Reciprocal System Research Society

Posted: **Sun Dec 07, 2025 6:48 pm**

In DFT-10a we showed that energy quantization for hydrogen follows from a simple geometric rule:

$\oint d\Theta = 2\pi n \quad\Longrightarrow\quad \oint p\,dx = 2\pi n\,\hbar$

This alone gave:

$E_n = -\frac{13.6\ \text{eV}}{n^2}$

In this supplement we do the same for orbital angular momentum.

We’ll show that:

$L^2 = \hbar^2\,\ell(\ell+1), \qquad L_z = m\,\hbar$

are not postulates about operators, but the only S-frame values compatible with the dual-frame geometry of scalar motion.

The structure is:

How orbital direction arises from T-frame geometry
Why $L_z = m\hbar$ follows from azimuthal single-valuedness
Why $L^2 = \hbar^2\,\ell(\ell+1)$ follows from global S² compatibility
Example: the $\ell=1$ states
How this addresses the reviewer’s “quantization not derived” criticism

1. Orbital direction as a projection of T-frame structure

We start with the same setup as DFT-10a:

$\left\Vert \frac{d\sigma}{d\lambda} \right\Vert = \kappa$

Scalar progression $\sigma(\lambda)$ is projected as:

$F_S : \sigma(\lambda) \mapsto x^\mu(\lambda)$

and:

$F_T : \sigma(\lambda) \mapsto \Theta(\lambda) = (\theta^1,\theta^2,\theta^3)$

For a bound atomic orbital, the direction of motion is what matters for angular momentum. The S-frame encodes this through a unit vector on a 2-sphere:

$\mathbf{\hat{r}} = (\sin\vartheta \cos\varphi,\, \sin\vartheta \sin\varphi,\, \cos\vartheta)$

In DFT terms, angles 𝜗 and 𝜑 are not arbitrary coordinates—they arise from effective T-frame combinations:

$\vartheta = \vartheta(\theta^1,\theta^2,\theta^3), \qquad \varphi = \varphi(\theta^1,\theta^2,\theta^3)$

So the angular part of orbital structure is really the image of some T-frame phase pattern on a compact manifold that, after quotienting out a global phase, looks like $S^2$ . That’s the geometric content behind the standard “angular part of the wavefunction lives on the sphere”.

The key fact: closed loops on this sphere must correspond to closed T-frame phase loops, up to integer windings. Quantization is just the statement that these loops are compatible.

2. Why $L_z = m\hbar$ arises from azimuthal consistency

Consider a loop at fixed polar angle $\vartheta$ , circling the z-axis once:

$\varphi: 0 \to 2\pi,\quad \vartheta = \text{const}$

In T-frame terms, going once around in $\varphi$ corresponds to some net change in a T-frame phase combination, say:

$\Phi_z(\lambda) = a_1 \theta^1 + a_2 \theta^2 + a_3 \theta^3$

Single-valuedness of the S-frame amplitude requires that after one full loop, the total phase advance be an integer multiple of $2\pi$ :

$\Delta \Phi_z = \oint d\Phi_z = 2\pi m, \qquad m \in \mathbb{Z}$

As in 10a, S-frame action is proportional to T-frame phase:

$\oint \mathbf{L} \cdot d\mathbf{\varphi} = \hbar \oint d\Phi_z = 2\pi m\,\hbar$

But for a fixed circle of latitude, $\mathbf{L} \cdot d\mathbf{\varphi}$ reduces to the z-component:

$\oint \mathbf{L} \cdot d\mathbf{\varphi} = \oint L_z \, d\varphi = L_z \int_0^{2\pi} d\varphi = 2\pi L_z$

Equating the two:

$2\pi L_z = 2\pi m\,\hbar \quad\Longrightarrow\quad L_z = m\,\hbar$

So the familiar azimuthal quantization is nothing more than:

A closed loop in the S-frame ( $\varphi \to \varphi+2\pi$ )
Matching a closed loop in T-frame phase space
With the same phase→action conversion $\hbar$ found in DFT-10a

No operator postulate, no “eigenvalues by fiat”—just loop consistency.

3. Why $L^2 = \hbar^2\,\ell(\ell+1)$ follows from global S² compatibility

So far we’ve quantized the component $L_z$ . Now we ask: why must the magnitude $L = |\mathbf{L}|$ take only discrete values $\hbar\sqrt{\ell(\ell+1)}$ ?

The key is that the angular amplitude on the sphere must be:

Single-valued
Smooth
Normalizable over $S^2$
Compatible with the rotational symmetry of the scalar embedding

The natural geometric operator encoding this is the Laplace–Beltrami operator on the 2-sphere, which in standard spherical coordinates reads:

$\nabla_{\Omega}^2 = \frac{1}{\sin\vartheta}\frac{\partial}{\partial\vartheta}\left(\sin\vartheta \frac{\partial}{\partial\vartheta}\right) + \frac{1}{\sin^2\vartheta}\frac{\partial^2}{\partial\varphi^2}$

In DFT language, this is the S-frame operator that measures how the probability density spreads over angular directions, given the underlying rotational symmetry of the T-frame. The only scalar angular functions compatible with global rotational symmetry and smoothness are the eigenfunctions of $\nabla_\Omega^2$ .

These are the spherical harmonics $Y_\ell^m(\vartheta,\varphi)$ , satisfying:

$\nabla_{\Omega}^2 Y_\ell^m = -\ell(\ell+1)\,Y_\ell^m, \qquad \ell = 0,1,2,\dots,\quad m=-\ell,\dots,\ell$

The S-frame quantum operator for orbital angular momentum is defined geometrically by:

$L^2 = -\hbar^2 \nabla_{\Omega}^2$

so that:

$L^2 Y_\ell^m = \hbar^2\,\ell(\ell+1)\,Y_\ell^m$

This is the standard result, but here is the DFT interpretation:
The curvature of angular probability distributions on $S^2$ is intrinsically quantized because $S^2$ is compact and rotationally symmetric.
The allowed angular patterns are the spherical harmonics, characterized by integer $\ell$ .
The eigenvalues $\ell(\ell+1)$ are the unique spectrum of the Laplacian on $S^2$ .

Thus orbital angular momentum magnitude is quantized because the only smooth, globally compatible angular distributions are those of integer $\ell$ . The $\hbar^2$ appears exactly as in DFT-10a: it is the frame-conversion factor between T-frame phase curvature and S-frame action/rotation.

This gives:

$L^2 = \hbar^2\,\ell(\ell+1),\qquad L_z = m\,\hbar$

as direct geometric consequences of:

Closed phase loops
Compact angular manifold $S^2$
Rotational symmetry
Phase→action conversion via $\hbar$

4. Example: the $\ell = 1$ states

Take the first nontrivial orbital sector $\ell = 1$ . The spherical harmonics $Y_1^m$ with $m=-1,0,1$ correspond to the familiar “p” orbitals.

From the above:

$L^2 = \hbar^2\,1(1+1) = 2\,\hbar^2$

So every $Y_1^m$ has:

$|\mathbf{L}| = \sqrt{2}\,\hbar$

while the z-component can only be:

$L_z = -\hbar,\ 0,\ \hbar$

Geometrically in DFT:

The T-frame winding structure selects an $\ell=1$ angular sector on $S^2$ .
The azimuthal loop condition forces three distinct $m$ branches for how the phase closes around the z-axis.
The S-frame sees three different “orientations” for the same magnitude $L = \sqrt{2}\hbar$ —exactly $2\ell+1 = 3$ states.

Nothing in this argument used operator axioms; everything came from loop closure and smoothness on a compact angular manifold.

Reciprocal System Research Society

DFT-10b: Angular Momentum Quantization as a Property of S/T-Frame Geometry

DFT-10b: Angular Momentum Quantization as a Property of S/T-Frame Geometry