A Higher-Dimensional Geometric Analogy for Non-Continuous Quantum Electron Transitions: Exploring Kaluza–Klein Compactification and the ER=EPR Conjecture as a Speculative Framework

The discrete, non-continuous character of electron transitions between atomic energy levels is fully accounted for by standard non-relativistic quantum mechanics: transitions are changes in the coefficients of a state expanded in the energy eigenbasis, not motion of a localized particle through space, so no classical trajectory needs to be recovered. This paper does not claim to solve an open problem in quantum mechanics. Instead, it explores, as a speculative geometric analogy, whether a Kaluza-Klein-style compactified fifth dimension could be pictured as encoding the "jump" between orbitals, and whether the ER=EPR conjecture offers a suggestive (but currently unproven) language for connecting such a picture to Planck-scale quantum gravity.