Golden Quantum Oscillator and Binet-Fibonacci Calculus

Abstract

The Binet-Fibonacci formula for Fibonacci numbers is treated as a q-number (and q-operator) with Golden ratio bases $q=\varphi$ and $Q=-1/\varphi$. 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.

19 pages, 1 figure

References in corpus (3)

• Physical Significance of q Deformation and Many-Body Interactions in Nuclei
• Vortex Images and q-Elementary Functions
• Solitons in Maximally Entangled Two Qubit Phase Space

Cited by in corpus (8)

• Fractals, coherent states and self-similarity induced noncommutative geometry
• Quantum Calculus of Fibonacci Divisors and Infinite Hierarchy of Bosonic-Fermionic Golden Quantum Oscillators
• Non-Commutative Q-Binomial Formula
• Bernoulli-Fibonacci Polynomials
• Quantum Coin Flipping, Qubit Measurement and Generalized Fibonacci Numbers
• Golden Binomials and Carlitz Characteristic Polynomials
• Geometric Probabilities and Fibonacci Numbers for Maximally Random n-Qubit Quantum Information States
• On Apostol-Type Mersenne-Bernoulli and Mersenne-Euler Polynomials