Atomic vibrational Approach

The classical Bohr model of the atom describes electron orbits as quantized energy levels, aconcept that laid the foundation for modern quantum mechanics. However, an alternativeapproach can be taken by considering atomic structure through vibrational mechanics,where electron motion is modeled using elastic interactions rather than purely electrostaticforces. In this paper, we propose a vibrational analogy for atomic and planetary motion,treating the electron's orbit as a harmonic oscillator with an efective elastic constant derivedfrom fundamental interactions.Building on this analogy, we extend the model to gravitational orbits, considering planetarymotion as governed by vibrational energy states. The Earth-Sun system is described as aquantized vibrational system where gravitational interactions are replaced by efective elasticforces. Our analysis shows that using vibrational quantization principles leads to stable,discrete orbital states, resembling quantum mechanical energy levels. By comparing orbitalenergy expressions in classical, vibrational, and quantum mechanical models, we find thatthe vibrational framework provides a natural bridge between classical mechanics andquantum behavior, with potential implications for atomic physics, planetary dynamics, andeven quantized gravity theories.