TITLE:
The Hydrogen 1s Electron as an Event 3 Process: A Gamma Distribution (3, 1) Derived from Bohr Velocity Time-Conversion
AUTHORS:
Motohisa Osaka
KEYWORDS:
Wave Function, Poisson Process, Exponential Function, Multiple Universes
JOURNAL NAME:
Applied Mathematics,
Vol.17 No.6,
June
16,
2026
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,
4π
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 (
~
e
?r
), and the spatial volume element (
~
r
2
) become mutually balanced, giving rise to the Gamma-distribution structure.