WKB Approximation with Conformable Operator

Abstract

In this paper, the WKB method is extended to be applicable for conformable Hamiltonian systems where the concept of conformable operator with fractional order α is used. The WKB approximation for the α-wavefunction is derived when the potential is slowly varying in space. The paper is furnished with some illustrative examples to demonstrate the method. The quantities of the conformable form are found to be in exact agreement with traditional quantities when α = 1.

Solution of the Conformable Angular Equation of the Schrodinger Equation

“…This definition is a natural extension of the usual derivative and satisfies the standard properties of the traditional derivative i.e the derivative of the product and the derivative of the quotient of two functions and satisfies the chain rule. The conformable calculus has many applications in several fields, for example in physics , it was used in quantum mechanics to study The effect of fractional calculus on the formation of quantum-mechanical operators [17], and an extension of the approximate methods used in quantum mechanics was made [18][19][20], and the of conformable harmonic oscillator is quantized using the annihilation and creation operators [21], besides, the effect of deformation of special relativity studied by conformable derivative [22], and the conformable Laguerre and associated Laguerre differential equations using conformable Laplace transform are solved [23]. In this work, the conformable Schrodinger equation is separated into two parts radial which depends on the knowing the potential and angular part which we solved and we obtained the conformable spherical harmonic.…”

WKB approximation under Killingbeck potential and its application to double heavy meson mass spectrum

“…In this case, numerical methods and approximate analytical techniques are required to obtain the solution of quantum system [1][2][3]. Wenzel-Kramers-Brillouin (WKB) approximation is a semiclassical approximation method, which can be used to analyze Schrödinger equation and successfully deals with some significant and vital problems [4][5][6][7],for example, the theory of electromagnetic waves [8]. The Bohr-Sommerfeld quantization rule, and the WKB approximation base on the Schrödinger equation, are both expected to give good results of energy eigenvalues in limit of large quantum numbers, in accordance with the Correspondence Principle [9].…”