Energy Gradient Model (EGM) Derivation of the Stable Equilibrium State of Hydrogen Atom 1s State — Physical Interpretation of Bohr Force Balance and Centrifugal Force Based on Energy Gradient Distribution
Based on the Energy Gradient Model (EGM), this paper starts from the basic postulate of energy density distribution and strictly derives the stable equilibrium condition of the hydrogen atom 1s state. The results show that the stable orbit of the electron in the ground state is not determined by artificially assumed quantization conditions, but results from the balance between the gradient force generated by the spatial distribution of the energy cloud and the Coulomb force. At the Bohr radius a_0, the magnitudes of the gradient force, Coulomb force, and Bohr centrifugal force are exactly equal, achieving self-consistent equilibrium. This model explains the physical mechanism of Bohr orbital stability from the nature of energy distribution, provides a microscopic energetic foundation for the classical force balance condition, and offers a new unified description for the physical origin of quantum stable states.
Based on the Energy Gradient Model (EGM), this paper starts from the basic postulate of energy density distribution and strictly derives the stable equilibrium condition of the hydrogen atom 1s state. The results show that the stable orbit of the electron in the ground state is not determined by artificially assumed quantization conditions, but results from the balance between the gradient force generated by the spatial distribution of the energy cloud and the Coulomb force. At the Bohr radius a_0, the magnitudes of the gradient force, Coulomb force, and Bohr centrifugal force are exactly equal, achieving self-consistent equilibrium. This model explains the physical mechanism of Bohr orbital stability from the nature of energy distribution, provides a microscopic energetic foundation for the classical force balance condition, and offers a new unified description for the physical origin of quantum stable states.
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.
This paper establishes a self-consistent framework for the kinetic energy distribution of the hydrogen atom ground state (1s) based on the **Energy Gradient Model (EGM)**. By clarifying the distinction between the global total kinetic energy and the local probabilistic energy density, we correct the conceptual ambiguities arising from the traditional assumption of a continuous energy field. The core proposition is that the energy density is fundamentally the total kinetic energy distributed according to the quantum probability cloud. The energy density is proportional to the squared modulus of the wavefunction, with the proportionality constant being the ground-state total kinetic energy of 13.6 eV, independent of spatial position. The full-space integral of the energy density naturally converges to the total kinetic energy, with no divergence or contradictions. This work provides a clear physical picture, a complete formula system, and numerical verification, offering a consistent and intuitive foundation for the EGM.
This chapter illustrates Niels Bohr's ideas in some detail. In the formulation of his model, Bohr assumed that the electron in the hydrogen atom moves under the influence of the Coulomb attraction between it and the positively charged nucleus, as assumed in classical mechanics. The Bohr model was very successful in explaining the discrete atomic spectra of one-electron atoms. The Bohr model is, of course, a theoretical model of an atom. Using Bohr's quantization postulate, find the linear velocity of the electron's sphere. To find the radius of the orbit, Bohr postulated that the angular momentum of the electron is quantized, i.e., it can only take values that are integer multiples of ħ. The chapter begins with considering the classical equation of motion for the electron in a circular orbit, which is based on Newton's laws of motion and Coulomb's law of electric force.
This paper proposes the Energy Gradient Model (EGM), which establishes a unified description of the energy density field, gravitational field and quantum wave function evolution by introducing the gradient force and gradient potential. The core hypothesis of the model is that the gradient force is proportional to the energy density gradient, and the energy density gradient is proportional to the square of the absolute value of the quantum wave function. Based on this hypothesis, the classical Schrödinger equation is modified, such that the gradient potential coupling is automatically introduced when the energy density distribution is inhomogeneous, and the model reverts to classical quantum mechanics in a homogeneous field. Based on the experiments of the Mozi Quantum Science Experimental Satellite and ground-based micro-precision experiments, the total energy difference is rigorously defined as the sum of the absolute value of the gravitational potential energy difference and the absolute value of the Earth's rotational kinetic energy difference, and the energy difference-distance constant C is derived. The macroscopic experiment yields C_{macro} \approx 0.0235 J/(kg·m), and the microscopic experiment yields C_{micro} \approx 1.6\times10^{-10} J/(kg·m), with the ratio of the two being about an order of 10^8, which is consistent with the theoretical prediction of EGM. This paper also clarifies the scope of application and limitations of the EGM, which realizes the unification of macroscopic gravity and microscopic quantum systems, and provides a new theoretical framework for quantum entanglement, gravitational gradient effects and the physical origin of fundamental physical constants.
The Energy Gradient Model 2.0 (EGM 2.0) provides a clear, intuitive, and experimentally consistent framework for describing microscopic particle dynamics. Unlike traditional quantum mechanics, which relies on abstract wavefunction penetration and probabilistic interpretations, EGM 2.0 interprets quantum behavior as energy‑gradient‑driven evolution of electron clouds. The model unifies quantum tunneling and quantum transition under a single physical mechanism distinguished by a critical structural stability threshold. This paper presents a concise introduction to the core concepts, fundamental assumptions, and basic formalism of EGM 2.0. Keywords: Energy Gradient Model, EGM 2.0, electron cloud, energy gradient, quantum tunneling, quantum transition
The existence of line spectra can be explained bymeans of the first “quantum”model of the atom, developed by Bohr in 1913. Although the Bohr model of the hydrogen atom was eventually replaced, it yields the correct values for the observed spectral lines, and gives a substantial insight into the structure of atoms in general. The following derivation has the objective of obtaining the energy levels of the Bohr atom. If we can obtain the energy levels, we can reproduce the hydrogen atom spectra. The derivation proceeds with three major elements: first, use force balance to relate the velocity to the radius of the electron orbit, then use a quantum assumption to get the radius, then solve for the energies.