Numerical Methods in Quantum Propagation

Go. Torres-Vega

A numerical propagation scheme based on dual Lanczos transformation theory is introduced for the propagation of quantum mechanical wave functions in coordinate and phase spaces. This method is applied to the numerical propagation of the coherent state moving in the kicked harmonic oscillator potential. We also make comparisons with classical analogs and we make some comments regarding the advantage of working directly in phase space.

和夫 斉藤

Corey Jason Trahan, Róbert E. Wyatt, Bill Poirier

In a previous publication [J. Chem. Phys. 118, 9911 (2003)], the derivative propagation method (DPM) was introduced as a novel numerical scheme for solving the quantum hydrodynamic equations of motion (QHEM) and computing the time evolution of quantum mechanical wave packets. These equations are a set of coupled, nonlinear partial differential equations governing the time evolution of the real-valued functions C and S in the complex action, S=C(r,t) + iS(r,t)/Planck's over 2pi, where Psi(r,t)=exp(S). Past numerical solutions to the QHEM were obtained via ensemble trajectory propagation, where the required first- and second-order spatial derivatives were evaluated using fitting techniques such as moving least squares. In the DPM, however, equations of motion are developed for the derivatives themselves, and a truncated set of these are integrated along quantum trajectories concurrently with the original QHEM equations for C and S. Using the DPM quantum effects can be included at various orders of approximation; no spatial fitting is involved; there is no basis set expansion; and single, uncoupled quantum trajectories can be propagated (in parallel) rather than in correlated ensembles. In this study, the DPM is extended from previous one-dimensional (1D) results to calculate transmission probabilities for 2D and 3D wave packet evolution on coupled Eckart barrier/harmonic oscillator surfaces. In the 2D problem, the DPM results are compared to standard numerical integration of the time-dependent Schrodinger equation. Also in this study, the practicality of implementing the DPM for systems with many more degrees of freedom is discussed.

Wolfgang Schweizer

Go. Torres-Vega, John H. Frederick

A recently proposed phase-space representation for quantum mechanics provides a formal means for investigating the correspondence between classical and quantum mechanics. To explore the implications of this new representation, we introduce a numerical method for propagating an initial wave function directly in phase space. We illustrate the usefulness of the proposed method by analyzing two simple quantum-mechanical systems in phase space.

Tao Pang