WavePacket: A Matlab package for numerical quantum dynamics. I: Closed quantum systems and discrete variable representations

Abstract

WavePacket is an open-source program package for the numerical simulation of quantummechanical dynamics. It can be used to solve time-independent or time-dependent linear Schrödinger and Liouville-von Neumann-equations in one or more dimensions. Also coupled equations can be treated, which allows to simulate molecular quantum dynamics beyond the Born-Oppenheimer approximation. Optionally accounting for the interaction with external electric fields within the semiclassical dipole approximation, WavePacket can be used to simulate experiments involving tailored light pulses in photo-induced physics or chemistry. The graphical capabilities allow visualization of quantum dynamics 'on the fly', including Wigner phase space representations. Being easy to use and highly versatile, WavePacket is well suited for the teaching of quantum mechanics as well as for research projects in atomic, molecular and optical physics or in physical or theoretical chemistry. The present Part I deals with the description of closed quantum systems in terms of Schrödinger equations. The emphasis is on discrete variable representations for spatial discretization as well as various techniques for temporal discretization. The upcoming Part II will focus on open quantum systems and dimension reduction; it also describes the codes for optimal control of quantum dynamics. The present work introduces the MATLAB version of WavePacket 5.2.1 which is hosted at the Sourceforge platform, where extensive Wiki-documentation as well as worked-out demonstration examples can be found. Programming language: Matlab Supplementary material: n. a. Journal reference of previous version: n. a. Does the new version supersede the previous version?: n. a. Reasons for the new version: n. a. Summary of revisions: n. a.* Nature of problem: Schrödinger's equations are of fundamental importance in non-relativistic quantum mechanics of distinguishable particles. The solutions of the time-independent equation (TISE) are wavefunctions in coordinate space, the absolute squares of which are usually interpreted as probability density. The time-dependent equation (TDSE) describes the dynamics of a quantum system evolving in time. It plays a crucial role for the simulation, understanding, and prediction of modern experiments in atomic, molecular and optical physics where systems are driven by temporally shaped external fields. Solution method:All numerical methods compiled in WavePacket are based on discrete variable representations.Currently implemented are Gauss-Hermite, Gauss-Legendre and FFT-based schemes. The TISE is solved either by direct diagonalization or by propagation in imaginary time. For the TDSE there is a choice between second order differencing, operator splitting and Chebychev polynomial methods. Additional comments including Restrictions and Unusual features:The WavePacket program package is rather easy and intuitive to use, providing visualization of quantum dynamics 'on the fly'. It is mainly intended for low-dimensional systems, typically not exceeding three t...

Electronic structure and time-dependent description of rotational predissociation of LiH

“…1 3.015 20288 1407 Boutalib 1992 (theory) 6 3.007 20174 The peaks in the absorption spectrum (FIG. 3) are obtained by solving the time dependent Schrödinger equation 52 (Eq. 12) with a Gaussian-shaped wavepacket Φ initially centered at 6.15 a 0 and possessing the half-width equal to 0.95 a 0 .…”

Conditional quasi-exact solvability of the quantum planar pendulum and of its anti-isospectral hyperbolic counterpart

“…The corresponding wavefunctions can be calculated from equation (38). For the A 1 symmetry, we obtain [54]. [54].…”

“…For the A 1 symmetry, we obtain [54]. [54]. with the requisite eigenvectors given as rows (the numbering of which starts from 0) of the following matrices: …”

“…Circles show analytic eigenenergies, which coincide for the trigonometric and hyperbolic systems. The numerical values were obtained with WavePacket software [54].…”

“…Hence, the approach outlined in Section 3.3, based on the tridiagonal matrices (17)- (19), is not suitable for generating states beyond the range of the analytic solutions of the hyperbolic system. Instead, we use the Fourier Grid Hamiltonian (FGH) approach [57,58] implemented in the qm bound program of the WavePacket software package [54]. Within the energy ranges considered here, well-converged energies are obtained using 1024 equally spaced grid points.…”