The effect of fractional calculus on the formation of quantum‐mechanical operators

Chung, Won Sang; Zare, Soroush; Hassanabadi, H.; Maghsoodi, E.

doi:10.1002/mma.6445

Cited by 16 publications

(23 citation statements)

References 13 publications

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“…The conformable time‐independent Schrodinger equation is 12 :

H_{α} false({\overset{x}{true}}_{α}, {\overset{p}{true}}_{α} false) {normalΨ}^{α} false(x, t false) = ((), \frac{{\overset{p}{true}}_{α}^{2}}{2 m^{α}} + V_{α} false({\overset{x}{true}}_{α} false)) {normalΨ}^{α} false(x, t false) = E^{α} {normalΨ}^{α} false(x, t false),

and the energy eigenvalues are

E_{n}^{α} = (⟨⟩, H_{α}) = ⟨ {normalΨ}^{α} (||, H_{α}) {normalΨ}^{α} ⟩ .

…”

Section: Theory Of Conformable Variational Methodsmentioning

confidence: 99%

“…Thus we have

⟨ Ψ^{α} | δ Ψ^{α} ⟩ + ⟨ δ Ψ^{α} | Ψ^{α} ⟩ = 0 .

After substituting in equation (), we obtain

δ ⟨H_{α}⟩ = 0 .

In the next section, we furnish this work with three illustrative examples. We will need the following definition of the inner product in the conformable derivative sense 12

⟨ ψ_{α} | ϕ_{α} ⟩ = {true\int}_{- \infty}^{\infty} \frac{ψ_{α}^{*} ϕ_{α}}{x^{1 - α}} d x .

And, it is worth to mention that if a function f ( x ) has an extreme value at x = x 0 , then

D_{x}^{α} f (x_{0}) = x_{0}^{1 - α} \frac{d}{d x} f (x_{0}) = 0 .

…”

Section: Theory Of Conformable Variational Methodsmentioning

confidence: 99%

“…To read more about conformable derivative, its properties, and its various applications, we refer you to previous works 6–11 . For instance, in Chung et al, 12 the deformation of the ordinary quantum mechanics using the conformable calculus is studied. The α ‐Hamiltonian operator was proposed and conformable Schrodinger equation was constructed.…”

Section: Introductionmentioning

confidence: 99%

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Extension of the variational method to conformable quantum mechanics

Al-Masaeed

Rabei

Al-Jamel

2021

Math Methods in App Sciences

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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.

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“…The conformable time‐independent Schrodinger equation is 12 :

H_{α} false({\overset{x}{true}}_{α}, {\overset{p}{true}}_{α} false) {normalΨ}^{α} false(x, t false) = ((), \frac{{\overset{p}{true}}_{α}^{2}}{2 m^{α}} + V_{α} false({\overset{x}{true}}_{α} false)) {normalΨ}^{α} false(x, t false) = E^{α} {normalΨ}^{α} false(x, t false),

and the energy eigenvalues are

E_{n}^{α} = (⟨⟩, H_{α}) = ⟨ {normalΨ}^{α} (||, H_{α}) {normalΨ}^{α} ⟩ .

…”

Section: Theory Of Conformable Variational Methodsmentioning

confidence: 99%

“…Thus we have

⟨ Ψ^{α} | δ Ψ^{α} ⟩ + ⟨ δ Ψ^{α} | Ψ^{α} ⟩ = 0 .

After substituting in equation (), we obtain

δ ⟨H_{α}⟩ = 0 .

In the next section, we furnish this work with three illustrative examples. We will need the following definition of the inner product in the conformable derivative sense 12

⟨ ψ_{α} | ϕ_{α} ⟩ = {true\int}_{- \infty}^{\infty} \frac{ψ_{α}^{*} ϕ_{α}}{x^{1 - α}} d x .

And, it is worth to mention that if a function f ( x ) has an extreme value at x = x 0 , then

D_{x}^{α} f (x_{0}) = x_{0}^{1 - α} \frac{d}{d x} f (x_{0}) = 0 .

…”

Section: Theory Of Conformable Variational Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Extension of the variational method to conformable quantum mechanics

Al-Masaeed

Rabei

Al-Jamel

2021

Math Methods in App Sciences

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“…Definition The α− Lorentz transformations between two inertial frames S and S ′ are defined as [14]:…”

Section: Theorymentioning

confidence: 99%

“…In [13], a new conformable fractional mechanics using the fractional addition was proposed and new definitions for the fractional velocity fractional acceleration are given. In [14], deformation of quantum mechanics due to the inclusion of conformable fractional derivative is presented and investigated with some physical illustrative examples. In [15], the Hamiltonian for the conformable harmonic oscillator is constructed using fractional operators termed α-creation and α-annihilation operators.…”

Section: Introductionmentioning

confidence: 99%

The effect of deformation of special relativity by conformable derivative

Al-Jamel,

Al-Masaeed,

Rabei

et al. 2021

Preprint

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In the article, the deformation of special relativity within the frame of conformable derivative is formulated. Within this context, the two postulates of the theory were re-stated. And, the addition of velocity laws were 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, and the conformable Gordon-Klein equation. The current formalism may be applicable when using special relativity in a nonlinear or dispersive medium.

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