Coherent States of the Conformable Quantum Oscillator
Abstract
1. Introduction
2. Basics of Conformable Quantum Mechanics in Dimensions
Fundamentals of Conformable Quantum Mechanics
- The states of a conformable quantum system are described by complex functions . At any , belongs to the Hilbert space of the quadratic integrable functions on endowed with the inner product with respect to the integration measure given by the following inner product:
- The time evolution of the conformable quantum system is described by the conformable time-dependent Schrödinger equation of the formHere, the conformable Hamiltonian operator for a particle of conformable mass parameter in the stationary potential is defined by the following relation:The conformable derivates and are reviewed in the Appendix A. We consider in this paper the case , which is the original derivative proposed in [9] for a single variable and which was generalized in [13,14] for multi-variables; see Equations (A1) and (A2).
- The observables of the conformable system are given by Hermitian operators that act on the Hilbert space , which are constructed from the physical quantities of the systemThe eigenvalues of the operator correspond to the measured values of the observable in the eigenstates . Note that the conformable systems represent just mathematical physics models that generalize the known quantum systems, with no physical system known to obey the conformable quantum mechanics up to now. Therefore, the concept of measurability should be understood in this abstract generalization sense.
3. Conformable Quantum Oscillator (CQO)
4. Conformable Coherent States (CCSs)
4.1. Conformable Uncertainty
4.2. Conformable Coherent States
4.3. Energy Expectation Value in CCSs
4.4. Time Evolution and Spatial Translation of CCS
4.5. Commentaries on Mathematical and Physical Properties of CCSs
- Coordinate: since .
- Measure: since .
- Operators: and .
- Wavefunction:
- Energy: .
4.6. CCSs as Eigenvectors of Annihilation Operator
5. Discussion
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
Appendix A. Basic Conformable Calculus Relations
Appendix B. Proof of the Cauchy–Buniakowsky–Schwarz Inequality
Appendix C. Closure of the Space of Functions
Appendix D. Proof of the Variance
Appendix E. Derivation of the Conformable Uncertainty Principle
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Property | Standard () | Conformable () |
---|---|---|
Coordinate | x | |
, | ||
Measure | ||
Annihilation eigenstate | ||
Uncertainty product | ||
Zero-point energy | ||
Normalization |
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Godinho, C.F.d.L.; Porto, C.M.; Rodriguez, M.C.; Vancea, I.V. Coherent States of the Conformable Quantum Oscillator. Dynamics 2025, 5, 26. https://doi.org/10.3390/dynamics5030026
Godinho CFdL, Porto CM, Rodriguez MC, Vancea IV. Coherent States of the Conformable Quantum Oscillator. Dynamics. 2025; 5(3):26. https://doi.org/10.3390/dynamics5030026
Chicago/Turabian StyleGodinho, Cresus Fonseca de Lima, Claudio Maia Porto, Marcos Cardoso Rodriguez, and Ion Vasile Vancea. 2025. "Coherent States of the Conformable Quantum Oscillator" Dynamics 5, no. 3: 26. https://doi.org/10.3390/dynamics5030026
APA StyleGodinho, C. F. d. L., Porto, C. M., Rodriguez, M. C., & Vancea, I. V. (2025). Coherent States of the Conformable Quantum Oscillator. Dynamics, 5(3), 26. https://doi.org/10.3390/dynamics5030026