Bridging Macroscopic and Quantum Scales: The Exact Conversion of 66.23 Terameters to Electron Compton Wavelength

1. Introduction

The interplay between large-scale astronomical distances and quantum mechanical scales is essential in unifying physical theories. The electron Compton wavelength, defined as:λe=hmec≈2.426×10−12 metersλe​=me​ch​≈2.426×10−12meters

(where hh is Planck’s constant, meme​ is the electron mass, and cc is the speed of light) serves as a fundamental unit in quantum electrodynamics. Conversely, terameters (1 Tm = 10¹² m) describe cosmological or interstellar distances. Converting 66.23 Tm to λₑ establishes a direct relationship between these scales.

2. Theoretical Framework

2.1 Dimensional Analysis

The conversion requires expressing 66.23 Tm in terms of λₑ:66.23 Tm=N×λe66.23Tm=N×λe​

Solving for NN:N=66.23×1012 mλeN=λe​66.23×1012m​

Substituting λe≈2.426×10−12 mλe​≈2.426×10−12m:N=66.23×10122.426×10−12≈2.730×1025N=2.426×10−1266.23×1012​≈2.730×1025

Thus, 66.23 Tm ≈ 2.730 × 10²⁵ electron Compton wavelengths.

2.2 Verification via Fundamental Constants

To ensure accuracy, we re-express the calculation using the 2019 CODATA values:λe=hmec=6.62607015×10−34 Js(9.1093837015×10−31 kg)(2.99792458×108 m/s)λe​=me​ch​=(9.1093837015×10−31kg)(2.99792458×108m/s)6.62607015×10−34Js​λe≈2.42631023867×10−12 mλe​≈2.42631023867×10−12m

Recalculating NN:N=66.23×10122.42631023867×10−12≈2.7299×1025N=2.42631023867×10−1266.23×1012​≈2.7299×1025

The result remains consistent within experimental precision.

3. Implications

3.1 Cosmological Significance

• Hubble Scale Comparisons: The Hubble length (~14.4 Tm) is of similar magnitude, suggesting that quantum-to-cosmological scale conversions may reveal underlying symmetries.
• Large Number Hypotheses: The ratio N∼1025N∼1025 echoes Dirac’s large number hypothesis, hinting at deep connections between micro- and macro-physics.

3.2 Quantum Metrology

• Precision Tests: Such conversions test the consistency of fundamental constants across scales.
• Interferometry Applications: Bridging Tm and λₑ could enhance gravitational wave detector calibrations.

4. Conclusion

The exact conversion of 66.23 Tm to 2.730×10252.730×1025 electron Compton wavelengths demonstrates the mathematical unity of physical scales. Future work could explore similar conversions for other quantum lengths (e.g., Bohr radius) and their role in theoretical models.

References

1. CODATA (2019). Fundamental Physical Constants. NIST.
2. Dirac, P. A. M. (1937). Nature, 139, 323.
3. Greiner, W. (2000). Quantum Electrodynamics. Springer.