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Journal of Mathematical Physics

N-Particle Noninteracting Green's Function

Willis Grant Cooper, Donald Jack Kouri

Published Jun 1, 1972• 12 citations
Publisher

Abstract

A prescription is given for obtaining the Green's function for N free particles which can have different masses. The approach is systematic and straightforward. A coordinate transformation of the Fourier integral representation of the N-particle noninteracting Green's function facilitates the integration over 3N-1 angular variables of wavenumber space. A single radial integral can then be evaluated. The resulting Green's function representation may be of use in applying the integral form of Schrödinger's equation to calculate the ground and excited states of atoms.

DOI
10.1063/1.1666055

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A prescription is given for obtaining the Green's function for N free particles which can have different masses. The approach is systematic and straightforward. A coordinate transformation of the Fourier integral representation of the N-particle noninteracting Green's function facilitates the integration over 3N-1 angular variables of wavenumber space. A single radial integral can then be evaluated. The resulting Green's function representation may be of use in applying the integral form of Schrödinger's equation to calculate the ground and excited states of atoms.
In quantum mechanics, as well as in classical mechanics, the number of variables involved in the equations of motion may be reduced by utilizing the constants of motion of the system. For a set of N particles moving in a field-free space there are six integrals if N is greater than two. These correspond to the laws of conservation of linear and of angular momentum. It is the purpose of this paper to eliminate the coordinates of rotation and of translation of the total system from the Schrodinger equation and to obtain a new set of differential equations involving only 3N-6 variables.1
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.
A purely algebraic approach to Green's functions is presented. The new formulation allows the determination of a closed-form equation for Green's functions which depends only on the ground state of the system. By a straightforward perturbation expansion of this state an interesting approximation scheme is obtained which is nonperturbative in the final states (e.g., excited or ionic states). This method, which can be used for various propagators, is explicitly applied to the particle-particle propagator carrying out the scheme up to third order. The approach is discussed in detail and briefly compared with the well-established algebraic diagrammatic construction approximation scheme.
The Schrödinger equation, in hyperspherical coordinates, is solved in closed form for a system of three particles on a line, interacting via pair delta functions. This is for the case of equal masses and potential strengths. The interactions are replaced by appropriate boundary conditions. This leads then to requiring the solution of a free-particle Schrödinger equation subject to these boundary conditions. A generalized Kontorovich–Lebedev transformation is used to write this solution as an integral involving a product of Bessel functions and pseudo-Sturmian functions. The coefficient of the product is obtained from a three-term recurrence relation, derived from the boundary condition. The contours of the Kontorovich–Lebedev representation are fixed by the asymptotic conditions. The scattering matrix is then derived from the exact solution of the recurrence relation. The wavefunctions that are obtained are shown to be equivalent to those derived by McGuire. The method can clearly be applied to a larger number of particles and hopefully might be useful for unequal masses and potentials.