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The integrated particle current produced by a slow periodic variation of the potential of a Schr\"odinger equation is evaluated. It is shown that in a finite torus the integral of the current over a period can vary continuously, but in an infinite periodic system with full bands it must have an integer value. This quantization of particle transport is used to classify the energy gaps in a one-dimensional system with competing or incommensurate periods. It is also used to rederive Prange's results for the fractional charge of a soliton.
D. J. Thouless (Sun,) studied this question.
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