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Fractal structures appearing in solutions of certain quantum problems are investigated. We prove the previously announced results concerning the existence and properties of fractal states for the Schrödinger equation in the infinite one-dimensional well. In particular, we show that for this problem, there exist solutions in the form of fractal quantum carpets: the probability density P(x,t) forms a fractal surface with dimension Dxy, while its cross-sections Pt(x) and Px(t) typically form fractal graphs with dimensions Dx and Dt, respectively, where Dxy=2+Dx/2 and Dt=1+Dx/2 (almost everywhere).
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