Quantization of fractional harmonic oscillator using creation and annihilation operators

In this article, the Hamiltonian for the conformable harmonic oscillator used in the previous study [Chung WS, Zare S, Hassanabadi H, Maghsoodi E. The effect of fractional calculus on the formation of quantum-mechanical operators. Math Method Appl Sci. 2020;43(11):6950?67.] is written in terms of fractional operators that we called ? -creation and ? -annihilation operators. It is found that these operators have the following influence on the energy states. For a given order ? , the ? -creation operator promotes the state while the ? -annihilation operator demotes the state. The system is then quantized using these creation and annihilation operators and the energy eigenvalues and eigenfunctions are obtained. The eigenfunctions are expressed in terms of the conformable Hermite functions. The results for the traditional quantum harmonic oscillator are found to be recovered by setting ? = 1 .