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

The radial probability distribution of the hydrogen 1s orbital is obtained by multiplying the probability density ${\left|\psi \right|}^2$ by the surface-area element of a sphere with radius r, $4\pi r^2$ . 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 r², 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 ($~{\text{e}}^{?r}$ ), and the spatial volume element ($~r^2$ ) become mutually balanced, giving rise to the Gamma-distribution structure.