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Partition function versus temperature for some values of j for a generalized harmonic oscillator with f (p) = 1 + βP 2j .
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In this paper, we study two generalized uncertainty principles (GUPs) including [X,P] = iℏ(1 + βP2j) and [X,P] = iℏ(1 + βP2 + kβ2P4) which imply minimal measurable lengths. Using two momentum representations, for the former GUP, we find eigenvalues and eigenfunctions of the free particle and the harmonic oscillator in terms of generalized trigonome...
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... to Fig. 1, for all temperatures, we obtain Z(β = 0) > Z j (β = 0). Also, at fixed GUP parameter and fixed temperature, Z j is a monotonic increasing function of j. Based on previous studies in the GUP framework, the number of microstates decrease due to the increasing of the volume of the fundamental cell in phase space which affects ...
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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...
Citations
... Many investigations have been done within this approach, including studies of the square potential well [26][27][28][29][30][31][32][33][34][35][36][37][38][39][40], quantum scattering [41], toy models of quantum field theory [42], time evolution by means of Euclidean path-integral [25,43], noncommutative quantum theories [44], and the Casimir effect [45][46][47]. ...
This note aims to elucidate certain aspects of the quasi-position representation frequently used in the investigation of one-dimensional models based on the generalized uncertainty principle (GUP). We specifically focus on two key points: (i) Contrary to recent claims, the quasi-position operator can possess physical significance even though it is non-Hermitian, and (ii) in the quasi-position representation, operators associated with the position, such as the potential energy, also behave as a derivative operator on the quasi-position coordinate, unless the method of computing expectation values is modified. The development of both points revolves around the observation that the position and quasi-position operators share the same set of eigenvalues and are connected through a non-unitary canonical transformation. This outcome may have implications for widely referenced constraints on GUP parameters.
... In a minimal-length scenario the study of an infinite square-well requires some special care since many mathematically possible solutions can be found 4 and the boundary conditions to be imposed on the wave function of the particle and its derivatives at the walls of the infinite square-well are not clearly fixed. Some authors have bypassed that problem neglecting the hyperbolic solutions by stating that they are not physical solutions since they would not satisfy the boundary conditions at the walls of the infinite square-well [22,23,24]. But, as it is well-known, the boundary conditions must be satisfied by the general solution, that is, the linear combination of the possible linearly independent solutions. ...
... A way to get around this problem is to solve the finite square-well potential, whose the boundary conditions are well fixed, and then to take the limit of the potential going to infinity 6 . This way we can check the results have been found in the literature [22,23,24,32]. With this goal in mind, we solve a symmetric finite square-well in a minimal-length scenario in Section 2 (in fact, the calculations can be found in the Appendix) and, in Section 3, we take the limit of the potential going to infinity in order to find the solution for an infinite square-well in that scenario. ...
... The previous results are in disagreement with ones in the literature [22,23,24,32], in which have been found no change in the eigenfunctions up to the first order in β. This is because those authors have presumed the discontinuity of the first derivative of the solutions (eigenfunctions) at the walls of the infinite square-well as for ordinary quantum mechanics. ...
One of the most widely problem studied in quantum mechanics is of an infinite square-well potential. In a minimal-length scenario its study requires additional care because the boundary conditions at the walls of the well are not well fixed. In order to avoid this we solve the finite square-well potential whose the boundary conditions are well fixed, even in a minimal-length scenario, and then we take the limit of the potential going to infinity to find the eigenfunctions and the energy equation for the infinite square-well potential. Although the first correction for the energy eigenvalues is the same as one found in the literature, our result shows that the eigenfunctions have the first derivative continuous at the square-well walls what is in disagreement with those previous work. That is because in the literature the authors have neglected the hyperbolic solutions and have assumed the discontinuity of the first derivative of the eigenfunctions at the walls of the infinite square-well which is not correct. As we show, the continuity of the first derivative of the eigenfunctions at the square-well walls guarantees the continuity of the probability current density and the unitarity of the time evolution operator.
... Since GUP is not unique, there exist a lot of possibilities for constructing the GUPcorrected Heisenberg algebra. The general form of the GUP is known to take the following form [47][48][49][57][58][59] : ...
... So, if we set (59) in (52), we obtain ...
In this paper, using Kempf and Mangano’s generalized uncertainty principle (GUP) [A. Kempf and G. Mangano, Phys. Rev. D 55, 7909 (1997)], we introduce a GUP which is called GUP*. The minimum measurable length predicted by the GUP* leads to the existence of a maximum momentum, in agreement with various previous studies. Then, we investigate some deformed calculus including both exponential and differential cases. Finally, as some applications of this GUP*, we discuss some quantum mechanical problems and obtain the results.
... where F (P ) is the deforming map giving the minimal length. Also, some dif- ferent choices of F (P ) lead to the following modified communication relations as 1,2,13,38,40,42 follows: ...
In this paper, using the new type of D-dimensional nonperturbative Generalized Uncertainty Principle (GUP) which has predicted both a minimal length uncertainty and
a maximal observable momentum, first, we obtain the maximally localized states and
express their connections to [P. Pedram, Phys. Lett. B 714, 317 (2012)]. Then, in the
context of our proposed GUP and using the generalized Schrodinger equation, we solve
some important problems including particle in a box and one-dimensional hydrogen
atom. Next, implying modified Bohr–Sommerfeld quantization, we obtain energy spectra of quantum harmonic oscillator and quantum bouncer. Finally, as an example, we
investigate some statistical properties of a free particle, including partition function and
internal energy, in the presence of the mentioned GUP.
... After some calculations, following the procedure of Ref. 33, we have Now, obeying the completeness relation, namely, ...
In this paper, we investigate non-relativistic anti-Snyder model in momentum representation and
obtain quantum mechanical eigenvalues and eigenfunctions. Using this framework, first, in one
dimension, we study a particle in a box and the harmonic oscillator problems. Then, for more
investigations, in three dimensions, the quantum mechanical eigenvalues and eigenfunctions of a
free particle problem and the radius of the neutron star are obtained.
... In order to study the effects of minimal length scale, a lot of studies have been done such as (1+1)-dimensional Dirac oscillator within Lorentz-covariant deformed algebra, 18 (2 + 1)-dimensional Dirac equation in a constant magnetic field with a minimal length uncertainty, 19,20 gravitational quantum well, 21 quantum mechanical solutions with minimal length uncertainty 22 and the classical limit of the minimal length uncertainty. 23 Also, according to the mean free path in thermodynamics, the effects of a minimal length scale are inevitable. ...
In this paper, using a deformed algebra [X,P] = iℏ/(1 − α2P2) which is originated from various theories of gravity, we study thermodynamical properties of the classical and extreme relativistic gases in canonical ensembles. In this regards, we exactly calculate the modified partition function, Helmholtz free energy, internal energy, entropy, heat capacity and the thermal pressure which conclude to the familiar form of the equation of state for the ideal gas. The advantage of applying this algebra is not only considering all natural cutoffs but also its structure is similar to the other effective quantum gravity models such as polymer, Snyder and noncommutative space–time frameworks. Moreover, after obtaining some thermodynamical quantities including internal energy and entropy, we conclude at high temperature limits due to the decreasing of the number of microstates, these quantities reach to maximal bounds which do not exist in standard cases and it concludes that at the presence of gravity for both micro-canonic and canonic ensembles, the internal energy and the entropy tend to these upper bounds.
According to a number of arguments in quantum gravity, both model-dependent and model-independent, Heisenberg’s uncertainty principle is modified when approaching the Planck scale. This deformation is attributed to the existence of a minimal length. The ensuing models have found entry into the literature under the term Generalized Uncertainty Principle (GUP). In this work, we discuss several conceptual shortcomings of the underlying framework and critically review recent developments in the field. In particular, we touch upon the issues of relativistic and field theoretical generalizations, the classical limit and the application to composite systems. Furthermore, we comment on subtleties involving the use of heuristic arguments instead of explicit calculations. Finally, we present an extensive list of constraints on the model parameter β, classifying them on the basis of the degree of rigour in their derivation and reconsidering the ones subject to problems associated with composites.
According to a number of arguments in quantum gravity, both model-dependent and model-independent, Heisenberg's uncertainty principle is modified when approaching the Planck scale. This deformation is attributed to the existence of a minimal length. The ensuing models have found entry into the literature under the term Generalized Uncertainty Principle (GUP). In this work, we discuss several conceptual shortcomings of the underlying framework and critically review recent developments in the field. In particular, we touch upon the issues of relativistic and field theoretical generalizations, the classical limit and the application to composite systems. Furthermore, we comment on subtleties involving the use of heuristic arguments instead of explicit calculations. Finally, we present an extensive list of constraints on the model parameter $\beta$, classifying them on the basis of the degree of rigour in their derivation and reconsidering the ones subject to problems associated with composites.
A good hundred years after the necessity for a quantum theory of gravity was acknowledged by Albert Einstein, the search for it continues to be an ongoing endeavour. Nevertheless, the field still evolves rapidly as manifested by the recent rise of quantum gravity phenomenology supported by an enormous surge in experimental precision. In particular, the minimum length paradigm ingrained in the program of generalized uncertainty principles (GUPs) is steadily growing in importance. The present thesis is aimed at establishing a link between modified uncertainty relations, derived from deformed canonical commutators, and curved spaces - specifically, GUPs and nontrivial momentum space as well as the related extended uncertainty principles (EUPs) and curved position space. In that vein, we derive a new kind of EUP relating the radius of geodesic balls, assumed to constrain the wave functions in the underlying Hilbert space, with the standard deviation of the momentum operator. This result is gradually generalized to relativistic particles in curved spacetime in accordance with the 3+1 decomposition. In a sense pursuing the inverse route, we find an explicit correspondence between theories yielding a GUP and quantum dynamics set on non-Euclidean momentum space. Quantitatively, the coordinate noncommutativity translates to momentum space curvature in the dual description, allowing for an analogous transfer of constraints from the literature. Finally, we find a formulation of quantum mechanics which proves consistent on the arbitrarily curved cotangent bundle. Along these lines, we show that the harmonic oscillator can not be used as a means to distinguish between curvature in position and momentum space, thereby providing an explicit instantiation of Born reciprocity in the context of curved spaces.
In this paper, we introduce the generalized Legendre transformation for the GUP Hamiltonian. From this, we define the non-canonical momentum. We interpret the momentum in GUP as the non-canonical momentum. We construct the GUP Lagrangian for some GUP models.
