# Abstract

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

## 1 Introduction

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. [13] called the conformable derivative. This definition is suggested as a natural extension of the usual derivative in the following senses. Given a function
*f* with order

*f*of order

The quantization of fractional system is of prime importance in physics. Rabei et al. [27] 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. [8]. However, the quantization of fractional singular Lagrangian systems using WKB approximation is studied by Rabei and Horani [28].

Recently, the deformation of the ordinary quantum mechanics based on the idea of fractional calculus is considered by Chung et al. [29]. They adopted the conformable fractional calculus, which depends on the basic limit definition of the derivative. The authors proposed the

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

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

## 2 The conformable quantum mechanics

Recently, Chung et al. [29] 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. [29], the fractional Schrodinger equation takes the form

*A*for a system in the state

## 3 Quantization of conformable harmonic oscillator

The Hamiltonian for the conformable harmonic oscillator is given as ref. [29]:

### 3.1
α
-creation operator
a
ˆ
α
†
and
α
-annihilation operator
a
ˆ
α

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

### 3.2 The eigenfunction and eigenvalue

First of all let us operate by the fractional Hamiltonian on

To calculate wave function for ground state

*n*-excited state, one finds that the energy eigenvalues are

*n*-excited state (

*n*th-excited state is then

### 3.3 Eigenfunctions in terms of Hermite polynomial

The eigenfunctions for the excited state equation (35) can be expressed in terms of the Hermite polynomials as follows. First, we rewrite equation (16) as

*Y*, we have

*n*-excited state in terms of Hermite polynomials is then given as

### 3.4 Eigenfunctions in terms of conformable Hermite polynomial

The eigenfunctions for the excited state equation (35) can be expressed in terms of the Hermite polynomials as follows. First, we rewrite equation (16)

*Y*, we have

*n*-times.

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

# Acknowledgment

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.

## Appendix A

## Proof

Using number operator

## Appendix B

## Proof

for

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**Received:**2021-03-10

**Revised:**2021-04-26

**Accepted:**2021-05-04

**Published Online:**2021-07-20

© 2021 Mohamed Al-Masaeed *et al*., published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.