Progress of Theoretical Physics

A New Approach to Green's Function of a Particle in One Dimension

Hirotsugu Matsuda

Published Apr 1, 1962• 12 citations

Abstract

A new method for solving the Schrödinger equation of a particle moving in one dimension in a given static potential is presented in this paper. We start from the observation that the density matrix corresponds to a Laplace transform of Green's function, and satisfies a diffusion-type equation which may be regarded as representing a certain limit of a discrete Markoffian random walk process. Our method of solving the Schrödinger equation is to divide the space occupied by the system into non-overlapping regions, in each of which the Schrödinger equation can be solved. We then introduce a path concept so as to synthesize the solutions for the various regions and thereby formulate the one-particle Green's function. As an application of this formulation, the energy bands of a one-dimensional infinite system are treated. The condition for the existence of impurity band is derived, which qualitatively supports Anderson's theory on the absence of diffusion in certain random lattices.

DOI: 10.1143/ptp.27.811

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