wave packets. Applying a time dependent variational principle, the time evolution of the parameters of the coupled Gaussian wave packets can be calculated from a set of ordinary differential equations. Unfortunately, the set of equations is ill behaved in most practical applications, depending on the number of propagated Gaussian wave packets, and methods for regularization are needed. We present a general method for regularization based on applying adequate nonholonomic inequality constraints to the evolution of the parameters, keeping the equations of motion well behaved. The power of the method is demonstrated for a nonintegrable system with two degrees of freedom.
The wave packet dynamics in atomic systems and in Bose-Einstein condensates is investigated by means of a time-dependent variational principle. The wave packets are assumed to be parametrized by a set of time-dependent parameters. The time evolution of the parameters of the coupled Gaussian wave packets can be calculated from a set of ordinary differential equations, as obtained from the time-dependent variational principle. Unfortunately, the set of equations is ill-behaved in most practical applications, depending on the number of propagated Gaussian wave packets, and methods for regularization are needed. A general method for regularization based on applying adequate nonholonomic inequality constraints to the evolution of the parameters, keeping the equations of motion well-behaved is presented. The power of the method is demonstrated for a non-integrable system with two degrees of freedom. The Gaussian wave packet (GWP) method is applied to the three-dimensional hydrogen atom. The regularization based on the introduction of Kustaanheimo-Stiefel coordinates and a fictitious time is performed. The regularization implies a transition from the three-dimensional physical position space to a four-dimensional parameter space, spanned by the Kustaanheimo-Stiefel coordinates. The regularization is accompanied by a restriction on physically allowed wave functions. The Coulomb potential is transformed to a harmonic potential and GWPs are the exact solutions, provided they fulfill the restriction. The effect of the restriction on the four-dimensional GWP is discussed and it is shown that the GWPs can satisfy the restriction if the Gaussian parameter space is reduced in a certain way. The exact analytic evolution of the restricted GWP is presented, and the expansion of a localized initial wave function in the restricted Gaussian basis set and its analytic propagation in the fictitious time are shown. Symmetry subspaces with conserved magnetic quantum number m and with conserved angular momentum quantum numbers l,m are treated separately. The method is also applied to the non-integrable H atom in in a homogeneous magnetic field and in perpendicular external electric and magnetic fields. The evolution of the wave packets is determined by the constrained time-dependent variational principle. The numerical results are compared to numerically exact values and show excellent agreement. Another class of systems where the variational Gaussian wave packet method yields good results are cold gases. In this thesis Bose-Einstein condensates are investigated, where in addition to the common short-range contact interaction two different long-range interactions, viz. a laser induced gravity-like 1/r-interaction or a magnetic dipole-dipole interaction are present. The dynamics as resulting from the time-dependent extended Gross-Pitaevskii equation for Bose-Einstein condensates with attractive 1/r-interaction is investigated with both the GWP method and numerically exact calculations. It is shown that these condensates exhibit signatures known from the nonlinear dynamics of autonomous Hamiltonian systems. The two stationary solutions created in a tangent bifurcation at a critical value of the scattering length are identified as elliptical and hyperbolical fixed points, corresponding to stable and unstable stationary states of the condensate. The stable stationary state is surrounded by elliptical islands, corresponding to condensates periodically oscillating in time, whereas condensate wave functions in the unstable region undergo a collapse within finite time. For negative scattering lengths below the tangent bifurcation no stationary solutions exist, i.e., the condensate is always unstable and collapses. The dynamics of condensates with inter-atomic magnetic dipole-dipole interaction is investigated in the mean field limit using a Gaussian trial function. The anisotropy of the magnetic dipole-dipole interaction breaks the spherical symmetry and for ordered dipoles an effectively two-dimensional system with cylindrical symmetry is obtained. Special attention is payed to the regularity of the dynamics and it is shown that a transition from regular to chaotic motion takes place with increasing energy where regions of regular and chaotic motion coexist. It is shown that stable modes exist at energies high above the saddle point energy, i.e. at energies where a collapse of the condensate is expected.
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.
A new method is proposed for computing the time evolution of quantum mechanical wave packets. Equations of motion for the real-valued functions C and S in the complex action S=C(r,t)+iS(r,t)/ℏ, with ψ(r,t)=exp(S), involve gradients and curvatures of C and S. In previous implementations of the hydrodynamic formulation, various time-consuming fitting techniques of limited accuracy were used to evaluate these derivatives around each fluid element in an evolving ensemble. In this study, equations of motion are developed for the spatial derivatives themselves and a small set of these are integrated along quantum trajectories concurrently with the equations for C and S. Significantly, quantum effects can be included at various orders of approximation, no spatial fitting is involved, there are no basis set expansions, and single quantum trajectories (rather than correlated ensembles) may be propagated, one at a time. Excellent results are obtained when the derivative propagation method is applied to anharmonic potentials involving barrier transmission.
We have recently published a new semiclassical method, generalized Gaussian wave packet dynamics, which extends Gaussian wave packet dynamics into complex phase space. Although we were able to give an accurate formulation of the method, we had at the time of writing that paper only an intuitive, heuristic understanding of the deeper causes which make the method work. A more mathematical understanding was needed. To close this gap we show in this paper the equivalence of the new method with a first order expansion of ℏ of the Schrödinger equation. We further prove that the new method is equivalent to the stationary phase approximation, using the usual WKB formula for the propagator. The latter equivalence enables us to show that all the symmetry properties of time-dependent quantum mechanics also hold in the new semiclassical theory. Finally, we provide some elaboration of the method, and clarify several issues that were not discussed before. With this new insight we are able to formulate a simple rule for the calculation of semiclassical wave functions that contain contributions from more than one branch. This corrects for the divergence of semiclassical wave functions near caustics, a problem that we encountered in the preceding paper.
Nondispersive wave packets in a fictitious time variable are calculated analytically for the field-free hydrogen atom. As is well known by means of the Kustaanheimo-Stiefel transformation the Coulomb problem can be converted into that of a four-dimensional harmonic oscillator, subject to a constraint. This regularization makes use of a fictitious time variable, but arbitrary Gaussian wave packets in that time variable in general violate that constraint. The set of ``restricted Gaussian wave packets'' consistent with the constraint is constructed and shown to provide a complete basis for the expansion of states in the original three-dimensional coordinate space. Using that expansion arbitrary localized Gaussian wave packets of the hydrogen atom can be propagated analytically and exhibit a nondispersive periodic behavior as functions of the fictitious time. Restricted wave packets with and without well-defined angular momentum quantum numbers are constructed. They will be used as trial functions in time-dependent variational computations for the hydrogen atom in static external fields in the subsequent paper [T. Fab\ifmmode \check{c}\else \v{c}\fi{}i\ifmmode \check{c}\else \v{c}\fi{}, J. Main, and G. Wunner, Phys. Rev. A 79, 043417 (2009)].
It is shown that application of the time-dependent variational principle to trial wave functions of a particular form results in a description of the quantum system as a classical system consisting of two mutually interacting parts: the ordinary classical analog of the quantum system and a (generally finite) number of additional degrees of freedom. This holds out the prospect of simulating quantum dynamics by coupling the corresponding classical system to additional degrees of freedom, and it is pointed out that such calculations have, in effect, already been performed. Since the trial wave functions that qualify for this treatment may be arbitrarily accurate, there is no limitation to the accuracy of the resulting classical description. In a particular case, the canonical coordinates and momenta for all degrees of freedom of the overall classical system are identified and investigated. In general, certain parameters appearing in the trial wave functions treated in this work may become redundant at certain times, leading to singularities in the Hamiltonian equations of motion. It is shown, however, that in certain such cases, a classical formalism for the quantum dynamics remains valid if the Hamiltonian function is suitably modified. Calculations applying the present Hamiltonian formalism are carried out that demonstrate that it has better convergence properties than an alternative, nonvariational, Hamiltonian technique suggested by previous work. These numerical studies also suggest that the observed spreading of wave packets and development of complicated structure in wave functions may be interpreted classically in terms of energy transfer among the various degrees of freedom in the overall classical system.