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 α \alpha -creation and α \alpha -annihilation operators. It is found that these operators have the following influence on the energy states. For a given order α \alpha , the α \alpha -creation operator promotes the state while the α \alpha -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 α = 1 \alpha =1 .

In this work, the variational method is developed within the framework of conformable quantum mechanics where conformable derivative is implemented. Three related theorems are presented and proved. We furnished the work with three illustrative examples. It is observed that the conformable ground state energy in each example tends to the ground state energy of the corresponding traditional Hamiltonian when α=1.

In this paper, the perturbation theory is extended to be applicable for systems containing conformable derivative of fractional order [Formula: see text]. This is needed as an essential and powerful approximation method for describing systems with conformable differential equations that are difficult to solve analytically. The work here is derived and discussed for the conformable Hamiltonian systems that appears in the conformable quantum mechanics. The required [Formula: see text]-corrections for the energy eigenvalues and eigenfunctions are derived. To demonstrate this extension, three illustrative examples are given, and the standard values obtained by the traditional theory are recovered when [Formula: see text].

In this paper, the conformable Laguerre and associated Laguerre differential equations are solved using the Laplace transform. The solution is found to be in exact agreement with that obtained using the power series method. In addition some properties and some recursion relations of the Laguerre and associated Laguerre functions are discussed and proved. Then, the conformable Rodriguez’s formula and generating function are propose

In this paper, the deformation of special relativity within the frame of conformable derivative is formulated. Within this context, the two postulates of the theory are re-stated. Then, the addition of velocity laws are derived and used to verify the constancy of the speed of light. The invariance principle of the laws of physics is demonstrated for some typical illustrative examples, namely, the conformable wave equation, the conformable Schrodinger equation, the conformable Klein-Gordon equation, and conformable Dirac equation. The current formalism may be applicable when using special relativity in a nonlinear or dispersive medium.