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Journal of Computational Physics

Hagedorn wavepackets and Schrödinger equation with time-dependent, homogeneous magnetic field

Vasile Catrinel Grădinaru, Oliver Rietmann

Published Jul 29, 2021• 9 citations
Publisher

Abstract

Certain generalized coherent states, so-called Hagedorn wavepackets, have been used to numerically solve the standard Schrödinger equation. We extend this approach and its recent enhancements to the magnetic Schrödinger equation for a time-dependent, spatially homogeneous magnetic field. We explain why Hagedorn wavepackets are naturally compatible with the aforementioned physical system. In numerical experiments we examine the order of convergence and the preservation of norm and energy. We use this method to simulate a Penning trap as proposed in recent work on quantum computing .

DOI
10.1016/j.jcp.2021.110581

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Certain generalized coherent states, so-called Hagedorn wavepackets, have been used to numerically solve the standard Schrödinger equation. We extend this approach and its recent enhancements to the magnetic Schrödinger equation for a time-dependent, spatially homogeneous magnetic field. We explain why Hagedorn wavepackets are naturally compatible with the aforementioned physical system. In numerical experiments we examine the order of convergence and the preservation of norm and energy. We use this method to simulate a Penning trap as proposed in recent work on quantum computing .
The semiclassical (or Hagedorn) wavepackets depending on a fixed set of parameters are an orthonormal L 2 -basis of generalized coherent states. They have been used to solve numerically the time-dependent Schrödinger equation in its semiclassical formulation, yet their localization property makes them inefficient in case of non-local phenomena such as quantum tunneling. In order to overcome this difficulty, we use simultaneously several bases with different parameters. We propose an algorithm to expand a given wavefunction in terms of multiple families of Hagedorn wavepackets; each family can then be accurately and efficiently propagated using modern semiclassical time-splittings.
Abstract Hagedorn functions are carefully constructed generalizations of Hermite functions to the setting of many-dimensional squeezed and coupled harmonic systems. Wavepackets formed by superpositions of Hagedorn functions have been successfully used to solve the time-dependent Schrödinger equation exactly in harmonic systems and variationally in anharmonic systems. To evaluate typical observables, such as position or kinetic energy, it is sufficient to consider orthonormal Hagedorn functions with a single Gaussian center. Instead, we derive various relations between Hagedorn bases associated with different Gaussians, including their overlaps, which are necessary for evaluating quantities nonlocal in time, such as the time correlation functions needed for computing spectra. First, we use the Bogoliubov transformation to obtain the commutation relations between the ladder operators associated with different Gaussians. Then, instead of using numerical quadrature, we employ these commutation relations to derive exact recurrence relations for the overlap integrals between Hagedorn functions with different Gaussian centers. Finally, we present numerical experiments that demonstrate the accuracy and efficiency of our algebraic method as well as its suitability for treating problems in spectroscopy and chemical dynamics.
In the present paper we consider the semiclassical magnetic Schrödinger equation, which describes the dynamics of charged particles under the influence of an electro-magnetic field. The solution of the time-dependent Schrödinger equation is approximated by a single Gaussian wave packet via the time-dependent Dirac–Frenkel variational principle. For the approximation we use ordinary differential equations of motion for the parameters of the variational solution and extend the second-order Boris algorithm for classical mechanics to the quantum mechanical case. In addition, we propose a modified version of the classical fourth-order Runge–Kutta method. Numerical experiments explore parameter convergence and geometric properties. Moreover, we benchmark against the analytical solution of the Penning trap.
In the present paper we consider the semiclassical magnetic Schrödinger equation, which describes the dynamics of charged particles under the influence of a electro-magnetic field. The solution of the time-dependent Schrödinger equation is approximated by a single Gaussian wave packet via the time-dependent Dirac--Frenkel variational principle. For the approximation we use ordinary differential equations of motion for the parameters of the variational solution and extend the second-order Boris algorithm for classical mechanics to the quantum mechanical case. In addition, we propose a modified version of the classical fourth order Runge--Kutta method. Numerical experiments explore parameter convergence and geometric properties. Moreover, we benchmark against the analytical solution of the Penning trap.
The derivation of a time-dependent Schrödinger equation (TDSE) from a time-independent Schrödinger equation (TISE) in the coherent state representation is considered for the special case of a simple coupled atom-field system described by the soluble Jaynes-Cummings model. The derivation shows why, from the outset, a linear combination of energy eigenstates, instead of a single state, must be used in order to obtain a TDSE for general states. Moreover, this study leads to a method of solving a TDSE by simply solving a TISE.
We review an explicit approach to obtaining numerical solutions of the Schrödinger equation that is conceptionally straightforward and capable of significant accuracy and efficiency. The method and its efficacy are illustrated with several examples. Because of its explicit nature, the algorithm can be readily extended to systems with a higher number of spatial dimensions. We show that the method also generalizes the staggered-time approach of Visscher and allows for the accurate calculation of the real and imaginary parts of the wave function separately.