Triadic Energy Balance (TEB) in Quantum Harmonic Oscillator Arrays
Author: D.J. Box
Date: February 20, 2025
(TEB)
Report
We report a novel triadic energy relationship in a system of six coupled quantum harmonic oscillators, where specific occupation number configurations—denoted as Alpha, Omega, and Neyen—satisfy the balance ᴱAlpha + ᴱOmega = ²ᴱNeyen. This relationship emerges from sequences derived modulo 9 and suggests a symmetry-driven resonance condition. The discovery points to potential applications in quantum computing, condensed matter physics, and energy-efficient state transitions in quantum systems.
Introduction
Symmetries and conservation laws are foundational in physics, often revealing underlying principles that govern complex systems. In quantum mechanics, harmonic oscillators serve as a fundamental model for understanding vibrational modes, photons, and quantum fields. Here, we explore a system of six identical quantum harmonic oscillators and uncover a triadic energy balance among three distinct occupation number configurations: Alpha, Omega, and Neyen. These configurations, derived from sequences with modular arithmetic properties, exhibit an exact energy relationship:
ᴱAlpha + ᴱOmega = ²ᴱNeyen.
This finding suggests a deeper symmetry that could inspire new approaches to manipulating quantum states with precision and efficiency.
ᴱAlpha + ᴱOmega = ²ᴱNeyen.
This finding suggests a deeper symmetry that could inspire new approaches to manipulating quantum states with precision and efficiency.
Methods
Sequences and Occupation Numbers
We consider three sequences of six integers each, interpreted as occupation numbers 𝑛ᵢ for a system of six quantum harmonic oscillators:
We consider three sequences of six integers each, interpreted as occupation numbers 𝑛ᵢ for a system of six quantum harmonic oscillators:
- Alpha: ( [1, 3, 7, 6, 4, 9] )
- Omega: ( [8, 6, 2, 3, 5, 0] )
- Neyen: ( [1, 2, 4, 8, 7, 5] )
These sequences exhibit specific properties:
1. Complementarity: For each position ( i ), Alphaᵢ + Omegaᵢ = 9, indicating a modulo 9 relationship.
2. Cyclicity: The Neyen sequence follows powers of 2 modulo 9: 2ᴷ mod9 for 𝑘 = 0 to 5, with a period of 6.
1. Complementarity: For each position ( i ), Alphaᵢ + Omegaᵢ = 9, indicating a modulo 9 relationship.
2. Cyclicity: The Neyen sequence follows powers of 2 modulo 9: 2ᴷ mod9 for 𝑘 = 0 to 5, with a period of 6.
Energy Calculation
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Results
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Remarkably, these energies satisfy:
ᴱAlpha + ᴱOmega = 33ℎ𝑓₀ + 27ℎ𝑓₀ = 60ℎ𝑓₀ = 2×30ℎ𝑓₀ = ²ᴱNeyen
This triadic relationship holds exactly, suggesting a balanced energy distribution across the configurations.
Discussion
The triadic energy balance ᴱAlpha + ᴱOmega = ²ᴱNeyen reveals a symmetry in the quantum oscillator array. The complementarity of Alpha and Omega, combined with the cyclical nature of Neyen, points to a resonance condition that could facilitate energy-efficient transitions between states. In particular:
- The Neyen configuration may act as a resonant bridge, enabling transitions between Alpha and Omega states while conserving total energy.
- This principle could be exploited in quantum technologies, such as quantum computing, where energy-balanced state manipulations are crucial for error correction and coherence preservation.
- In condensed matter physics, this relationship might inspire the design of materials with tailored vibrational properties, enhancing thermal or acoustic transport.
Conclusion
We have demonstrated a triadic energy balance in a quantum harmonic oscillator array, governed by sequences with modulo 9 properties. This finding suggests a novel symmetry that could have practical applications in quantum technologies and materials science. Future work should focus on experimental verification, generalization to other quantum systems, and exploration of the underlying mathematical framework. This discovery opens a new avenue for understanding and manipulating quantum states through symmetry-driven principles.
References
- Messiah, A. Quantum Mechanics. Dover Publications, 1999.
- Cohen-Tannoudji, C., Diu, B., & Laloë, F. Quantum Mechanics. Wiley, 1977.
- Sachdev, S. Quantum Phase Transitions. Cambridge University Press, 2011.
Thank you S. Mendez for insightful discussions and support.
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Triadic Energy Balance Law - “Patent Pending"
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