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![Plot of λ versus x = m E with β = 0.1 for the ordinary case (brown), Refs. [20,21] (purple) and GUP* (pink)](publication/331618579/figure/fig2/AS:735081496854528@1552268494216/Plot-of-l-versus-x-m-E-with-b-01-for-the-ordinary-case-brown-Refs-20-21.png)
Plot of λ versus x = m E with β = 0.1 for the ordinary case (brown), Refs. [20,21] (purple) and GUP* (pink)
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![Fig. 1 Plot of y = =ψ λ ( p)|ψ λ ( p) versus x = λ−λ ¯ h for Ref. [14]...](publication/331618579/figure/fig1/AS:735081496842240@1552268494187/Plot-of-y-ps-l-pps-l-p-versus-x-l-l-h-for-Ref-14-brown-Refs-20-21_Q320.jpg)
In this paper we present a new higher order generalized (gravitational) uncertainty principle (GUP*) which has the maximal momentum as well as the minimal length. We discuss the position representation and momentum representation. We also discuss the position eigenfunction and maximal localization states. As examples we discuss one dimensional box...
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... relation shows that similar to the ordinary quantum mechanics, the quasiposition wave function of a momentum eigenstate ψ ¯ p ( p) = δ( p − ¯ p) with energy E = ¯ p 2 /2m is still a plane wave but with a modified dispersion relation where λ or d (E) is the ordinary wavelength which is obtained from the limit β → 0. Figure 2 shows the plot of λ versus x = m E with β = 0.1 for the ordinary case (bown), Refs. ...
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Citations
... The implications of these modified Heisenberg commutations relations over another simple one-dimensional system, specifically on the infinite square-well potential, can be found in [13][14][15][16][17][18][19][20][21]. ...
... For the step function potential (15), the time-independent Schrödinger in the x > 0 region is ...
... For the step-function potential (15), the Schrödinger equation for x > 0 and β > 0 is ...
This paper explores the implications of modifying the canonical Heisenberg commutation relations over two simple systems, such as the free particle and the tunnel effect generated by a step-like potential. The modified commutation relations include position-dependent and momentum-dependent terms analyzed separately. For the position deformation case, the corresponding free wave functions are sinusoidal functions with a variable wave vector k(x). In the momentum deformation case, the wave function has the usual sinusoidal behavior, but the energy spectrum becomes non-symmetric in terms of momentum. Tunneling probabilities depend on the deformation strength for both cases. Also, surprisingly, the quantum mechanical model generated by these modified commutation relations is related to the Black–Scholes model in finance. In fact, by taking a particular form of a linear position deformation, one can derive a Black–Scholes equation for the wave function when an external electromagnetic potential is acting on the particle. In this way, the Scholes model can be interpreted as a quantum-deformed model. Furthermore, by identifying the position coordinate x in quantum mechanics with the underlying asset S, which in finance satisfies stochastic dynamics, this analogy implies that the Black–Scholes equation becomes a quantum mechanical system defined over a random spatial geometry. If the spatial coordinate oscillates randomly about its mean value, the quantum particle’s mass would correspond to the inverse of the variance of this stochastic coordinate. Further, because this random geometry is nothing more than gravity at the microscopic level, the Black–Scholes equation becomes a possible simple model for understanding quantum gravity.
... Its possible implications for entanglement and the Hamilton-Jacobi equation are analysed in [8][9][10][11][12]. In particular, the analysis of the implications of the modified Heisenberg commutation relations of one-dimensional systems, and specifically the infinite square-well potential, can be found in [13][14][15][16][17][18][19][20]. ...
... The studies developed in references [13][14][15][16][17][18][19][20] ...
We explore some consequences of modifying the usual Heisenberg commutation relations of two simple systems: first, the one-dimensional quantum system given by the infinite square-well potential, and second, the case of a gas of N non-interacting particles in a box of volume V, which permit obtaining analytical solutions. We analyse two possible cases of modified Heisenberg commutation relations: one with a linear and non-linear dependence on the position and another with a linear and quadratic dependence on the momentum. We determine the eigenfunctions, probability densities, and energy eigenvalues for the one-dimensional square well for both deformation cases. For linear and non-linear x deformation dependence, the wave functions and energy levels change substantially when the weight factor associated with the modification term increases. Here, the energy levels are rescaled homogeneously. Instead, for linear and quadratic momentum p deformation dependence, the changes in the energy spectrum depend on the energy level. However, the probability densities are the same as those without any modification. For the non-interacting gas, the position deformation implies that the ideal gas state equation is modified, acquiring the form of a virial expansion in the volume, whereas the internal energy is unchanged. Instead, the ideal gas state equation remains unchanged at the lowest order in β for the momentum modification case. However, the temperature modifies the internal energy at the lowest order in β. Thus, this study indicates that gravity could generate forces on particles by modifying the Heisenberg commutation relations. Therefore, gravitation could be the cause of the other three forces of nature.
... We see that in such 1 dimensional case a higher order terms would appear, see e.g. [61] for similar feature arising in the case of GUP. Expanding the RHS of (57) in deformation parameters we can see that the results of the previous section (30) can be applied and the invariant phase space volume element would amount to: ...
Modifications in quantum mechanical phase space lead to changes in the Heisenberg uncertainty principle, these can result in the Generalized Uncertainty Principle (GUP) or the Extended Uncertainty Principle (EUP), which introduce quantum gravitational effects at small and large distances, respectively. A combination of GUP and EUP, the Extended Generalized Uncertainty Principle (EGUP or GEUP), further generalizes these modifications by incorporating noncommutativity in both coordinates and momenta. This paper examines the impact of EGUP on the Liouville theorem in statistical physics and density of states within non-relativistic quantum mechanics framework. We find a weighted invariant phase space volume element in the cases of Snyder-de Sitter and Yang models, presenting how EGUP alters the density of states, potentially affecting physical (thermodynamical) properties.
... With the emergence of deformed theories based on generalized Heisenberg commutation relations [1][2][3][4][5][6], new developments were made in the evaluation of the related propagator using the path integral formalism and which required some subtle techniques. The latter focuses on the natural construction in relation to the notion of trajectory and its discretization by contrast to the simplicity associated with the wave function obeying the Schrödinger equation describing the same dynamics. ...
... Therefore we get the following standard result where there is two types of propagation, one with positive energy propagation and the other with negative energy propagation where¯s Y = Y + 3 and¯s F = F + . 3 The corresponding wave functions are respectively ...
In this paper, we extend the path integral formalism for the Dirac oscillator in (1+1) dimension by replacing the spatial derivative with the Dunkl derivative. Utilizing representations in position space-time coordinates, we precisely calculate the propagator, expressed in terms of generalized Hermite polynomials. The energy eigenvalues of the electron, along with their corresponding wave functions, are determined. In special cases, we can precisely evaluate the non-relativistic energy eigenvalues and wave functions, even in the absence of Dunkl parameters.
... The generalized uncertainty principles has been discussed widely in the literature [15][16][17][18][19][20][21][22]. Quantum gravity theories lead to a minimal length scale which can result from a modification of the Heisenberg uncertainty principle ∆ ∆ ≥ ℏ 2 to a generalized uncertainty principle (GUP) [15,[23][24][25][26] where ( , ∆ ) is some function of the the GUP parameter and the uncertainty in momentum ∆ . In [4], we employ a first order approximation of (31) ∆ ∆ ≥ . ...
... Similar to the GUP-corrected Carnot cycle let us look at ∆ . From (25) and (26), ...
We study the effects of the generalized uncertainty principle (GUP) on the efficiency of quantum heat engines based on a particle in an infinite square well using the partition function approach. In particular, we study the Carnot and Otto heat engines. For the system we used, the GUP-corrected efficiencies turned out to be lower than efficiencies without the GUP effects. However, as expected, GUP effects increase as the temperature of the cold heat bath decreases and as the width of the potential well decreases.
... As shown in literature, there is a great interest in the study of the effects the GUP and its uses [37][38][39][40][41][42][43][44]. It is also important to point out that there are various deformed Heisenberg algebra scenarios up to the higher-orders as GUP * , namely gravitational generalized uncertainty principle [45]. Here are some references for an overview [46][47][48]. ...
In this article, we discuss the classic limit and investigate the Ehrenfest’s theorem of the Dirac equation in the context of minimal uncertainty in momentum within a noncommutative setting, and examine its CPT symmetry and Lorentz symmetry violation. Also, we study the non-relativistic limit of this Dirac system, which leads to obtain a deformed Schrödinger–Pauli equation. Besides we check if this obtained equation still show explicitly the gyromagnetic factor of the electron. Interestingly, the overlap and congruence aspects of the classical and non-relativistic limits of the Dirac equation are clarified. The effects of both minimal uncertainty in momentum and noncommutativity on the Ehrenfest’s theorem and non-relativistic limit are well examined. Knowing that with both the linear Bopp–Shift and ⋆product, the noncommutativity is inserted.
... [10][11][12][13][14][15][16][17][18]. In particular, considering that the effect of generalized uncertainty principle (GUP hereafter, which is a kind of quantum gravity model [19][20][21][22][23][24][25]) can modify the properties of the universe and change its thermodynamic state, Das et al. [26] used the KMM model model and ADV model of GUP to investigate BAU, their results demonstrated that the universe in the GUP framework at non-thermal equilibrium, leading to a non-zero factor η for measuring the the number of baryonic matter exceeding the antibaryonic matter. Subsequently, Ref. [27] have extended this approach to the case of higher-order GUP and showed that the higher order quantum gravity (QG) correction terms have a significant effect on the baryon asymmetry. ...
In this manuscript, we explore the baryon asymmetry of the universe by employing a novel higher-order generalized uncertainty principle (GUP) that maintains a minimum length for both positive and negative GUP parameters. Our results demonstrate that the influence of the GUP noticeably modifies the Friedmann equations, leading to a transformation in the characteristics of the pressure and density of the Universe, and subsequently disrupting its thermal equilibrium. Additionally, by amalgamating the adapted Friedmann equations with the conventional theory of gravitational baryogenesis, one can derive a non-zero factor of baryon asymmetry , indicating that the quantity of matter in the universe surpasses that of antimatter. Finally, we also utilized astronomical observations to constrain the bounds for both the positive and negative GUP parameters.
... 19 The constant coefficient β is known as the quadratic GUP parameter, whose exact value is debated by several previous studies. [20][21][22][23][24][25] Another approach, consistent with string theory, black hole physics, and DSR is called the linear GUP, with modified uncertainty principle ∆x∆p ∼ (1 − α∆p) and phase space volume (1−αp) −4 d 3 xd 3 p. 13 Numerous other approaches exist in the literature, such as the linear-quadratic GUP, 26 various higher-order GUP approaches, [27][28][29][30][31][32][33][34][35][36] and extended GUP for nonzero cosmological constant. 37 For simplicity, this paper focuses on the quadratic GUP alone. ...
In quantum gravity phenomenology, the effect of the generalized uncertainty principle (GUP) on white dwarf structure has been given much attention in recent literature. However, these studies assume a zero temperature equation of state (EoS), excluding young white dwarfs whose initial temperatures are substantially high. To that cause, this paper calculates the Chandrasekhar EoS and resulting mass-radius relations of finite temperature white dwarfs modified by the quadratic GUP, an approach that extends Heisenberg’s uncertainty principle by a quadratic term in momenta. The EoS was first approximated by treating the quadratic GUP parameter as perturbative, causing the EoS to exhibit expected thermal deviations at low pressures, and conflicting behaviors at high pressures, depending on the order of approximation. We then proceeded with a full numerical simulation of the modified EoS, and showed that in general, finite temperatures cause the EoS at low pressures to soften, while the quadratic GUP stiffens the EoS at high pressures. This modified EoS was then applied to the Tolman–Oppenheimer–Volkoff equations and its classical approximation to obtain the modified mass-radius relations for general relativistic and Newtonian white dwarfs. The relations for both cases were found to exhibit the expected thermal deviations at small masses, where low-mass white dwarfs are shifted to the high-mass regime at large radii, while high-mass white dwarfs acquire larger masses, beyond the Chandrasekhar limit. Additionally, we find that for sufficiently large values of the GUP parameter and temperature, we obtain mass-radius relations that are completely removed from the ideal case, as high-mass deviations due to GUP and low-mass deviations due to temperature are no longer mutually exclusive.
... It is worth mentioning that in literature there are various deformed Heisenberg algebra scenarios up to the higher-orders. As examples, we mention Ref. [59], where Chung and Hassanabadi proposed a new higherorder GUP (GUP * ) formalism. They examined the algebraic structure of the GUP * formalism and showed that it is different from Pedram's approach. ...
In this paper, by using the canonical Foldy–Wouthuysen transformation for relativistic spin-1/2 particles, we investigate the non-relativistic limit and determine the basic properties of the Dirac equation in the presence of a minimal uncertainty in momentum within a noncommutative phase-space. This leads to obtaining a deformed Schrödinger– Pauli equation; there, we examine the effect of both the noncommutativity in phase-space and the minimal uncertainty in momentum on the non-relativistic limit of the Dirac equation. However, with both Bopp– Shift linear transformation, and the Moyal– Weyl product (⋆product), we introduce the noncommutativity in phase-space. We as well test the efficacy and behavior of the Foldy– Wouthuysen transformation in deriving the non-relativistic limit under certain conditions and influences.
... Papers [9][10][11][12] discussed measurements in which one may be able to probe the effects of quantum gravity. The implications of the deformed forms of the commutation relation have attracted large attention [13][14][15][16][17][18]. In particular, researchers analyzed the consequences for the harmonic oscillator [17,18], the free particle, and potentials with infinitely sharp boundaries [14]. ...
... where the variables x and x are both restricted to the range −∆x/2, +∆x/2 and η * (x) = x|η * . One uses Equation (16) under the assumption that the origins of both the x and κ axes are placed into centers of the two bins for which the optimality is reached. Surely, this holds for the ordinary commutation relation. ...
... Surely, this holds for the ordinary commutation relation. To analyze consequences of the generalized uncertainty principle, one needs also to examine Equation (16) in more detail than it was made in the paper [33]. Substituting x = ξ∆x, s = ∆x∆κ/(2π) and µ 2 = sλ, one rewrites Equation (16) as ...
The emergence of a minimal length at the Planck scale is consistent with modern developments in quantum gravity. This is taken into account by transforming the Heisenberg uncertainty principle into the generalized uncertainty principle. Here, the position-momentum commutator is modified accordingly. In this paper, majorization uncertainty relations within the generalized uncertainty principle are considered. Dealing with observables with continuous spectra, each of the axes of interest is divided into a set of non-intersecting bins. Such formulation is consistent with real experiments with a necessarily limited precision. On the other hand, the majorization approach is mainly indicative for high-resolution measurements with sufficiently small bins. Indeed, the effects of the uncertainty principle are brightly manifested just in this case. The current study aims to reveal how the generalized uncertainty principle affects the leading terms of the majorization bound for position and momentum measurements. Interrelations with entropic formulations of this principle are briefly discussed.
