2026-07-17
Geometric Origin of the Rydberg Law
Why do the energies of the hydrogen atom obey the law 1/n2? In standard quantum mechanics, this dependence follows from the solution of the Schrödinger equation for the Coulomb field. In this paper, we propose a geometric approach: the n level is formed by an internal splitting of the phase space into n2 equivalent components. Normalizing their common phase scale reduces the angle of an individual component as 1/n, and the energy, to a first approximation, turns out to be quadratic in this angle.
The quadratic approximation is then continued to a full orthogonal projection. As a result, the ordinary Rydberg law arises as the first term of a more general cosine dependence. This construction applies to the internal state of a stationary bound system; the external motion of the atom is not considered here.
\[\tag{1} \boxed{ N_n=n^2 \quad\longrightarrow\quad \theta_n=\frac{\alpha_{\mathrm{fs}}}{n} \quad\longrightarrow\quad B_n=\mu c^2\bigl(\sec\theta_n-1\bigr) }. \] Algebraic splitting yields the number of components n2. The transition from this number to the phase angle requires the conservation of the general quadratic phase norm, and the transition from the angle to the total energy requires the rules of orthogonal projection. These two rules are the physical postulates of the proposed model.
1. Geometric Level Splitting
The geometric construction of atomic states is discussed in detail in the article "Atomic Orbitals as Projections of Split Phase Space". Here, we use only the result necessary for the energy derivation.
A level with principal quantum number n corresponds to n−1 binary splittings in each of two independent internal directions. After permutation symmetrization, the dimensions of these directions are equal to
\[\tag{2} \dim\mathcal A_n=n, \qquad \dim\mathcal B_n=n. \] Their joint space contains
\[\tag{3} \boxed{ N_n =\dim(\mathcal A_n*\mathcal B_n) =n\cdot n=n^2 }. \] These components are grouped into angular subspaces of dimensions 1, 3, 5, and further:
\[\tag{4} \boxed{ N_n=n^2 =\sum_{l=0}^{n-1}(2l+1) =1+3+5+\ldots +(2n-1) }. \] After spatial projection, they correspond to states with
\[\tag{5} l=0,1,\ldots,n-1, \qquad m=-l,-l+1,\ldots,l. \] Thus, the number n2 is the dimension of the internal space of spatial orbitals of the level. Spin orientations are not included in the number \(N_n\).
2. Normalization of the Phase Angle
Let the full internal phase scale of the first level correspond to the dimensionless angle
\[\tag{6} \theta_1=\alpha_{\mathrm{fs}}. \] When a level is split, this scale does not disappear, but is distributed between \(N_n\) mutually orthogonal components. Therefore, we assume the conservation of the general quadratic phase norm:
\[\tag{7} \boxed{ \sum_{q=1}^{N_n}\theta_{n,q}^{,2} =\theta_1^2 =\alpha_{\mathrm{fs}}^2 }. \] Before taking into account additional interactions, components of the same level are geometrically equivalent. Therefore,
\[\tag{8} \theta_{n,1}=\theta_{n,2}=\ldots=\theta_{n,N_n}=\theta_n. \] Then the normalization condition takes the form
\[\tag{9} N_n\theta_n^2=\alpha_{\mathrm{fs}}^2. \] Taking into account \(N_n=n^2\), we obtain the main geometric result:
\[\tag{10} \boxed{ \theta_n =\frac{\alpha_{\mathrm{fs}}}{\sqrt{N_n}} =\frac{\alpha_{\mathrm{fs}}}{n} }. \] The square in the Rydberg law arises here in two related stages: the number of orthogonal components increases as n2, and the normalized phase angle of an individual component decreases as 1/n.
Condition (7) means preserving the norm of the internal phase state, rather than dividing the physical energy of one electron between n2 different orbitals. The components form a space of possible projections of the same level.
3. The 1/n² Law in First Approximation
For a small phase angle, the change in the energy of the orthogonal projection begins with a quadratic term. Therefore, to a first approximation, we write the binding energy modulus as
\[\tag{11} B_n^{(2)}=\frac12\mu c^2\theta_n^2, \] where \(\mu\) is the reduced mass of the electron and nucleus:
\[\tag{12} \mu=\frac{m_eM}{m_e+M}. \] Substituting the geometric angle (10) yields
\[\tag{13} \boxed{ B_n^{(2)} =\frac{\mu c^2\alpha_{\mathrm{fs}}^2}{2n^2} }. \] Let's define the energy scale of the first level:
\[\tag{14} E_{\mathrm R}=\frac12\mu c^2\alpha_{\mathrm{fs}}^2. \] Then
\[\tag{15} \boxed{ B_n^{(2)} =\frac{E_{\mathrm R}}{n^2} =\frac{E_{\mathrm R}}{N_n} }. \] The energy level relative to the ionization limit has a negative sign:
\[\tag{16} \boxed{ \mathcal E_n^{(2)}=-B_n^{(2)} =-\frac{\mu c^2\alpha_{\mathrm{fs}}^2}{2n^2} }. \] Thus, the well-known dependence 1/n2 follows directly from the splitting geometry:
\[\tag{17} \boxed{ n-1\text{ splittings} \longrightarrow N_n=n^2 \longrightarrow \theta_n^2=\frac{\alpha_{\mathrm{fs}}^2}{N_n} \longrightarrow B_n^{(2)}=\frac{E_{\mathrm R}}{n^2} }. \] 4. Complete Orthogonal Projection
Formula (11) is an approximation for small \(\theta_n\). To obtain the complete geometric expression, we consider the rectangular projection of the internal energy.
Let \(E_n^{\mathrm{geom}}\) be the auxiliary total value of the internal phase state, and \(\mu c^2\) be its constant base projection. Then
\[\tag{18} \boxed{ E_n^{\mathrm{geom}}\cos\theta_n=\mu c^2 }. \] Hence
\[\tag{19} E_n^{\mathrm{geom}}=\frac{\mu c^2}{\cos\theta_n}. \] We define the modulus of the binding energy as the difference between the total internal value and its base projection:
\[\tag{20} B_n=E_n^{\mathrm{geom}}-\mu c^2. \] Using formulas (19) and (10), we obtain
\[\tag{21} \boxed{ B_n =\mu c^2 \left[ \frac{1}{\cos\theta_n}-1 \right] =\mu c^2 \left[ \frac{1}{\cos(\alpha_{\mathrm{fs}}/n)}-1 \right] }. \] The quantity \(E_n^{\mathrm{geom}}\) is not called the total observable energy of a bound electron. It is an auxiliary geometric quantity, the difference between which and the general base projection \(\mu c^2\) defines the positive coupling modulus. The observed level relative to the ionization limit is
\[\tag{22} \boxed{\mathcal E_n=-B_n}. \] At \(n\to\infty\), the phase angle and binding energy vanish:
\[\tag{23} \lim_{n\to\infty}\theta_n=0, \qquad \lim_{n\to\infty}B_n=0, \qquad \lim_{n\to\infty}\mathcal E_n=0. \] The 1/n2 law by itself does not determine a unique nonlinear continuation. The cosine form (21) follows precisely from the additional rectangular projection rule (18). Therefore, the leading term has a more general status than the proposed higher corrections.
5. Transition to the 1/n² law
The expansion of the secant for a small argument has the form
\[\tag{24} \frac1{\cos x} =1+\frac{x^2}{2} +\frac{5x^4}{24} +\frac{61x^6}{720} +\cdots. \] Substituting \(x=\alpha_{\mathrm{fs}}/n\) into formula (21), we find
\[\tag{25} B_n =\mu c^2 \left[ \frac{\alpha_{\mathrm{fs}}^2}{2n^2} +\frac{5\alpha_{\mathrm{fs}}^4}{24n^4} +\frac{61\alpha_{\mathrm{fs}}^6}{720n^6} +\cdots \right]. \] The first term exactly matches the geometric approximation (13):
\[\tag{26} \boxed{ B_n =\frac{\mu c^2\alpha_{\mathrm{fs}}^2}{2n^2} +O\!\left(\frac{\alpha_{\mathrm{fs}}^4}{n^4}\right) }. \] Therefore, the full projection does not replace the original geometric law, but continues it:
\[\tag{27} \boxed{ \frac{E_{\mathrm R}}{n^2} \quad\longrightarrow\quad \mu c^2 \left[ \frac1{\cos(\alpha_{\mathrm{fs}}/n)}-1 \right] }. \] 6. Generalized Rydberg Law
Consider the transition of an electron from the upper level \(n_2\) to the lower level \(n_1\), where
\[\tag{28} n_2>n_1. \] Since \(B_{n_1}>B_{n_2}\), the lower state is more strongly bound. The energy of the emitted photon is equal to the difference in the coupling moduli:
\[\tag{29} hf=B_{n_1}-B_{n_2}. \] Substituting the full expression (21), we obtain the generalized law:
\[\tag{30} \boxed{ hf =\mu c^2 \left[ \frac1{\cos(\alpha_{\mathrm{fs}}/n_1)} - \frac1{\cos(\alpha_{\mathrm{fs}}/n_2)} \right] }. \] Taking into account \(f=c/\lambda\) we find the wave form:
\[\tag{31} \boxed{ \frac1\lambda =\frac{\mu c}{h} \left[ \frac1{\cos(\alpha_{\mathrm{fs}}/n_1)} - \frac1{\cos(\alpha_{\mathrm{fs}}/n_2)} \right] }. \] As a first approximation, formula (30) becomes
\[\tag{32} hf \approx \frac{\mu c^2\alpha_{\mathrm{fs}}^2}{2} \left( \frac1{n_1^2}-\frac1{n_2^2} \right). \] We define the Rydberg constant for a nucleus of mass M:
\[\tag{33} \boxed{ R_M=\frac{\mu c\alpha_{\mathrm{fs}}^2}{2h} }. \] Then we obtain the usual Rydberg law:
\[\tag{34} \boxed{ \frac1\lambda \approx R_M \left( \frac1{n_1^2}-\frac1{n_2^2} \right) }. \] As \(M\to\infty\), the reduced mass tends to the masselectron, and
\[\tag{35} \boxed{ R_\infty=\frac{m_ec\alpha_{\mathrm{fs}}^2}{2h} }. \] 7. Higher geometric corrections and output status
Taking into account the first additional term, the spectral transition energy is
\[\tag{36} \begin{aligned} hf\approx{}& \frac{\mu c^2\alpha_{\mathrm{fs}}^2}{2} \left( \frac1{n_1^2}-\frac1{n_2^2} \right) \\[1mm] &+ \frac{5\mu c^2\alpha_{\mathrm{fs}}^4}{24} \left( \frac1{n_1^4}-\frac1{n_2^4} \right) +\cdots. \end{aligned} \] The first term reproduces the established Rydberg formula. The second and subsequent terms are the proper corrections to the chosen cosine projection. They depend only on n, so they do not, by themselves, remove the degeneracy between states with different l and m and cannot be automatically identified with the fine structure of the real spectrum.
| Construction Element | Status |
|---|---|
| Two symmetric directions of dimension n | result of orbital geometry |
| Number of components \(N_n=n^2\) | mathematical consequence of independent multiplication |
| Conservation \(\sum_q\theta_{n,q}^2=\alpha_{\mathrm{fs}}^2\) | phase normalization postulate |
| Angle \(\theta_n=\alpha_{\mathrm{fs}}/n\) | consequence of normalization of equivalent components |
| Law \(B_n^{(2)}=E_{\mathrm R}/n^2\) | consequence of the quadratic energy of a small angle |
| Projection \(E_n^{\mathrm{geom}}\cos\theta_n=\mu c^2\) | geometric postulate of the complete model |
| Cosine function \(B_n=\mu c^2(\sec\theta_n-1)\) | consequence of the accepted rule projections |
| Ordinary Rydberg law | leading term of the expansion |
| Higher cosine corrections | hypothesis requiring experimental verification |
The resulting sequence is of the form
\[\tag{37} \boxed{ \begin{aligned} n-1\text{ splits} &\longrightarrow N_n=n^2, \\[1mm] N_n\theta_n^2=\alpha_{\mathrm{fs}}^2 &\longrightarrow \theta_n=\frac{\alpha_{\mathrm{fs}}}{n}, \\[1mm] B_n^{(2)}=\frac12\mu c^2\theta_n^2 &\longrightarrow B_n^{(2)}=\frac{E_{\mathrm R}}{n^2}, \\[1mm] E_n^{\mathrm{geom}}\cos\theta_n=\mu c^2 &\longrightarrow B_n=\mu c^2(\sec\theta_n-1), \\[1mm] B_{n_1}-B_{n_2} &\longrightarrow hf. \end{aligned}} \] Thus, the 1/n2 dependence is associated with the number of internal phase components of the level, and the more general cosine law is associated with the full geometry of their orthogonal energy projection. A rigorous dynamic derivation of the phase normalization rules and full projection remains a task for further development of the model.
Materials used
- Wikipedia. Rydberg's Formula.
- NIST. Atomic Data for Hydrogen.
- NIST. Atomic Spectra Database.