Exploring Quantum States: Proton Resonances and Beyond
Overview
This page explores a novel model that represents the proton as a quantized vortex in a superfluid, solving the proton radius puzzle and predicting quantum state energies for proton resonances. We extend this analysis to include all known \( N \) (nucleon) and \( \Delta \) resonances and compare them with the model's predictions for various quantum numbers \( n \). Additionally, we examine how this model might apply to other particles, such as the Higgs boson, mesons, and other bosons, presenting their masses for context.
The Proton as a Quantized Vortex
The proton radius puzzle highlights a discrepancy between traditional measurements (~0.877 fm) and muonic hydrogen measurements (~0.841 fm). This model proposes the proton as a circular quantized vortex with:
- Mass \( m = m_p \) (938 MeV/c²)
- Velocity \( v = c \) at the vortex radius
- Quantum number \( n = 4 \) for the ground state
Using the quantized circulation equation:
\[ v \cdot 2\pi r = \frac{h}{m} n \]
The radius is:
\[ r_p = \frac{4 \hbar}{m_p c} \approx 0.841 \, \text{fm} \]
For excited states, the energy is modeled as:
\[ E_n = 45.52 + 55.78 n^2 \, \text{MeV} \]
This formula is fitted to match \( E_4 = 938 \, \text{MeV} \) (ground state) and \( E_5 = 1440 \, \text{MeV} \) (Roper resonance), allowing predictions for higher \( n \).
Proton \( N \) and \( \Delta \) Resonances
The model predicts quantum state energies for \( n = 4 \) to \( n = 15 \), which we compare to all known \( N \) and \( \Delta \) resonances. Below is a chart showing these predictions alongside experimental data.
**Chart Explanation:**
- Blue Points: Model predictions for \( n = 4 \) to \( 15 \), calculated as \( E_n = 45.52 + 55.78 n^2 \).
- Red Points: Known \( N \) resonances, plotted at \( n_{\text{res}} = \sqrt{(E_{\text{res}} - 45.52) / 55.78} \).
- Green Points: Known \( \Delta \) resonances, plotted similarly.
The model captures several key resonances (e.g., \( N(938) \) at \( n=4 \), \( N(1440) \) near \( n=5 \), \( \Delta(1232) \) between \( n=4 \) and \( n=5 \)), though not all align perfectly due to additional quantum numbers (e.g., spin, parity) not yet incorporated.
Higgs, Mesons, and Bosons
Extending the model to other particles like the Higgs boson, mesons, and other bosons requires adapting the quantum number \( n \) or the energy formula. Due to the vast energy scale differences (e.g., Higgs at 125 GeV vs. proton resonances at ~1-3 GeV), a separate chart is provided below.
**Notes:**
- The logarithmic scale accommodates the range from ~135 MeV (pion) to ~125 GeV (Higgs).
- Speculative model extension might assign lower \( n \) or different parameters for lighter particles, but this requires further theoretical development.
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