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arXiv:1107.4389 · Journal of Physics A Mathematical and Theoretical

Golden quantum oscillator and Binet–Fibonacci calculus

Oktay K. Pashaev, Şengül Nalcı

Published Dec 1, 2011• 58 citations
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Abstract

The Binet formula for Fibonacci numbers is treated as a q -number and a q -operator with Golden ratio bases q = φ and Q = −1/φ, 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.

DOI
10.1088/1751-8113/45/1/015303

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FINDING: The arXiv paper (2410.04169v2) introduces a "golden oscillator" — a quantum harmonic oscillator whose energy spectrum is quantized via Fibonacci divisors and quantum calculus with bases φ (golden ratio) and ψ (silver ratio), yielding a supersymmetric hierarchy of fermion-boson entangled states. The YouTube results are standard QHO pedagogy (E_n = (n+½)ħω) and add no novel mathematics. MATH: - Standard QHO: E_n = ħω(n + ½), n ∈ ℕ₀. Partition function Z = Σ exp(−βE_n) = 1/[2 sinh(βħω/2)]. - Golden oscillator spectrum (from abstract): E_n ∝ [n]_φ = (φ^n − (−φ)^−n)/(φ − (−φ)^−1) = (φ^n − (−φ)^−n)/√5 — the Fibonacci number F_n. - Quantum calculus: [n]_q = (1 − q^n)/(1 − q). For q = φ², [n]_{φ²} = (φ^{2n} − 1)/(φ² − 1) = F_{2n}/φ (using φ² = φ+1, φ²−1 = φ). - Silver ratio: ψ = 1+√2 ≈ 2.414; [n]_{ψ²} = (ψ^{2n} − 1)/(ψ² − 1) = Pell numbers / (2ψ). - Binet formula for Fibonacci divisor number operator: N_φ = (φ^N − (−φ)^−N)/√5, acting on Fock space. - Supersymmetric hierar Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

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The Binet formula for Fibonacci numbers is treated as a q -number and a q -operator with Golden ratio bases q = φ and Q = −1/φ, 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.
FINDING: The arXiv paper (2410.04169v2) introduces a "golden oscillator" — a quantum harmonic oscillator whose energy spectrum is quantized via Fibonacci divisors and quantum calculus with bases φ (golden ratio) and ψ (silver ratio), yielding a supersymmetric hierarchy of fermion-boson entangled states. The YouTube results are standard QHO pedagogy (E_n = (n+½)ħω) and add no novel mathematics. MATH: - Standard QHO: E_n = ħω(n + ½), n ∈ ℕ₀. Partition function Z = Σ exp(−βE_n) = 1/[2 sinh(βħω/2)]. - Golden oscillator spectrum (from abstract): E_n ∝ [n]_φ = (φ^n − (−φ)^−n)/(φ − (−φ)^−1) = (φ^n − (−φ)^−n)/√5 — the Fibonacci number F_n. - Quantum calculus: [n]_q = (1 − q^n)/(1 − q). For q = φ², [n]_{φ²} = (φ^{2n} − 1)/(φ² − 1) = F_{2n}/φ (using φ² = φ+1, φ²−1 = φ). - Silver ratio: ψ = 1+√2 ≈ 2.414; [n]_{ψ²} = (ψ^{2n} − 1)/(ψ² − 1) = Pell numbers / (2ψ). - Binet formula for Fibonacci divisor number operator: N_φ = (φ^N − (−φ)^−N)/√5, acting on Fock space. - Supersymmetric hierar Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
FINDING: Quantum probability interference terms acquire golden-ratio phase angles in N=2 supersymmetric "golden oscillator" hierarchies, linking Fibonacci divisor calculus to fermion-boson entanglement. | MATH: φ = (1+√5)/2 ≈ 1.618; silver ratio δ_S = 1+√2 ≈ 2.414; Binet form F_n = (φⁿ − ψⁿ)/√5, ψ = −1/φ ≈ −0.618; two-base quantum calculus q = φ, q' = δ_S; energy spectrum of golden oscillators E_n ∝ φⁿ; Fibonacci divisor number operator acting on Fock space. | CONNECTION: φ and its conjugate 1/φ = 0.618 appear as dual bases; 0.618 = 1 − 0.382; φ² = 2.618 = φ + 1; the golden oscillator ladder spacing is geometric with ratio φ, matching the continued-fraction fixed point φ = 1 + 1/(1 + 1/(1+…)) — the "most irrational" number, which suppresses resonant decoherence in quasi-periodic systems (Aubry-André / Fibonacci lattice). | DEPTH: 7 FINDING: Golden ratio observed empirically inside a quantum system where standard theory did not predict it (Meridian Labs report). | MATH: No explicit equ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
FINDING: The only substantive result in this set is a quantum-calculus construction linking Golden/Silver ratios to supersymmetric oscillator hierarchies and fermion–boson entanglement; the CHSH–√2–φ connection is not present in the retrieved sources. | MATH: φ = (1+√5)/2 = 1.6180339887…, satisfying φ² = φ + 1 and φ = √(1+√(1+√(1+…))); silver ratio δ_S = 1+√2 = 2.4142135623…, satisfying δ_S² = 2δ_S + 1; Binet form F_n = (φⁿ − ψⁿ)/√5 with ψ = −1/φ = −0.6180339887…; two-base quantum calculus with bases q = φ and q = δ_S acting on Fibonacci-divisor number operator in Fock space. | CONNECTION: φ and δ_S are the positive roots of x² − x − 1 = 0 and x² − 2x − 1 = 0 respectively — i.e., the two simplest quadratic irrationals of the form x² − nx − 1 = 0 (n = 1, 2). These are the eigenvalues of the SL(2,ℤ) matrices [[0,1],[1,n]], generators of the modular group; φ corresponds to the pentagonal/icosahedral root system H₃ (5-fold symmetry), δ_S to the octagonal/√2 lattice (D₄-related, 8-fold). Th Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
FINDING: Generalized Fibonacci formulas are proposed as a universal scaling model for quantum oscillators, with transmission coefficients scaling as φ⁻², while standard QHO theory (Hermite-Gaussian eigenfunctions) shows no inherent Fibonacci structure. | MATH: Fibonacci recurrence Fₙ₊₁ = Fₙ + Fₙ₋₁; golden ratio φ = (1+√5)/2 ≈ 1.6180339887; φ⁻² = 2−φ ≈ 0.381966; QHO energy Eₙ = ħω(n+½); transmission coefficient T(E) = 1/(1 + (V₀²/4E(V₀−E)) sinh²(κa)) for rectangular barrier — no φ appears in standard QHO. | CONNECTION: φ⁻² = 0.381966 ≈ 0.382 (the complementary golden ratio); φ⁻¹ = 0.6180339; φ⁻² = 1 − φ⁻¹; these are the two fundamental ratios of the golden section. If a quantum transmission coefficient scales as φ⁻², it directly ties barrier penetration to the golden ratio's self-similar structure — a geometric harmony between exponential decay (tunneling) and logarithmic spiral growth. | DEPTH: 6 — The claim is intriguing but the evidence is thin: the YouTube sources are non-peer-revie Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
FINDING: Fibonacci anyons have quantum dimension equal to the golden ratio φ, enabling universal topological quantum computation via non-Abelian braiding. | MATH: Quantum dimension d=ϕ=1+52≈1.618 d = \phi = \frac{1+\sqrt{5}}{2} \approx 1.618 d=ϕ=21+5​​≈1.618; Fibonacci anyon fusion rules: τ×τ=1+τ \tau \times \tau = 1 + \tau τ×τ=1+τ; modular tensor category (MTC) classification yields Fibonacci category as rank-2 MTC with S S S-matrix S=1ϕ+2(1ϕϕ−1) S = \frac{1}{\sqrt{\phi+2}} \begin{pmatrix} 1 & \phi \\ \phi & -1 \end{pmatrix} S=ϕ+2​1​(1ϕ​ϕ−1​); topological twist θτ=e4πi/5 \theta_\tau = e^{4\pi i/5} θτ​=e4πi/5. | CONNECTION: Golden ratio φ appears as quantum dimension; related ratios: ϕ−1=0.618 \phi^{-1} = 0.618 ϕ−1=0.618, ϕ2=2.618 \phi^2 = 2.618 ϕ2=2.618; fusion rules mirror Fibonacci recurrence; Binet's formula Fn=(ϕn−(−ϕ)−n)/5 F_n = (\phi^n - (-\phi)^{-n})/\sqrt{5} Fn​=(ϕn−(−ϕ)−n)/5​ links anyon counting to Fibonacci numbers; no direct base-60 or crystallographic symmetry, but φ appears in Penrose tilings (5-fold symmetry) and icosahedral quasicrystals. | DEPTH: 9 — Direct embedding of golden ratio into quantum Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com