Abstract
The Schrodinger equation of a particle moving in one dimension in a given static potential is solved. The density matrix corresponds to a Laplace transform of Green's function and satisfies a diffusion-type equation whlch may be regarded as representing a certain limit of a discrete Markoffian random walk process. The space occupied by the system is divided into nonoverlapping regions; in each, the Sehrodinger equation can be solved. A path concept is introduced to synthesrze the solutions for the regions and formulate the one- particle Green's function. The energy bands of a one-dimensional infinite system are discussed. The condition for the existence of the impurity band is derived, which qualitatively supports Anderson's theory on the absence of diffusion in certain random lattices.
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