TITLE:
Solid Angle and the Fine-Structure Constant: A Geometric View of Hydrogen Length Scales
AUTHORS:
Zaki Harari
KEYWORDS:
Fine-Structure Constant, Hydrogen Atom, Lyman Limit, Bohr Radius, Solid Angle, Dimensional Analysis, Atomic Units, Hydrogenic Ions
JOURNAL NAME:
Journal of Modern Physics,
Vol.17 No.8,
August
17,
2026
ABSTRACT: The Lyman-limit wavelength
λ
Ly
(
∞
)
≡1/
R
∞
(defined for infinite nuclear mass) and the Bohr radius
a
0
are two of the most basic length scales in atomic physics. We show that their ratio satisfies the exact identity
λ
Ly
(
∞
)
a
0
=
4π
α
=
Ω
sph
α
, where
α≈1/
137
is the fine-structure constant and
Ω
sph
=4π
sr is the solid angle of a complete sphere. We write
λ
Ly
for
λ
Ly
(
∞
)
throughout, and
λ
Ly
(
real
)
where the finite-proton-mass correction is discussed. The identity follows from CODATA 2022 definitions in four lines of algebra. The constant
α
is the ratio of the electron’s ground-state speed to
c
; a factor 2π arises from
h=2πℏ
; a factor 2 arises from the Coulomb virial theorem,
〈 T 〉=−
E
total
. These two factors are structurally identical to the azimuthal and polar integrals whose product gives the solid angle of a sphere. We stress that this is a formal, convention-dependent analogy: it does not imply angular integration over the hydrogen wavefunction or the photon’s emission pattern, and it is tied to expressing photon energy as
hν
rather than
ℏω
; in the latter convention the factor 2π disappears entirely while the physics is unchanged. The result is verified numerically to better than one part in 108 using CODATA 2022 data; because
λ
Ly
,
a
0
,
R
∞
, and
α
are themselves algebraically linked through their CODATA definitions, this is best read as a consistency check on those definitions, not an independent test. We also present the hierarchy of the three electromagnetic length scales of hydrogen (
r
e
,
a
0
,
λ
Ly
), separated by powers of
α
and 4π, and discuss the leading finite-proton-mass correction (+0.054%) and extensions to hydrogenic ions and atomic units. No new physics is proposed. The paper packages standard textbook definitions into a single exact, precision-testable identity with a transparent (if convention-dependent) geometric reading, offered as a pedagogical and diagnostic tool.