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Annals of Physics

Fractional corresponding operator in quantum mechanics and applications: A uniform fractional Schrödinger equation in form and fractional quantization methods

Xiao Zhang, Chaozhen Wei, Yingming Liu, Maokang Luo

Published Jul 12, 2014• 33 citations
Publisher
DOI
10.1016/j.aop.2014.07.007

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In this paper, we construct a fractional time corresponding operator to derive the uniform fractional time Schrödinger equation, which implies the relationship between the fractionaloperatorandtheenergy. Weverifythreebasicprinciplesoftheuniformfractional time Schrödinger equation: (a) Unitarity of the time evolution operator; (b) Existence of stationary energy spectra; (c) Probability conservation law. We apply the uniform equation to study three typical particles: free particle, an infinite potential well and a rectangular barrier. This uniform equation brings some typical physical phenomena, including (1) non-symmetry basic wave functions; (2) continuous energy spectrum for a free particle and an infinite potential well; (3) The barrier penetration in the rectangular barrier. These phenomena were not found in previous researches and the standard quantum mechanics, which will attract further applications in various fields of science.
BACKGROUND: Sequel to the work on the quantization of nonconservative systems using fractional calculus and quantization of a system with Brownian motion, which aims to consider the dissipation effects in quantum-mechanical description of microscale systems. RESULTS: The canonical quantization of a system represented classically by one-dimensional Fick's law, and the diffusion equation is carried out according to the Dirac method. A suitable Lagrangian, and Hamiltonian, describing the diffusive system, are constructed and the Hamiltonian is transformed to Schrodinger's equation which is solved. An application regarding implementation of the developed mathematical method to the analysis of diffusion, osmosis, which is a biological application of the diffusion process, is carried out. Schrödinger's equation is solved. CONCLUSIONS: The plot of the probability function represents clearly the dissipative and drift forces and hence the osmosis, which agrees totally with the macro-scale view, or the classical-version osmosis.