Propagation Of Gaussian Wave Packets Research Articles

Speed-up and slow-down of a quantum particle

We study non-relativistic propagation of Gaussian wave packets in one-dimensional Eckart potential, a barrier, or a well. In the picture used, the transmitted wave packet results from interference between the copies of the freely propagating state with different spatial shifts (delays), x', induced by the scattering potential. The Uncertainty Principle precludes relating the particle’s final position to the delay experienced in the potential, except in the classical limit. Beyond this limit, even defining an effective range of the delay is shown to be an impracticable task, owing to the oscillatory nature of the corresponding amplitude distribution. Our examples include the classically allowed case, semiclassical tunnelling, delays induced in the presence of a virtual state, and scattering by a low barrier. The properties of the amplitude distribution of the delays, and its pole representation are studied in detail.

Exact transparent boundary condition for the multidimensional Schrödinger equation in hyperrectangular computational domain.

In this paper an exact transparent boundary condition for the multidimensional Schrödinger equation in a hyperrectangular computational domain is proposed. It is derived as a generalization of exact transparent boundary conditions for two-dimensional (2D) and 3D equations reported before. An exact fully discrete (i.e., derived directly from the finite-difference scheme used) 1D transparent boundary condition is also proposed. Several numerical experiments using an improved unconditionally stable numerical implementation in the 3D space demonstrate propagation of Gaussian wave packets in free space and penetration of a particle through a 3D spherically asymmetrical barrier. The application of the multidimensional transparent boundary condition to the dynamics of the 2D system of two noninteracting particles is considered. The proposed boundary condition is simple, robust, and can be useful in the field of computational quantum mechanics, when an exact solution of the multidimensional Schrödinger equation (including multiparticle problems) is required.

Simulation of 1D Gaussian Wave Packet and 2D Waves in Square Membrane Using Excel Spreadsheet with Visual Basic for Teaching Activity

The aim of this study is to demonstrate 1D Gaussian wave packet propagation and 2D wave deflection in a square membrane using the excel interactive spreadsheets with the Visual Basic for Application (VBA) programming. It is based on the analytic solution of Gaussian packet wave propagation in x direction and 2D wave propagation in square membrane. The vibration modes of 2D wave are determined by using differential equation and initial conditions. The simulation based on the spread sheets functions and facilities has been explored to show the 1D and 2D moving objects in transitional processes. This instrument is very useful both in teaching activity and learning process of wave physics. Visualizing of the vibration of wave in the square membrane show a very clear manner of m and n vibration modes of the wave. There for, in the Gaussian packet waves with the superposition processes between those waves show that the amplitude shifted in a different manner.

Propagation of Gaussian wave packets in complex media and application to fracture characterization

Propagation of Gaussian wave packets in complex media and application to fracture characterization

Strong scarring of logarithmic quasimodes

We consider a semiclassical (pseudo)differential operator on a compact surface (M,g), such that the Hamiltonian flow generated by its principal symbol admits a hyperbolic periodic orbit γ at some energy E0. For any ϵ>0, we then explicitly construct families of quasimodes of this operator, satisfying an energy width of order ϵh|logh| in the semiclassical limit, but which still exhibit a strong scar on the orbit γ, i.e. that these states have a positive weight in any microlocal neighbourhood of γ. We pay attention to optimizing the constants involved in the estimates. This result generalizes a recent result of Brooks \cite{Br13} in the case of hyperbolic surfaces. Our construction, inspired by the works of Vergini et al. in the physics literature, relies on controlling the propagation of Gaussian wavepackets up to the Ehrenfest time.

A Gaussian Wave Packet Propagation Approach to Vibrationally Resolved Optical Spectra at Non-Zero Temperatures

An effective time dependent approach based on a method that is similar to the Gaussian wave packet propagation (GWP) technique of Heller is developed for the computation of vibrationally resolved electronic spectra at finite temperatures in the harmonic, Franck-Condon/Hertzberg-Teller approximations. Since the vibrational thermal density matrix of the ground electronic surface and the time evolution operator on that surface commute, it is possible to write the spectrum generating correlation function as a trace of the time evolved doorway state. In the stated approximations, the doorway state is a superposition of the harmonic oscillator zero and one quantum eigenfunctions and thus can be propagated by the GWP. The algorithm has an O(N(3)) dependence on the number of vibrational modes. An application to pyrene absorption spectrum at two temperatures is presented as a proof of the concept.

Electronically nonadiabatic wave packet propagation using frozen Gaussian scattering.

We present an approach, which allows to employ the adiabatic wave packet propagation technique and semiclassical theory to treat the nonadiabatic processes by using trajectory hopping. The approach developed generates a bunch of hopping trajectories and gives all additional information to incorporate the effect of nonadiabatic coupling into the wave packet dynamics. This provides an interface between a general adiabatic frozen Gaussian wave packet propagation method and the trajectory surface hopping technique. The basic idea suggested in [A. D. Kondorskiy and H. Nakamura, J. Chem. Phys. 120, 8937 (2004)] is revisited and complemented in the present work by the elaboration of efficient numerical algorithms. We combine our approach with the adiabatic Herman-Kluk frozen Gaussian approximation. The efficiency and accuracy of the resulting method is demonstrated by applying it to popular benchmark model systems including three Tully's models and 24D model of pyrazine. It is shown that photoabsorption spectrum is successfully reproduced by using a few hundreds of trajectories. We employ the compact finite difference Hessian update scheme to consider feasibility of the ab initio "on-the-fly" simulations. It is found that this technique allows us to obtain the reliable final results using several Hessian matrix calculations per trajectory.

Perturbative wave-packet spawning procedure for non-adiabatic dynamics in diabatic representation

I present a new formulation of wave-packet spawning procedure based on a second order perturbation theory expression for population transfer between different diabatic electronic states. The employed perturbation theory (PT) expansion is based on an assumption that diabatic states can be represented locally with their Taylor series up to quadratic terms in nuclear coordinates (local harmonic approximation). The corresponding local harmonic basis of vibrational states makes infinite summation over excited states in PT expressions possible, and thus, it provides a complete basis set expression for the population transfer. This allows me to detect when a finite basis set expansion employed in variational wave packet propagation does not adequately describe the interstate population transfer. Also, it suggests a rigorous criterion for basis set expansion (spawning). The proposed procedure is illustrated for the variational multiconfigurational Gaussian wave packet method applied to 1D and 2D model examples, and it also can be extended to direct on-the-fly dynamics with any Gaussian wave packet propagation method.

An action principle for complex quantum trajectories

In a recent paper [B. Poirier, Chem. Phys. 370, 4 (2010)], a formulation of quantum mechanics was presented for which the usual wavefunction and Schrödinger equation are replaced with an ensemble of real-valued trajectories satisfying a principle of least action. It was found that the resultant quantum trajectories are those of Bohmian mechanics. In this paper, analogous ideas are applied to Bohmian Mechanics with Complex Action (BOMCA). The standard BOMCA trajectories as previously defined are found not to satisfy an action principle. However, an alternate set of complex equations of motion is derived that does exhibit this desirable property, and an approximate numerical implementation is presented. Exact analytical results are also presented, for Gaussian wavepacket propagation under quadratic potentials.

Initial phase and free-particle wave packet evolution

The evolution of the free-particle wave function in one dimension is the same as scalar Fresnel diffraction from a one-dimensional structure. Quantum mechanics courses often explore the propagation of Gaussian wave packets, but the diffractionlike mathematics is sufficiently tractable to investigate the propagation of other wave packets, both numerically and analytically. More importantly, the diffraction analogy facilitates the development of an intuitive understanding of the role that the initial phase plays in free-particle wave packet evolution. This article considers some of the effects of the initial phase function on the subsequent evolution of free-particle wave packets in the position representation. These considerations reinforce the idea that the classical mechanics limit embodied in the correspondence principle and formalized in the Ehrenfest theorem is necessarily an incomplete representation of quantum behavior.

From slow to superluminal propagation: Dispersive properties of surface plasmon polaritons in linear chains of metallic nanospheroids

We consider propagation of surface plasmon polaritons (SPPs) in linear periodic chains (LPCs) of prolate and oblate metallic spheroids. We show that the SPP group velocity can be efficiently controlled by varying the aspect ratio of the spheroids. For sufficiently small aspect ratios, a gap appears in the first Brillouin zone of the chain lattice in which propagating modes do not exist. Depending on the SPP polarization, the gap extends to certain intervals of the Bloch wave number $q$. Thus, for transverse polarization, no propagating SPPs exist with wave numbers $q$ such that ${q}_{c}^{\ensuremath{\perp}}<|q|<\ensuremath{\pi}/h$, $h$ being the chain period. For longitudinally polarized SPPs, the gap spans the interval $|q|<{q}_{c}^{\ensuremath{\parallel}}$. Here ${q}_{c}^{\ensuremath{\perp}}$ and ${q}_{c}^{\ensuremath{\parallel}}$ are different constants, which depend on the chain parameters, spheroid aspect ratio, and its orientation with respect to the chain axis. The dependence of the dispersion curves on the spheroid aspect ratio leads to a number of interesting effects. In particular, bandwidth of SPPs that can propagate in an LPC can be substantially increased by utilizing prolate or oblate spheroids. When $q$ is close to a critical value, so that $|q\ensuremath{-}{q}_{c}^{\ensuremath{\perp}}|⪡\ensuremath{\pi}/h$ or $|q\ensuremath{-}{q}_{c}^{\ensuremath{\parallel}}|⪡\ensuremath{\pi}/h$, the decay length of the SPPs is dramatically increased. In addition, the dispersion curves acquire a very large positive or negative slope. This can be used to achieve superluminal group velocity for realistic chain parameters. We demonstrate superluminal propagation of Gaussian wave packets in numerical simulations. Both theory and simulations are based on Maxwell equations with account of retardation and, therefore, are fully relativistic.

Semiclassical Propagation of Gaussian Wave Packets

We analyze the semiclassical evolution of Gaussian wave packets in chaotic systems. We show that after some short time a Gaussian wave packet becomes a primitive WKB state. From then on, the state can be propagated using the standard time-dependent WKB scheme. Complex trajectories are not necessary to account for the long-time propagation. The Wigner function of the evolving state develops the structure of a classical filament plus quantum oscillations, with phase and amplitude being determined by geometric properties of a classical manifold.

Propagation of Gaussian wave packets in thin periodic quantum waveguides with a nonlocal nonlinearity

We consider the nonlinear Schrodinger equation with an integral Hartree-type nonlinearity in a thin quantum waveguide and study the propagation of Gaussian wave packets localized in the spatial variables. In the case of periodically varying waveguide walls, we establish the relation between the behavior of wave packets and the spectral properties of the auxiliary periodic problem for the one-dimensional Schrodinger equation. We show that for a positive value of the nonlinearity parameter, the integral nonlinearity prevents the packet from spreading as it propagates. In addition, we find situations such that the packet is strongly focused periodically in time and space.

Time propagation of constrained coupled Gaussian wave packets

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.

Semiclassical Dynamics of Electron Transfer at Metal Surfaces

In this Letter, we demonstrate that nonadiabatic dynamics of molecular scattering from metal surfaces can be efficiently simulated by semiclassical Gaussian wave packet propagation on a local complex potential. The method relies on the wideband limit decoupling of the nuclear equations of motion on different electronic states. If the continuum diabatic potential surfaces are assumed to be parallel, the number of Gaussian wave packets spawned scales at most linearly with propagation time, allowing efficient propagation of nuclear dynamics.

A regular semiclassical approximation for the propagation of wave packets with complex trajectories

The semiclassical propagation of Gaussian wave packets by complex classical trajectories involves multiple contributing and noncontributing solutions interspersed by phase space caustics. Although the phase space caustics do not generally lie exactly on the relevant trajectories, they might strongly affect the semiclassical evolution depending on their proximity to them. In this paper, we derive a third-order regular semiclassical approximation which correctly accounts for the caustics and which is finite everywhere. We test the regular formula for the potential V( x)= 1/x 2 , where the complex classical trajectories and phase space caustics can be computed analytically. We make a detailed analysis of the structure of the complex functions involved in the saddle point approximations and show how the changes in the steepest descent integration contour control both the contributing and noncontributing trajectories and the type of Airy function that appears in the regular approximation.

Gaussian resolutions for equilibrium density matrices

The Gaussian wavepacket propagation method of Hellsing et al. [Chem. Phys. Lett. 122 (1985) 303] for the computation of equilibrium density matrices ρ̂T is revisited and modified. The variational principle applied to the ‘imaginary time’ Schrödinger equation provides the equations of motion for Gaussians in a resolution of ρ̂T, described by their width matrix, center and scale factor, all treated as dynamical variables. The method is computationally very inexpensive, has favorable scaling with the system size and, with the current implementation, is surprisingly accurate in a wide temperature range, even for cases involving quantum tunneling. Incorporation of symmetry constraints, such as reflection or particle statistics, is discussed as well.

Estimation of purity in terms of correlation functions

We prove a rigorous inequality that estimates the purity of a reduced density matrix of a composite quantum system in terms of cross correlation of the same state and an arbitrary product state. Various immediate applications of our result are proposed, in particular, concerning Gaussian wave-packet propagation under classically regular dynamics.

Multidimensional quantum propagation with the help of coupled coherent states

A previous initial value coupled coherent state (CCS) representation is applied to Gaussian wave packet propagation on multidimensional Henon Heiles potentials. Solutions of the time-dependent integro-differential Schrödinger equation are obtained in a basis of trajectory guided Frozen Gaussian Coherent States, with Monte Carlo sampling to ensure a unique capability for propagating multidimensional wave functions. Results, which are obtained for up to 14 D, are compared with those derived by the Herman–Kluk semiclassical initial value representation (IVR) wave packet method.

Description of tunneling with the help of coupled frozen Gaussians

A previous initial value coherent state representation is applied to Gaussian wave packet propagation on one-dimensional and two-dimensional double well potentials. The method uses a basis of trajectory guided frozen Gaussian coherent states, sampled from a Monte Carlo ensemble, for numerical solutions of the quantum time dependent integro-differential Schrödinger equation. Accurate tunneling splittings are obtained. Comparisons are also made with corresponding results obtained by the Herman–Kluk semiclassical initial value representation wave packet method.