NUVO Planck’s Constant h, hbar and Fine Structure Constant From First Principles

⚛️ NUVO Planck’s Constant h

In standard quantum mechanics, Planck’s constant h and the reduced constant ℏ\hbar are introduced axiomatically to set the scale of quantization. The fine structure constant α then emerges as a dimensionless ratio involving charge, light speed, and ℏ\hbar.

In NUVO Theory, these constants are not fundamental starting points. Instead, they are derived from scalar geometric principles using only one empirical input: the hydrogen ground state energy of 13.6 eV. From this, NUVO reconstructs h, ℏ\hbar, and α\alphaα by interpreting orbital energy as a consequence of geometric arc advance.

🔁 Orbital Advance in NUVO

NUVO predicts that for each orbit, the electron advances by an additional arc length:

\(
\Delta s = 2\pi r_e
\)

This advance occurs in coordinate space and results in an energy surplus — not because of radiation or momentum, but due to scalar geometric expansion.

Each cycle adds this fixed spatial increment to the total path length, contributing additional kinetic energy to the electron’s motion.

🔋 Energy from Arc Advance

To remain coherent in orbit, the electron must supply enough energy to match this geometric advance, which implies an extra energy cost per orbit. Over many revolutions, this builds up a geometric phase deficit that defines the binding energy of the hydrogen atom.

Using the only empirical input — the known ground state energy of the hydrogen atom:

\(
E_b = 13.6 \ \text{eV}
\)

NUVO calculates how many orbits are required for the total arc advance to reconstruct a photon matching that energy. This leads directly to a derivation of the photon’s energy-wavelength relation:

\(
E = h f
\)

But here, h arises as a geometric consequence, not a starting constant.

🧮 Deriving hhh from Scalar Geometry

Let:

The classical electron radius be \( r_e = \frac{e^2}{4\pi \varepsilon_0 m_e c^2} \)
Each orbital advance be \( 2\pi r_e \)
The number of orbits needed for coherence be \( N = \frac{1}{\alpha^2} \) (shown on the quantization page)

Then the total arc length over one coherence cycle is:

\(
L = N \cdot 2\pi r_e = \frac{2\pi r_e}{\alpha^2}
\)

Since the electron is moving at nearly the speed of light in the coordinate frame, the effective frequency of that coherence cycle is:

\(
f = \frac{c}{L} = \frac{c \alpha^2}{2\pi r_e}
\)

Now multiply the empirical energy per photon:

\(
h = \frac{E}{f} = \frac{13.6 \ \text{eV}}{f} = \frac{13.6 \cdot 2\pi r_e}{c \alpha^2}
\)

This directly yields:

\(
h = \frac{2\pi r_e \cdot 13.6 \ \text{eV}}{c \alpha^2}
\)

From this, we can also derive:

\(
\hbar = \frac{h}{2\pi}
\)

And rearranging the expression gives a purely geometric view of the fine structure constant:

\(
\alpha^2 = \frac{2\pi r_e \cdot 13.6 \ \text{eV}}{c h}
\)

🧠 Summary

Planck’s constant \( h \) is not fundamental — it is emergent from geometry.
The electron’s orbital advance of \( 2\pi r_e \) per cycle encodes an energy surplus.
Over a coherence interval, this builds up the exact energy needed to emit a photon of 13.6 eV.
Using only that empirical energy value and NUVO’s scalar geometry, we derive:
- \( h \) and \( \hbar \)
- The orbital frequency
- The fine structure constant \( \alpha \) as a geometric closure ratio