Application of Scale Relativity to the Problem of a Particle in a Simple Harmonic Oscillator Potential

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.

Modeling the Hydrogeochemical Processes and Source of Ions in the Groundwater of Aquifers within Kasra-Nukhaib Region (West Iraq)

Hydrogeological investigation of groundwater has been performed within Kasra- Nukhaib district (west Iraq). The physicochemical analyses of the groundwater samples collected from Um Erdhuma-Tayarat aquifer are used in determination of hydrogeochemical processes, hydrochemical facies, hydrochemical ratio, and saturation indices supported by phreeqc software. The monitoring network of groundwater quality consists of twelve physiochemical variables in twenty four water wells were determined, aiming to examine the hydrogeo-chemical phenomena which are benefit in groundwater development and support the plans of future uses (exploitation) and groundwater management. A hydrogeological model was prepared to examine qualitative evaluation of aquifers media related to geochemical processes. Spatial hydrochemical bi-models were achieved for quantitative interpretation. Various graphical plots such as Piper, Durov, and Gibbs diagrams are used in proving different geochemical processes. The results are correlated with standards classifications to deduce the hydrogeo-chemical phenomena. The results showed that the groundwater of Um Erdhuma-Tayarat aquifer has a property of Carbonate weathering represents the major hydrogeochemical processes. In addition, ion exchange and reverse ion exchange were two possible processes of water-rock interaction (e.g. dolomitization processes) within the hydrogeologic system. Also, the evaporation process and mixing action of recharge vadose waters, and ancient trapped fossil waters have a moderate effect on the evolution of groundwater quality. An increased salt content is observed in groundwater at different static water levels indicating mixing with various sources. The mechanism controlling groundwater chemistry is originally related to geogenic process. In addition, anthropogenic activities have not significantly altered the geochemical nature of groundwater in the aquifers system.