2026-08-13
From the geometric state of the electron to the orbit of the hydrogen atom
The classical radius of an electron is almost nineteen thousand times smaller than the radius of the first orbit of a hydrogen atom. At first glance, these are two completely different scales: one is associated with a free electron, the other with the bound state of an atom. However, there is a simple sequence between them, built entirely on the fine structure constant [1]. It is sufficient to double the radius by a factor of \(1/\alpha_{\mathrm{fs}}\) to move from the classical electron radius first to the reduced Compton length, and then to the Bohr radius.
But the radius alone does not determine the state of motion. Along with it, it is necessary to trace the frequency, linear velocity, energy, and magnetic moment. It is here that the main result arises: during the complete transition from the electron's internal geometric state to an electron in the first orbit of hydrogen, the radius increases by a factor of \(1/\alpha_{\mathrm{fs}}^2\), the frequency decreases by a factor of \(1/\alpha_{\mathrm{fs}}^3\), and the velocity decreases by a factor of \(1/\alpha_{\mathrm{fs}}\).
This construction has a broader purpose than simply comparing a few known radii and frequencies. It is proposed to develop the concept of Wave Electricity, in which the electrical properties of a particle are associated with its internal periodic process, and the electrical interaction is associated with the change and propagation of the wave state generated by it. Therefore, before describing the electric force, wave, and particle interactions, it is necessary to establish how the electron's initial internal frequency is mapped into external space and how its radius, velocity, and energy change in the bound atomic state.
This paper considers a geometric interpretation of known electron and atomic scales. The algebraic relationships between physical constants are exact, while their combination into a sequential transformation process represents the hypothesis of the proposed model.
1. Problem Statement
We will consider three sequentially bound states:
\[\tag{1} \text{internal geometric state of an electron} \;\longrightarrow\; \text{observed projection of a free electron} \;\longrightarrow\; \text{electron in the first orbit of hydrogen}. \] The task is not only to obtain the correct values of the radii. It is necessary to show how all the main characteristics of the periodic motion are consistently transformed at each stage:
\[\tag{2} R,\qquad \Omega,\qquad v=R\Omega,\qquad \hbar\Omega,\qquad \mu. \] Here \(R\) denotes the radius of the corresponding state, \(\Omega\) is its angular frequency, \(v\) is the linear velocity, \(\hbar\Omega\) is the energy scale associated with the frequency, and \(\mu\) is the magnetic moment of the annular motion.
2. Notations and Initial Quantities
We introduce notations that will be used hereafter without abbreviations. The classical radius of an electron is
\[\tag{3} r_e=2{,}8179403\cdot10^{-15}\, \text{m}. \] Fine structure constant:
\[\tag{4} \alpha_{\mathrm{fs}}=7{.}2973526\cdot10^{-3}, \qquad \frac{1}{\alpha_{\mathrm{fs}}}\approx137{.}036. \] We define the internal geometric angular frequency of an electron using the classical radius:
\[\tag{5} \boxed{ \omega_e=\frac{c}{r_e} } =1{.}0638709\cdot10^{23}\, \text{s}^{-1}. \] Since \(\omega_e\) is an angular frequency, it should be measured in radians per second or, since the radian is dimensionless, in \(\text{s}^{-1}\). The frequency in cycles per second would be \(f_e=\omega_e/(2\pi)\).
The rest energy of an electron is denoted by \(E_C\):
\[\tag{6} \boxed{ E_C=m_ec^2 } =8{,}1871058\cdot10^{-14}\, \text{J}. \] The subscript \(C\) here indicates the Compton energy scale. The quantity \(E_C\) should not be confused with the Coulomb potential energy.
The following relations hold between the selected quantities
\[\tag{7} r_e\omega_e=c, \] \[\tag{8} \boxed{ E_C=\alpha_{\mathrm{fs}}\hbar\omega_e }, \qquad \hbar\omega_e=\frac{E_C}{\alpha_{\mathrm{fs}}}. \] Formula (8) shows that the internal geometric energy \(\hbar\omega_e\) is \(1/\alpha_{\mathrm{fs}}\) times greater than the observed rest energy. Therefore, the internal frequency \(\omega_e\) is not directly identified with the Compton frequency of the electron.
3. Internal Geometric State of the Electron
The initial state of the model is a periodic process with radius \(r_e\) and angular frequency \(\omega_e\). Its linear geometric velocity is determined by the product of the radius and the angular frequencyFrequency:
\[\tag{9} v_g=r_e\omega_e=c. \] This equality does not mean that the observed electron is a classical material point rotating in a circle of radius \(r_e\) at the speed of light. We are talking about an internal geometric process of the model, the characteristics of which are then mapped into external observable quantities.
The energy scale of the internal process is
\[\tag{10} E_g=\hbar\omega_e =\frac{\hbar c}{r_e} =\frac{E_C}{\alpha_{\mathrm{fs}}}. \] Thus, the initial geometric state is defined by the set
\[\tag{11} \boxed{ \left( r_e,\, \omega_e,\, c,\, \frac{E_C}{\alpha_{\mathrm{fs}}} \right). } \] 4. Observed projection of a free electron
To move from the inner geometric region to the outer observable region, the model adopts a conjugate transformation of radius and frequency:
\[\tag{12} \boxed{ R_{\mathrm{obs}}=\frac{R_g}{\alpha_{\mathrm{fs}}}, \qquad \Omega_{\mathrm{obs}}=\alpha_{\mathrm{fs}}\omega_g. } \] The frequency decreases in the same proportion as the spatial scale increases. Therefore, the product of radius and frequency is preserved:
\[\tag{13} R_{\mathrm{obs}}\Omega_{\mathrm{obs}} =\frac{R_g}{\alpha_{\mathrm{fs}}} \alpha_{\mathrm{fs}}\omega_g =R_g\omega_g. \] Applying this rule to the initial electron state, we obtain the observed radius
\[\tag{14} \boxed{ \frac{r_e}{\alpha_{\mathrm{fs}}} =\frac{\hbar}{m_ec} =\bar\lambda_C, } \] that is, the reduced Compton length of the electron [2]. The observed frequency is
\[\tag{15} \boxed{ \omega_C=\alpha_{\mathrm{fs}}\omega_e =\frac{m_ec^2}{\hbar}. } \] Checking the speed yields
\[\tag{16} \bar\lambda_C\omega_C =\frac{r_e}{\alpha_{\mathrm{fs}}} \alpha_{\mathrm{fs}}\omega_e =r_e\omega_e =c. \] The energy of the observed frequency becomes equal to the rest energy of the electron \(E_C\):
\[\tag{17} \boxed{ \hbar\omega_C =\alpha_{\mathrm{fs}}\hbar\omega_e =E_C =m_ec^2. } \] The frequency decrease in formula (15) can be described as a consequence of the dilation of internal geometric time. When observing a periodic source in the transverse direction, the same decrease manifests itself as the transverse Doppler effect. In this sense, time dilation is the transformation mechanism, and the transverse Doppler shift is its observable frequency manifestation.
The increase in radius in formula (12) is not the usual transverse length dilation in special relativity. It is a separate conjugate geometric transformation of the model, chosen so that the product \(R\Omega\) is preserved when representing the internal process.
5. Formation of the Electron State in the Hydrogen Orbit
The next step relates the observed scale of a free electron to the first orbit of the hydrogen atom. The reduced Compton length and the Bohr radius differ by another factor, \(1/\alpha_{\mathrm{fs}}\):
\[\tag{18} \boxed{ a_0 =\frac{\bar\lambda_C}{\alpha_{\mathrm{fs}}} =\frac{r_e}{\alpha_{\mathrm{fs}}^2} =\frac{\hbar}{m_ec\alpha_{\mathrm{fs}}}. } \] However, the second transition differs fundamentally from the first. During the transition to the orbital state, the velocity is no longer maintained equal to \(c\). For the first orbit of hydrogen, it becomes equal to
\[\tag{19} \boxed{ v_B=\alpha_{\mathrm{fs}}c. } \] The orbital angular frequency is determined by the usual kinematic relation:
\[\tag{20} \omega_B=\frac{v_B}{a_0}. \] After substituting formulas (18) and (19), we obtain
\[\tag{21} \omega_B =\frac{\alpha_{\mathrm{fs}}c} {r_e/\alpha_{\mathrm{fs}}^2} =\alpha_{\mathrm{fs}}^3\frac{c}{r_e}. \] Therefore, relative to the original internal frequency:
\[\tag{22} \boxed{ \omega_B=\alpha_{\mathrm{fs}}^3\omega_e. } \] A relative to the observed Compton frequency:
\[\tag{23} \boxed{ \omega_B=\alpha_{\mathrm{fs}}^2\omega_C. } \] Thus, the second transformation has the form
\[\tag{24} \boxed{ \bar\lambda_C \longrightarrow \frac{\bar\lambda_C}{\alpha_{\mathrm{fs}}}=a_0, \qquad \omega_C \longrightarrow \alpha_{\mathrm{fs}}^2\omega_C=\omega_B, \qquad c \longrightarrow \alpha_{\mathrm{fs}}c. } \] 6. Three Sequential States
Let's collect the obtained results in a single table. It shows not three independent sets of quantities, but rather the sequence of a single geometric construction: from the internal state of the electron through the observed free state to the state of the electron in the first orbit of hydrogen.
| Magnitude | Internal geometric state of the electron | Observed projection of a free electron | Electron in the first orbit of hydrogen |
|---|---|---|---|
| Radius of state | \(r_e\) | \(\dfrac{r_e}{\alpha_{\mathrm{fs}}}=\bar\lambda_C\) | \(\dfrac{r_e}{\alpha_{\mathrm{fs}}^2}=a_0\) |
| Angular Frequency | \(\omega_e\) | \(\alpha_{\mathrm{fs}}\omega_e=\omega_C\) | \(\alpha_{\mathrm{fs}}^3\omega_e=\omega_B\) |
| Linear Velocity | \(c\) | \(c\) | \(\alpha_{\mathrm{fs}}c=v_B\) |
| Periodic Energy process | \(\hbar\omega_e=\dfrac{E_C}{\alpha_{\mathrm{fs}}}\) | \(\hbar\omega_C=E_C=m_ec^2\) | \(\hbar\omega_B=E_C\alpha_{\mathrm{fs}}^2\) |
| Product of radius and frequencies | \(r_e\omega_e=c\) | \(\bar\lambda_C\omega_C=c\) | \(a_0\omega_B=\alpha_{\mathrm{fs}}c\) |
| Magnetic moment of circular motion | \(\dfrac{e}{2}r_e c=\alpha_{\mathrm{fs}}\mu_B\) | \(\dfrac{e}{2}\dfrac{r_e}{\alpha_{\mathrm{fs}}} c=\mu_B\) | \(\dfrac{e}{2}\dfrac{r_e}{\alpha_{\mathrm{fs}}^2} (c\alpha_{\mathrm{fs}})=\mu_B\) |
where: \(\mu_B\) is the Bohr magnetron [3].
The first transition increases the radius and decreases the frequency in inverse proportion, so the speed remains equal to \(c\). In the second transition, the radius increases by another factor of \(1/\alpha_{\mathrm{fs}}\), but the frequency decreases by a factor of \(1/\alpha_{\mathrm{fs}}^2\). Therefore, the product of the radius and frequency, i.e., the linear velocity, decreases by a factor of \(1/\alpha_{\mathrm{fs}}\).
It is especially important that the intermediate scale
\[\tag{25} \frac{r_e}{\alpha_{\mathrm{fs}}}=\bar\lambda_C \] plays a dual role in the circuit. It is the observed spatial scale of a free electron and, simultaneously, the geometric initial scale for the subsequent transition to a hydrogen orbital.
7. Complete Transition from a Geometric Electron to a Hydrogen Orbital
If the intermediate state is temporarily omitted, the complete transition can be represented by a single table.
| Magnitude | Initial Geometric Value | Complete Transition Ratio | Orbital Value hydrogen |
|---|---|---|---|
| Radius | \(r_e\) | \(1/\alpha_{\mathrm{fs}}^2\) | \(a_0=r_e/\alpha_{\mathrm{fs}}^2\) |
| Angular frequency | \(\omega_e\) | \(\alpha_{\mathrm{fs}}^3\) | \(\omega_B=\alpha_{\mathrm{fs}}^3\omega_e\) |
| Linear speed | \(c\) | \(\alpha_{\mathrm{fs}}\) | \(v_B=\alpha_{\mathrm{fs}}c\) |
| Energy of periodic Process | \(E_C/\alpha_{\mathrm{fs}}\) | \(\alpha_{\mathrm{fs}}^3\) | \(E_C\alpha_{\mathrm{fs}}^2\) |
| Magnetic moment of the model | \(\alpha_{\mathrm{fs}}\mu_B\) | \(1/\alpha_{\mathrm{fs}}\) | \(\mu_B\) |
In compact form, the result is written like this:
\[\tag{26} \begin{array}{rcl} r_e &\longrightarrow& \dfrac{r_e}{\alpha_{\mathrm{fs}}^2}=a_0, \\[2mm] \omega_e &\longrightarrow& \alpha_{\mathrm{fs}}^3\omega_e=\omega_B, \\[2mm] c &\longrightarrow& \alpha_{\mathrm{fs}}c=v_B, \\[2mm] \dfrac{E_C}{\alpha_{\mathrm{fs}}} &\longrightarrow& E_C\alpha_{\mathrm{fs}}^2. \end{array} \] 8. Why the third power appears in frequency
A decrease in frequency by a factor of \(1/\alpha_{\mathrm{fs}}^3\) may seem surprising, since the radius only increases by a factor of \(1/\alpha_{\mathrm{fs}}^2\). The reason becomes obvious if we start with the kinematic definition of angular frequency:
\[\tag{27} \Omega=\frac{v}{R}. \] The ratio of the final orbital frequency to the initial internal frequency is
\[\tag{28} \frac{\omega_B}{\omega_e} =\frac{v_B}{c}\frac{r_e}{a_0}. \] The radius adds a factor
\[\tag{29} \frac{r_e}{a_0}=\alpha_{\mathrm{fs}}^2, \] and decreasing the speed adds another factor:
\[\tag{30} \frac{v_B}{c}=\alpha_{\mathrm{fs}}. \] Therefore
\[\tag{31} \boxed{ \frac{\omega_B}{\omega_e} =\alpha_{\mathrm{fs}} \alpha_{\mathrm{fs}}^2 =\alpha_{\mathrm{fs}}^3. } \] In other words, two powers of the fine structure constant are associated with an increase in radius, and the third power arises from a decrease in linear velocity. If the velocity remained equal to \(c\),The final frequency would be equal to \(\alpha_{\mathrm{fs}}^2\omega_e\). But such a state would no longer correspond to the velocity of an electron in the first Bohr orbit.
9. Energy and the Meaning of Frequency Scales
Three different frequencies and three corresponding energy scales appear in the sequence under consideration. They must be strictly distinguished.
Internal geometric energy:
\[\tag{32} E_g=\hbar\omega_e=\frac{E_C}{\alpha_{\mathrm{fs}}}. \] Observed rest energy of a free electron:
\[\tag{33} E_C=\hbar\omega_C=m_ec^2. \] Orbital frequency energy scale:
\[\tag{34} \boxed{ E_{\mathrm{orb}} =\hbar\omega_B =E_C\alpha_{\mathrm{fs}}^2. } \] The last equality follows directly from the frequency transformation:
\[\tag{35} \hbar\omega_B =\hbar\alpha_{\mathrm{fs}}^3\omega_e =\alpha_{\mathrm{fs}}^2 \left(\alpha_{\mathrm{fs}}\hbar\omega_e\right) =E_C\alpha_{\mathrm{fs}}^2. \] In this case, the electron's rest energy does not disappear and remains equal to \(E_C\). The quantity \(E_{\mathrm{orb}}\) characterizes an additional energy scale of the bound motion and does not replace the total energy of the electron.
In the Bohr model, for the ground state, the kinetic energy is equal to half this scale:
\[\tag{36} K_1=\frac12E_C\alpha_{\mathrm{fs}}^2. \] The total binding energy relative to the ionization boundary has the value
\[\tag{37} \mathcal E_1=-\frac12E_C\alpha_{\mathrm{fs}}^2. \] Therefore, the quantity \(\hbar\omega_B=E_C\alpha_{\mathrm{fs}}^2\) cannot be called the total energy of the ground state of an atom. It defines the orbital energy scale, while the total binding energy contains an additional coefficient \(-1/2\).
The relationship between the orbital energy scale and the Coulomb potential energy and its spatial gradient is a separate topic and is not discussed in detail in this paper.
10. Magnetic Moment during Geometric Unfolding
Let's consider the magnetic moment of the periodic annular motion of a charge. If a charge \(e\) makes one revolution with an angular frequency \(\Omega\), the current is
\[\tag{38} I=\frac{e\Omega}{2\pi}. \] Area of a ring of radius \(R\):
\[\tag{39} S=\pi R^2. \] Then the magnetic moment:
\[\tag{40} \boxed{ \mu=IS =\frac{e\Omega R^2}{2} =\frac{evR}{2}. } \] For the initial internal geometric state, we obtain
\[\tag{41} \mu_g =\frac{ecr_e}{2}. \] Since
\[\tag{42} r_e=\alpha_{\mathrm{fs}}\frac{\hbar}{m_ec}, \] then
\[\tag{43} \mu_g =\alpha_{\mathrm{fs}} \frac{e\hbar}{2m_e} =\alpha_{\mathrm{fs}}\mu_B. \] After the transition to the observed free electron, the radius becomes equal to \(\bar\lambda_C=r_e/\alpha_{\mathrm{fs}}\), and the velocity remains equal to \(c\). Therefore
\[\tag{44} \mu_{\mathrm{free}} =\frac{ec}{2} \frac{r_e}{\alpha_{\mathrm{fs}}} =\frac{e\hbar}{2m_e} =\mu_B. \] For an electron in the first orbit of hydrogen, the radius increases again, but the velocity decreases:
\[\tag{45} \mu_{\mathrm H} =\frac{e}{2} \left(\alpha_{\mathrm{fs}}c\right) \frac{r_e}{\alpha_{\mathrm{fs}}^2} =\frac{ecr_e}{2\alpha_{\mathrm{fs}}} =\mu_B. \] Thus, in the ring geometric model, the observed free electron and the electron in the first orbit of hydrogen produce the same magnetic moment:
\[\tag{46} \boxed{ \mu_{\mathrm{free}}=\mu_{\mathrm H}=\mu_B. } \] This equality is ensured by the compensation of two changes: upon transition to the hydrogen orbit, the radius increases by \(1/\alpha_{\mathrm{fs}}\) times, and the velocity decreases by \(1/\alpha_{\mathrm{fs}}\) times. Therefore, the product \(vR\), which is part of the magnetic moment formula, is conserved.
The obtained result pertains to the annular motion of the charge within the geometric model. In standard quantum mechanics, the ground state of hydrogen \(1s\) has an orbital quantum number \(l=0\) and zero orbital magnetic moment. Therefore, the internal annular motion described here cannot be identified with the standard quantum orbital moment of the electron without additional justification.
11. Numerical Verification
We substitute the numerical values of the fundamental constants. The results for three consecutive states are shown in the table.
| Magnitude | Geometric electron | Observed free electron | Electron in hydrogen orbital |
|---|---|---|---|
| Radius, m | \(2{,}81794\cdot10^{-15}\) | \(3{,}86159\cdot10^{-13}\) | \(5{,}29177\cdot10^{-11}\) |
| Angular frequency, \(\text{с}^{-1}\) | \(1{,}06387\cdot10^{23}\) | \(7{,}76344\cdot10^{20}\) | \(4{,}13414\cdot10^{16}\) |
| Linear velocity, m/s | \(2{,}99792\cdot10^8\) | \(2{,}99792\cdot10^8\) | \(2{,}18769\cdot10^6\) |
| Energy scale, J | \(1{,}12193\cdot10^{-11}\) | \(8{,}18711\cdot10^{-14}\) | \(4{,}35974\cdot10^{-18}\) |
| Magnetic moment of the model, J/T | \(6.767\cdot10^{-26}\) | \(9.27401\cdot10^{-24}\) | \(9.27401\cdot10^{-24}\) |
The intermediate radius exactly matches the reduced Compton length:
\[\tag{47} \frac{r_e}{\alpha_{\mathrm{fs}}} =3{,}86159\cdot10^{-13}\, \text{m} =\bar\lambda_C. \] The final radius is the same as the Bohr radius:
\[\tag{48} \frac{r_e}{\alpha_{\mathrm{fs}}^2} =5.29177\cdot10^{-11}\, \text{m} =a_0. \] Velocity on the first orbit:
\[\tag{49} \alpha_{\mathrm{fs}}c =2.18769\cdot10^6\, \text{m/s}. \] Orbital angular frequency:
\[\tag{50} \omega_B =\alpha_{\mathrm{fs}}^3\omega_e =4{.}13414\cdot10^{16}\, \text{s}^{-1}. \] Corresponding energy scale:
\[\tag{51} \hbar\omega_B =E_C\alpha_{\mathrm{fs}}^2 =4{.}35974\cdot10^{-18}\, \text{J} \approx27{.}2114\, \text{eV}. \] The total binding energy of the ground state of hydrogen is half this value minus:
\[\tag{52} \mathcal E_1 =-\frac12\hbar\omega_B \approx-13{.}6057\, \text{eV}. \] 12. Known Identities and Hypotheses of the Model
To correctly evaluate the result, it is necessary to separate the exact relationships between known constants from their new geometric interpretation.
Regarding the Known Identities include:
\[\tag{53} r_e=\alpha_{\mathrm{fs}}\bar\lambda_C, \qquad a_0=\frac{\bar\lambda_C}{\alpha_{\mathrm{fs}}}, \] \[\tag{54} a_0=\frac{r_e}{\alpha_{\mathrm{fs}}^2}, \qquad v_B=\alpha_{\mathrm{fs}}c, \] \[\tag{55} E_C=m_ec^2, \qquad \mu_B=\frac{e\hbar}{2m_e}. \] From these equalities, the frequency dependencies follow algebraically.
\[\tag{56} \omega_C=\alpha_{\mathrm{fs}}\omega_e, \qquad \omega_B=\alpha_{\mathrm{fs}}^3\omega_e, \] if the initial frequency is defined as \(\omega_e=c/r_e\), and the orbital frequency is defined as \(\omega_B=v_B/a_0\).
The hypotheses of the geometric model are:
- the existence of an internal geometric region of the electron with its own time and spatial scale;
- interpretation \(\omega_e=c/r_e\) as the internal geometric frequency;
- mapping the internal frequency to the observable frequency with a coefficient of \(\alpha_{\mathrm{fs}}\);
- conjugate increase in radius by a factor of \(1/\alpha_{\mathrm{fs}}\);
- interpretation of the classical, Compton, and Bohr radii as successive scales of a single geometric transformation;
- description of the magnetic moment through the annular motion of the internal charge.
This distinction is fundamental. The coincidence of the numerical scales does not yet prove the physical mechanism of the transition, but it shows that the proposed sequence does not contradict the fundamental algebraic relationships between the parameters of the electron and the hydrogen atom.
13. Discussion
The resulting picture combines three characteristic spatial scales:
\[\tag{57} r_e, \qquad \bar\lambda_C, \qquad a_0. \] Each subsequent radius is \(1/\alpha_{\mathrm{fs}}\) times larger than the previous one:
\[\tag{58} r_e \xrightarrow{\ 1/\alpha_{\mathrm{fs}}\ } \bar\lambda_C \xrightarrow{\ 1/\alpha_{\mathrm{fs}}\ } a_0. \] However, the frequency sequence is not completely symmetrical to the radii sequence:
\[\tag{59} \omega_e \xrightarrow{\, \alpha_{\mathrm{fs}}\ } \omega_C \xrightarrow{\, \alpha_{\mathrm{fs}}^2\ } \omega_B. \] At the first transition, the radius and frequency change inversely, so the velocity remains equal to \(c\). At the second transition, the frequency gains an additional factor \(\alpha_{\mathrm{fs}}\), reflecting the decrease in the velocity of the associated motion:
\[\tag{60} c\longrightarrow c\longrightarrow\alpha_{\mathrm{fs}}c. \] This is why a complete transition cannot be described solely by space dilation and an inverse decrease in frequency. The formation of an atomic state involves an additional kinematic change—a decrease in orbital velocity.
The magnetic moment of the annular motion is simultaneously conserved between the observed free state and the orbital state. In the second transition, the decrease in velocity is exactly compensated by the increase in radius:
\[\tag{61} v_Ba_0 =\left(\alpha_{\mathrm{fs}}c\right) \frac{\bar\lambda_C}{\alpha_{\mathrm{fs}}} =c\bar\lambda_C. \] This relation also reproduces the Bohr angular momentum condition:
\[\tag{62} m_ev_Ba_0 =m_ec\bar\lambda_C =\hbar. \] Thus, the radius, frequency, velocity, energy scale, angular momentum, and magnetic moment form a consistent set. The novelty of the proposed interpretation lies not in the individual formulas—many of which are known—but in their unification into a single sequence of geometric states.
14. Conclusion
This paper considers the transition from the internal geometric state of an electron with radius \(r_e\) and frequency \(\omega_e=c/r_e\) to the observed free electron, and then to an electron in the first orbit of the hydrogen atom.
The first transformation increases the radius by a factor of \(1/\alpha_{\mathrm{fs}}\) and decreases the frequency by the same proportion. As a result, the classical radius transforms into the reduced Compton length, the intrinsic frequency transforms into the Compton frequency, and the energy \(\hbar\omega_e\) transforms into the observed rest energy \(E_C=m_ec^2\). The product of the radius and frequency remains equal to \(c\).
The second transformation increases the radius by another factor of \(1/\alpha_{\mathrm{fs}}\), but the frequency decreases by a factor of \(1/\alpha_{\mathrm{fs}}^2\). The additional factor is due to the decrease in velocity from \(c\) to \(\alpha_{\mathrm{fs}}c\). The result is the Bohr radius, the Bohr orbital velocity, and the corresponding angular frequency.
\[\tag{63} \begin{array}{rclcl} r_e &\xrightarrow{\,1/\alpha_{\mathrm{fs}},}& \bar\lambda_C &\xrightarrow{\,1/\alpha_{\mathrm{fs}},}& a_0, \\[2mm] \omega_e &\xrightarrow{\,\alpha_{\mathrm{fs}},}& \omega_C &\xrightarrow{\,\alpha_{\mathrm{fs}}^2,}& \omega_B, \\[2mm] c &\longrightarrow& c &\longrightarrow& \alpha_{\mathrm{fs}}c, \\[2mm] \dfrac{E_C}{\alpha_{\mathrm{fs}}} &\longrightarrow& E_C &\longrightarrow& E_C\alpha_{\mathrm{fs}}^2. \end{array} \] The complete transition from the geometric state of the electron to the hydrogen orbital is therefore characterized by three coefficients:
\[\tag{64} \boxed{ R\times\frac{1}{\alpha_{\mathrm{fs}}^2}, \qquad \Omega\times\alpha_{\mathrm{fs}}^3, \qquad v\times\alpha_{\mathrm{fs}}. } \] The proposed scheme shows that the known electron and atomic scales can be represented as a consistent geometric unfolding. In this case, the precise algebraic relationships between the constants are separated from the hypothesis of the physical existence of an internal geometric region. The next task is to relate this transformation to the dynamics of the full model operator and independently substantiate the mechanism for the formation of the bound state.
The resulting sequence allows us to impart an additional geometric meaning to the fine-structure constant. Within the framework of the proposed model, it acts not only as a well-known dimensionless characteristic of electromagnetic interaction, but also as a transition coefficient between the internal geometric state of the electron and its external manifestations. It links adjacent spatial scales, determines the fraction of the internal frequency and energy that manifests itself during observation, and, in the bound state, specifies the ratio of the electron's orbital velocity to the speed of light.
Thus, the fine-structure constant characterizes the degree to which the electron's internal geometry manifests itself in external space. The same value determines the scaling of the radius, the decrease in the observed frequency and energy, and the deceleration of the bound motion. In this sense, it represents a universal coefficient of consistency between the electron's internal periodic process and its observed free and atomic states.