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The Journal of Chemical Physics

Nested interaction representations in time dependent quantum mechanics

David J. Tannor, Anna Besprozvannaya, Carl J. Williams

Published Feb 15, 1992• 38 citations
Publisher

Abstract

Two significant developments in the formulation of the equations of motion in the interaction representation (IR) are introduced in this paper. These developments have direct bearing on the efficient propagation in the interaction representation. The first of these developments involves the concept of sequential, or ‘‘nested’’ interaction transformations on the Hamiltonian and the wave function. Two such nested IRs are described. The first is a momentum- or ‘‘P-adapted’’ IR which, in addition to removing wave packet spreading, systematically maintains the average momentum 〈P〉 at zero. This facilitates a grid which not only has a minimal range, but whose points are maximally sparse. Evaluating the Hamiltonian operator in the P-adapted interaction representation involves only one simple algorithmic modification to the ordinary IR: replacing eiH0t/ℏV(R)e−iH0t /ℏ→eiH0t/ℏV(R+〈P〉t/m) eiH0t/ℏ, i.e., using the potential energy function which is downstream from the wave function by the classical propagation distance. A PR-adapted IR is also defined, which systematically maintains both the average position 〈R〉 and momentum 〈P〉 at zero. The PR-adapted IR also involves only one modification to the ordinary IR: replacing eiH0t/ℏV(R)e−iH0t /ℏ→eiH0t/ℏV(R+〈R〉 +〈P〉t/m)eiH0t/ℏ. The second formal development consists of the application of the operator equation eiH0t/ℏV(R)e−iH0t /ℏ=V(R+Pt/m) in conjunction with the nested IRs. This formal identity, which is essentially the Heisenberg representation for V(R), provides a convenient route to propagation in the interaction representation without reconstructing the Schrödinger wave function as an intermediate. The time propagation in all representations is performed using an iterative Lanczos reduction scheme combined with a second- order Magnus expansion. Numerical results for the exponential and the 1/R potential are presented, and illustrate that the new representations can easily lead to a savings of an order of magnitude in the size of the grid required for the propagation.

DOI
10.1063/1.461997

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Two significant developments in the formulation of the equations of motion in the interaction representation (IR) are introduced in this paper. These developments have direct bearing on the efficient propagation in the interaction representation. The first of these developments involves the concept of sequential, or ‘‘nested’’ interaction transformations on the Hamiltonian and the wave function. Two such nested IRs are described. The first is a momentum- or ‘‘P-adapted’’ IR which, in addition to removing wave packet spreading, systematically maintains the average momentum 〈P〉 at zero. This facilitates a grid which not only has a minimal range, but whose points are maximally sparse. Evaluating the Hamiltonian operator in the P-adapted interaction representation involves only one simple algorithmic modification to the ordinary IR: replacing eiH0t/ℏV(R)e−iH0t /ℏ→eiH0t/ℏV(R+〈P〉t/m) eiH0t/ℏ, i.e., using the potential energy function which is downstream from the wave function by the classical propagation distance. A PR-adapted IR is also defined, which systematically maintains both the average position 〈R〉 and momentum 〈P〉 at zero. The PR-adapted IR also involves only one modification to the ordinary IR: replacing eiH0t/ℏV(R)e−iH0t /ℏ→eiH0t/ℏV(R+〈R〉 +〈P〉t/m)eiH0t/ℏ. The second formal development consists of the application of the operator equation eiH0t/ℏV(R)e−iH0t /ℏ=V(R+Pt/m) in conjunction with the nested IRs. This formal identity, which is essentially the Heisenberg representation for V(R), provides a convenient route to propagation in the interaction representation without reconstructing the Schrödinger wave function as an intermediate. The time propagation in all representations is performed using an iterative Lanczos reduction scheme combined with a second- order Magnus expansion. Numerical results for the exponential and the 1/R potential are presented, and illustrate that the new representations can easily lead to a savings of an order of magnitude in the size of the grid required for the propagation.
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.
Abstract Two methods for the numerical integration of the time‐dependent Schrödinger equation with given initial conditions (initial wave packet) are presented. The first method (method A) is based on the Schrödinger representation of the quantum‐dynamical system while the second one (method B) is based upon the intermediate representation. In both cases the quantum dynamical equation is transformed into a system of Hamilton‐Jacobi type equations of motion as occurring in multi particle classical dynamics, i.e. standard molecular dynamics techniques can be applied for the integration. The dynamics of a minimum uncertainty Gaussian wave packet in a strongly anharmonic oscillator is taken as an example.
The general, N state matrix representation of the time-dependent Schrödinger equation is equivalent to an N degree of freedom classical Hamiltonian system. We describe how classical mechanical methods and ideas can be applied towards understanding and modeling exact quantum dynamics. Two applications are presented. First, we illustrate how qualitative insights may be gained by treating the two state problem with a time-dependent coupling. In the case of periodic coupling, Poincaré surfaces of section are used to view the quantum dynamics, and features such as the Floquet modes take on interesting interpretations. The second application illustrates computational implications by showing how Liouville’s theorem, or more generally the symplectic nature of classical Hamiltonian dynamics, provides a new perspective for carrying out numerical wave packet propagation. We show how certain simple and explicit symplectic integrators can be used to numerically propagate wave packets. The approach is illustrated with an application to the problem of a diatomic molecule interacting with a laser, although it and related approaches may be useful for describing a variety of problems.
It is shown that application of the time-dependent variational principle to trial wave functions of a particular form results in a description of the quantum system as a classical system consisting of two mutually interacting parts: the ordinary classical analog of the quantum system and a (generally finite) number of additional degrees of freedom. This holds out the prospect of simulating quantum dynamics by coupling the corresponding classical system to additional degrees of freedom, and it is pointed out that such calculations have, in effect, already been performed. Since the trial wave functions that qualify for this treatment may be arbitrarily accurate, there is no limitation to the accuracy of the resulting classical description. In a particular case, the canonical coordinates and momenta for all degrees of freedom of the overall classical system are identified and investigated. In general, certain parameters appearing in the trial wave functions treated in this work may become redundant at certain times, leading to singularities in the Hamiltonian equations of motion. It is shown, however, that in certain such cases, a classical formalism for the quantum dynamics remains valid if the Hamiltonian function is suitably modified. Calculations applying the present Hamiltonian formalism are carried out that demonstrate that it has better convergence properties than an alternative, nonvariational, Hamiltonian technique suggested by previous work. These numerical studies also suggest that the observed spreading of wave packets and development of complicated structure in wave functions may be interpreted classically in terms of energy transfer among the various degrees of freedom in the overall classical system.
The time-dependent form of the Lippmann–Schwinger integral equation is used as the basis for a novel wave-packet propagation scheme. The method has the advantage over a previous integral equation treatment in that it does not require extensive matrix inversions involving the potential. This feature will be important when applications are made to systems where in some degrees of freedom the potential is expressed in a basis expansion. As was the case for the previous treatment, noniterated and iterated versions of the equations are given; the iterated equations, which are much simpler in the present new scheme than in the old, eliminate a matrix inversion that is required for solving the earlier noniterated equations. In the present noniterated equations, the matrix to be inverted is a function of the kinetic energy operator and thus is diagonal in a Bessel function basis set (or a sine basis set, if the centrifugal potential operator is incorporated into an effective potential). Transition amplitudes for various orbital angular momentum quantum numbers can be obtained from: (1) Fourier transform of the amplitude density from the time to the energy domain, and (2) direct analysis of the scattered wave packet. The approach is illustrated by an application to a standard potential scattering model problem.
This chapter begins by developing the relation between the quantum theory and the linear algebra. It discusses the most fundamental postulates of the theory and provides a phenomenological development of the Schrödinger wave equation (SWE) although the results might be gleaned from the classical Lagrangian and Hamiltonian with the Poisson bracket relations. Afterward some simple examples for the infinitely and finitely deep wells help clarify the basic postulates. The simple harmonic oscillator has perhaps the most sweeping implications and applications for the quantum theory. For this reason, we present the most applicable formalism for the Harmonic oscillator that uses the operator approach rather than the classical method of partial differential equations. The angular momentum and spin are presented for the study of the atom and for applications for nanodevices, and quantum computing. Next the representation formalism clarifies the distinction between the dynamical operators and the wave functions and their interrelations. The chapter discusses both timeindependent and time-dependent perturbation theories. The density operator combines the quantum theory with the classical theory for finding the behavior of a system. The density operator represents one of the basic concepts in the quantum theory. The remainder of the chapter introduces the more advanced material meant to introduce the quantum field theory starting with the second quantization and the propagator.