Pisot \(q\)-coherent states quantization of the harmonic oscillator. (English) Zbl 1266.81111
Summary: We revisit the quantized version of the harmonic oscillator obtained through a \(q\)-dependent family of coherent states. For each \(q\), \(0<q<1\), these normalized states form an overcomplete set that resolves the unity with respect to an explicit measure. We restrict our study to the case in which \(q^{-1}\) is a quadratic unit Pisot number, since then the \(q\)-deformed integers form Fibonacci-like sequences of integers. We then examine the main characteristics of the corresponding quantum oscillator: localization in the configuration and in the phase spaces, angle operator, probability distributions and related statistical features, time evolution and semi-classical phase space trajectories.
MSC:
81R50 | Quantum groups and related algebraic methods applied to problems in quantum theory |
81R30 | Coherent states |
81S05 | Commutation relations and statistics as related to quantum mechanics (general) |
17B37 | Quantum groups (quantized enveloping algebras) and related deformations |