Golden Quantum Oscillator and Binet-Fibonacci Calculus
Oktay K. Pashaev, Sengul Nalci
TL;DR
This paper introduces a quantum harmonic oscillator model based on Fibonacci numbers and the Golden ratio, revealing a spectrum governed by Fibonacci sequence and Golden ratio properties.
Contribution
It develops a novel Golden quantum oscillator framework using Fibonacci-based calculus, linking quantum spectra to Fibonacci numbers and Golden ratio.
Findings
Oscillator spectrum equals Fibonacci numbers.
Energy level ratios follow the Golden sequence.
Asymptotic states exhibit Golden ratio properties.
Abstract
The Binet-Fibonacci formula for Fibonacci numbers is treated as a q-number (and q-operator) with Golden ratio bases and . Quantum harmonic oscillator for this Golden calculus is derived so that its spectrum is given just by Fibonacci numbers. Ratio of successive energy levels is found as the Golden sequence and for asymptotic states it appears as the Golden ratio. This why we called this oscillator as the Golden oscillator. By double Golden bosons, the Golden angular momentum and its representation in terms of Fibonacci numbers and the Golden ratio are derived.
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