2026-07-17
Geometric Origin of the Generalized Rydberg Law in the I-Basis
Introduction
In this paper, a generalized Rydberg law [1] is derived in a new complex-extended idempotent basis (I-basis), linking the frequency of a spectral transition to the geometry of a particle's internal states. Discrete energy levels are not introduced as an independent quantum postulate: they arise from the closure condition of the internal periodic motion, which leads to the appearance of an integer parameter \( n \).
On this basis, the dependence of the particle's energy on the geometric angle of its internal motion is consistently established, the transition energy between two states is determined, and a generalized formula for spectral frequencies is obtained. The classical Rydberg law is further considered as a special case of this more general geometric dependence.
The model is based on the particle's internal vector, constructed using the basis \(\{\ep,i\ep,\em,i\em\}\). Its rotational component determines the internal state, and the total geometric angle of motion determines the observed velocity and total energy of the particle. The transition between two admissible states is accompanied by the release of the energy difference in the form of a wave with frequency \(f\).
The initial properties of the u-basis, the relationship between internal rotation and norm conservation, and the geometric origin of mass were discussed in previous works:
- The Schrödinger Equation and the Origin of Mass in a New Idempotent Basis;
- Geometry of the Hyperbolic Unit.
1. Algebraic Foundation of the U-Basis
Consider two complementary idempotents \(\ep\) and \(\em\) satisfying the relations
\[ \tag{1} \ep^2=\ep, \qquad \em^2=\em, \qquad \ep\em=0, \qquad \ep+\em=1. \] The complex extension of each idempotent direction forms a four-dimensional real basis.
\[ \tag{2} \left\{ \ep, \;i\ep, \;\em, \;i\em \right\}. \] We define the internal state of the particle as a dimensionless vector.
\[ \tag{3} J(t) = \ep + \em e^{-i\omega t}. \] The first component remains constant, while the second rotates in the complex plane \(\{\em,i\em\}\). Expanding the exponential gives
\[ \tag{4} J(t) = \ep + \em\cos\omega t - i\em\sin\omega t. \] 2. Mass as a characteristic of internal rotation
The derivative of the internal vector is equal to
\[ \tag{5} \frac{dJ}{dt} = -i\omega\em e^{-i\omega t}. \] Its norm is constant:
\[ \tag{6} \left\| \frac{dJ}{dt} \right\| = \omega. \] The mass of a particle is determined by the rate of change of its internal state:
\[ \tag{7} m = \frac{\hbar}{c^2} \left\| \frac{dJ}{dt} \right\|. \] Substituting expression (6) yields the relationship
\[ \tag{8} mc^2 = \hbar\omega. \] Thus, the mass is related not to the absolute value of the internal phase, but to the constant angular velocity of its change.
3. The Total Motion Vector
The total velocity vector of a particle can be represented as
\[ \tag{9} V(t) = c e^{i\theta}J(t), \] where \(\theta\) is the geometric angle that determines the distribution of motion between the internal and observed components. The norm of the total vector is preserved:
\[ \tag{10} \left|V(t)\right| = c. \] The observed velocity is the projection of the total motion:
\[ \tag{11} v = c\sin\theta. \] Hence
\[ \tag{12} \beta = \frac{v}{c} = \sin\theta. \] The geometric analog of the Lorentz factor is:
\[ \tag{13} \gamma = \frac{1}{\cos\theta} = \frac{1}{\sqrt{1-\beta^2}}. \] The total energy of the particle is then determined by the expression
\[ \tag{14} E(\theta) = \gamma mc^2 = \frac{mc^2}{\cos\theta}. \] 4. Transition Energy between States
Each admissible state of the particle corresponds to a specific geometry of its internal motion. In the previous section, it was shown that the total energy of such a state is determined by the angle \( \theta \) and is given by formula (14). Consequently, the transition between two stable states is accompanied by a change in the total energy of the particle.
Let a particle transition from state \( 1 \) to state \( 2 \). If the energy of the first state is greater than the energy of the second, then the resulting energy difference must be transferred to the surrounding space. Assuming that this energy propagates as an electromagnetic wave, we obtain
\[ \tag{15} hf = E_1-E_2. \] Now we substitute formula (14) into this expression for each of thestates. Then the transition energy takes the form
\[ \tag{16} hf = mc^2 \left( \frac{1}{\cos\theta_1} - \frac{1}{\cos\theta_2} \right). \] The resulting formula shows that the frequency of the emitted or absorbed wave is determined solely by the change in the internal state of the particle. Thus, the wave does not arise as an independent physical object, but as a consequence of the transition between two geometrically different configurations of internal motion.
Until this point, only the geometry of the particle was considered. Formula (16) is the first to link this geometry to a propagating wave. Consequently, the wave process appears as a direct consequence of a change in the internal state of the particle.
However, expression (16) still does not explain the origin of the line spectrum. If the angles \( \theta_1 \) and \( \theta_2 \) could take arbitrary values, the frequency would also change continuously. Therefore, the next step is to show that the internal motion allows only a discrete set of stable states. It is this condition that leads to the appearance of the integer parameter \( n \) and allows us to derive the generalized Rydberg formula.
5. Closed Internal States
We introduce the phase parameter \(\varphi\) and consider the family of internal states.
\[ \tag{17} J_n(\varphi) = \ep + \em e^{-in\varphi}. \] After a full rotation \(\varphi=2\pi\), the internal vector must return to its original state:
\[ \tag{18} J_n(2\pi) = J_n(0). \] For the rotational component, this means
\[ \tag{19} e^{-i2\pi n} = 1. \] The condition is satisfied for positive integer values
\[ \tag{20} n = 1,2,3,\ldots \] Thus, the natural number \(n\) arises as the number of internal phase revolutions in one complete cycle of the parameter \(\varphi\). The discreteness of the state is a consequence of the topological closure condition.
6. Conservation of Mass in Discrete States
The time derivative of the vector \(J_n\) is equal to
\[ \tag{21} \frac{dJ_n}{dt} = -in\dot{\varphi}_n \em e^{-in\varphi_n}. \] Its norm is determined by the product of the number of revolutions and the angular velocity of the phase parameter:
\[ \tag{22} \left\| \frac{dJ_n}{dt} \right\| = n\left|\dot{\varphi}_n\right|. \] Since the mass of a single particle should not depend on the state number, the value of (22) should remain constant:
\[ \tag{23} n\left|\dot{\varphi}_n\right| = \Omega = \text{const}. \] Hence
\[ \tag{24} \left|\dot{\varphi}_n\right| = \frac{\Omega}{n}. \] Therefore, as the number of internal revolutions increases, the rate of change of the general phase parameter decreases inversely proportional to \(n\).
7. Velocity and Angle of a Discrete State
We assume that the observed velocity of a particle is proportional to the rate of change of the phase parameter:
\[ \tag{25} v_n \propto \left|\dot{\varphi}_n\right|. \] Then from expression (24) it follows that
\[ \tag{26} v_n = \frac{v_1}{n}. \] For the ground state, we adopt the characteristic velocity
\[ \tag{27} v_1 = \alpha_{\mathrm{fs}}c, \] where \(\alpha_{\mathrm{fs}}\) is the fine structure constant, which we obtained previously from the Lorentz factor. Therefore,
\[ \tag{28} v_n = \frac{\alpha_{\mathrm{fs}}c}{n}. \] Taking into account the geometric relationship \(v_n=c\sin\theta_n\), we obtain
\[ \tag{29} \sin\theta_n = \frac{\alpha_{\mathrm{fs}}}{n}. \] The permissible angles of motion are therefore determined by the discrete series
\[ \tag{30} \theta_n = \arcsin \left( \frac{\alpha_{\mathrm{fs}}}{n} \right). \] 8. The exact energy of a discrete state
From expression (29) it follows
\[ \tag{31} \cos\theta_n = \sqrt{ 1- \frac{\alpha_{\mathrm{fs}}^2}{n^2} }. \] The Lorentz factor for state \(n\) is
\[ \tag{32} \gamma_n = \frac{1}{\cos\theta_n} = \frac{1}{ \sqrt{1-\alpha_{\mathrm{fs}}^2/n^2} } . \] After multiplying the numerator and denominator by \(n\), we obtain the equivalent form
\[ \tag{33} \gamma_n = \frac{n}{ \sqrt{ n^2- \alpha_{\mathrm{fs}}^2 } }. \] The total energy of the state is
\[ \tag{34} E_n = mc^2\gamma_n = mc^2 \frac{n}{ \sqrt{ n^2- \alpha_{\mathrm{fs}}^2 } }. \] When \(n\to\infty\), the angle \(\theta_n\to0\), the factor \(\gamma_n\to1\), and the energy \(E_n\to mc^2\). Therefore, \(mc^2\) is the limiting energy of the free state in a given geometricalscheme.
9. Generalized Rydberg Law
Substituting the discrete energies (34) into the transition law (16), we obtain the exact spectral formula
\[ \tag{35} hf = mc^2 \left( \frac{n_1}{ \sqrt{ n_1^2- \alpha_{\mathrm{fs}}^2 } } - \frac{n_2}{ \sqrt{ n_2^2- \alpha_{\mathrm{fs}}^2 } } \right). \]
For positive energy to be emitted, the initial state is assumed to have higher energy. With the adopted numbering, this corresponds to the condition \(n_1 < n_2\).
Since \(f=c/\lambda\), formula (35) can be written as
\[ \tag{36} \frac{1}{\lambda} = \frac{mc}{h} \left( \frac{n_1}{ \sqrt{ n_1^2- \alpha_{\mathrm{fs}}^2 } } - \frac{n_2}{ \sqrt{ n_2^2- \alpha_{\mathrm{fs}}^2 } } \right). \] Expression (36) can be viewed as a generalized Rydberg law, containing an exact geometric dependence on the fine structure constant.
10. Transition to the classical Rydberg law
Since \(\alpha_{\mathrm{fs}}\ll1\), we expand the factor \(\gamma_n\) in a series:
\[ \tag{37} \gamma_n = \left( 1- \frac{\alpha_{\mathrm{fs}}^2}{n^2} \right)^{-1/2}. \] Using the binomial expansion, we obtain
\[ \tag{38} \gamma_n = 1 + \frac{\alpha_{\mathrm{fs}}^2}{2n^2} + \frac{3\alpha_{\mathrm{fs}}^4}{8n^4} + \frac{5\alpha_{\mathrm{fs}}^6}{16n^6} + \cdots. \] As a first approximation
\[ \tag{39} E_n \approx mc^2 + \frac{mc^2\alpha_{\mathrm{fs}}^2}{2n^2}. \] The constant part \(mc^2\) cancels out when calculating the energy difference. Therefore
\[ \tag{40} hf \approx \frac{mc^2\alpha_{\mathrm{fs}}^2}{2} \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right). \] Moving from frequency to wavelength, we obtain
\[ \tag{41} \frac{1}{\lambda} = R \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right), \] where
\[ \tag{42} R = \frac{mc\alpha_{\mathrm{fs}}^2}{2h}. \] Formulas (41) and (42) coincide in structure with the classical Rydberg law and its constant in the approximation of an infinitely heavy nucleus.
11. Accounting for Nuclear Motion
For a real atom, the electron and nucleus move relative to their common center of mass. Therefore, instead of the electron mass, it is necessary to use the reduced mass.
\[ \tag{43} \mu = \frac{m_eM}{m_e+M}, \] where \(M\) is the mass of the nucleus. Then the exact geometric formula takes the form
\[ \tag{44} hf = \mu c^2 \left( \frac{n_1}{ \sqrt{ n_1^2- \alpha_{\mathrm{fs}}^2 } } - \frac{n_2}{ \sqrt{ n_2^2- \alpha_{\mathrm{fs}}^2 } } \right), \] and the Rydberg constant for a given nucleus is equal to
\[ \tag{45} R_M = \frac{\mu c\alpha_{\mathrm{fs}}^2}{2h}. \] As \(M\to\infty\), the reduced mass tends to the electron mass, and expression (45) transforms into formula (42).
12. Higher Geometric Corrections
The exact formula (35) contains not only the leading term of order \(\alpha_{\mathrm{fs}}^2\), but also an infinite sequence of higher corrections. Taking into account the next term in the expansion, we have
\[ \tag{46} hf \approx \frac{mc^2\alpha_{\mathrm{fs}}^2}{2} \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) + \frac{3mc^2\alpha_{\mathrm{fs}}^4}{8} \left( \frac{1}{n_1^4} - \frac{1}{n_2^4} \right) + \cdots. \] The first term reproduces the usual Rydberg law. The subsequent terms are the model's own geometric corrections and arise directly from the exact expression for \(\gamma_n\).
The relative scale of the first correction is determined by a value of the order of \(\alpha_{\mathrm{fs}}^2\), so it is significantly smaller than the main spectral term.
13. The Geometric Meaning of the Quantum Number
In the model under consideration, the number \(n\) has several interconnected interpretations. It determines the number of internal phase revolutions, reduces the phase parameter velocity, specifies the observed particle velocity, and fixes the geometric angle of the state.
\[ \tag{47} n \longrightarrow \left|\dot{\varphi}_n\right| = \frac{\Omega}{n} \longrightarrow v_n = \frac{\alpha_{\mathrm{fs}}c}{n} \longrightarrow \sin\theta_n = \frac{\alpha_{\mathrm{fs}}}{n}. \] The discreteness of energy, therefore, is not a separate requirement, but the result of the combined action of three conditions: the closure of the internal trajectory, the conservation of mass, and the geometric relationship between the velocity and the angle of motion.
14. Boundary states
At \(n=1\), the velocity and angle are maximum:
\[ \tag{48} v_1 = \alpha_{\mathrm{fs}}c, \qquad \theta_1 = \arcsin\alpha_{\mathrm{fs}}. \] As \(n\) increases, the velocity decreases, the angle tends to zero, and the energy approaches \(mc^2\):
\[ \tag{49} \lim_{n\to\infty}v_n = 0, \qquad \lim_{n\to\infty}\theta_n = 0, \qquad \lim_{n\to\infty}E_n = mc^2. \] This limit state can be interpreted as the separation of the particle from the bound discrete structure, when the geometric addition to the energy disappears.
15. Final system of relations
The main results of the model can be presented in a compact form:
\[ \tag{50} J_n(\varphi) = \ep + \em e^{-in\varphi}, \qquad n = 1,2,3,\ldots \] \[ \tag{51} n\left|\dot{\varphi}_n\right| = \Omega, \qquad v_n = \frac{\alpha_{\mathrm{fs}}c}{n}. \] \[ \tag{52} \sin\theta_n = \frac{\alpha_{\mathrm{fs}}}{n}, \qquad \gamma_n = \frac{n}{ \sqrt{ n^2- \alpha_{\mathrm{fs}}^2 } }. \] \[ \tag{53} E_n = mc^2 \frac{n}{ \sqrt{ n^2- \alpha_{\mathrm{fs}}^2 } }. \] \[ \tag{54} hf = mc^2 \left( \frac{n_1}{ \sqrt{ n_1^2- \alpha_{\mathrm{fs}}^2 } } - \frac{n_2}{ \sqrt{ n_2^2- \alpha_{\mathrm{fs}}^2 } } \right). \] \[ \tag{55} \frac{1}{\lambda} \approx R \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right). \] Conclusions
This paper proposes a geometric construction of discrete particle states in the u-basis. The natural quantum number arises from the internal phase closure condition, and the inverse dependence of the velocity on \(n\) follows from the conservation of the norm of the derivative of the internal vector and, consequently, the particle mass.
The relationship between the observed velocity and the geometric angle leads to the relation \(sin\theta_n=\alpha_{\mathrm{fs}}/n\). Based on this, the exact factor \(gamma_n\) and a discrete energy series are obtained, which do not require the preliminary introduction of the classical formula for the energy levels of the hydrogen atom.
The difference in the exact energies of the two states forms the generalized Rydberg law. To a first approximation in the fine structure constant, it transforms into the conventional spectral formula with the Rydberg constant \(R=mc\alpha_{\mathrm{fs}}^2/(2h)\). Replacing the electron mass with the reduced mass naturally takes into account the motion of the nucleus.
The exact formula also contains higher-order terms in \(\alpha_{\mathrm{fs}}\). Within the model, these are interpreted as geometric corrections arising from the full, rather than approximate, expression for the state energy.
Thus, the Rydberg law is associated with three fundamental geometric principles: the closure of internal motion, conservation of mass, and the projective nature of the observed velocity. The I-basis allows us to combine these principles into a single system and obtain a spectral dependence as a consequence of the particle's internal geometry.
Materials used
- Wikipedia. Rydberg's Formula.