The theory of scale relativity (ScR) is based on the extension of the principle of relativity of motion to include relativity of scale. Fractal space-time is the basis of this new theory as formulated by Nottale. Applications of this theory encompass diverse fields from microphysics, cosmology and complex systems. Previously, direct numerical simulations using this theory in quantum physics as performed by Hermann have resulted in the appearance of the correct quantum behavior without resorting to the Schrödinger equation. However these simulations were performed for only one limited case of a particle in an infinite one-dimensional square well. Hence, there is a need for more such applications using this theory to establish its validity in the quantum domain.
In the present work, and along the lines of Hermann, ScR theory is applied to other standard one-dimensional quantum mechanical problems. These problems are: a particle in a finite one-dimensional square well, a particle in a simple harmonic oscillator (SHO) potential and a particle in a one-dimensional double-well potential. Some mathematical problems that arise when obtaining the solution to these problems were overcome by utilizing a novel mathematical connection between ScR theory and the well-known Riccati equation. Then, computer programmes were written using the standard MATLAB 7 code to numerically simulate the behavior of the quantum particle in the above potentials utilizing the solutions of the fractal equations of motion obtained from ScR theory. Several attempts were made to fix some of the parameters in the numerical simulations to obtain the best possible results in a practical computer CPU time within the limited local computer facilities.
Comparison of the present results for the particle probability density in the three potentials with the corresponding results obtained from conventional quantum mechanics by solving the Schrödinger equation, shows very good 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 ScR theory can be used as a basis for describing the quantum behavior without reference to conventional quantum mechanics. Hence, it can also be concluded that the fractal nature of space-time, which is the basis of ScR theory, is the origin of the quantum behavior observed in these problems.
More applications to potentials in more than one-dimension, including asymmetric potentials, would give greater confidence along these lines. Also, the novel mathematical connection between ScR theory and the Riccati equation, that was previously used in quantum mechanics without reference to ScR theory, needs further investigation in future work.
سعيد الراشد | Saeed AlRashid | 4613