2026-08-03
Geometric construction of the orbits of the hydrogen atom
Part 1. First orbit
When an electron enters an atom, it's easy to imagine it starting its history anew: the nucleus creates a field, an orbit arises within this field, and only then do additional properties emerge in the resulting state. In the proposed model, the order of events is different. The electron does not enter the atom as a structureless mathematical point. Even before the atom's formation, it represents a closed wave process with its own frequency, spatial scale, and internal split geometry.
This slight change to the original picture raises an important question. If the internal motion of a free electron is already divided into two closely related geometric branches, should they disappear upon atomic formation? Or is the first outer orbit built on top of the existing internal structure and therefore preserves it? In the second possibility, a simple geometric path emerges from the internal state of a single particle to the splitting of atomic levels and spectral lines.
In this first part, we will examine precisely this particular and most illustrative case: we will trace two closely related states of an electron, transfer them to the first outer bound state, and show how a spatial difference can become an energetic one, and then a spectral one. The general harmonic spectrum, the numbering of states, and the quantitative formula for fine splitting will be introduced in the second part.
1. Electron before the atom
The model is based on an operator that combines two independent phase motions:
\[\tag{1} J(a,b) = \j^{a}(-\j)^{b} = \ep e^{i\pi b} + \em e^{i\pi a}. \] The parameter a describes the internal state of the particle, and the parameter b describes its external motion. The idempotent components e+ and e− are orthogonal, so two motions can simultaneously belong to the same state without mixing:
\[\tag{2} \ep^{2}=\ep, \qquad \em^{2}=\em, \qquad \ep\em=0, \qquad \ep+\em=1. \] For a free electron, whose center does not move in external space, we can set b = 0. Then the internal rotation remains:
\[\tag{3} J_e(t) = \j^{a(t)}, \qquad a(t)=\frac{\omega_e t}{\pi}. \] This corresponds to an internal frequency \(\omega_e\) and a characteristic spatial scale \(r_e\). They are related by the wave invariant already used in the model:
\[\tag{4} \boxed{ \omega_e r_e=c }. \] This formula is convenient to read literally. Over a time corresponding to a change in the internal phase, the wave process traverses its closed spatial scale. Frequency and radius are not two independent characteristics: an increase in one inevitably decreases the other, so that their product remains equal to the wave propagation velocity.
Here, the word "radius" does not mean that the electron is declared to be a small classical sphere. The quantity \(r_e\) characterizes the geometric scale of the internal wave closure. The electron itself in this picture is not an object moving along a circle drawn inside it, but a stable closed process.
2. Previously Obtained Two Branches of Internal Motion
At this stage, we do not need to re-explain the emergence of two internal branches from the properties of split geometry. Such a two-branch electron structure was constructed earlier. Here we accept it as the initial result already obtained and trace what happens to it during the formation of an atomic bound state.
The internal motion of the electron is a single closed trajectory, successively passing along two close radial branches. Their radii are located on opposite sides of the average internal scale \(r_e\):
\[\tag{5} r_{+} = r_e+\Delta r, \qquad r_{-} = r_e-\Delta r. \] Therefore, the average radius of internal motion is
\[ r_e = \frac{r_{+}+r_{-}}{2}, \] and the geometric distance between the branches is
\[\tag{6} \boxed{ r_{+}-r_{-} = 2\Delta r }. \] These are not two independent orbits or two different electrons. Both branches belong to a single continuous internal trajectory. After traversing one branch, the internal state moves to the second leaf, and a full return to the original configuration is completed only after traversing both branches.
\[ r_{+} \;\longrightarrow\; r_{-} \;\longrightarrow\; r_{+}. \] Split geometry here doesn't create two branches anew. It provides a two-componentA complex internal space in which the already constructed two-sheet trajectory can be represented mathematically. Idempotent components serve to distinguish the two sheets and describe the transition between them, but the radii \(r_{+}\) and \(r_{-}\) themselves belong to the geometric model of the electron's internal motion.
Thus, by the time an atom is formed, the electron already has a certain internal structure:
\[ \boxed{ \text{electron} = \text{single internal wave} + \text{two close geometric branches} }. \] The main question of this article can now be formulated more precisely: does the distance \(2\Delta r\) disappear when adding external closed motion, or is the two-branch structure of the electron transferred to the geometry of the bound atomic state?
3. The Appearance of the First Outer Orbit
Now let's place the electron in a bound state with a proton. External motion, described by the second factor of the operator, is added to the internal rotation:
\[\tag{7} J_H(t) = \j^{a(t)}(-\j)^{b(t)}. \] This is not a simple replacement of internal motion by external motion. Formula (7) contains both processes simultaneously. The exponent a does not disappear after the appearance of b; Consequently, the formation of an atomic orbital does not erase the electron's internal state. The external geometry is built upon the existing internal one.
In the previous article, it was shown that the combined action of two mutually orthogonal relativistic motions creates a coefficient \(\alpha_{\mathrm{fs}}^2\). Due to this, the internal scale of the electron is transformed into the scale of the first external bound state:
\[\tag{8} R_1 = \frac{r_e}{\alpha_{\mathrm{fs}}^2}. \] The resulting value coincides with the Bohr radius:
\[\tag{9} \boxed{ R_1=a_0 }. \] This creates a new closed geometry, far exceeding the internal scale of the electron. But it's important to see what exactly has increased. The center of the internal process has gained the ability to perform external closed motion around the nucleus. The internal process itself, however, has not been destroyed. The electron remains an electron even in orbit: its natural frequency, split structure, and internal closure mechanisms continue to enter the complete state.
A simple analogy can be used. If a small rotating wheel is mounted on a larger rotating platform, the platform's motion will not cancel the wheel's rotation. A composite motion with two scales will emerge. This analogy is not a physical model of the electron, but it clearly conveys the main geometric principle: the new external closure preserves the nested internal closure.
4. The electron transfers its splitting to a bound state.
Since the outer orbital is constructed not from an abstract point, but from an already structured electron, its two inner branches must correspond to two close variants of the complete bound state:
\[\tag{10} r_{-},r_{+} \quad\longrightarrow\quad R_1^{(-)},R_1^{(+)}. \] It is convenient to represent the main orbit \(R_1=a_0\) as a geometric mean line, and the preserved internal structure as small displacements relative to it:
\[\tag{11} R_1^{(-)} = R_1+\delta R_{-}, \qquad R_1^{(+)} = R_1+\delta R_{+}. \] Then the distance between the branches is
\[\tag{12} \begin{aligned} \Delta R_1 &= R_1^{(+)}-R_1^{(-)} \\ &= \delta R_{+}-\delta R_{-}. \end{aligned} \] The key assumption of this article is that the small displacements \(\delta R_{-}\) and \(\delta R_{+}\) are not regenerated in the atom. They are an external geometric manifestation of the two internal branches of the electron. In other words, the first orbit inherits the particle's internal splitting.
However, the two branches should not be thought of as two classical circles along which the same ball simultaneously flies. In the model, these are two acceptable ways of closing the complete wave state. When measured, the bound system exhibits one of them, but mathematically both belong to the space of allowed states.
In this paper, the quantity \(\Delta R_1\) is left in its general form for now. Its construction through internal harmonics, as well as its generalization to an arbitrary atomic level, will be carried out in the second part. Here, it is necessary to first follow the physical chain and understand how a small spatial difference can become visible in the spectrum at all.
5. Geometrical difference turns into energy difference
The mere presence of two close radii does not mean that the spectrometer will see two lines. For this to happen, the two geometric states must correspond to different energies. This relationship arises because the energy of a bound state depends on its position in the spectrum.spatial distribution of energy.
Let's denote this dependence by \(E(R)\). Then the two branches correspond to
\[\tag{13} E_1^{(-)} = E\!\left(R_1^{(-)}\right), \qquad E_1^{(+)} = E\!\left(R_1^{(+)}\right). \] Their energy splitting is determined by the difference
\[\tag{14} \Delta E_1 = E_1^{(+)}-E_1^{(-)}. \] Since the distance between the branches is small compared to the radius of the external state, the energy function can be considered near \(R_1\). As a first approximation, we obtain
\[\tag{15} \boxed{ \Delta E_1 \approx \left. \frac{\partial E}{\partial R} \right|_{R_1} \Delta R_1 }. \] Formula (15) is especially important for understanding the mechanism. The energy gradient does not create two branches. They were already present in the previously constructed two-branch geometry of the electron's internal motion and were preserved during the formation of the external bound state. The gradient plays a different role: it translates the existing geometric distance \(\Delta R_1\) into an observed energy difference \(\Delta E_1\).
This relationship can be compared to two nearby points on a slope. The distance between the points is determined by the geometry, and the difference in their heights is determined by the slope of the slope. If the slope is horizontal, even widely spaced points are at the same height. If there is a gradient, a small horizontal shift creates a vertical difference. In an atom, the role of "height" is played by energy.
For a hydrogen-like bound state, the energy dependence in the Bohr approximation can be written as
\[\tag{16} E(R) = -\frac{e^2}{8\pi\varepsilon_0R}. \] Its derivative is nonzero:
\[\tag{17} \frac{\partial E}{\partial R} = \frac{e^2}{8\pi\varepsilon_0R^2}. \] Therefore, two closely spaced geometric branches should indeed have slightly different energies. However, formulas (16)–(17) are used here only to demonstrate the principle. A complete quantitative derivation of the fine structure requires the correct construction of the branches at the nth level and will be given separately.
6. Why One Spectral Line Becomes Two
Energy splitting becomes visible when an atom transitions from a split state to a lower state with energy \(E_f\). For each initial branch, Planck's condition is satisfied:
\[\tag{18} \begin{aligned} h\nu_1 &= E_{n,j_1}-E_f, \\ h
u_2 &= E_{n,j_2}-E_f, \qquad n>1. \end{aligned} \] Subtracting one equality from the other, we obtain
\[\tag{19} \boxed{ h\left|\nu_1-\nu_2\right| = \left|E_{n,j_1}-E_{n,j_2}\right| = \left|\Delta E_{n;j_1j_2}\right| }. \] This is why the spectrometer registers not an abstract "radius splitting," but two close frequencies. Each allowed branch of the complete state has its own energy and therefore emits or absorbs a photon of its own frequency.
The entire causal chain now fits on a single line:
\[\tag{20} \boxed{ \text{two internal branches of an electron} \;\longrightarrow\; \Delta R \;\longrightarrow\; \Delta E \;\longrightarrow\; \Delta\nu }. \] In a normal spectrum, two frequencies can be so close that the naked eye perceives them as a single color. However, an instrument with sufficient resolution reveals that the original line consists of two components. Thus, the invisible internal structure of the bound state gains a measurable manifestation.
7. An important clarification about the first orbital
Here, it is necessary to carefully distinguish between the first outer bound state and the first observed spectral doublet. In the Bohr picture, the scale \(R_1=a_0\) corresponds to the ground state of the hydrogen atom. In modern spectroscopic notation, this is the \(1s\) state, for which the orbital quantum number is zero.
Therefore, the \(1s\) state itself does not form the usual spin-orbit doublet \(P_{1/2},P_{3/2}\). This does not contradict the proposed mechanism. The first outer orbital is needed primarily as a new closed geometry in which the internal electron branches are preserved and become part of the bound system. Their clear spectral separation is observed in those excited states where the geometry allows for multiple values of the total angular momentum.
In other words, the first orbital creates the possibility of transferring internal structure into the atom, but is not required to exhibit it as a familiar doublet. The observed splitting should be sought, for example, in \(P\) states, which have two close branches \(P_{1/2}\) and \(P_{3/2}\).
8. Where is such splitting observed?
The fine structure of atomic spectra was known long before the proposed model appeared. Our goal is not to rediscover the existence of doublets, but to propose a unified geometric mechanism for them. Several haCharacteristic examples are given in the table. For more details, see here [1-2].
| Atom or State | Near Branches | Observed Manifestation | Role in This Article |
|---|---|---|---|
| Hydrogen, \(2P\) | \(2P_{1/2}\) and \(2P_{3/2}\) | Fine Splitting of One Excited Level | Simplest One-Electron Example |
| Hydrogen, \(3P\) | \(3P_{1/2}\) and \(3P_{3/2}\) | Two closely related energy branches | Shows repetition of the mechanism at the next level |
| Hydrogen, \(3D\) | \(3D_{3/2}\) and \(3D_{5/2}\) | Splitting of the \(D\) state | Shows that the phenomenon is not limited to the \(P\) branches |
| Sodium, \(3P\) | \(3P_{1/2}\) and \(3P_{3/2}\) | Yellow lines \(D_1\) and \(D_2\) | The most obvious optical example of a doublet |
| Other states with \(l>0\) | Several allowed values of \(j\) | Doublets and more complex multiplets | Points the way to a common spectrum of states |
The sodium doublet is especially striking. The two intense yellow lines of neutral sodium have wavelengths of approximately 589.0 and 589.6 nm. They arise during transitions from two closely related states \(3P_{3/2}\) and \(3P_{1/2}\) to a common lower state \(3S_{1/2}\). What the eye perceives as the familiar yellow light of sodium, the spectrometer splits into two lines.
For hydrogen, the situation is particularly important theoretically: it is a single-electron system, so it is easier to separate the internal electron structure from multi-electron effects. For alkali atoms, including sodium, the qualitative picture of the doublet remains very clear, but an accurate calculation must already take into account the shielding of the nucleus and the interaction of the outer electron with the inner electron shells.
Therefore, the listed examples are not yet quantitative proof of the model, but a comparison of the proposed geometric mechanism with a actually observed type of phenomenon. Numerical comparison should be based on the general splitting formula; We'll move on to this in the second part.
9. What exactly was obtained in the first part
Now we can separate the original model results from the new assumption and from the experimentally known consequence.
| Position | Status |
| \(\omega_e r_e=c\) | Initial wave invariant of the model |
| \(R_1=r_e/\alpha_{\mathrm{fs}}^2=a_0\) | Result of the previous geometric construction of the Bohr atom |
| The internal state does not disappear when external motion appears | Follows from the product \(\j^a(-\j)^b\), preserving both factors. |
| Two internal branches transform into two branches of the bound state. | The main geometric hypothesis of this article. |
| \(\Delta E\approx(\partial E/\partial R)\Delta R\) | Linear transformation of small geometric separation into energy separation. |
| \(h\Delta\nu=\Delta E\) | Relationship between energy and spectral Splits |
| Hydrogen and Sodium Doublets | Observed Examples of the Phenomenon to Which the Model Is Compared |
This division is crucial. The first part does not yet prove that the entire fine structure of atoms has been completely deduced from split geometry. It establishes a more specific but important result: internal splitting need not be lost upon atom formation, but can be transferred to an external bound state and become observable due to the energy gradient.
10. What to Obtain in Part Two
To transform the discovered mechanism into a general theory, several more questions must be answered. Why do internal states form a discrete family? What integer should be used to number the harmonics? How does the internal spatial scale transfer from the first orbit to an arbitrary level \(n\)? Why do the atomic branches differ by a value containing \(j+1/2\)? And finally, does the geometric construction reproduce the known dependence of the fine structure on \(\alpha_{\mathrm{fs}}^4\) and \(n^{-3}\)?
These questions require a transition from the two selected branches to the full spectrum of allowed closures. In the second part, we will introduce frequencies \(\omega_\kappa\), their corresponding spatial scales \(\rho_\kappa\), atomic levels \(R_n\), and the relationship of the harmonic index to the total angular momentum of state.
Thus, the first part provides the physical plot, and the second should provide its general mathematics. First, we saw how one particle transfers its internal structure to an atom; then we must show why this structure gives rise to precisely the discrete order detected in spectroscopy.
Conclusion
The main conclusion can be formulated simply: an atomic orbital is not created from a structureless point. In the proposed model, the electron is already a closed wave process and has an internal split geometry. When external motion around the nucleus is added to internal rotation, the previous structure does not disappear, because both motions remain within a single operator \(J(a,b)\).
Therefore, two closely related internal states of the electron can continue as two closely related branches of a bound state. Their spatial separation is transformed by the energy gradient into an energy difference, and this difference manifests itself as two emission or absorption frequencies:
\[\tag{21} \boxed{ \begin{aligned} \text{internal electron splitting} &\longrightarrow \text{bound state splitting} \\ &\longrightarrow \text{energy difference} \\ &\longrightarrow \text{spectral doublet}. \end{aligned} } \] The most familiar visual image of this result is two yellow sodium lines instead of one. But behind this simple observation lies a general question: why does nature allow strictly defined, rather than arbitrary, branches of the state? The answer requires moving from a partial doublet to a spectrum of internal harmonics. This will be the subject of the second part.
Related materials: "Geometric Origin of the Square of the Fine Structure Constant and the Parameters of the Bohr Atom", "Particle Mass as a Geometric Projection".
Materials Used
- Reference data on sodium spectral lines and hydrogen levels. NIST Table for Na I.
- Reference data on sodium spectral lines and hydrogen levels. H I level base.