Golden quantum oscillator and Binet-Fibonacci calculus

The Binet formula for Fibonacci numbers is treated as a q-number and a q-operator with Golden ratio bases q = and Q = −1 , and the corresponding Fibonacci or Golden calculus is developed. A quantum harmonic oscillator for this Golden calculus is derived so that its spectrum is given only by Fibonacc...

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Published inJournal of physics. A, Mathematical and theoretical Vol. 45; no. 1; pp. 15303 - 23
Main Authors Pashaev, Oktay K, Nalci, Sengul
Format Journal Article
LanguageEnglish
Published Bristol IOP Publishing 13.01.2012
IOP
Subjects
Angular momentum
Asymptotic properties
Binet-Fibonacci formula
Bosons
Calculus
Exact sciences and technology
Fermions
Fibonacci numbers
Golden ratio
Mathematical analysis
Oscillators
Physics
quantum oscillator
Representations
Operator
Quantum oscillator
Fermions
Angular momentum
Entangled states
Bosons
Harmonic oscillators
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Summary:The Binet formula for Fibonacci numbers is treated as a q-number and a q-operator with Golden ratio bases q = and Q = −1 , and the corresponding Fibonacci or Golden calculus is developed. A quantum harmonic oscillator for this Golden calculus is derived so that its spectrum is given only by Fibonacci numbers. The ratio of successive energy levels is found to be the Golden sequence, and for asymptotic states in the limit n → ∞ it appears as the Golden ratio. We call this oscillator the Golden oscillator. Using double Golden bosons, the Golden angular momentum and its representation in terms of Fibonacci numbers and the Golden ratio are derived. Relations of Fibonacci calculus with a q-deformed fermion oscillator and entangled N-qubit states are indicated.
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ISSN:1751-8113
1751-8121
DOI:10.1088/1751-8113/45/1/015303