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In the present work, Scale Relativity (SR) is applied to a particle in a simple harmonic oscillator (SHO) potential. This is done by utilizing a novel mathematical connection between SR approach to quantum mechanics and the well-known Riccati equation. Then, computer programs were written using the standard MATLAB 7 code to numerically simulate the behavior of the quantum particle utilizing the solutions of the fractal equations of motion obtained from SR method. Comparison of the results with the conventional quantum mechanics probability density is shown to be in very precise agreement. This agreement was improved further for some cases by utilizing the idea of thermalization of the initial particle state and by optimizing the parameters used in the numerical simulations such as the time step and number of coordinate divisions. It is concluded from the present work that SR method can be used as a basis for description the quantum behavior without reference to conventional formulation of quantum mechanics. Hence, it can also be concluded that the fractal nature of space-time implied by SR, is at the origin of the quantum behavior observed in these problems. The novel mathematical connection between SR and the Riccati equation, which was previously used in quantum mechanics without reference to SR, needs further investigation in future work.
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篇名 Application of Scale Relativity to the Problem of a Particle in a Simple Harmonic Oscillator Potential
来源期刊 量子信息科学期刊(英文) 学科 数学
关键词 Simple HARMONIC OSCILLATOR Scale RELATIVITY Numerical Simulations FRACTAL SPACE-TIME
年,卷(期) 2017,(3) 所属期刊栏目
研究方向 页码范围 77-88
页数 12页 分类号 O1
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Simple
HARMONIC
OSCILLATOR
Scale
RELATIVITY
Numerical
Simulations
FRACTAL
SPACE-TIME
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量子信息科学期刊(英文)
季刊
2162-5751
武汉市江夏区汤逊湖北路38号光谷总部空间
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142
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0
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