A numerical method for fractional Schrödinger equation of Lennard-Jones potential

Al‐Raeei, Marwan; El-Daher, Moustafa Sayem

doi:10.1016/j.physleta.2019.07.019

Cited by 24 publications

(10 citation statements)

References 39 publications

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“…[46], [48], [49]. As can be shown in Tables (10)(11)(12)(13)(14)(15), the ro-vibrational energy spectra of all selected DMs rise as the vibrational and rotational quantum numbers increase. Importantly, one can see that our estimates are perfectly consistent with prior works that used other techniques.…”

Section: Resultsmentioning

confidence: 89%

The Generalized Fractional NU Method for the Diatomic Molecules in the Deng-Fan Model

Abu-Shady¹,

Abdel-Karim²,

Khokha³

2022

Preprint

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A solution of the fractional N -dimensional radial Schrödinger equation (SE) with the Deng-Fan potential (DFP) is investigated by the generalized fractional NU method. The analytical formulas of energy eigenvalues and corresponding eigenfunctions for the DFP are generated. Furthermore, the current results are applied to several diatomic molecules (DMs) for the DFP as well as the shifted Deng-Fan potential (SDFP). For both the DFP and its shifted potential, the effect of the fractional parameter (δ) on the energy levels of various DMs is examined numerically and graphically. We found that the energy eigenvalues are gradually improved when the fractional parameter increases. The energy spectra of various DMs are also evaluated in three-dimensional space and higher dimensions. It's worthy to note that the energy spectrum raises as the number of dimensions increases. In addition, the dependence of the energy spectra of the DFP and its shifted potential on the reduced mass, screening parameter, equilibrium bond length, rotational and vibrational quantum numbers is illustrated. To validate our findings, we estimate the energy levels of the DFP and SDFP at the classical case (δ = 1) for various DMs and found that they are entirely compatible with earlier studies.

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Section: Resultsmentioning

confidence: 89%

The Generalized Fractional NU Method for the Diatomic Molecules in the Deng-Fan Model

Abu-Shady¹,

Abdel-Karim²,

Khokha³

2022

Preprint

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“…Al-Raeei and El-Daher relied on the definition of Riemann–Liouville fractional derivative with a numerical technique to solve the space-dependent fractional SE for the Coulomb potential [ 10 ], Van Der Walls potential [ 11 ], Lennard-Jones potential [ 12 ] and Morse potential [ 13 ].…”

Section: Introductionmentioning

confidence: 99%

“…In addition, they employed the power series approach to investigate the solution of the fractional Klein-Gordon (KG) equation with fractional scalar and vector and potentials [9]. a e-mail: dr.abushady@gmail.com b e-mail: emad.mohammed@eng.modern-academy.edu.eg (corresponding author) c e-mail: tabdelkarim63@yahoo.com Al-Raeei and El-Daher relied on the definition of Riemann-Liouville fractional derivative with a numerical technique to solve the space-dependent fractional SE for the Coulomb potential [10], Van Der Walls potential [11], Lennard-Jones potential [12] and Morse potential [13].…”

Section: Introductionmentioning

confidence: 99%

The generalized fractional NU method for the diatomic molecules in the Deng–Fan model

2022

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A solution of the fractional N-dimensional radial Schrödinger equation (SE) with the Deng–Fan potential (DFP) is investigated by the generalized fractional Nikiforov–Uvarov (NU) method. The analytical formulas of energy eigenvalues and corresponding eigenfunctions for the DFP are generated. Furthermore, the current results are applied to several diatomic molecules (DMs) for the DFP as well as the shifted Deng–Fan potential (SDFP). For both the DFP and its shifted potential, the effect of the fractional parameter ($${\delta }$$ δ ) on the energy levels of various DMs is examined numerically and graphically. We found that the energy eigenvalues are gradually improved when the fractional parameter increases. The energy spectra of various DMs are also evaluated in three-dimensional space and higher dimensions. It is worthy to note that the energy spectrum raises as the number of dimensions increases. In addition, the dependence of the energy spectra of the DFP and its shifted potential on the reduced mass, screening parameter, equilibrium bond length and rotational and vibrational quantum numbers is illustrated. To validate our findings, the energy levels of the DFP and SDFP are estimated at the classical case ($${\delta =1}$$ δ = 1 ) for various DMs and found that they are entirely compatible with earlier studies. Graphical abstract In this study, a new algorithm of the generalized fractional Nikiforov–Uvarov method is employed to obtain new solutions to the fractional N-dimensional radial Schrödinger equation with the Deng–Fan potential. In addition, the results are applied to several diatomic molecules. The impact of the fractional parameter on the energy levels of various diatomic molecules is investigated. We found that the energy of the diatomic molecule is more bounded at lower fractional parameter values than in the classical case.

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“…The linear fractional Schrödinger equation or the fractional Schrödinger equation is a type of the linear Schrödinger equations used in the fractional quantum mechanics and this equation use one of the physical potentials to give the probability. The fractional Schrödinger equation in the general space representation form with a space fractional parameter α is given in the linear formalism as follows [ 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 ]:

Where h is Planck constant, Ψ( r ,t) is the wave function of the system in space-representation, j is the imaginary unit and

is Hamiltonian operator in the fractional type which is defined by the following formula:

K α is a coefficient, U ( r ) is the interaction potential of the system and

is the space fractional operator which is given by:

…”

Section: Introductionmentioning

confidence: 99%

“…The linear fractional Schr€ odinger equation or the fractional Schr€ odinger equation is a type of the linear Schr€ odinger equations used in the fractional quantum mechanics and this equation use one of the physical potentials to give the probability. The fractional Schr€ odinger equation in the general space representation form with a space fractional parameter α is given in the linear formalism as follows [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19]:…”

Section: Introductionmentioning

confidence: 99%

Numerical simulation of the space dependent fractional Schrödinger equation for London dispersion potential type

Al‐Raeei

El-Daher

2020

Heliyon

Self Cite

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In this study, we apply the definition of one of the fractional derivatives definitions of increasing values of the variable, which is the fractional derivative of Riemann-Liouville, and the numerical-integral methods to find numerical solutions of the fractional Schrödinger equation with the time-independent form for Van Der Walls potential type. We use the dimensionless formalism of the fractional Schrödinger equation in the space-dependent form in case of London dispersion potential in the stationary state. The solutions are found for multiple values of the space-dependent fractional Schrödinger equation parameter with a certain value of the energy. We find that the numerical solutions are physically acceptable for some values of the space dependent fractional parameter of the fractional Schrödinger equation but are not physically acceptable for others for a specific case. The numerical solutions can be applied for the systems that obey London dispersion potential type, which is resulted from the polarization of the instantaneous multi-poles of two moieties, such as soft materials systems and fluids of the inert gases.

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A numerical method for fractional Schrödinger equation of Lennard-Jones potential

Cited by 24 publications

References 39 publications

The Generalized Fractional NU Method for the Diatomic Molecules in the Deng-Fan Model

The Generalized Fractional NU Method for the Diatomic Molecules in the Deng-Fan Model

The generalized fractional NU method for the diatomic molecules in the Deng–Fan model

Numerical simulation of the space dependent fractional Schrödinger equation for London dispersion potential type

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