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arXiv:2303.09464 · Journal of Computational Physics

The discrete Green's function method for wave packet expansion via the free Schrödinger equation

Jan-Frederik Mennemann, Sebastian Erne, I. E. Mazets, Norbert J. Mauser

Published May 23, 2024• 2 citations
arXivHTMLPDF

Abstract

We consider the expansion of wave packets governed by the free Schrödinger equation. This seemingly simple task plays an important role in simulations of various quantum experiments, especially in the field of matter-wave interferometry. The initial tight confinement of quantum particles results in a very fast expansion of the wave function at later times which significantly complicates an efficient and precise numerical evaluation. In many practical cases the expansion time is too short for the validity of the stationary phase approximation and too long for an efficient application of Fourier collocation-based methods. We develop an alternative method based on a discretization of the free-particle propagator. This simple approach yields highly accurate results which readily follows from the exceptionally fast convergence of the trapezoidal rule approximation of integrals involving smooth, rapidly decaying functions. We discuss and analyze our approach in detail and demonstrate how to estimate the numerical error in the one-dimensional setting. Furthermore, we show that by exploiting the separability of the Green's function, the numerical effort of the multi-dimensional approximation is considerably reduced. Our method is very fast, highly accurate, and easy to implement on modern hardware.

DOI
10.1016/j.jcp.2024.113131

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We consider the expansion of wave packets governed by the free Schrödinger equation. This seemingly simple task plays an important role in simulations of various quantum experiments, especially in the field of matter-wave interferometry. The initial tight confinement of quantum particles results in a very fast expansion of the wave function at later times which significantly complicates an efficient and precise numerical evaluation. In many practical cases the expansion time is too short for the validity of the stationary phase approximation and too long for an efficient application of Fourier collocation-based methods. We develop an alternative method based on a discretization of the free-particle propagator. This simple approach yields highly accurate results which readily follows from the exceptionally fast convergence of the trapezoidal rule approximation of integrals involving smooth, rapidly decaying functions. We discuss and analyze our approach in detail and demonstrate how to estimate the numerical error in the one-dimensional setting. Furthermore, we show that by exploiting the separability of the Green's function, the numerical effort of the multi-dimensional approximation is considerably reduced. Our method is very fast, highly accurate, and easy to implement on modern hardware.
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.
The physics of special relativity is applied to a wave-packet model of a particle without the usual de Broglie-like assumption of Planck's rule, E = hv. The velocity of the particle is equated to the group-velocity of the wave-packet to associate the particle with its wave-packet. With such an identification it is demonstrated using special relativity that the momentum four-vector, P = [p, E/c2] of the particle is parallel and proportional to a central wave-number four-vector, K0 = [k0. ω0/c2], of the particle's wave-packet. The constant of proportionality, relating these two four-vectors, has the dimension of action and by diffraction experiments it has the value h/2π where h is Planck's constant. This refinement of the de Broglie wave-like nature of a particle suggests that special relativity underlies quantum physics more deeply than heretofore believed. The above quantum rules are used to find the wave-packet function in the one-dimensional case of a non-relativistic free particle and a particle in a conservative force field of potential V (x). This wave-packet function is shown to satisfy an integral time-evolution equation which relates the wave function at time t0 to the function at a slightly later time t. The resulting integral equation is equivalent to the propagator equation or path-integral hypothesised by Feynman. This fact also demonstrates, using the method of Feynman, that the wave-packet function satisfies Schrödinger's partial differential equation. The wave-packet function of a particle in a force field is directly shown to satisfy a linear integral equation which relates the wave-packet at one instant of time to its values at all later times. The kernel or propagator of this integral equation is also equivalent to Feynman's path integral. A computation demonstrates that the position of a moving particle in a simple quadratic potential well does not become more uncertain with the passage of time. This contrasts with the behaviour of a moving free particle which as a function of time becomes more and more difficult to accurately locate.
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To model the decay of a quasibound state we use the modified two-potential approach introduced by Gurvitz and Kalbermann [1,2]. This method has proved itself useful in the past for calculating the decay width and the energy shift of an isolated quasistationary state [5]. We follow the same approach in order to propagate the wave-packet in time with the ultimate goal of extracting the momentum-distribution of emitted particles. The advantage of the method is that it provides the time-dependent wave function in a simple semi-analytic form. We intend to apply this method to the modeling of metastable states for which no direct integration of the time-dependent Schrödinger equation is available today.
We develop methodology for performing time dependent quantum mechanical calculations by representing the wave function as a sum of Gaussian wave packets (GWP) each characterized by a set of parameters such as width, position, momentum, and phase. The problem of computing the time evolution of the wave function is thus reduced to that of finding the time evolution of the parameters in the Gaussians. This parameter motion is determined by minimizing the error made by replacing the exact wave function in the time dependent Schrödinger equation with its Gaussian representation approximant. This leads to first order differential equations for the time dependence of the parameters, and those describing the packet position and the momentum of each packet have some resemblance with the classical equations of motion. The paper studies numerically the strategy needed to achieve the best GWP representation of time dependent processes. The issues discussed are the representation of the initial wave function, the numerical stability and the solution of the differential equations giving the evolution of the parameters, and the analysis of the final wave function. Extensive comparisons are made with an approximate method which assumes that the Gaussians are independent and their width is smaller than the length scale over which the potential changes. This approximation greatly simplifies the calculations and has the advantage of a greater resemblance to classical mechanics, thus being more intuitive. We find, however, that its range of applications is limited to problems involving localized degrees of freedom that participate in the dynamic process for a very short time. Finally we give particular attention to the notion that the GWP representation of the wave function reduces the dynamics of one quantum degree of freedom to that of a set of pseudoparticles (each represented by one packet) moving according to a pseudoclassical (i.e., classical-like) mechanics whose ‘‘phase space’’ is described by a position and momentum as well as a complex phase and width.
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.