2026-08-03
Geometric construction of the orbits of the hydrogen atom
Part 2. Higher Orbits and Subtle Splitting
In the first part, we traced a qualitative chain: two close branches of the electron's internal motion are preserved during the formation of an external bound state, their geometric separation is transformed into an energy difference, and this difference becomes a splitting of the spectral line. This result explains the mechanism by which the doublet appears, but it still doesn't answer three main questions. Why does an atom allow only certain branches, not just any branches? How are these branches numbered? And why does the magnitude of their energy splitting depend simultaneously on the principal quantum number \(n\) and the total angular momentum \(j\)?
Now let's move from two visual branches to the overall spectrum of allowed states. For this, we will need three scales already obtained in the model: the electron's internal frequency \(\omega_e\), the classical electron radius \(r_e\), and the radius of the first Bohr orbit \(a_0\). Their relationship has a particularly simple form:
\[\tag{1} r_e = \alpha_{\mathrm{fs}}^2a_0. \] It is precisely this small scale ratio that will yield the order of fine splitting \(\alpha_{\mathrm{fs}}^4\). The first factor \(\alpha_{\mathrm{fs}}^2\) is already contained in the energy of the atomic level, and the second will arise from a small geometric shift of the orbital branch relative to the main radius.
This article continues part one and builds on the previously obtained geometric derivation of the parameters of the Bohr atom. Our goal is not to repeat the standard relativistic calculation, but to show which spatial structure can generate the same splitting dependence on \(n\) and \(j\).
1. From Two Branches to a Spectrum
The word "spectrum" is usually associated with the set of observable frequencies of light. But before an optical spectrum can arise, the bound system itself must possess a set of allowed internal states. Therefore, here we will distinguish between two related concepts: the spectrum of electron geometric modes and the spectrum of frequencies emitted during transitions between atomic levels.
Let's denote the set of allowed internal frequencies by
\[\tag{2} \mathcal S_{\omega} = \left\{ \omega_1,\omega_2,\omega_3,\ldots \right\}. \] This is not an arbitrary set of numbers. The allowed frequency must ensure the closure of the internal wave simultaneously with the closure of the external atomic state. If the phase does not return to its original value after a complete cycle, such a state does not form a stable geometric configuration.
The simplest closure condition yields an integer number of internal revolutions:
\[\tag{3} \omega_{\kappa} = \kappa\omega_e, \qquad \kappa=1,2,3,\ldots \] where \(\omega_e\) is the carrier frequency of the internal motion of a free electron, and \(\kappa\) is the number of its complete phase revolutions per cycle of the composite state. The integer nature of \(\kappa\) appears not as an additional energy quantization rule, but as a standard condition for the closure of a periodic trajectory:
\[\tag{4} \omega_{\kappa}T_e = 2\pi\kappa, \qquad T_e = \frac{2\pi}{\omega_e}. \] It is important not to confuse the frequency \(\omega_{\kappa}\) with the new natural frequency of another electron. The particle's mass is still related to the fundamental intrinsic frequency \(\omega_e\). Harmonics describe the modes of phase matching of the already composite system "internal electron motion - external atomic state."
\[\tag{5} m_ec^2 \longleftrightarrow \omega_e, \qquad \text{atomic mode} \longleftrightarrow \omega_{\kappa}. \] 2. Inverse Spectrum of Spatial Scales
For the internal motion of an electron, a relationship between the characteristic frequency and radius was previously obtained. It expresses the constancy of the linear wave scale:
\[\tag{6} \omega_e r_e = c. \] If the same geometric relationship holds for each resolved harmonic, then
\[\tag{7} \omega_{\kappa}\rho_{\kappa} = c. \] Substituting \(\omega_{\kappa}=\kappa\omega_e\) into this expression, we obtain the inverse spectrum of internal spatial scales:
\[\tag{8} \boxed{ \rho_{\kappa} = \frac{r_e}{\kappa} }. \] The greater the number of phase revolutions, the smaller the corresponding radial scale. This is a common property of a closed wave: increasing the spatial frequency by a factor of \(\kappa\) reduces the length of one of its elements by the same factor. Therefore, the frequency spectrum and the length spectrum are directed oppositely:
\[\tag{9} \omega_e, 2\omega_e, 3\omega_e,\ldots \quad\longleftrightarrow\quad r_e, \frac{r_e}{2}, \frac{r_e}{3},\ldots \] For two modes, the difference in their characteristic scales is
\[\tag{10} \boxed{ \Delta\rho_{\kappa_1\kappa_2} = r_e \left( \frac{1}{\kappa_1} - \frac{1}{\kappa_2} \right) }. \] This transforms the two branches of the first part into a common family. Now, each allowed branch has its own integer \(\kappa\), and the geometric distance between any two branches is determined by the difference of their reciprocals.
3. Principal Quantum Number and Outer Radius
The index \(\kappa\) numbers the internal harmonics, but it should not replace the principal quantum number. The number \(n\) has a different geometric meaning: it specifies the outer atomic level and the number of phase links required to close the electron wave at this level.
For a hydrogen-like atom, the fundamental radius of the \(n\)th level in the Bohr approximation is
\[\tag{11} R_n = \frac{n^2a_0}{Z}, \] where \(Z\) is the charge number of the nucleus. Let's consider hydrogen first, that is, \(Z=1\):
\[\tag{12} R_n = n^2a_0. \] A single value of \(n\) specifies a common large radius, while different values of \(kappa\) distinguish between closely spaced branches within this level. Therefore, it is convenient to write the total radius as
\[\tag{13} R_{n,\kappa} = R_n+\delta R_{n,\kappa}, \qquad \left|\delta R_{n,\kappa}\right| \ll R_n. \] Here \(R_n\) describes the basic geometry of the atom, and \(\delta R_{n,\kappa}\) is a small displacement of a specific branch. Thus, the two indices answer different questions:
| Index | Geometric meaning | What it changes |
|---|---|---|
| \(n\) | Number of phase links of the outer atomic state | Principal radius \(R_n=n^2a_0\) and level energy |
| \(\kappa\) | Number of internal harmonic revolutions | Minor position of a branch within a single level |
4. Why the factor n appears
In the first part, we considered only the fact of the transfer of internal splitting to the outer bound state. For the general level, its phase length must be taken into account. At the \(n\)th level, the closed state consists of \(n\) successively matched phase links. Each link carries the same small radial contribution \(\rho_{\kappa}\).
If after each link the inner and outer phases are connected in the same orientation, these contributions add coherently. Then the total branch shift is the sum of \(n\) identical elements:
\[\tag{14} \delta R_{n,\kappa} = -\sum_{q=1}^{n}\rho_{\kappa}. \] The minus sign means that the correction in question is directed toward a smaller radius relative to the nonrelativistic Bohr scale. Using formula (8), we obtain
\[\tag{15} \boxed{ \delta R_{n,\kappa} = -n\rho_{\kappa} = -\frac{nr_e}{\kappa} }. \] The factor \(n\) is not added here to ensure consistency with the energy formula. It arises because the internal scale is transferred not to one arbitrarily chosen location in the orbit, but to all \(n\) phase links of the closed state. If the contributions of different links were mutually compensated, the linear factor \(n\) would not appear, and the well-known dependence of fine splitting on \(n\) would not be reproduced.
For two branches of the same level, we have
\[\tag{16} R_{n,\kappa_1} = R_n - \frac{nr_e}{\kappa_1}, \qquad R_{n,\kappa_2} = R_n - \frac{nr_e}{\kappa_2}. \] Therefore, their geometric separation is equal to
\[\tag{17} \boxed{ \Delta R_{n;\kappa_1\kappa_2} = nr_e \left( \frac{1}{\kappa_1} - \frac{1}{\kappa_2} \right) }. \] Furthermore, the magnitude of this difference will be important for the splitting value, so the order of the indices can be chosen arbitrarily.
5. From Geometric Shift to Energy
Now we will use the same principle introduced in the first part. The energy gradient does not create branches, but merely transforms the existing distance between them into an observable energy difference.
For the circular hydrogen state, the total Bohr energy as a function of radius is written as
\[\tag{18} E(R) = -\frac{1}{2} \frac{e^2}{4\pi\varepsilon_0R}. \] Its derivative with respect to radius is
\[\tag{19} \frac{\partial E}{\partial R} = \frac{1}{2} \frac{e^2}{4\pi\varepsilon_0R^2}. \] Using identities
\[\tag{20} \frac{e^2}{4\pi\varepsilon_0} = m_ec^2\alpha_{\mathrm{fs}}^2a_0, \qquad r_e = \alpha_{\mathrm{fs}}^2a_0. \] At the point \(R_n=n^2a_0\), the energy gradient takes the form
\[\tag{21} \left. \frac{\partial E}{\partial R} \right|_{R_n} = \frac{m_ec^2\alpha_{\mathrm{fs}}^2} {2n^4a_0}. \] Since the branch offsets are extremely small compared to \(R_n\), the first term of the expansion is sufficient:
\[\tag{22} E_{n,\kappa} \approx E_n + \left. \frac{\partial E}{\partial R} \right|_{R_n} \delta R_{n,\kappa}. \] Substituting formulas (15) and (21), we obtain a branch-dependent correction
\[\tag{23} \delta E_{n,\kappa} = -\frac{m_ec^2\alpha_{\mathrm{fs}}^4} {2n^3\kappa}. \] Here the geometry of both small factors is clearly visible. The quantity \(\alpha_{\mathrm{fs}}^2\) is part of the fundamental atomic gradient, and another \(\alpha_{\mathrm{fs}}^2\) is contained in the ratio of the electron's inner radius to the Bohr radius:
\[\tag{24} \alpha_{\mathrm{fs}}^2 \times \frac{r_e}{a_0} = \alpha_{\mathrm{fs}}^2 \times \alpha_{\mathrm{fs}}^2 = \alpha_{\mathrm{fs}}^4. \] The energy difference between two branches of the same level is therefore equal to
\[\tag{25} \boxed{ \left| \Delta E_{n;\kappa_1\kappa_2} \right| = \frac{m_ec^2\alpha_{\mathrm{fs}}^4} {2n^3} \left| \frac{1}{\kappa_1} - \frac{1}{\kappa_2} \right| }. \] This is the central result of the paper. The \(1/n^3\) dependence arose from the product of two geometric factors: the gradient at the radius \(R_n=n^2a_0\) yields \(1/n^4\), and the coherent accumulation of a small displacement over \(n\) phase units returns one factor \(n\).
6. Relationship of the Harmonic Index to the Atomic Momentum
Until now, \(\kappa\) was introduced as the positive number of the internal harmonic. To correlate the geometric branches with the usual classification of atomic states, we establish a correspondence.
\[\tag{26} \boxed{ \kappa = j+\frac{1}{2} }. \] Since the total angular momentum of an electron takes half-integer values
\[\tag{27} j = \frac12, \frac32, \frac52,\ldots, \] the index \(\kappa\) automatically becomes an integer:
\[\tag{28} \kappa = 1,2,3,\ldots \] Thus, the half-integer classification of observed atomic states is associated with the integer number of revolutions of the internal harmonic. For example, the two branches of the \(2P\) state have a simple interpretation:
\[\tag{29} 2P_{1/2} \longleftrightarrow \kappa_1=1, \qquad 2P_{3/2} \longleftrightarrow \kappa_2=2. \] Therefore, their geometric factor is
\[\tag{30} \left| \frac{1}{\kappa_1} - \frac{1}{\kappa_2} \right| = 1-\frac12 = \frac12. \] After the substitution \(\kappa=j+1/2\), formula (25) takes the familiar form:
\[\tag{31} \boxed{ \left| \Delta E_{n;j_1j_2} \right| = \frac{m_ec^2\alpha_{\mathrm{fs}}^4} {2n^3} \left| \frac{1}{j_1+\frac12} - \frac{1}{j_2+\frac12} \right| }. \] This dependence describes the leading difference in the fine structure between states with the same \(n\) but different \(j\). In the geometric model, the denominator \(j+1/2\) ceases to be merely a formal quantum combination: it becomes the number of closed revolutions of the corresponding internal harmonic.
7. Comparison with the full fine structure formula
In the standard expansion of the energy of the hydrogen atom to order \(\alpha_{\mathrm{fs}}^4\), the level is written as
\[\tag{32} E_{n,j} \approx -\frac{m_ec^2\alpha_{\mathrm{fs}}^2}{2n^2} - \frac{m_ec^2\alpha_{\mathrm{fs}}^4}{2n^4} \left( \frac{n}{j+\frac12} - \frac34 \right). \] The part of this formula that depends on \(j\) completely coincides with the geometric result (23):
\[\tag{33} -\frac{m_ec^2\alpha_{\mathrm{fs}}^4}{2n^3} \frac{1}{j+\frac12}. \] However, formula (32) also contains a term that is the same for all \(j\) for a fixed \(n\):
\[\tag{34} \delta E_n^{\mathrm{common}} = \frac{3m_ec^2\alpha_{\mathrm{fs}}^4}{8n^4}. \] The proposed radial construction does not yet derive this common correction. However, when calculating the distance between two branches of the same level, it cancels out, so splitting formula (31) is obtained in full. In other words, the model at this stage explains the relative positions of the branches, not the entire absolute relativistic correction to the level center.
\[\tag{35} \delta E_n^{\mathrm{common}} - \delta E_n^{\mathrm{common}} = 0. \] This difference is fundamental. The coincidence of the formula for the spacing between branches does not yet mean that a complete theory of fine structure has already been obtained. This would require a separate geometric origin for the common shift, and then accounting for the motion of the nucleus, radiative corrections, and other small effects.
8. Numerical Verification for Hydrogen
Let's check the central formula on several states of the hydrogen atom. We use the leading geometric result (25). For convenience, we compare not only the energy but also the wavenumber \(\Delta E/(hc)\) and frequency \(\Delta\nu=\Delta E/h\).
| Interval | \(\kappa_1,\kappa_2\) | Model, cm\(^{-1}\) | Model, GHz | NIST data, cm\(^{-1}\) |
|---|---|---|---|---|
| \(2P_{1/2}-2P_{3/2}\) | 1, 2 | 0.36523 | 10.9493 | 0.3659 |
| \(3P_{1/2}-3P_{3/2}\) | 1, 2 | 0.10822 | 3.2442 | 0.1084 |
| \(3D_{3/2}-3D_{5/2}\) | 2, 3 | 0.03607 | 1.0814 | 0.0361 |
The tabulated level values are taken from the database [1]. A slight difference is expected: formula (25) contains only the leading term of order \(\alpha_{\mathrm{fs}}^4\), while the actual levels include the reduced mass, Lamb shift, nuclear recoil, and higher-order corrections.
For the most famous example, the \(2P\) doublet, the calculation is particularly transparent:
\[\tag{36} \left| \Delta E_{2P} \right| = \frac{m_ec^2\alpha_{\mathrm{fs}}^4}{2\cdot2^3} \left(1-\frac12\right) = \frac{m_ec^2\alpha_{\mathrm{fs}}^4}{32}. \] Numerically this gives
\[\tag{37} \left| \Delta E_{2P} \right| \approx 4{,}5283\times10^{-5} \;\mathrm{eV}, \] \[\tag{38} \Delta\nu_{2P} \approx 10{,}9493 \;\mathrm{GHz}. \] Thus, the plot reproduces not only the functional dependence on quantum numbers, but also the correct scale of the observed interval.
9. Hydrogen-like and multi-electron atoms
For a hydrogen-like ion with a nuclear charge of \(Z\), the fundamental radius decreases by a factor of \(Z\):
\[\tag{39} R_n^{(Z)} = \frac{n^2a_0}{Z}. \] The Coulomb energy contains another factor of \(Z\), so the gradient at this radius increases as \(Z^3\):
\[\tag{40} \left. \frac{\partial E^{(Z)}}{\partial R} \right|_{R_n^{(Z)}} = \frac{m_ec^2\alpha_{\mathrm{fs}}^2Z^3} {2n^4a_0}. \] If we keep the internal branch shift exactly the same as in hydrogen, then the geometric construction predicts a dependence of \(Z^3\). Meanwhile, the leading fine structure of hydrogen-like ions scales as \(Z^4\). To reproduce it, the radial transport must also gain a factor of \(Z\):
\[\tag{41} \delta R_{n,\kappa}^{(Z)} = -\frac{nZr_e}{\kappa}. \] Only under this additional condition does the formula take the form
\[\tag{42} \left| \Delta E_{n;\kappa_1\kappa_2}^{(Z)} \right| \approx \frac{m_ec^2(Z\alpha_{\mathrm{fs}})^4}{2n^3} \left| \frac{1}{\kappa_1} - \frac{1}{\kappa_2} \right|. \] In this article, relation (41) is not derived from the operator \(J(a,b)\). Therefore, formula (42) is not a final result of the model, but a condition for its future generalization to \(Z>1\). Geometrically, it remains to be shown why the nuclear charge should increase the internal scale transfer by a factor of \(Z\). For hydrogen, this additional question is absent, since \(Z=1\).
For alkali and other multielectron atoms, the qualitative idea of two branches is preserved, as is clearly seen from the spectral line doublets. However, the hydrogen formula cannot be applied literally to them: nuclear shielding, electron wave penetration into inner shells, and electron interactions alter the radial energy. Therefore, the yellow sodium doublet remains a good illustration of the phenomenon itself, but not a direct numerical verification of hydrogen geometry.
10. What is a conclusion and what is a hypothesis in the construction?
To avoid confusing different levels of the model, we list the status of the main steps.
| Position | Status in the current model |
|---|---|
| \(a_0=r_e/\alpha_{\mathrm{fs}}^2\), \(R_n=n^2a_0\) | Supporting spatial relationships obtained or used previously |
| \(\omega_\kappa=\kappa\omega_e\) | Condition for integer closure of the inner Harmonics |
| \(\rho_\kappa=r_e/\kappa\) | A corollary of the invariant \(\omega_\kappa\rho_\kappa=c\) |
| \(\delta R_{n,\kappa}=-n\rho_\kappa\) | A new geometric hypothesis on small-scale coherent transfer on \(n\) phase links |
| \(\kappa=j+1/2\) | Comparison of the integer harmonic index with the observed branch classification |
| Formula (25) | Mathematical consequence of the listed geometric conditions |
| General correction \(3/(4n)\) | Not yet derived in the present construction |
| Generalization to \(Z>1\) | Requires a separate derivation of the dependence \(\delta R_{n,\kappa}^{(Z)}=-nZr_e/\kappa\) |
The strongest point of the construction is that after choosing one geometric transfer rule, there is no need to separately postulate the dependence \(\alpha_{\mathrm{fs}}^4/n^3\). It emerges from the already existing scales and energy gradient. The weakest point is also clearly visible: the rule is coherentThe summation over \(n\) links should subsequently be obtained directly from the full operator \(J(a,b)\), and not remain only a spatial interpretation.
11. General geometric chain
Now the entire transition from the internal dynamics of the electron to the atomic spectrum can be written in a single sequence:
\[\tag{43} \boxed{ \begin{aligned} \omega_e &\longrightarrow \omega_{\kappa}=\kappa\omega_e \longrightarrow \rho_{\kappa}=\frac{r_e}{\kappa} \\ &\longrightarrow \delta R_{n,\kappa}=-\frac{nr_e}{\kappa} \longrightarrow E_{n,\kappa} \\ &\longrightarrow \Delta E_{n;\kappa_1\kappa_2} \longrightarrow \Delta\nu. \end{aligned} }. \] In this chain, each link plays its own role. Frequency determines the number of internal revolutions; the frequency and radius invariant transform them into an inverse length spectrum; the external closure of the \(n\) level accumulates the small scale; the energy gradient converts the geometric distance into an energy difference; the transition between levels makes this difference observable in the spectrum.
Conclusion
In the first part, two orbital branches were introduced as a preserved manifestation of the previously constructed two-branch structure of the electron. In the second part, this picture is generalized to an entire family of allowed states. The integer harmonic \(\kappa\) creates an inverse spatial scale \(r_e/\kappa\), and the closure of the outer level of \(n\) phase links transfers it to a radial shift \(-nr_e/\kappa\).
After this, fine splitting no longer arises as a separate spectroscopic formula, but as a direct effect of the energy gradient on two close geometric branches. The relation \(r_e=\alpha_{\mathrm{fs}}^2a_0\) automatically creates the order \(\alpha_{\mathrm{fs}}^4\), and the combination of the radius \(R_n=n^2a_0\) with accumulation over \(n\) links yields the dependence \(1/n^3\).
After comparing \(\kappa=j+1/2\), the resulting energy difference coincides with the leading formula for the fine splitting of hydrogen:
\[\tag{44} \boxed{ \left| \Delta E_{n;j_1j_2} \right| = \frac{m_ec^2\alpha_{\mathrm{fs}}^4}{2n^3} \left| \frac{1}{j_1+\frac12} - \frac{1}{j_2+\frac12} \right| }. \] It is precisely the interval between the branches that is obtained. The overall shift of the level center, as well as radiative and nuclear corrections, remain beyond the scope of the present construction. Nevertheless, the main result has already been determined: the internal harmonics of the electron, the spatial splitting of the orbits, and the observed fine structure can be represented as successive manifestations of a single geometric system.
Let us also make a terminological remark. Following Bohr's picture, we call the external bound motion with radius \(R_1=a_0\) the first atomic orbit; In the modern quantum description, it corresponds to the orbital \(1s\). We call the internal closed motion of an electron on the scale \(r_e\) the zero orbit. This term is introduced within the framework of the proposed model and does not denote the atomic orbital with \(n=0\), which does not exist in standard quantum mechanics. The index "zero" merely indicates the place of this motion in the geometric sequence: it exists in the electron even before the formation of the atom and serves as the internal basis from which the first external orbit is then constructed.
Materials Used
- Reference data on sodium spectral lines and hydrogen levels. NIST: Energy Levels of Neutral Hydrogen.