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Physical Review Letters

Power Series Expansion for the Time Evolution Operator with a Harmonic-Oscillator Reference System

Alexander N. Drozdov

Published Dec 11, 1995• 12 citations
Publisher

Abstract

A simple framework for accurate solution of a general class of one-dimensional Fokker-Planck and/or Schr\"odinger equations is presented. The main idea is representing the propagator in the form $P(x, t|{x}_{0})={P}_{0}(x, t|{x}_{0})\mathrm{exp}[W(x, t|{x}_{0})]$ and expanding the exponent $W$ in a power series in a given function of $t$, where ${P}_{0}$ is the exact solution of a reference harmonic-oscillator problem. The expansion coefficients are analytically evaluated from recursive relations. This approach is shown to be a dramatic improvement over the standard Taylor series expansion for the propagator in that just a few terms of the present expansion are sufficient to attain a very accurate description in the whole time domain.

DOI
10.1103/physrevlett.75.4342

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A simple framework for accurate solution of a general class of one-dimensional Fokker-Planck and/or Schr"odinger equations is presented. The main idea is representing the propagator in the form P(x,t∣x0)=P0(x,t∣x0)exp[W(x,t∣x0)]P(x, t|{x}_{0})={P}_{0}(x, t|{x}_{0})\mathrm{exp}[W(x, t|{x}_{0})]P(x,t∣x0​)=P0​(x,t∣x0​)exp[W(x,t∣x0​)] and expanding the exponent WWW in a power series in a given function of ttt, where P0{P}_{0}P0​ is the exact solution of a reference harmonic-oscillator problem. The expansion coefficients are analytically evaluated from recursive relations. This approach is shown to be a dramatic improvement over the standard Taylor series expansion for the propagator in that just a few terms of the present expansion are sufficient to attain a very accurate description in the whole time domain.
The coordinate matrix element of the time evolution operator, exp[−iĤt/ℏ], is determined by expanding (its exponent) in a power series in t. Recursion relations are obtained for the expansion coefficients which can be analytically evaluated for any number of degrees of freedom. Numerical application to the tunneling matrix element in a double well potential and to the reactive flux correlation function for a barrier potential show this approach to be a dramatic improvement over the standard short time approximation for the propagator. Its use in a Feynman path integral means that fewer ‘‘time slices’’ in the matrix product exp[(−i/ℏ)ΔtĤ]N, Δt=t/N, will be required. The first few terms in the present expansion constitute a fully quantum version of the short time propagator recently obtained by us using semiclassical methods [Chem. Phys. Lett. 151, 1 (1988)].
This paper compares the efficiency and accuracy of different second-order approximations for the single-step propagator obtained by means of the power series expansion formalism and the split operator method. The former is based on a harmonic reference system, while the latter allows for the use of physically motivated (anharmonic) zeroth-order representations. Three typical examples---a system-bath Hamiltonian, a H\'enon-Heiles anharmonic resonating system, and a Fokker-Planck chaotic model---are considered in the present testing. The examples cover a variety of situations with strongly anharmonic coupling. Although no system-specific reference system is involved in the power series representation of the propagator, it quite accurately describes the dynamics of very anharmonic processes in the entire time domain even though the coupling is strong. Another appealing feature of the approach is that it is essentially analytical and therefore does not require any computational effort to implement. This makes the power series expansion technique particularly attractive for efficiently treating many-body problems. In contrast, numerical implementation of the split operator method can be arduous as a general multidimensional calculation, while its utility is in general restricted to the separable limit, when the coupling is almost turned off. In that limit the method indeed provides accurate results over a broad range of t. With increasing coupling, however, the efficiency of the method deteriorates very rapidly regardless of the particular choice of the zeroth-order representation.
In 1926, E.Schrödinger found an analytical solution for the time evolution of the one-dimensional harmonic oscillator Schrödinger Equation. This solution is not in the form of a summation of energy eigenstate wave functions, which provides a new perspective on the time evolution in quantum mechanics. This paper mainly analyzes this analytical time evolution solution and presents two derivation methods. Then the paper further generalizes to obtain more non-series analytical solutions for time evolution oscillator, and finds the recursive relation to obtain these analytical solutions. Finally, the educational significance and applications of these analytical solutions are discussed.
The one-dimensional Fokker-Planck equation is solved by the Asymptotic Taylor Expansion Method for the time-dependent probability density of a particle. Using an ansatz wave function, one obtains the series expansion of the solution for the Schrödinger and it allows one to find out the eigen functions and eigen energies of the states to the evaluation of the probability. The eigen energies of some certain kind of Bistable potentials are calculated for some certain potential parameters. The probability function is determined and graphed for potential parameters. The numerical results are compared with existing literature, and a conclusion about the advantages and disadvantages on the method is given.
In this paper we obtain a propagator of path integral for a harmonic oscillator and a driven harmonic oscillator by using the power series expansion. It is shown that our result for the harmonic oscillator is more exact than the previous one obtained with other approximation methods. By using the same method, we obtain a propagator of path integral for the driven harmonic oscillator, which does not have any exact expansion. The more exact propagators may improve the path integral results for these systems.
We construct an explicit solution of the Cauchy initial value problem for the n-dimensional Schrödinger equation with certain time-dependent Hamiltonian operator of a modified oscillator. The dynamical SU(1,1) symmetry of the harmonic oscillator wave functions, Bargmann’s functions for the discrete positive series of the irreducible representations of this group, the Fourier integral of a weighted product of the Meixner–Pollaczek polynomials, a Hankel-type integral transform, and the hyperspherical harmonics are utilized in order to derive the corresponding Green function. It is then generalized to a case of the forced modified oscillator. The propagators for two models of the relativistic oscillator are also found. An expansion formula of a plane wave in terms of the hyperspherical harmonics and solution of certain infinite system of ordinary differential equations are derived as by-products.