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Prog. Theor. Phys. Vol. 27 No. 4 (1962) pp. 811-836

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

Hirotsugu Matsuda

Department of Chemistry, Faculty of Science, Kyoto University, Kyoto

(Received November 23, 1961)

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.

URL : http://ptp.ipap.jp/link?PTP/27/811/
DOI : 10.1143/PTP.27.811

[ Full Text PDF : FREE ACCESS (1599K) ] Citation: Plain Text BiBTeX EndNote(RIS) RSS

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Citing Article(s) :

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6. Progress of Theoretical Physics Vol. 35 No. 5 (1966) pp. 759-776 :

   Note on Electronic State of Random Lattice. III

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7. Progress of Theoretical Physics Vol. 40 No. 4 (1968) pp. 734-757 :

   A Systematic Approach to the Problems of Random Lattices. I

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8. Progress of Theoretical Physics Vol. 47 No. 4 (1972) pp. 1124-1139 :

   A Systematic Approach to the Problems of Random Lattices. II

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