Verification of the Core Hypothesis of the Energy Gradient Model (EGM) in the 1s State of Hydrogen Atom Energiegradientenmodell (EGM): Verifikation der Kernhypothese im 1s-Zustand des Wasserstoffatoms

The Energy Gradient Model (EGM) takes the proportionality between gradient force and energy density gradient as its core hypothesis, the theoretical foundation for constructing a unified description of gravitational and quantum mechanical systems. This paper selects the hydrogen atom 1s state—a typical spherical symmetric, stable quantum system conforming to EGM's application scope—as the research object. Based on the EGM-modified Schrödinger equation, the EGM gradient potential U_EGM of the hydrogen atom 1s state is rigorously derived by substituting the spherical symmetric Laplace operator and hydrogen atom Coulomb potential; the gradient force F_EGM is further obtained according to EGM's gradient force definition. Combined with the hydrogen atom 1s state wave function, the energy density ρ_E and its spatial gradient ∇ρ_E are calculated in accordance with the EGM definition ρ_E=|ψ|². The proportional relationship between F_EGM and ∇ρ_E is quantitatively verified at the Bohr radius, and the accurate value, unit and physical meaning of the proportional constant K are given. Results show that F_EGM and ∇ρ_E at the Bohr radius satisfy the strict proportional relationship F_EGM=K·∇ρ_E; K is a fixed scalar, with the unit of cubic meter (m³) in the International System of Units and a numerical value of about -1.014×10^-39 m³. The sign difference between F_EGM and ∇ρ_E is a necessary result derived from the EGM definition chain, consistent with the model's theoretical logic. Compared with the approximate calculation method based on the hydrogen atom 1s-2s energy transition, the derivation method based on the EGM-modified Schrödinger equation avoids inherent errors in energy density calculation during the transition process, and the obtained K value has higher precision and theoretical rigor. This study completes the strict theoretical verification of the EGM core hypothesis in the typical quantum system of the hydrogen atom 1s state, which not only proves the rationality of EGM's basic assumption, but also provides a reliable theoretical derivation method for the subsequent application of EGM in more quantum systems, and lays a solid foundation for the further development of EGM in the unification of gravity and quantum mechanics.