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). This law relates 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:
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 the particle transition from state \( 1 \) to state \( 2 \). If enIf 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 the states. 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 arises not as an independent physical object, but as a consequence of the transition between two geometrically different configurations of internal motion.
Up to this point, only the particle's geometry has been 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 particle's internal state.
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
The internal state of a particle is determined by the rotational component of the vector
\[ \tag{17} J(t) = \ep + \em e^{-i\omega t}. \] Let the main internal cycle have a period
\[ \tag{18} T = \frac{2\pi}{\omega}. \] A stable state can be closed not only in one fundamental period, but also in an integer number of such periods:
\[ \tag{19} T_n = nT, \qquad n = 1,2,3,\ldots \] The natural number \(n\) thus determines the order of closure of the internal periodic motion. Unclosed states do not return the complete internal vector to its original configuration and do not form a steady state.
6. Frequency of a Discrete State
The angular frequency of a state closing in time \(T_n\) is equal to
\[ \tag{20} \omega_n = \frac{2\pi}{T_n}. \] Taking into account expression (19), we obtain
\[ \tag{21} \omega_n = \frac{2\pi}{nT} = \frac{\omega}{n}. \] Consequently, different closure orders correspond to subharmonic modes of internal motion. As \(n\) increases, the complete internal cycle takes longer, and its effective angular frequency decreases inversely proportional to \(n\).
7. Discrete State Angle
Let the ground state correspond to a characteristic geometric angle
\[ \tag{22} \theta_1 = \alpha_{\mathrm{fs}}, \] where \(\alpha_{\mathrm{fs}}\) is the fine structure constant, the geometric interpretation of which was considered previously.
Since the geometric angle is determined by the accumulated phase of the internal motion, for a state with frequency \(\omega_n=\omega/n\) we obtain
\[ \tag{23} \theta_n = \frac{\theta_1}{n}. \] Therefore, the admissible angles form a discrete series.
\[ \tag{24} \theta_n = \frac{\alpha_{\mathrm{fs}}}{n}, \qquad n = 1,2,3,\ldots \] Unlike the previous derivation, here dividing the angle by \(n\) follows directly from the increase in the closure period and the decrease in the angular frequency of the internal state.
8. The Exact Energy of a Discrete State
The Lorentz factor for the state \(n\) is determined by the formula
\[ \tag{25} \gamma_n = \frac{1}{\cos\theta_n}. \] Substituting the discrete angle (24), we obtain
\[ \tag{26} \gamma_n = \frac{1}{ \cos \left( \alpha_{\mathrm{fs}}/n \right) }. \] The total energy of the state is
\[ \tag{27} E_n = mc^2\gamma_n = \frac{mc^2}{ \cos \left( \alpha_{\mathrm{fs}}/n \right) }. \] 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 this geometric scheme.
9. Generalized Rydberg Law
Substituting the discrete energies (27) into the transition law (16), we obtain the exact spectral formula
\[ \tag{28} hf = mc^2 \left( \frac{1}{ \cos \left( \alpha_{\mathrm{fs}}/n_1 \right) } - \frac{1}{ \cos \left( \alpha_{\mathrm{fs}}/n_2 \right) } \right). \] For positive energy emission, the initial state is assumed to have higher energy. With the adopted numbering, this corresponds to the condition \(n_1
Since \(f=c/\lambda\), formula (28) can be written as
\[ \tag{29} \frac{1}{\lambda} = \frac{mc}{h} \left( \frac{1}{ \cos \left( \alpha_{\mathrm{fs}}/n_1 \right) } - \frac{1}{ \cos \left( \alpha_{\mathrm{fs}}/n_2 \right) } \right). \]
Expression (29) can be viewed as a generalized Rydberg law, containing an exact geometric dependence on the discrete angles of the internal state.
10. Transition to the Classical Rydberg Law
Since \(\alpha_{\mathrm{fs}}\ll1\), we expand the small-angle secant in a series:
\[ \tag{30} \frac{1}{\cos x} = 1 + \frac{x^2}{2} + \frac{5x^4}{24} + \frac{61x^6}{720} + \cdots. \] For a discrete state \(x=\alpha_{\mathrm{fs}}/n\), therefore
\[ \tag{31} \gamma_n = 1 + \frac{\alpha_{\mathrm{fs}}^2}{2n^2} + \frac{5\alpha_{\mathrm{fs}}^4}{24n^4} + \frac{61\alpha_{\mathrm{fs}}^6}{720n^6} + \cdots. \] As a first approximation
\[ \tag{32} E_n \approx mc^2 + \frac{mc^2\alpha_{\mathrm{fs}}^2}{2n^2}. \] The constant part \(mc^2\) cancels out when calculating the energy difference. Therefore
\[ \tag{33} hf \approx \frac{mc^2\alpha_{\mathrm{fs}}^2}{2} \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right). \] Moving from frequency to wavelength, we obtain
\[ \tag{34} \frac{1}{\lambda} = R \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right), \] where
\[ \tag{35} R = \frac{mc\alpha_{\mathrm{fs}}^2}{2h}. \] Formulas (34) and (35) 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.
\[ \tag{36} \mu = \frac{m_eM}{m_e+M}, \] where \(M\) is the mass of the nucleus. Then the exact geometric formula takes the form
\[ \tag{37} hf = \mu c^2 \left( \frac{1}{ \cos \left( \alpha_{\mathrm{fs}}/n_1 \right) } - \frac{1}{ \cos \left( \alpha_{\mathrm{fs}}/n_2 \right) } \right), \] and the Rydberg constant for this kernel is
\[ \tag{38} R_M = \frac{\mu c\alpha_{\mathrm{fs}}^2}{2h}. \] As \(M\to\infty\), the reduced mass approaches the electron mass, and expression (38) transforms into formula (35).
12. Higher Geometric Corrections
The exact formula (28) 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
\[ \tag{39} hf \approx \frac{mc^2\alpha_{\mathrm{fs}}^2}{2} \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) + \frac{5mc^2\alpha_{\mathrm{fs}}^4}{24} \left( \frac{1}{n_1^4} - \frac{1}{n_2^4} \right) + \cdots. \] 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 \(\gamma_n=\sec(\alpha_{\mathrm{fs}}/n)\).
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 \(n\) determines the order of closure of the internal motion, increases its period, reduces the angular frequency, and fixes the geometric angle of the state.
\[ \tag{40} n \longrightarrow T_n = nT \longrightarrow \omega_n = \frac{\omega}{n} \longrightarrow \theta_n = \frac{\alpha_{\mathrm{fs}}}{n}. \] The discreteness of energy, therefore, is not a separate requirement, but a result of the closure of the internal trajectory through an integer number of fundamental periods and the geometric dependence of energy on the angle of motion.
14. Boundary states
At \(n=1\), the angle and energy are maximum:
\[ \tag{41} \theta_1 = \alpha_{\mathrm{fs}}, \qquad E_1 = \frac{mc^2}{ \cos\alpha_{\mathrm{fs}} }. \] As \(n\) increases, the angle tends to zero, and the energy approaches \(mc^2\):
\[ \tag{42} \lim_{n\to\infty}\theta_n = 0, \qquad \lim_{n\to\infty}\gamma_n = 1, \qquad \lim_{n\to\infty}E_n = mc^2. \] This limit state can be interpreted as the separation of the particle from the bounddiscrete structure, when the geometric addition to the energy disappears.
15. The final system of relations
The main results of the model can be presented in a compact form:
\[ \tag{43} T_n = nT, \qquad n = 1,2,3,\ldots \] \[ \tag{44} \omega_n = \frac{\omega}{n}, \qquad \theta_n = \frac{\alpha_{\mathrm{fs}}}{n}. \] \[ \tag{45} \gamma_n = \frac{1}{ \cos \left( \alpha_{\mathrm{fs}}/n \right) }. \] \[ \tag{46} E_n = \frac{mc^2}{ \cos \left( \alpha_{\mathrm{fs}}/n \right) }. \] \[ \tag{47} hf = mc^2 \left( \frac{1}{ \cos \left( \alpha_{\mathrm{fs}}/n_1 \right) } - \frac{1}{ \cos \left( \alpha_{\mathrm{fs}}/n_2 \right) } \right). \] \[ \tag{48} \frac{1}{\lambda} \approx R \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right). \] Conclusions
This paper proposes a geometric construction of discrete energy states of a particle in the i-basis. The natural quantum number arises not as an independent postulate, but as a consequence of the condition of closure of the internal periodic motion through an integer number of fundamental cycles.
Increasing the period of a state according to the law \( T_n=nT \) leads to a decrease in its angular frequency: \( \omega_n=\omega/n \). Since the geometric angle of a state is related to the frequency of the internal motion, the admissible states form a discrete series \( \theta_n=\alpha_{\mathrm{fs}}/n \).
On this basis, the exact geometric factor \( \gamma_n=1/\cos(\alpha_{\mathrm{fs}}/n) \) and the corresponding energy series \( E_n=mc^2/\cos(\alpha_{\mathrm{fs}}/n) \) were obtained. As \( n\to\infty \) the geometric angle tends to zero, and the energy of the state approaches the limiting value \( mc^2 \).
The difference in the energies of two admissible states leads to the generalized spectral formula \[ hf = mc^2 \left( \frac{1}{ \cos \left( \alpha_{\mathrm{fs}}/n_1 \right) } - \frac{1}{ \cos \left( \alpha_{\mathrm{fs}}/n_2 \right) } \right). \] It links the emission or absorption frequency directly to a change in the geometry of the particle's internal state.
When expanding the exact dependence in terms of a small parameter \( \alpha_{\mathrm{fs}} \) the constant part of the energy \( mc^2 \) cancels out in the level difference, and the first variable term takes on the dependence \( 1/n^2 \). As a result, the generalized formula transforms into the classical Rydberg law: \[ \frac{1}{\lambda} \approx R \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right), \qquad R = \frac{mc\alpha_{\mathrm{fs}}^2}{2h}. \]
Thus, the classical dependence \( E_n\sim1/n^2 \) refers not to the entire energy of a state, but to its variable geometric part relative to the limiting level \( mc^2 \). The inverse square law arises as a first approximation of the more general dependence \( 1/\cos(\alpha_{\mathrm{fs}}/n) \).
Using the reduced mass instead of the electron mass allows for a natural account of the motion of the nucleus relative to the common center of mass. The exact formula also contains higher-order terms in \( \alpha_{\mathrm{fs}} \), which, within the framework of the model under consideration, represent proper geometric corrections.
The quantum number \( n \) acquires a direct geometric meaning in this scheme. It characterizes the order of closure of the internal motion, determines the duration of a complete cycle, its effective frequency, and the corresponding geometric angle of the state: \[ n \longrightarrow T_n=nT \longrightarrow \omega_n=\frac{\omega}{n} \longrightarrow \theta_n=\frac{\alpha_{\mathrm{fs}}}{n} \longrightarrow E_n. \]
A change in the energy state can therefore be interpreted as a transition between different stable regimes of internal periodicity. The spectral line arises from the energy difference between two such regimes, while the line spectrum is a manifestation of discrete regimes of the particle's closed internal motion.
This interpretation does not require a direct connection between the quantum number and the spatial orbital number. The possibility of constructing a model of atomic states without introducing a set of classical stationary orbits represents a separate direction for further research.
Thus, the generalized Rydberg law is linked to two fundamental geometric principles: the existence of stable closed regimes of internal motion and the dependence of the particle's total energy on the geometry of this motion. The I-basis combines these principles into a single mathematical framework and allows one to obtain the spectral dependence as a consequence of the particle's internal dynamics.
Materials used
- Wikipedia. Rydberg's Formula.