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Physical Review A

Green's functions for the rapid computation of nonrelativistic wave functions

Douglas W. McColm, Glen W. Erickson

Published Sep 1, 1982• 2 citations
Publisher

Abstract

A multiparticle Green's function satisfying an extended form of the Schr\"odinger equation is obtained as a function of the hyperspherical radius in the multidimensional coordinate space. This Green's function is designed for application to quantum-mechanical bound-state problems. Use of the quadratic Fourier-Laplace transform of this function is proposed as a means of simplifying the computational process. A computational formalism employing this transform is developed.

DOI
10.1103/physreva.26.1173

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A multiparticle Green's function satisfying an extended form of the Schr"odinger equation is obtained as a function of the hyperspherical radius in the multidimensional coordinate space. This Green's function is designed for application to quantum-mechanical bound-state problems. Use of the quadratic Fourier-Laplace transform of this function is proposed as a means of simplifying the computational process. A computational formalism employing this transform is developed.
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Abstract The Green function method is a powerful technique for solving the initial value problem in quantum mechanics. Once the Green function is solved the whole wavefunction evolution is represented in a concise form and can be evaluated conveniently by numerical integration. We present a method for constructing the Green functions systematically which is different from the conventional methods of eigenfunction expansion or path integration. By using variable changing, function substitution, and Fourier transforms, the time dependent Schrödinger equations can be simplified and the solutions for the simplified equations can be easily derived. We then obtain the Green functions for the original equations by the reverse transforms. The method is demonstrated for the linear potential, the harmonic oscillator, the centrifugal potential, and the centripetal barrier oscillator, where the Green function for the centripetal barrier oscillator has not been solved previously by conventional methods. The method and examples illustrated in this paper can be utilised to strengthen undergraduate courses on quantum mechanics and/or partial differential equation.
This article is devoted to the theoretical investigation of processes in macroscopic systems based on wave particle duality and quantum statistical distributions using the Green’s function method. In particular, it pro vides a detailed description of the properties and characteristics of the Green’s function for an ideal Fermi liquid, single- and two-particle Green’s functions, as well as Green’s functions at limiting temperatures and for the Dirac and Schrödinger equations. The study analyzes the application of the Green’s function method to processes in many-particle quantum systems and to solving equations that describe the constituent particles of such systems. For a given operator and set of boundary conditions, a Green’s function can be defined, al lowing the solution of the corresponding equation to be expressed in integral form. A distinctive feature of this method is the application of mathematical-physics tools, such as Fourier methods, integral transforms, special functions, and mathematical analysis, to problems in quantum physics governed by the principles of quantum and statistical mechanics. The relevance of this study stems from the application of the Green’s function method to the analysis of quantum phenomena. This approach is considered one of the effective methods for solving partial differential equations describing various physical processes. It involves transform ing the original differential equation into an equivalent integral equation involving Green’s functions, which can then be solved. The transformation involves passing to the momentum representation and applying Fouri er transforms using methods of complex analysis.