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Physical Review A

Method for evaluating one-dimensional path integrals

Michael Nauenberg, Fred Kuttner, Miguel A. Furman

Published Mar 1, 1976• 17 citations
Publisher

Abstract

A practical method is developed for calculating numerically one-dimensional path integrals for an arbitrary potential $V(x)$. For the oscillator potential $V(x)=\frac{k{x}^{2}}{2}$ the well-known analytic solution is obtained. To illustrate the numerical convergence of this method the path integral for the Ginzburg-Landau potential, $V(x)=\ensuremath{-}\frac{A{x}^{2}}{2}+\frac{B{x}^{4}}{4}$, is calculated over a range of the positive constants $A$ and $B$ and compared with numerical solutions of the Schr\"odinger equation.

DOI
10.1103/physreva.13.1185

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A practical method is developed for calculating numerically one-dimensional path integrals for an arbitrary potential V(x)V(x)V(x). For the oscillator potential V(x)=kx22V(x)=\frac{k{x}^{2}}{2}V(x)=2kx2​ the well-known analytic solution is obtained. To illustrate the numerical convergence of this method the path integral for the Ginzburg-Landau potential, V(x)=\ensuremath−Ax22+Bx44V(x)=\ensuremath{-}\frac{A{x}^{2}}{2}+\frac{B{x}^{4}}{4}V(x)=\ensuremath−2Ax2​+4Bx4​, is calculated over a range of the positive constants AAA and BBB and compared with numerical solutions of the Schr"odinger equation.
So far we have computed path integrals for potentials V which are linear in the variable x: V(x) = cx for some constant c. Consider now a quadratic potential: V(x) = cx2. Specifically, consider V(x) =C9; 2 mx 2/2 where m is the mass of a simple harmonic oscillator with frequency v = ω/2π. As discussed in Chapter 2 the equation of motion is 1 x(t)=Acos⁡(ωt−δ)=Asin⁡(ωt+Φ),x(t) = A\cos (\omega t - \delta ) = A\sin (\omega t + \Phi ),x(t)=Acos(ωt−δ)=Asin(ωt+Φ),
The work of Reading and Sigel on the solution of the one-dimensional Schr\"odinger equation for a particle moving in the presence of a collection of $\ensuremath{\delta}$-function potentials of arbitrary strength and position is generalized so that any arbitrary potential V(x)V(x)V(x) can be handled. A close connection is shown to exist between the methods described in this note and those, involving Jost functions, which have been developed for handling the radial Schr\"odinger equation in three dimensions.
In this paper, one of the numerical solution method of one- particle, one dimensional timeindependentSchrodinger equation are presented that allows one to obtain accurate bound state eigenvalues and functions for an arbitrary potential energy function V(x).For each case, we draw eigen functions versus the related reduced variable for the correspondingenergies. The paper ended with a comparison of the result obtained by the numerical solutions withthose obtained via the analytical solutions. The agreement between the results obtained by analyticalsolution method and numerical solution is represents the top Numerov method for numerical solutionSchrodinger equation with different potentials energy.