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.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.

arXiv:2509.13005Sep 16, 20250 citations

Gaussian Wave Packets

Anatoli V. Babin, Alexander Figotin

Theoretical and mathematical physicsJan 1, 20161 citation

Wave packet construction in two-dimensional quantum billiards: Blueprints for the square, equilateral triangle, and circular cases

M. A. Doncheski, S. Heppelmann, Richard W. Robinett +1 more

We present quasi-analytical and numerical calculations of Gaussian wave packet solutions of the Schr\\"odinger equation for two-dimensional infinite well and quantum billiard problems with equilateral triangle, square, and circular footprints. These cases correspond to N=3, N=4, and $N \\to \\infty$ regular polygonal billiards and infinite wells, respectively. In each case the energy eigenvalues and wavefunctions are given in terms of familiar special functions. For the first two systems, we obtain closed form expressions for the expansion coefficients for localized Gaussian wavepackets in terms of the eigenstates of the particular geometry. For the circular case, we discuss numerical approaches. We use these results to discuss the short-time, quasi-classical evolution in these geometries and the structure of wave packet revivals. We also show how related half-well problems can be easily solved in each of the three cases.We present quasi-analytical and numerical calculations of Gaussian wave packet solutions of the Schr\\"odinger equation for two-dimensional infinite well and quantum billiard problems with equilateral triangle, square, and circular footprints. These cases correspond to N=3, N=4, and $N \\to \\infty$ regular polygonal billiards and infinite wells, respectively. In each case the energy eigenvalues and wavefunctions are given in terms of familiar special functions. For the first two systems, we obtain closed form expressions for the expansion coefficients for localized Gaussian wavepackets in terms of the eigenstates of the particular geometry. For the circular case, we discuss numerical approaches. We use these results to discuss the short-time, quasi-classical evolution in these geometries and the structure of wave packet revivals. We also show how related half-well problems can be easily solved in each of the three cases.

Jul 9, 200320 citations

Wave Packet Solutions of the Schrdinger Equation for Simple Harmonic Oscillator

Dong Shao

By solving exactly the time dependent Schrdinger equation for simple harmonic oscillator, we have obtained respectively the wave packet solutions associated with the displaced ground state and the displaced excited eigenstates, and then demonstrated the centroid of the wave packet oscillates according to the classical law and that the wave packet form remains unchanged. For simple harmonic oscillator with the force constant depending on time the wave packets with the same structures as that of the usual simple harmonic oscillator can still be found to move in such a way that the centroids move according to the classical law, but their wave packets may deform in time. Depending on how the force constant varies with time, the wave packets can spread, contract or pulsate.By solving exactly the time dependent Schrdinger equation for simple harmonic oscillator, we have obtained respectively the wave packet solutions associated with the displaced ground state and the displaced excited eigenstates, and then demonstrated the centroid of the wave packet oscillates according to the classical law and that the wave packet form remains unchanged. For simple harmonic oscillator with the force constant depending on time the wave packets with the same structures as that of the usual simple harmonic oscillator can still be found to move in such a way that the centroids move according to the classical law, but their wave packets may deform in time. Depending on how the force constant varies with time, the wave packets can spread, contract or pulsate.

Journal of Sichuan Teachers CollegeJan 1, 20020 citations

N \\to \\infty

regular polygonal billiards and infinite wells, respectively. In each case the energy eigenvalues and wavefunctions are given in terms of familiar special functions. For the first two systems, we obtain closed form expressions for the expansion coefficients for localized Gaussian wavepackets in terms of the eigenstates of the particular geometry. For the circular case, we discuss numerical approaches. We use these results to discuss the short-time, quasi-classical evolution in these geometries and the structure of wave packet revivals. We also show how related half-well problems can be easily solved in each of the three cases.