Schrödinger Equation vs. Hamzah Equation: A Comparative Analysis of Superiority.

All Articles are Available: Orcid ID: https://orcid.org/my-orcid?orcid=0009-0009-3175-8563 Science Open ID: https://www.scienceopen.com/user/2c98a8bc-b8bb-49b3-9c91-2f2986a7e16e Safe Creative register the work titled "The Theory of Intelligent Evolution, the Hamzah Equation, and the Quantum Civilisation". Safe Creative registration #2504151474836. .................................................................................................................................................................................................. Specialized Abstract – Schrödinger Equation in the ψ–Hamzah Model The Schrödinger equation, the core of non-relativistic quantum mechanics, provides a wave-like description of the behavior of subatomic particles in the form of a wave function. As a fundamental principle, it explains interference patterns, tunneling, and energy dissipation in quantum systems. However, its structural limitations become evident when confronted with nonlinear phenomena, memory effects, and complex informational fields. The ψ–Hamzah model, grounded in fractional derivatives of variable orders, memory field space, and the fractal–cognitive ψ-time, offers a novel reformulation of the Schrödinger equation. This reformulation not only preserves probabilistic behavior but also accommodates the understanding of non-local dynamics, informational changes, and intelligent–quantum phenomena. In this paper, we first introduce the classical Schrödinger equation in detail, and then, by converting it into its ψ–Hamzah version, attempt to perform numerical analysis, memory-based modeling, and the extraction of multi-scalar behaviors. This model provides more accurate results in the realms of quantum chaos, quantum in noisy environments, and biological–quantum processes, and is also applicable in quantum artificial intelligence platforms and cognitive modeling. A comparative analysis of the classical and ψ–Hamzah models across ten main numerical–analytical criteria, along with a phenomenological analysis of the awareness–field–time structure differences, will ultimately lead to a civilizational statement, showing that ψ–Hamzah is not only a physical reformulation but a new scientific paradigm for understanding reality and advancing future quantum sciences.