Erratum: “Bound states for multiple Dirac-δ wells in space-fractional quantum mechanics” [J. Math. Phys. 55, 012106 (2014)]

AbstractThis paper investigates whether the framework of fractional quantum mechanics can broaden our perspective of black hole thermodynamics. Concretely, we employ a space-fractional derivative (Riesz in Acta Math 81:1, 1949) as our main tool. Moreover, we restrict our analysis to the case of a Schwarzschild configuration. From a subsequently modified Wheeler–DeWitt equation, we retrieve the corresponding expressions for specific observables. Namely, the black hole mass spectrum, M, its temperature T, and entropy, S. We find that these bear consequential alterations conveyed through a fractional parameter, $$\alpha $$ α . In particular, the standard results are recovered in the specific limit $$\alpha =2$$ α = 2 . Furthermore, we elaborate how generalizations of the entropy-area relation suggested by Tsallis and Cirto (Eur Phys J C 73:2487, 2013) and Barrow (Phys Lett B 808:135643, 2020) acquire a complementary interpretation in terms of a fractional point of view. A thorough discussion of our results is presented.

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Broadening quantum cosmology with a fractional whirl

Modern Physics Letters A ◽

10.1142/s0217732321400058 ◽

2021 ◽

pp. 2140005

Author(s):

S. M. M. Rasouli ◽

S. Jalalzadeh ◽

P. V. Moniz

Keyword(s):

Quantum Mechanics ◽

Fractional Calculus ◽

Quantum Cosmology ◽

Cosmological Solution ◽

Isotropic Universe ◽

Fractional Quantum ◽

Stiff Matter ◽

Standard Quantum

We start by presenting a brief summary of fractional quantum mechanics, as means to convey a motivation towards fractional quantum cosmology. Subsequently, such application is made concrete with the assistance of a case study. Specifically, we investigate and then discuss a model of stiff matter in a spatially flat homogeneous and isotropic universe. A new quantum cosmological solution, where fractional calculus implications are explicit, is presented and then contrasted with the corresponding standard quantum cosmology setting.

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WKB FORMALISM AND A LOWER LIMIT FOR THE ENERGY EIGENSTATES OF BOUND STATES FOR SOME POTENTIALS

Modern Physics Letters A ◽

10.1142/s0217732307022979 ◽

2007 ◽

Vol 22 (35) ◽

pp. 2675-2687 ◽

Cited By ~ 1

Author(s):

LUIS F. BARRAGÁN-GIL ◽

ABEL CAMACHO

Keyword(s):

Quantum Mechanics ◽

Gravitational Field ◽

Lower Bound ◽

Harmonic Oscillator ◽

Approximation Method ◽

Bound States ◽

The Other ◽

Homogeneous Gravitational Field ◽

Definition Of ◽

Ground Energy

In this work the conditions appearing in the so-called WKB approximation formalism of quantum mechanics are analyzed. It is shown that, in general, a careful definition of an approximation method requires the introduction of two length parameters, one of them always considered in the textbooks on quantum mechanics, whereas the other is usually neglected. Afterwards we define a particular family of potentials and prove, resorting to the aforementioned length parameters, that we may find an energy which is a lower bound to the ground energy of the system. The idea is applied to the case of a harmonic oscillator and also to a particle freely falling in a homogeneous gravitational field, and in both cases the consistency of our method is corroborated. This approach, together with the so-called Rayleigh–Ritz formalism, allows us to define an energy interval in which the ground energy of any potential, belonging to our family, must lie.

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Fractional quantum mechanics of open quantum systems

Applications in Physics, Part B ◽

10.1515/9783110571721-011 ◽

2019 ◽

pp. 257-278 ◽

Cited By ~ 2

Author(s):

Vasily E. Tarasov

Keyword(s):

Quantum Mechanics ◽

Open Quantum Systems ◽

Quantum Systems ◽

Fractional Quantum ◽

Open Quantum

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A review of de Broglie particle–wave mechanical systems

Mathematics and Mechanics of Solids ◽

10.1177/1081286520917201 ◽

2020 ◽

Vol 25 (10) ◽

pp. 1763-1777

Author(s):

James M Hill

Keyword(s):

Quantum Mechanics ◽

Dark Energy ◽

Special Relativity ◽

Wave Energy ◽

Particle Energy ◽

De Broglie Wave ◽

De Broglie Waves ◽

Energy Formula ◽

Work Done ◽

Math Phys

The existence of the so-called ‘dark’ issues of mechanics implies that our present accounting for mass and energy is incorrect in terms of applicability on a cosmological scale, and the question arises as to where the difficulty might lie. The phenomenon of quantum entanglement indicates that systems of particles exist that individually display certain characteristics, while collectively the same characteristic is absent simply because it has cancelled out between individual particles. It may therefore be necessary to develop theoretical frameworks in which long-held conservation beliefs do not necessarily always apply. The present paper summarises the formulation described in earlier papers (Hill, JM. On the formal origin of dark energy. Z Angew Math Phys 2018; 69:133-145; Hill, JM. Some further comments on special relativity and dark energy. Z Angew Math Phys 2019; 70: 5–14; Hill, JM. Special relativity, de Broglie waves, dark energy and quantum mechanics. Z Angew Math Phys 2019; 70: 131–153.), which provides a framework that allows exceptions to the law that matter cannot be created or destroyed. In these papers, it is proposed that dark energy arises from conventional mechanical theory, neglecting the work done in the direction of time and consequently neglecting the de Broglie wave energy [Formula: see text]. These papers develop expressions for the de Broglie wave energy [Formula: see text] by making a distinction between particle energy [Formula: see text] and the total work done by the particle [Formula: see text], that which accumulates from both a spatial physical force [Formula: see text] and a force [Formula: see text] in the direction of time. In any experiment, either particles or de Broglie waves are reported, so that only one of [Formula: see text] or [Formula: see text] is physically measured, and particles appear for [Formula: see text] and de Broglie waves occur for [Formula: see text], but in either event both a measurable and an immeasurable energy exists. Conventional quantum mechanics operates under circumstances such that [Formula: see text] vanishes and [Formula: see text] becomes purely imaginary. If both [Formula: see text] and [Formula: see text] are generated as the gradient of a potential, the total particle energy is necessarily conserved in the conventional manner.

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New scattering features in non-Hermitian space fractional quantum mechanics

Annals of Physics ◽

10.1016/j.aop.2018.07.008 ◽

2018 ◽

Vol 396 ◽

pp. 371-385 ◽

Cited By ~ 1

Author(s):

Mohammad Hasan ◽

Bhabani Prasad Mandal

Keyword(s):

Quantum Mechanics ◽

Fractional Quantum ◽

Hermitian Space

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Adaptation of Conformable Residual Power Series Scheme in Solving Nonlinear Fractional Quantum Mechanics Problems

Applied Sciences ◽

10.3390/app10030890 ◽

2020 ◽

Vol 10 (3) ◽

pp. 890 ◽

Cited By ~ 6

Author(s):

Mohammed Shqair ◽

Mohammed Al-Smadi ◽

Shaher Momani ◽

Essam El-Zahar

Keyword(s):

Quantum Mechanics ◽

Power Series ◽

Quantum Physics ◽

Taylor Series Expansion ◽

Fractional Quantum ◽

Analytical Technique ◽

Different Types ◽

Residual Power Series ◽

Residual Errors ◽

Residual Power

In this paper, the general state of quantum mechanics equations that can be typically expressed by nonlinear fractional Schrödinger models will be solved based on an attractive efficient analytical technique, namely the conformable residual power series (CRPS). The fractional derivative is considered in a conformable sense. The desired analytical solution is obtained using conformable Taylor series expansion through substituting a truncated conformable fractional series and minimizing its residual errors to extract a supportive approximate solution in a rapidly convergent fractional series. This adaptation can be implemented as a novel alternative technique to deal with many nonlinear issues occurring in quantum physics. The effectiveness and feasibility of the CRPS procedures are illustrated by verifying three realistic applications. The obtained numerical results and graphical consequences indicate that the suggested method is a convenient and remarkably powerful tool in solving different types of fractional partial differential models.

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