The Hydrogen 1s Electron as an Event 3 Process: A Gamma Distribution (3, 1) Derived from Bohr Velocity Time-Conversion

Abstract

The radial probability distribution of the hydrogen 1s orbital is obtained by multiplying the probability density | ψ | 2 by the surface-area element of a sphere with radius r , 4 π r 2 . By introducing a characteristic time through the Bohr velocity and interpreting distance as an effective waiting time, the resulting distribution can be mathematically reduced to a Gamma distribution with shape parameter 3. This structure is equivalent to the sum of three independent exponential waiting processes. The origin of the parameter “3” is traced to the factor r 2 , which arises from the three-dimensional spherical volume element. Thus, the Gamma-distribution structure is closely related to the dimensionality of space. Based on this observation, we propose a mathematical interpretation in which the electron reaching distance ( r ) corresponds to the accumulation of probability amplitudes along three independent spatial directions, ( x ), ( y ), and ( z ). In higher-dimensional spaces, the Coulomb potential is modified according to Gauss’s law, leading to the well-known “fall-to-the-center” behavior, where stable bound states at the Bohr radius can no longer be maintained. In such cases, the radial distribution no longer reduces to a Gamma distribution. These results suggest that only in three dimensions do the Coulomb potential ( ~ 1 / r ), the exponential wave function ( ~ e ? r ), and the spatial volume element ( ~ r 2 ) become mutually balanced, giving rise to the Gamma-distribution structure.