2026-09-02
Atomic orbitals as projections of split phase space
This article proposes a consistent geometric interpretation of atomic orbitals. The starting point is a binary orbital operator admitting two alternative phase branches. Several independent operators form distinct idempotent planes, and permutation symmetry extracts a finite number of physically distinguishable states from them. Two such symmetric directions of the n level create n2 common phase components, which are grouped into subspaces of dimensions 1, 3, 5, 7, and so on. After a single projection of the two-component phase state into three-dimensional space, these subspaces reproduce spherical harmonics and the familiar shapes of s-, p-, d-, and f-orbitals. The construction does not replace quantum mechanics, but rather formulates a possible internal geometric mechanism for the known orbital multiplicities.
1. The Problem of Geometric Construction
In a nonrelativistic hydrogen-like atom, the energy level with the principal quantum number n obeys the law
\[\tag{1} E_n=\frac{E_1}{n^2}. \] Within the framework of Wave Electricity, the energy modulus is associated with the internal phase frequency. Therefore, the energy law corresponds to a decrease in phase velocity:
\[\tag{2} \begin{aligned} \mathcal E_n&=\frac{\mathcal E_1}{n^2}, &\varpi_n&=\frac{\varpi_1}{n^2},\\ x_n&=\varpi_n t, &x_n&=\frac{\varpi_1t}{n^2}. \end{aligned} \] In the standard classification, for a given n, the following sublevels are allowed:
\[\tag{3} l=0,1,\ldots,n-1, \qquad m=-l,-l+1,\ldots,l. \] A sublevel with orbital number l contains 2l+1 spatial orbitals. Their total number at the level is
\[\tag{4} \boxed{ N_n=\sum_{l=0}^{n-1}(2l+1)=n^2 }. \] | Index | Meaning | Available Values |
|---|---|---|
| n | energy level and radial mode | 1, 2, 3, ... |
| l | angular sublevel type | 0, ..., n−1 |
| m | orientation of orbital shape | −l, ..., +l |
| σ | electron spin orientation | +, − |
Our task is to obtain both the square of n2 and its decomposition into successive odd layers from a single internal geometric rule:
\[\tag{5} \boxed{ n^2=1_s+3_p+5_d+7_f+\ldots+(2n-1) }. \] 2. Dynamic and Orbital Operators
In the concept of Wave Electricity, the complete phase state of a particle is described by the operator
\[\tag{6} J_{\mathrm{dyn}}(a,b) =\j^a(-\j)^b. \] Here, parameter a refers to the internal state of the particle, and parameter b refers to its external motion. In general, they are defined by the relations
\[\tag{7} a=\varpi t, \qquad \omega=\pi\varpi, \qquad b=\frac{\arcsin\beta}{\pi}, \qquad \beta=\frac{v}{c}. \] Further, where the difference with orbital operators is explicit, the full dynamical operator is abbreviated as \(J(a,b)\).
Constructing orbitals requires a different level of description. Let's introduce an elementary orbital operator:
\[\tag{8} J_r^{\mathrm{orb}}(\alpha,\beta) =\j_r^\alpha(-\j_r)^\beta, \] where the index r denotes an independent internal generator. The arguments α and β in formula (8) number the two branches of the internal orbital splitting and are not the velocity of the center of the atom.
The restrictions introduced below for the elementary operator \(J_r^{\mathrm{orb}}\) do not apply to the full dynamical operator \(J_{\mathrm{dyn}}(a,b)\). For a moving particle, the internal parameter a and the external parameter b can be simultaneously nonzero.
3. Elementary Binary State
Assume that the elementary orbital operator preserves one chosen phase branch. Then two alternatives are allowed for it:
\[\tag{9} J_r^{\mathrm{orb}}(\alpha,0)=\j_r^\alpha, \qquad J_r^{\mathrm{orb}}(0,\alpha)=(-\j_r)^\alpha. \] We write the elementary condition as
\[\tag{10} \boxed{\alpha\beta=0}. \] It means that the two branches of the same generator are alternatives, and not two simultaneously active components of an elementary orbital component. The states \(J_r^{\mathrm{orb}}(\alpha,-\alpha)\) or \(J_r^{\mathrm{orb}}(\alpha,\alpha)\) mathematically exist, but are not included in the chosen physical class of elementary orbital states.
This restriction does not prohibitt combine opposite branches of different independent generators. For example, the state
\[\tag{11} \j_r^\alpha *(-\j_s)^\beta, \qquad r\ne s, \] is valid: its factors belong to different internal phase planes.
4. Independent Generators and Common Planes
Let different splitting directions correspond to independent commuting hyperbolic units:
\[\tag{12} \j_r^2=1, \qquad \j_r\j_s=\j_s\j_r, \qquad \j_r\ne\pm\j_s \quad(r\ne s). \] Their independence means that the exponents of different generators do not add up:
\[\tag{13} \boxed{ \j_r^\alpha *\j_s^\beta \ne \j^{\alpha+\beta}, \qquad r\ne s }. \] The usual multiplication sign \(*\) here denotes the product within the extended algebra of independent generators. It preserves information about each factor and is equivalent to writing the joint state in the product of two independent spaces.
For two units \(\j_A\) and \(\j_B\), we define pairs of idempotents:
\[\tag{14} \ep_A^\pm=\frac{1\pm\j_A}{2}, \qquad \ep_B^\pm=\frac{1\pm\j_B}{2}. \] Their products form four joint projectors:
\[\tag{15} E_{++}=\ep_A^+\ep_B^+, \quad E_{+-}=\ep_A^+\ep_B^-, \quad E_{-+}=\ep_A^-\ep_B^+, \quad E_{--}=\ep_A^-\ep_B^-. \] They are idempotent, mutually orthogonal, and sum to one:
\[\tag{16} E_{\sigma\tau}E_{\mu\nu} =\delta_{\sigma\mu}\delta_{\tau\nu}E_{\sigma\tau}, \qquad \sum_{\sigma,\tau=\pm}E_{\sigma\tau}=1. \] After the complex expansion, each projector has its own phase plane.
\[\tag{17} \Pi_{\sigma\tau} =\operatorname{span}_{\mathbb R} \{E_{\sigma\tau},iE_{\sigma\tau}\}. \] Therefore, the product of two independent binary splittings creates not a single power of the original \(\j\), but four different joint planes:
\[\tag{18} \boxed{2*2=4}. \] This construction is a special case of multilevel idempotent splitting of phase planes.
5. Symmetry as a Restriction Rule
If one internal direction of level n is constructed from n−1 binary operators without additional conditions, 2n−1 combinations will arise. This number grows too quickly and does not correspond to the atomic structure. Therefore, we introduce the rule of permutation symmetry.
Permutation of equivalent binary operators within a single direction does not create a new physical state. Only the number of selected branches of each symbol remains distinguishable, not their order.
Let there be n−1 operators in one direction. Let \(N_+\) and \(N_-\) denote the number of positive and negative branches. Then
\[\tag{19} N_++N_-=n-1. \] The number \(N_-\) can take values from 0 to n−1, meaning there are exactly n different symmetrical compositions. Let's denote them by \(A_r^{(n)}\), where r is equal to the number of negative branches:
\[\tag{20} \boxed{ A_r^{(n)} =\frac{1}{\sqrt{\binom{n-1}{r}}} \operatorname{Sym} \left[(A^+)^{n-1-r}(A^-)^r\right], \quad r=0,\ldots,n-1 }. \] The sign \(\operatorname{Sym}\) denotes the sum of all different permutations of branches with the same coefficients. The normalization factor is chosen under the assumption that the primary channels are orthonormal.
Thus, the symmetric sector of the first direction has dimension
\[\tag{21} \dim\mathcal A_n=n. \] The independent second direction is constructed in the same way:
\[\tag{22} B_s^{(n)} =\frac{1}{\sqrt{\binom{n-1}{s}}} \operatorname{Sym} \left[(B^+)^{n-1-s}(B^-)^s\right], \qquad \dim\mathcal B_n=n. \] 6. Two directions and n² phase states
Compare one state of direction \(\mathcal A_n\) with one state of independent direction \(\mathcal B_n\):
\[\tag{23} Q_{rs}^{(n)} =A_r^{(n)}*B_s^{(n)}, \qquad r,s=0,1,\ldots,n-1. \] The number of possible pairs is
\[\tag{24} \boxed{ \dim(\mathcal A_n*\mathcal B_n) =n\cdot n=n^2 }. \] Thus, two independent directions, each constructed by symmetrizing n−1 binary splittings, create the required number of orbital components. Unlike a simple binary tree, the symmetry replaces the exponential multiplicity of 4n−1 with a square multiplicity of n2.
However, an individual component \(Q_{rs}^{(n)}\) is not necessarily a state with defined l and m. The physical orbital shape arises as a matched linear combination of the primary channels:
\[\tag{25} \boxed{ \Phi_{lm}^{(n)} =\sum_{r,s=0}^{n-1} C_{rs}^{\,lm}Q_{rs}^{(n)} }. \] The coefficients \(C_{rs}^{\,lm}\) determine the phase matching of the channels. If a binary pair is transformed as a two-component state, the symmetric sector of n−1 pairs has the geometric index
\[\tag{26} j_n=\frac{n-1}{2}. \] Combining two identical sectors yields all integer values from 0 to 2jn:
\[\tag{27} j_n*j_n =0\oplus1\oplus\ldots\oplus(n-1). \] Therefore, the joint space decomposes into subspaces
\[\tag{28} \boxed{ \mathcal O_n =\bigoplus_{l=0}^{n-1}\mathcal H_l, \qquad \dim\mathcal H_l=2l+1 }. \] By dimensions, we obtain
\[\tag{29} \boxed{ n*n =1\oplus3\oplus5\oplus\ldots\oplus(2n-1), \qquad n^2=\sum_{l=0}^{n-1}(2l+1) }. \] The algebraic independence of the generators creates primary channels, but does not in itself determine their expansion in l. For formulas (26)–(29), the rule is additionally adopted that binary branches are transformed as a two-component phase state, and physically distinguishable groups are identified according to the addition law of such states.
7. Level 1: Construction of the 1s orbital
When n=1, there are no additional orbital generators. The space of angular forms is one-dimensional:
\[\tag{30} \mathcal O_1=\mathcal H_0, \qquad 1^2=1_s. \] The angular function is constant:
\[\tag{31} \Phi_{1s}=1. \] The two spin orientations are determined not by an additional generator, but by two placements of the same internal phase \(x_1\) in the main idempotent planes of the full operator \(J\):
\[\tag{32} J_+(x_1)=J(x_1,0)=\j^{x_1}, \qquad J_-(x_1)=J(0,x_1)=(-\j)^{x_1}. \] The complete states of the first orbital are of the form
\[\tag{33} \begin{aligned} \Psi_{1s,+} &=R_{10}(r)J(x_1,0) =R_{10}(r)\j^{x_1},\\ \Psi_{1s,-} &=R_{10}(r)J(0,x_1) =R_{10}(r)(-\j)^{x_1}. \end{aligned} \] The states \(J(x_1,0)\) and \(J(0,x_1)\) describe two spin orientations, but not two different spatial orbitals. After spatial projection, both correspond to a single spherically symmetric form 1s.
8. Second Level: Construction of 2s and 2p Orbitals
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The second energy level is the first nontrivial example of orbital splitting. For n=1, there is only one spatial state, whereas for n=2, four independent phase channels must arise. In the proposed design, these are formed by the combined action of two independent binary operators. Each operator allows two alternative branches, so their product creates 2*2=4 primary states. These states are not yet separate atomic orbitals: only their consistent symmetric and antisymmetric combinations form one 2s and three 2p orbitals.
8.1. Two Independent Binary Operators
For n=2, each direction contains one binary operator. We denote the two independent generators by (j_A) and (j_B). We distribute the total orbital phase \(x_2\) evenly between them:
\[\tag{34} A_+=\j_A^{x_2/2}, \qquad A_-=(-\j_A)^{x_2/2}, \qquad B_+=\j_B^{x_2/2}, \qquad B_-=(-\j_B)^{x_2/2}. \] Their independent multiplication creates four primary phase channels:
\[\tag{35} \begin{aligned} Q_{++}&=\j_A^{x_2/2}*\j_B^{x_2/2},\\ Q_{+-}&=\j_A^{x_2/2}*(-\j_B)^{x_2/2},\\ Q_{-+}&=(-\j_A)^{x_2/2}*\j_B^{x_2/2},\\ Q_{--}&=(-\j_A)^{x_2/2}*(-\j_B)^{x_2/2}. \end{aligned} \] The indices store information about which branch is chosen in each of two independent directions.
8.2. Permutation Symmetry
Consider the permutation of the two directions \(A\leftrightarrow B\). The channels \(Q_{++}\) and \(Q_{--}\) remain unchanged, and the two mixed channels transform into each other:
\[\tag{36} Q_{+-}\longleftrightarrow Q_{-+}. \] From the four channels, three symmetric combinations and one antisymmetric combination are formed. We write three symmetric combinations as follows:
\[\tag{37} \begin{aligned} \Phi_{2p,+1}&=Q_{++},\\ \Phi_{2p,0}&=\frac{Q_{+-}+Q_{-+}}{\sqrt2},\\ \Phi_{2p,-1}&=Q_{--}. \end{aligned} \] They form a three-dimensionalcomponent sector with l=1 and m=+1, 0, −1. The remaining antisymmetric combination is one-dimensional:
\[\tag{38} \Phi_{2s} =\frac{Q_{+-}-Q_{-+}}{\sqrt2}. \] It corresponds to a scalar sector with l=0. In expanded notation, there are four orbital forms equal
\[\tag{39} \begin{aligned} \Phi_{2s}&=\frac{\j_A^{x_2/2}*(-\j_B)^{x_2/2}-(-\j_A)^{x_2/2}*\j_B^{x_2/2}}{\sqrt2},\\ \Phi_{2p,+1}&=\j_A^{x_2/2 }*\j_B^{x_2/2},\\ \Phi_{2p,0}&=\frac{\j_A^{x_2/2}*(-\j_B)^{x_2/2}+(-\j_A)^{x_2/2}*\j_B^{x_2/2}}{\sqrt2},\\ \Phi_{2p,-1}&=(-\j_A)^{x_2/2}*(-\j_B)^{x_2/2}. \end{aligned} \] This way the second level gets the correct structure
\[\tag{40} \boxed{2*2=1_s+3_p=4}. \] After adding two independent spin orientations, one 2s orbital yields two states, and three 2p orbitals yield six:
\[\tag{41} 2+6=8=2n^2 \qquad(n=2). \] 9. Third and Subsequent Levels
For n=3, each direction is constructed from two binary operators. After symmetrization, three states remain:
\[\tag{42} (++),\qquad \frac{(+-)+(-+)}{\sqrt2}, \qquad (--). \] Here, the signs denote the choice of branches of two independent generators, and the average expression is their symmetric sum. Combining two three-dimensional directions yields nine components, which are decomposed into three sublevels:
\[\tag{43} \boxed{3*3=1_s+3_p+5_d=9}. \] For n=4, each direction is constructed from three binary operators. Symmetric compositions
\[\tag{44} (+++),\qquad \operatorname{Sym}(++-), \qquad \operatorname{Sym}(+--), \qquad (---) \] form four states of one direction. Two directions give
\[\tag{45} \boxed{4*4=1_s+3_p+5_d+7_f=16}. \] | n | Binary operators in one direction | Joint components | Orbital sublevels |
|---|---|---|---|
| 1 | 0 | 1 | 1s |
| 2 | 1 | 4 | 2s + 2p |
| 3 | 2 | 9 | 3s + 3p + 3d> |
| 4 | 3 | 16 | 4s + 4p + 4d + 4f> |
| 5 | 4 | 25 | 5s + 5p + 5d + 5f + 5g> |
10. Two-Component Phase Space
Counting states does not yet determine the observed shape of an orbital. To get there, we represent the normalized internal angular state in an idempotent basis:
\[\tag{46} \Xi=u\ep+v\em, \] where
\[\tag{47} u=\rho_+e^{i\phi_+}, \qquad v=\rho_-e^{i\phi_-}, \qquad |u|^2+|v|^2=1. \] The common phase of both components does not change the observed orientation:
\[\tag{48} (u,v)\sim \left(e^{i\chi}u,e^{i\chi}v\right). \] Therefore, the amplitude distribution between the two planes and the relative phase remain geometrically significant.
\[\tag{49} \Delta\phi=\phi_--\phi_+. \] The normalized pair of complex amplitudes belongs to a three-dimensional sphere \(S^3\). Removing the common unobservable phase leaves a two-dimensional sphere of directions \(S^2\). This allows us to define an explicit mapping of the internal phase state into three-dimensional space.
11. Projection of the phase state in 3D
We define three real quantities:
\[\tag{50} X=u^*v+v^*u, \] \[\tag{51} Y=-i\left(u^*v-v^*u\right), \] \[\tag{52} Z=|u|^2-|v|^2. \] From normalization (47) it immediately follows that
\[\tag{53} X^2+Y^2+Z^2=1. \] Therefore, the vector
\[\tag{54} \mathbf n=(X,Y,Z) \] is a unit vector in ordinary three-dimensional space. We parameterize the internal amplitudes as
\[\tag{55} u=\cos\frac{\theta}{2}e^{-i\varphi/2}, \qquad v=\sin\frac{\theta}{2}e^{i\varphi/2}. \] Then formulas (50)–(52) become
\[\tag{56} X=\sin\theta\cos\varphi, \qquad Y=\sin\theta\sin\varphi, \qquad Z=\cos\theta. \] Thus, the two internal complex components define the direction in 3D:
\[\tag{57} \boxed{ (u,v)\overset{\mathcal P}{\longrightarrow}(X,Y,Z) }. \] The overall phase is lost during projection, the relative phase determines the azimuthal direction, and the difference in the squares of the amplitudes determines the position relative to the Z-axis. The primary channels \(Q_{rs}^{(n)}\) define the internal structure, their matched combinations form phase functions \(\Phi_{lm}^{(n)}(u,v)\), and the projection transforms these functions into spherical harmonics:
\[\tag{58} \boxed{ Q_{rs}^{(n)} \longrightarrow \Phi_{lm}^{(n)}(u,v) \overset{\mathcal P}{\longrightarrow} Y_l^m(\theta,\varphi) }. \] 12. Spatial Shapes of s-, p-, and d-Orbitals
12.1. Spherical s-Shape
The scalar sector l=0 corresponds to a constant angular function:
\[\tag{59} \Phi_s(u,v)=1. \] After projection, it is independent of angles:
\[\tag{60} \mathcal P[\Phi_s] \propto Y_0^0(\theta,\varphi) =\frac{1}{\sqrt{4\pi}}. \] Therefore, all s orbitals have spherical angular symmetry. The difference between 1s, 2s, 3s, and subsequent states is contained in the radial part.
12.2. Three p-forms
The three-component sector l=1 corresponds to three coordinates of the unit vector:
\[\tag{61} \Phi_{p_x}=X, \qquad \Phi_{p_y}=Y, \qquad \Phi_{p_z}=Z. \] | Orbital | Intrinsic phase function | 3D projection |
|---|---|---|
| px | u* v + v* u | sin θ cos φ |
| py | −i(u* v − v* u) | sin θ sin φ |
| pz | |u|² − |v|² | cos θ |
For example, for \(p_z\), the nodal plane arises when the amplitudes of the two internal components are equal:
\[\tag{62} |u|^2=|v|^2 \quad\Longrightarrow\quad Z=0. \] The two regions with \(Z>0\) and \(Z<0\) have opposite phases and, after projection, form two lobes of the orbital. Thus, the two-lobed shape arises as a projection of the alternating difference in the intensities of the internal components.
12.3. Five d-forms
For l=2, the angular functions are harmonic quadratic combinations of X, Y, and Z. Of the six symmetric quadratic combinations, the spherical trace \(X^2+Y^2+Z^2=1\) is eliminated, leaving five independent forms:
\[\tag{63} XY, \qquad XZ, \qquad YZ, \qquad X^2-Y^2, \qquad 3Z^2-1. \] | Orbital | Function after projection |
|---|---|
| dxy | XY |
| dxz | XZ |
| dyz | YZ |
| dx²−y² | X²−Y² |
| dz² | 3Z²−1 |
For example,
\[\tag{64} \Phi_{d_{z^2}}(u,v) =3\left(|u|^2-|v|^2\right)^2-1 \overset{\mathcal P}{\longrightarrow} 3\cos^2\theta-1. \] For \(d_{xy}\), the nodal conditions \(X=0\) and \(Y=0\) define two mutually perpendicular planes separating the four blades.
12.4. Arbitrary l and m
For general l, the phase forms are represented by harmonic polynomials of degree l in \(X(u,v)\), \(Y(u,v)\), \(Z(u,v)\). The space of such functions has dimension
\[\tag{65} \dim\mathcal H_l=2l+1. \] The unified projection rule is of the form
\[\tag{66} \boxed{ \mathcal P[\Phi_{lm}(u,v)] =Y_l^m(\theta,\varphi) }. \] 13. Radial part and full orbital
The constructed projection defines the angular geometry. The full spatial function requires an independent radial mode:
\[\tag{67} \psi_{nlm}(r,\theta,\varphi) =R_{nl}(r)Y_l^m(\theta,\varphi). \] The spin orientation is already contained in the choice of the full phase operator \(J(x_n,0)\) or \(J(0,x_n)\). Therefore, the internal states of the two orientations are written without additional spin multiplier:
\[\tag{68} \boxed{ \begin{aligned} \Psi_{nlm}^{(+)} &=R_{nl}(r)\,\Phi_{lm}(u,v)\,J(x_n,0) =R_{nl}(r)\,\Phi_{lm}(u,v)\,\j^{x_n},\\ \Psi_{nlm}^{(-)} &=R_{nl}(r)\,\Phi_{lm}(u,v)\,J(0,x_n) =R_{nl}(r)\,\Phi_{lm}(u,v)\,(-\j)^{x_n}. \end{aligned} } \] After projection:
\[\tag{69} \boxed{ \begin{aligned} \mathcal P[\Psi_{nlm}^{(+)}] &=R_{nl}(r)Y_l^m(\theta,\varphi)J(x_n,0),\\ \mathcal P[\Psi_{nlm}^{(-)}] &=R_{nl}(r)Y_l^m(\theta,\varphi)J(0,x_n). \end{aligned} } \] Thus, n defines the energy level and radial mode, l and m define the angular shape, and σ defines the spin orientation. The number of radial nodes in the standard problem is n−l−1, but the geometric origin of this rule from binary splitting is not yet deduced here.
The observed orbital surface is not an electron trajectory, but a surface of constant density.
\[\tag{70} \rho_{nlm}(r,\theta,\varphi) =|R_{nl}(r)|^2|Y_l^m(\theta,\varphi)|^2. \] 14. Spin and the Number of Electron States
Orbital generators \(\j_A,\j_B,\ldots\) describe the depthThe lateral splitting from which the functions \(\Phi_{lm}\) are formed. The spin orientation does not require another generator: it is specified by placing the internal phase \(x_n\) in one of the two main idempotent components of the same complete operator \(J\):
\[\tag{71} \boxed{ J_+(x_n)=J(x_n,0)=\j^{x_n}, \qquad J_-(x_n)=J(0,x_n)=(-\j)^{x_n} }. \] In the notation \(J(0,x_n)\), the quantity \(x_n\) remains an internal parameter. Rearranging the arguments means changing the spin orientation, not turning the internal phase into a parameter of external motion. Therefore, each spatial function \(\Phi_{lm}\) corresponds to two states: \(\Phi_{lm}J(x_n,0)\) and \(\Phi_{lm}J(0,x_n)\). The number of single-electron states of the level is equal to
\[\tag{72} \boxed{N_n^{\mathrm{electron}}=2n^2}. \] | Sublevel | Spatial Orbitals | Spin States |
|---|---|---|
| s | 1 | 2 |
| p | 3 | 6 |
| d | 5 | 10 |
| f | 7 | 14 |
15. Geometric Exclusivity Premise
The elementarity condition \(\alpha\beta=0\) means that one orbital generator selects only one of two alternative branches. In this sense, the construction contains local exclusivity of the internal phase state. Together with two different spin orientations, it creates the geometric premise that one spatial orbital has two different total states.
However, this does not yet constitute a derivation of the Pauli exclusion principle. The exclusion principle applies to two identical electrons and requires antisymmetry of their total joint state. The mathematical goal of a future derivation would be the equality
\[\tag{73} \Psi_q\wedge\Psi_q=0, \qquad q=(n,l,m,\sigma). \] Obtaining such an antisymmetry from the two-sheeted 4π-periodic geometry of the electron remains a separate problem. Therefore, in the present construction, the condition \(\alpha\beta=0\) is considered only as a prerequisite for exceptionality, but not as a proof of the Pauli exclusion clause.
16. What is obtained and what is assumed
| Construction element | Status |
|---|---|
| Independent commuting units \(\j_r^2=1\) | algebraic construction |
| Four joint planes of two binary generators | algebraic consequence of idempotent decomposition |
| Prohibition of mixed branches of one \(J_r^{\mathrm{orb}}\) | physical rule of the model |
| Selection of permutation-symmetric combinations | geometric postulate equivalence |
| The dimension of one direction, equal to n | a consequence of the symmetrization of n−1 binary states |
| Two directions give n2 components | a mathematical consequence of independent multiplication |
| Decomposition into 1, 3, 5, ..., 2n−1 | a consequence of the adopted law of transformation of two-component states |
| Projection \((u,v)\to(X,Y,Z)\) | an explicit mathematical mapping |
| Comparison with \(Y_l^m\) | geometric interpretation of orbital shapes |
| Relationship between the number of components n2 and the energy 1/n2 | central physical hypothesis |
| Radial nodes | not derived in the present construction |
| Pauli exclusion hypothesis | working hypothesis, not a complete derivation |
17. Conclusion
The proposed construction begins the construction of atomic orbitals with a single binary rule: the elementary orbital operator is in the state \(\j_r^\alpha\) or \((-\j_r)^\alpha\), retaining only one active branch. Independent generators form different joint idempotent planes, and symmetrization eliminates the difference between permutations of equivalent factors.
From n−1 binary operators, an n-dimensional symmetric direction is obtained. Independent multiplication of two such directions creates n2 primary phase components. Their consistent linear combinations form subspaces of dimensions 1, 3, 5, 7, and further, corresponding to the s-, p-, d-, and f-sublevels.
\[\tag{74} \boxed{ \begin{aligned} J_r^{\mathrm{orb}}(\alpha,0) \mid J_r^{\mathrm{orb}}(0,\alpha) &\longrightarrow \mathcal A_n*\mathcal B_n \\ &\longrightarrow n^2=\sum_{l=0}^{n-1}(2l+1) \\ &\longrightarrow \Phi_{lm}(u,v) \overset{\mathcal P}{\longrightarrow} Y_l^m(\theta,\varphi). \end{aligned} } \] Single projectThe \((u,v)\to(X,Y,Z)\) action transforms internal phase functions into spherical harmonics and thereby links the abstract splitting with the observed three-dimensional shapes of the orbitals. The decrease in phase velocity as 1/n2 acquires the expected geometric content: it corresponds to an increase in the number of admissible spatial projections to n2.
The main result is a unified sequence: independent binary states create symmetric n-dimensional directions; the two directions form n2 phase components; the components are grouped into layers 1, 3, 5, 7; and their projection reproduces the angular geometry of atomic orbitals. Radial nodes, precise energetic degeneracy removal, and a complete derivation of the Pauli exclusion principle remain tasks for further development of the model.