Semiclassical dynamics with quantum trajectories: Formulation and comparison with the semiclassical initial value representation propagator (original) (raw)

Semiclassical dynamics based on quantum trajectories

Chemical Physics Letters, 2002

We present a trajectory-based method that incorporates quantum effects in the context of Hamiltonian dynamics. It is based on propagation of trajectories in the presence of quantum potential within the hydrodynamic formulation of the Schr€ o odinger equation. The quantum potential is derived from the density approximated as a linear combination of gaussian functions. One-gaussian fit gives exact result for parabolic potentials, as do successful semiclassical methods. The limit of the large number of fitting gaussians and trajectories gives the full quantum-mechanical result. The method is systematically improvable from classical to fully quantum, as demonstrated on a transmission through the Eckart barrier.

Comparative Study of Semiclassical Approaches to Quantum Dynamics

International Journal of Modern Physics C - IJMPC, 2009

Quantum states can be described equivalently by density matrices, Wigner functions, or quantum tomograms. We analyze the accuracy and performance of three related semiclassical approaches to quantum dynamics, in particular with respect to their numerical implementation. As test cases, we consider the time evolution of Gaussian wave packets in different one-dimensional geometries, whereby tunneling, resonance, and anharmonicity effects are taken into account. The results and methods are benchmarked against an exact quantum mechanical treatment of the system, which is based on a highly efficient Chebyshev expansion technique of the time evolution operator.

Cellular dynamics: A new semiclassical approach to time-dependent quantum mechanics

The Journal of Chemical Physics, 1991

A new semiclassical approach that constructs the full semiclassical Green's function propagation of any initial wave function directly from an ensemble of real trajectories, without root searching, is presented. Each trajectory controls a cell of initial conditions in phase space, but the cell area is not constrained by Plan&s constant. The method is shown to be accurate for rather long times in anharmonic oscillators, indicating the semiclassical time-dependent Green's function is clearly worthy of more study. The evolution of wave functions in anharmonic potentials is examined and a spectrum from the semiclassical correlation function is calculated, comparing with exact fast Fourier transform results.

Beating the Efficiency of Both Quantum and Classical Simulations with a Semiclassical Method

Physical Review Letters, 2011

While rigorous quantum dynamical simulations of many-body systems are extremely difficult (or impossible) due to exponential scaling with dimensionality, the corresponding classical simulations ignore quantum effects. Semiclassical methods are generally more efficient but less accurate than quantum methods and more accurate but less efficient than classical methods. We find a remarkable exception to this rule by showing that a semiclassical method can be both more accurate and faster than a classical simulation. Specifically, we prove that for the semiclassical dephasing representation the number of trajectories needed to simulate quantum fidelity is independent of dimensionality and also that this semiclassical method is even faster than the most efficient corresponding classical algorithm. Analytical results are confirmed with simulations of fidelity in up to 100 dimensions with 2 1700 -dimensional Hilbert space.

Modified quantum trajectory dynamics using a mixed wave function representation

The Journal of Chemical Physics, 2004

Dynamics of quantum trajectories provides an efficient framework for description of various quantum effects in large systems, but it is unstable near the wave function density nodes where the quantum potential becomes singular. A mixed coordinate space/polar representation of the wave function is used to circumvent this problem. The resulting modified trajectory dynamics associated with the polar representation is nonsingular and smooth. The interference structure and the nodes of the wave function density are described, in principle, exactly in the coordinate representation. The approximate version of this approach is consistent with the semiclassical linearized quantum force method ͓S. Garashchuk and V. A. Rassolov, J. Chem. Phys. 120, 1181 ͑2004͔͒. This approach is exact for general wave functions with the density nodes in a locally quadratic potential.

Beating the efficiency of both quantum and classical simulations with semiclassics

2011

While rigorous quantum dynamical simulations of many-body systems are extremely difficult (or impossible) due to the exponential scaling with dimensionality, corresponding classical simulations completely ignore quantum effects. Semiclassical methods are generally more efficient but less accurate than quantum methods, and more accurate but less efficient than classical methods. We find a remarkable exception to this rule by showing that a semiclassical method can be both more accurate and faster than a classical simulation. Specifically, we prove that for the semiclassical dephasing representation the number of trajectories needed to simulate quantum fidelity is independent of dimensionality and also that this semiclassical method is even faster than the most efficient corresponding classical algorithm. Analytical results are confirmed with simulations of quantum fidelity in up to 100 dimensions with 2^1700-dimensional Hilbert space.

Semiclassical nonadiabatic dynamics using a mixed wave-function representation

The Journal of Chemical Physics, 2005

Nonadiabatic effects in quantum dynamics are described using a mixed polar/coordinate space representation of the wave function. The polar part evolves on dynamically determined potential surfaces that have diabatic and adiabatic potentials as limiting cases of weak localized and strong extended diabatic couplings. The coordinate space part, generalized to a matrix form, describes transitions between the surfaces. Choice of the effective potentials for the polar part and partitioning of the wave function enables one to represent the total wave function in terms of smooth components that can be accurately propagated semiclassically using the approximate quantum potential and small basis sets. Examples are given for two-state one-dimensional problems that model chemical reactions that demonstrate the capabilities of the method for various regimes of nonadiabatic dynamics.

Quantum wave packet dynamics with trajectories: reflections on a downhill ramp potential

Chemical Physics Letters, 2000

The quantum trajectory method was recently developed to solve the hydrodynamic equations of motion in the Lagrangian, moving-with-the-fluid, picture. In this approach, trajectories are integrated for fluid elements ͑''particles''͒ moving under the influence of the combined force from the potential surface and the quantum potential. To accurately compute the quantum potential and the quantum force, it is necessary to obtain the derivatives of a function given only the values on the unstructured mesh defined by the particle locations. However, in some regions of space-time, the particle mesh shows compression and inflation associated with regions of large and small density, respectively. Inflation is especially severe near nodes in the wave function. In order to circumvent problems associated with highly nonuniform grids defined by the particle locations, adaptation of moving grids is introduced in this study. By changing the representation of the wave function in these local regions ͑which can be identified by diagnostic tools͒, propagation is possible to much longer times. These grid adaptation techniques are applied to the reflected portion of a wave packet scattering from an Eckart potential.

Semiclassical approximation with zero velocity trajectories

Chemical Physics, 2007

We present a new semiclassical method that yields an approximation to the quantum mechanical wavefunction at a fixed, predetermined position. In the approach, a hierarchy of ODEs are solved along a trajectory with zero velocity. The new approximation is local, both literally and from a quantum mechanical point of view, in the sense that neighboring trajectories do not communicate with each other. The approach is readily extended to imaginary time propagation and is particularly useful for the calculation of quantities where only local information is required. We present two applications: the calculation of tunneling probabilities and the calculation of low energy eigenvalues. In both applications we obtain excellent agrement with the exact quantum mechanics, with a single trajectory propagation.