The Bohr Radius in the DRUMS Framework
1. Superfluid Field Representation
In DRUMS, matter emerges from a coherent superfluid field:
\[
\Psi(\mathbf{x},t) = \sqrt{\rho(\mathbf{x},t)} e^{i\theta(\mathbf{x},t)}
\]
The velocity field is:
\[
\mathbf{v} = \frac{\hbar}{m} \nabla \theta
\]
2. Bound States as Phase-Orbit Equilibria
An electron orbit corresponds to a stable circulating phase structure around a central defect (proton).
Quantization arises from single-valuedness of the phase:
\[
\oint \nabla \theta \cdot d\mathbf{l} = 2\pi n
\]
Thus velocity satisfies:
\[
m v r = n \hbar
\]
3. Effective Potential Balance
The electron experiences an effective central potential emerging from density curvature and interaction energy:
\[
V(r) \sim -\frac{\alpha \hbar c}{r}
\]
Force balance requires centripetal acceleration:
\[
\frac{m v^2}{r} = \frac{\alpha \hbar c}{r^2}
\]
4. Solving for Orbital Radius
Substitute quantized momentum:
\[
v = \frac{n \hbar}{m r}
\]
Insert into force balance:
\[
\frac{m}{r} \left( \frac{n^2 \hbar^2}{m^2 r^2} \right) = \frac{\alpha \hbar c}{r^2}
\]
Simplify:
\[
\frac{n^2 \hbar^2}{m r^3} = \frac{\alpha \hbar c}{r^2}
\]
Solve for \(r\):
\[
r_n = \frac{n^2 \hbar}{\alpha m c}
\]
5. Bohr Radius Emergence
The fundamental radius (n = 1):
\[
a_0 = \frac{\hbar}{\alpha m c}
\]
This matches the standard Bohr radius.
6. DRUMS Interpretation
In DRUMS, this radius is not arbitrary but arises from:
- Phase quantization (topological constraint)
- Balance of kinetic and interaction energy
- Coherence length of the superfluid medium
The fine-structure constant appears as a coupling strength between phase curvature and density response.
7. Energy Levels
Total energy:
\[
E = \frac{1}{2} m v^2 - \frac{\alpha \hbar c}{r}
\]
Substitute expressions:
\[
E_n = -\frac{1}{2} \frac{\alpha^2 m c^2}{n^2}
\]
This emerges directly from the same phase constraints.
8. Final Interpretation
The Bohr radius in the DRUMS framework emerges naturally from:
- Quantized phase circulation
- Superfluid velocity constraints
- Balance between kinetic flow and attractive interaction
It is therefore a direct geometric consequence of the underlying coherent field structure, not an imposed quantum rule.