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arXiv:2509.13005

Low-complexity approximations with least-squares formulation of the time-dependent Schr{ö}dinger equation

Mi-Song Dupuy, Virginie Ehrlacher, Clément Guillot

Published Sep 16, 2025• 0 citations
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Abstract

We propose new methods designed to numerically approximate the solution to the time dependent Schr{ö}dinger equation, based on two types of ansatz: tensors, and approximation by a linear combination of gaussian wave packets. In both cases, the method can be seen as a restricted optimization problem, which can be solved by adapting either the Alternating Least Square algorithm in the tensor case, or some greedy algorithm in the gaussian wavepacket case. We also discuss the efficiency of both approaches.

DOI
10.48550/arxiv.2509.13005

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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.
We show that one can obtain analytic solutions of the time-dependent Schrödinger equation that are more complex than the well-known oscillating coherent wave packet. Such Hermite-Gaussian or initially square wave packets exist for a free particle or for one subject to the harmonic oscillator potential. In either case, the Hermite-Gaussian packets retain their nodal structure even after long times. There is a single class of exact solutions for the system with oscillator constant K > 0, K = 0, or K < 0, leading to wave functions for the harmonic oscillator, the free particle, and the inverted oscillator, respectively.
We propose an algorithm for the numerical solution of the time-dependent Schrödinger equation for N >> 1bosons interacting via a two-body interaction potential in the mean-field scaling regime. Unlike existing numerical algorithms for quantum dynamics, the proposed algorithm has a computational complexity that is independent of the number N of particles. It is based on an ansatz for the wave function inspired by the Bogoliubov theory, which is an asymptotic theory for large N. In a first part, we follow Bogoliubov theory in evolving the condensate function by the Hartree equation, but we use the Dirac--Frenkel time-dependent variational principle to derive nonlinear equations of motion for the excitation functions in the Bogoliubov ansatz. By carefully exploiting the bosonic symmetry, the so obtained approximate wave functions are shown to have derivatives bounded independently of N apart from a time-dependent global phase factor which rotates with a frequency proportional to N. In the second part, we numerically approximate the wave function (modulo the phase) by a composite algorithm, which employs an appropriate splitting of the equations of motion and uses a dynamical low-rank approximation of the fundamental two-particle excitation function by linear combinations of products of single-particle functions. The low-rank excitation function is evolved in time by a basis update and Galerkin (BUG) integrator that is robust to small singular values of the coefficient matrix and allows for rank adaptivity. Numerical results are provided to assess the performance of the proposed algorithm.
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A method for the solution of the time-dependent Schrödinger equation in two dimensions is presented which is based on the hydrodynamic analogy to quantum mechanics. The continuum introduced in that analogy is approximated by a finite number of particles, whose trajectories are computed. The procedure is applied here to the dynamics of a Gaussian wave packet on a two-dimensional quadratic potential surface containing a saddle point. Comparison with the known analytical solution indicates that the method is capable of accurate results with substantially less computer time required than in methods previously employed. This particular check problem was chosen because of its close relationship to the wave-packet dynamics of chemical reaction rates.