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 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.
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.
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.
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
A concept of kinetic energy in quantum mechanics is analyzed. Kinetic energy is a non-zero positive value in many cases of bound states, when a wave function is a real-valued one and there are no visible motion and flux. This can be understood, using expansion of the wave function into Fourier integral, that is, on the basis of virtual plane waves. The ground state energy of a hydrogen atom is calculated in a special way, regarding explicitly all the terms of electrostatic and kinetic energies. The correct values of the ground state energy and the radius of decay are achieved only when the electrostatic energies of the electron and the proton (self-energies) are not taken into account. This proves again that self-action should be excluded in quantum mechanics. A model of a spherical ball with uniformly distributed charge of particles is considered. It is shown that for a neutral ball (with compensated electric charge) the electrostatic energy is a non-zero negative value in this model. This occurs because the self-energy of the constituting particles should be subtracted. So it shown that the energy of the electric field does not have to be of a positive value in any imaginable problem.
The Bohr model of the hydrogen atom, despite its heuristic success in predicting discrete energy levels, is fundamentally incompatible with the probabilistic interpretation of modern quantum mechanics. In this work, we present a pedagogical derivation of Bohr’s results directly from the time-independent Schrödinger equation in three dimensions. By applying two successive geometric constraints, first, a genuine dimensional reduction to a 2D planar system (not a naive restriction of the spherical Laplacian), and second, a radial localization to a fixed radius, we demonstrate that the resulting radial-azimuthal equation yields the quantized Bohr orbits. We show that the energy levels En = −13.6/n2 eV and orbital radii r[n] = n² ao emerge naturally from the wavefunction normalization combined with the de Broglie standing-wave condition. Special attention is given to the correct form of the 2D Laplacian, which differs from the 3D Laplacian restricted to θ = π/2 by a term (1/r) ∂/∂r. This approach provides a clear conceptual bridge between the diffuse quantum cloud and the classical planetary picture.