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 .
The fractional derivative extends the classical derivative by allowing the operators of differentiation to take fractional orders, and it has played an important role in physics, mathematics, and engineering sciences [1,2,3, 4,5,6,7, 8,9,10]. The definition of fractional derivative and fractional integral subject to several approaches such as Riemann–Liouville fractional, Caputo, Riesz, Riesz–Caputo, Weyl, Grünwald–Letnikov, Hadamard, and Chen fractional derivatives [1,2,3, 4,5,6, 7,8,9, 10,11,12]. Recently, a new definition of fractional derivative was presented by Khalil et al.  called the conformable derivative. This definition is suggested as a natural extension of the usual derivative in the following senses. Given a function , the conformable derivative of f with order is defined by ref. 
The quantization of fractional system is of prime importance in physics. Rabei et al.  discussed how to find the solution of the Schrodinger equation for some systems that have a fractional behavior in their Lagrangian and obey the WKB approximation. Besides, the canonical quantization of a system with Brownian motion is carried out using fractional calculus by Rabei et al. . However, the quantization of fractional singular Lagrangian systems using WKB approximation is studied by Rabei and Horani .
Recently, the deformation of the ordinary quantum mechanics based on the idea of fractional calculus is considered by Chung et al. . They adopted the conformable fractional calculus, which depends on the basic limit definition of the derivative. The authors proposed the -position operator and -momentum operator and they constructed the -Hamiltonian operator and fractional Schrodinger equation. Also by considering the fractional calculus, they have formulated the conformable quantum mechanics and they have discussed some physical examples such as the harmonic oscillator problem.
The harmonic oscillator problem is of great importance in quantum mechanics. The treatment of this problem using the algebraic method based on the creation and annihilation operators rather than solving the Schrodinger equation is well known in quantum mechanics. It plays a central role in modeling various physical phenomena as well as its importance in the canonical field quantization. It is then a natural step to extend the algebraic method within the frame of conformable quantum mechanics. The main purpose of this article is to treat the conformable harmonic oscillator using an algebraic method with newly defined operators that we call the -creation and -annihilation operators. Their names are justified by noting that the -Hamiltonian of the system is factored in terms of these operators and that they have the effect of promotion and demotion of the -states. It should be mentioned that this treatment is presumably needed to lay out the transition into any possible conformable quantum field theory.
This article is organized as follows. In Section 2, we present a brief review of the formulation of conformable quantum mechanics. In Section 3, we present and discuss the quantization of fractional harmonic oscillator using the -creation and -annihilation operators. In Section 4, we present our summary and conclusions of this work.
2 The conformable quantum mechanics
Recently, Chung et al.  proposed a formulation of the ordinary quantum mechanics in fractional form using the conformable derivative. Here, we present the main definitions and relations needed for our work. According to Chung et al. , the fractional Schrodinger equation takes the form
3 Quantization of conformable harmonic oscillator
The Hamiltonian for the conformable harmonic oscillator is given as ref. :
3.1 -creation operator and -annihilation operator
We will develop here a fractional algebraic method for solving equation (10). It involves the definition of two operators, namely, the fractional creation operator of order ( ) and fractional annihilation operator of order ( ). We start by rewriting the Hamiltonian equation (9) as
3.2 The eigenfunction and eigenvalue
First of all let us operate by the fractional Hamiltonian on , then we have
To calculate wave function for ground state , we stipulate that
3.3 Eigenfunctions in terms of Hermite polynomial
3.4 Eigenfunctions in terms of conformable Hermite polynomial
The eigenfunction for n-excited state in terms of conformable Hermite polynomials is then given as
4 Summary and conclusions
In this article, an algebraic method, using -creation operator and -annihilation operator , is established for the conformable harmonic oscillator. The Hamiltonian for the systems is written in terms of these operators. It is found that for a given order , the -creation operator has the effect of promoting the present state of the system while the -annihilation operator demotes the state. The system is quantized in terms of and 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 . The formulation of the harmonic oscillator using and may be useful in the formulation of conformable field quantization.
The authors would like to thank the anonymous reviewers for their careful reading of the manuscript and constructive feedback.
Conflict of interest: Authors state no conflict of interest.
Using number operator
for -times we get
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