# Fine-structure constant (spiral)

H atom n-shell energy levels emerge from a gravitational orbit
If we map the electron transition between energy levels in the H atom semi-classically as a gravitational orbit, we find a hyperbolic spiral appears. This spiral has interesting properties, namely if we measure the transition radius as $r=mr_{0}$ (m is a non-integer, r$_0$ as the Bohr radius), we find periodically $m=n^{2}$ where n is the principal quantum number, this periodicity occurring as a function of pi (a geometrical periodicity).
The number of steps required to reach each $m=n^{2}$ level corresponds to the transition frequencies of the electron, thus we can derive the Bohr model from this spiral (when $m=n^{2}$, the formulas are the same).
In order to map the electron as a gravitational orbit, the electron is treated as an oscillation between an electric state and a mass state; the electric state duration = particle frequency, the mass state duration = 1 unit of Planck time. In the mass state the particle is a point (i.e.: it can be assigned Cartesian co-ordinates). An orbit can then be simulated as a gravitational orbit by plotting these (mass) points.
During transition, the incoming photon is treated likewise, as the sum of oscillations (albeit without a mass state), where m corresponds to the number of photon oscillations, as each photon oscillation is added, the radius increases. During this process the electron continues orbiting the proton, a hyperbolic spiral path then emerging. As only the mass state has point co-ordinates, the transition is mapped as a series of steps, each step as 1 oscillation cycle, and so when mapped as a gravitational orbital transition, the path is semi-classical. The total number of steps giving the transition frequency.
In this context the Bohr model can be considered as a gravitational model (which is what the spiral model is), and so rather than the Schrödinger equation replacing the Bohr model, the actual electron orbit could be a combination of both models; the Schrödinger equation mapping the electron (electric) wave-state and the Bohr model mapping the electron (mass) point-state.
And so although the Bohr model does not include the wave-state, it is not an incorrect description of the atomic orbital, rather it is incomplete, there is a distinction. The same may be said of the Schrödinger equation.
Source code H-orbital.c
At these angles on the spiral, the (x, y) coordinates converge to give exact integer values (4, 9, 16, 25, 36 ...)
    $\varphi =(2)\pi ,\;r=4r_{0}$ (360°)
    $\varphi =(8/3)\pi ,\;r=9r_{0}$ (360+120°)
    $\varphi =(3)\pi ,\;r=16r_{0}$ (360+180°)
    $\varphi =(16/5)\pi ,\;r=25r_{0}$ (360+216°)
    $\varphi =(10/3)\pi ,\;r=36r_{0}$ (360+240°)
    $\varphi =(7/4)\pi ,\;r=49r_{0}$
    $\varphi =(7/2)\pi ,\;r=64r_{0}$ (360+270°)

#### Hyperbolic spiral

A hyperbolic spiral is a type of spiral with a pitch angle that increases with distance from its center. As this curve widens (radius r increases), it approaches an asymptotic line (the y-axis) with the limit set by a scaling factor a (as r approaches infinity, the y axis approaches a).
    $x=a^{2}{\frac {cos(\varphi )}{\varphi ^{2}}},\;y=a^{2}{\frac {sin(\varphi )}{\varphi ^{2}}},\;0<\varphi <4\pi$
    radius = ${\sqrt {}x^{2}+y^{2})r$
    $\varphi =(2)\pi ,\;4r$ (360°)
    $\varphi =(4/3)\pi ,\;9r$ (240°)
    $\varphi =(1)\pi ,\;16r$ (180°)
    $\varphi =(4/5)\pi ,\;25r$ (144°)
    $\varphi =(2/3)\pi ,\;36r$ (120°)

#### AI spiral rationale

AI was used to condense the derivation (summarize the method and give conclusions). This presumes familiarity with the Bohr model.
Chat GPT chatgpt.com/share/67ce62fc-8bf8-8012-8622-37a7a4fae6d6
    Conclusion : the stability of the n orbital shells (with $m=n^{2}$) arises because at these levels the spiral’s radial and angular increments are in perfect resonance. The electron’s semi-classical motion then traces a repeating, closed path. This geometrical resonance, which directly involves π and the circular nature of the motion, naturally leads to the quantized orbital shells without invoking the usual quantum-mechanical postulates.
Deepseek deepseek-spiral.pdf
    Conclusion : The levels $m=n^{2}$ are stable because the spiral’s geometry enforces a self-consistent synchronization between radial growth and angular displacement. This resonance prevents dispersion, creating discrete orbital shells without invoking quantum mechanics. The quantization of m into squares is a geometric constraint ensuring constructive interference in the spiral’s path, much like how integer multiples of wavelengths stabilize standing waves. The stability of $m=n^{2}$ levels arises from geometric resonance in the spiral, where radial expansion and angular progression harmonize to form closed or self-reinforcing paths. This ensures discrete, stable orbital shells.
Qwen chat.qwen.ai/s/9fe132a6-91d7-4ec6-8c82-ebf2e1b2b422
    Conclusion : The stability of $m=n^{2}$ orbits is a geometric necessity . The spiral’s radial and angular evolution align to create resonant, non-decaying paths at these specific radii. These orbits are "stable" because they satisfy the minimal condition for periodicity and constructive interference, ensuring the electron’s motion remains bounded without invoking quantum theory.
Claude claude.ai/share/355e21e4-d623-4810-962a-fb1892c2ef3f
    Conclusion : The stability of orbits at m = n² emerges from the geometric properties of the spiral itself, specifically from resonances in the relationship between angular rotation and radial expansion. This geometric perspective provides a semi-classical explanation for why certain orbital levels are preferred without invoking quantum mechanical principles.
Grok x.com/i/grok/share/2ERWIbPFpB0wlCsaqcGErGVX2
    Conclusion : The integer ( n ) introduces a quantization effect, and at these points, the spiral’s geometry—through the number of steps and the resulting angular position—creates a resonant or balanced configuration. This allows the electron, moving semi-classically, to occupy stable orbital shells, not because of quantum wavefunctions, but due to the inherent structure and symmetry of the spiral’s path at these discrete, integer-squared intervals.

#### Theory
##### Principal quantum number n

The H atom has 1 proton and 1 electron orbiting the proton, in the Bohr model (which approximates a gravitational orbit), the electron can be found at select radius (the Bohr radius) from the proton (nucleus), these radius represent the permitted energy levels (orbital regions) at which the electron may orbit the proton. Electron transition (to a higher energy level) occurs when an incoming photon provides the required energy (momentum). Conversely emission of a photon will result in electron transition to a lower energy level.
The principal quantum number n denotes the energy level for each orbital. As n increases, the electron is at a higher energy level and is therefore less tightly bound to the nucleus (as n increases, the electron orbit is further from the nucleus). Each shell can accommodate up to n$^2$ (1, 4, 9, 16 ... ) electrons. Accounting for two states of spin this becomes 2n$^2$ electrons. As these energy levels are fixed according to this integer n, the orbitals may be said to be quantized.

##### Oscillating electron Bohr orbital

Note: Here the electron is treated as an oscillation between an electric state and a mass state; the electric state duration = particle frequency, the mass state duration = 1 unit of Planck time. In the mass state the particle is a point (i.e.: it can be assigned Cartesian co-ordinates).
The basic orbital radius has 2 components, dimensionless (the fine structure constant alpha) and dimensioned (electron + proton wavelength);
wavelength $\lambda _{H}=\lambda _{p}+\lambda _{e}$
radius $r_{orbital}=2\alpha n^{2}(\lambda _{H})$
As a mass point, the electron orbits the proton at a fixed radius (the Bohr radius) in a series of steps (the duration of each step corresponds to the wavelength component). The distance travelled per step (per wave-point oscillation) equates to the distance between mass point states and is the inverse of the radius
length (per step) = $l_{orbital}={\frac {1}{r_{orbital}}}$
Duration = 1 step per wavelength and so velocity
velocity = $v_{orbital}={\frac {1}{2\alpha n}}$
Giving period of orbit
period = ${\frac {2\pi r_{orbital}}{v_{orbital}}}=2\pi 2\alpha 2\alpha n^{3}\lambda _{H}$
As we are not mapping the wavelength component, a base (reference) orbital (n=1)
    $t_{ref}=2\pi 4\alpha ^{2}$ = 471964.356...
The angle of rotation depends on the orbital radius
    $\beta ={\frac {1}{r_{orbital}{\sqrt {r_{orbital}}}{\sqrt {2\alpha }}}}$

##### Photon orbital model

The electron can jump between n energy levels via the absorption or emission of a photon. In the Photon-orbital model, the orbital (Bohr) radius is treated as a 'physical wave' akin to the photon albeit of inverse or reverse phase such that $orbital\;radius+photon=zero$ (cancel).
The photon can be considered as a moving wave, the orbital radius as a standing/rotating wave (trapped between the electron and proton), as such it is the orbital radius that absorbs or emits the photon during transition, in the process the orbital radius is extended or reduced (until the photon is completely absorbed/emitted). The electron itself has a `passive' role in the transition phase. It is the rotation of the orbital radius that pulls the electron, resulting in the electron orbit around the nucleus (orbital momentum comes from the orbital radius), and this rotation continues during the transition phase resulting in the electron following a spiral path.
The photon is actually 2 photons as per the Rydberg formula (denoted initial and final).
    $\lambda _{photon}=R.({\frac {1}{n_{i}^{2}}}-{\frac {1}{n_{f}^{2}}})={\frac {R}{n_{i}^{2}}}-{\frac {R}{n_{f}^{2}}}$
    $\lambda _{photon}=(+\lambda _{i})-(+\lambda _{f})$
The wavelength of the ($\lambda _{i}$) photon corresponds to the wavelength of the orbital radius. The (+$\lambda _{i}$) will then delete the orbital radius as described above (orbital + photon = zero), however the (-$\lambda _{f}$), because of the Rydberg minus term, will have the same phase as the orbital radius and so conversely will increase the orbital radius. And so for the duration of the (+$\lambda _{i}$) photon wavelength, the orbital radius does not change as the 2 photons cancel each other;
    $r_{orbital}=r_{orbital}+(\lambda _{i}-\lambda _{f})$
However, the ($\lambda _{f}$) has the longer wavelength, and so after the ($\lambda _{i}$) photon has been absorbed, and for the remaining duration of this ($\lambda _{f}$) photon wavelength, the orbital radius will be extended until the ($\lambda _{f}$) is also absorbed. For example, the electron is at the n = 1 orbital. To jump from an initial $n_{i}=1$ orbital to a final $n_{f}=2$ orbital, first the ($\lambda _{i}$) photon is absorbed ($\lambda _{i}+\lambda _{orbital}=zero$ which corresponds to 1 complete n = 1 orbit by the electron, the orbital phase), then the remaining ($\lambda _{f}$) photon continues until it too is absorbed (the transition phase).
    $t_{ref}\sim 2\pi 4\alpha ^{2}$
    $\lambda _{i}=1t_{ref}$
    $\lambda _{f}=4t_{ref}$ (n = 2)
After the ($\lambda _{i}$) photon is absorbed, the ($\lambda _{f}$) photon still has $\lambda _{f}=(n_{f}^{2}-n_{i}^{2})t_{ref}=3t_{ref}$ steps remaining until it too is absorbed.
This process does not occur as a single `jump' between energy levels by the electron, but rather absorption/emission of the photon takes place in discrete steps, each step corresponds to a unit of $r_{incr}$ (both photon and orbital radius may be considered as constructs from multiple units of this geometry);
    $r_{incr}=-{\frac {1}{2\pi 2\alpha r_{wavelength}}}$
In summary; the ($\lambda _{i}$) photon, which has the same wavelength as the orbital radius, deletes the orbital radius in steps
$r=r_{orbital}$
WHILE ($\lambda _{i}>0$)
    $r=r+r_{incr}$
    //$\lambda _{i}$ photon
Conversely, because of its minus term, the ($\lambda _{f}$) photon will simultaneously extend the orbital radius accordingly;
WHILE ($\lambda _{f}>0$)
    $r=r-r_{incr}$
    //$\lambda _{f}$ photon
The model assumes orbits also follow along a timeline z-axis
    $t_{orbital}=t_{ref}{\sqrt {1-{\frac {1}{(v_{orbital})^{2}}}}}$
The orbital phase has a fixed radius, however at the transition phase this needs to be calculated for each discrete step as the orbital velocity depends on the radius;
    $t_{transition}=t_{ref}{\sqrt {1-{\frac {1}{(v_{transition})^{2}}}}}$

### Transition

The following refers to an idealized Rydberg atom (a nucleus of point size, infinite mass and disregarding wavelength). In this example the electron transition starts at the initial (n$_i$ = 1) orbital
    $\varphi =0,\;r_{orbital}=2\alpha$
For each step during transition;
    $\beta ={\frac {1}{r_{orbital}{\sqrt {r_{orbital}}}{\sqrt {2\alpha }}}}$
    $\varphi =\varphi +\beta$
Setting t = step number (FOR t = 1 TO ...), we can calculate the radius r and $m$ at each step.
    $r=r_{orbital}+{\frac {t}{2\pi 2\alpha }}$ (number of increments t of $r_{incr}$)
    $m=1+{\frac {t}{2\pi 4\alpha ^{2}}}$ ($m$ as a function of t)
    $\varphi =4\pi {\frac {(m-{\sqrt {m}})}{m}}$ ($\varphi$ at any $m$)
We can then re-write ($m$ is only an integer at prescribed spiral angles);
    $\beta ={\frac {1}{{r_{orbital}}^{2}m^{(3/2)}}}$
Giving integer values at these spiral angles
    $\varphi =(2)\pi ,\;r=4r_{orbital}$ (360°)
    $\varphi =(8/3)\pi ,\;r=9r_{orbital}$ (360+120°)
    $\varphi =(3)\pi ,\;r=16r_{orbital}$ (360+180°)
    $\varphi =(16/5)\pi ,\;r=25r_{orbital}$ (360+216°)
    $\varphi =(10/3)\pi ,\;r=36r_{orbital}$ (360+240°)
    $\varphi =(7/4)\pi ,\;r=49r_{orbital}$
    $\varphi =(7/2)\pi ,\;r=64r_{orbital}$ (360+270°)

#### Bohr model derivation

The wavelength of the H atom
    $\lambda _{H}=\lambda _{p}+\lambda _{e}$
    $H={\frac {2c}{\lambda _{H}}}$
Frequency (Hz)
    $f_{H}={\frac {(n^{2}-1)}{n^{2}}}{\frac {H}{2\pi r_{0}^{2}}}$
We can replace wavelength with the Rydberg constant R and the proton-electron mass ratio μ
    $H={\frac {2c}{\lambda _{H}}}=8\pi c\alpha ^{2}R{\frac {\mu }{\mu +1}}$
This gives the Bohr model when $m=n^{2}$
    $f_{H}={\frac {(n^{2}-1)}{n^{2}}}Rc{\frac {\mu }{\mu +1}}$