Minimal Length Uncertainty Relation Research Articles

Algebraic solution and thermodynamic properties for the one- and two-dimensional Dirac oscillator with minimal length uncertainty relations

We study the quantum characteristics of the Dirac oscillator within the framework of Heisenberg's generalized uncertainty principle. This principle leads to the appearance of a minimal length of the order of the Planck length. Hidden symmetries are identified to solve the model algebraically. The presence of the minimal length leads to a quadratic dependence of the energy spectrum on the quantum number n, implying the hard confinement property of the system. Thermodynamic properties are calculated using the canonical partition function. The latter is well determined by the method based on Epstein's zeta function. The results reveal that the minimal length has a significant effect on the thermodynamic properties.

Two-dimensional Dirac oscillator in a magnetic field in deformed phase space with minimal-length uncertainty relations

Two-dimensional Dirac oscillator in a magnetic field in deformed phase space with minimal-length uncertainty relations

Two-dimensional Dirac oscillator in a magnetic field in deformed phase space with minimal-length uncertainty relations

Изучается динамика осциллятора Дирака в магнитном поле, при этом алгебра Гейзенберга строится в некоммутативном фазовом пространстве при условии существования минимальной длины. С помощью метода Никифорова-Уварова получены точные выражения для собственных энергий и соответствующих волновых функций в импульсном пространстве, которые выражаются через гипергеометрические функции.

Thermodynamic properties and algebraic solution of the N-dimensional harmonic oscillator with minimal length uncertainty relations

The algebraic solution of the deformed N-dimensional harmonic oscillator in the presence of a minimal length is determined. To reach this goal, we identify a hidden symmetry for the N free deformed harmonic oscillators. This allows us to algebraically determine the spectrum of the system. After determining the deformed partition function, we give some thermodynamics properties such as the mean energy, the mean free energy, the entropy, and the specific heat. The effect of the minimal length on all thermodynamic properties is very clear and considerable. In particular, from the curve of the entropy function no abrupt change has been identified for different values of the deformation parameter β. This means that the curvature observed in the specific heat curve does not present or indicate the existence of a phase transition. In the limit β = 0, the specific heat of a crystalline body is independent of the temperature and of the body considered for large values of the temperature.

Thermodynamics of the Schwarzschild and Reissner\u2013Nordstr\xf6m black holes under the Snyder\u2013de Sitter model

This paper examines the effects of a new form of the extended generalized uncertainty principle in the Snyder–de Sitter model on the thermodynamics of the Schwarzschild and Reissner–Nordström black holes. Firstly, we present a generalization of the minimal length uncertainty relation with two deformation parameters. Then we obtain the corrected mass–temperature relation, entropy and heat capacity for Schwarzschild black hole. Also we investigate the effect of the corrected uncertainty principle on the thermodynamics of the charged black holes. Our discussion of the corrected entropy involves a heuristic analysis of a particle which is absorbed by the black hole. Finally, we compare the thermodynamics of a charged black hole with the thermodynamics of a Schwarzschild black hole and with the usual forms, that is, without corrections to the uncertainty principle.

Propagator of Dirac oscillator in 2D with the presence of the minimal length uncertainty

In order to emphasize the role of quantum correction in two dimensions with the presence of the minimal length uncertainty relation, let us consider the case of relativistic Dirac oscillator in $ (2+1)$ -dimensional momentum space. At modified midpoint discretization interval $ (\bar{p}_{n} = (p_{n}+p_{n-1})/2)$ we can obtain the exact relativistic propagators, also the energy eigenvalues and their corresponding normalized radial momentum space eignefunctions are deduced.

Quantum scattering in one-dimensional systems satisfying the minimal length uncertainty relation

In quantum gravity theories, when the scattering energy is comparable to the Planck energy the Heisenberg uncertainty principle breaks down and is replaced by the minimal length uncertainty relation. In this paper, the consequences of the minimal length uncertainty relation on one-dimensional quantum scattering are studied using an approach involving a recently proposed second-order differential equation. An exact analytical expression for the tunneling probability through a locally-periodic rectangular potential barrier system is obtained. Results show that the existence of a non-zero minimal length uncertainty tends to shift the resonant tunneling energies to the positive direction. Scattering through a locally-periodic potential composed of double-rectangular potential barriers shows that the first band of resonant tunneling energies widens for minimal length cases when the double-rectangular potential barrier is symmetric but narrows down when the double-rectangular potential barrier is asymmetric. A numerical solution which exploits the use of Wronskians is used to calculate the transmission probabilities through the Pöschl–Teller well, Gaussian barrier, and double-Gaussian barrier. Results show that the probability of passage through the Pöschl–Teller well and Gaussian barrier is smaller in the minimal length cases compared to the non-minimal length case. For the double-Gaussian barrier, the probability of passage for energies that are more positive than the resonant tunneling energy is larger in the minimal length cases compared to the non-minimal length case. The approach is exact and applicable to many types of scattering potential.

Energy levels of one-dimensional systems satisfying the minimal length uncertainty relation

The standard approach to calculating the energy levels for quantum systems satisfying the minimal length uncertainty relation is to solve an eigenvalue problem involving a fourth- or higher-order differential equation in quasiposition space. It is shown that the problem can be reformulated so that the energy levels of these systems can be obtained by solving only a second-order quasiposition eigenvalue equation. Through this formulation the energy levels are calculated for the following potentials: particle in a box, harmonic oscillator, Pöschl–Teller well, Gaussian well, and double-Gaussian well. For the particle in a box, the second-order quasiposition eigenvalue equation is a second-order differential equation with constant coefficients. For the harmonic oscillator, Pöschl–Teller well, Gaussian well, and double-Gaussian well, a method that involves using Wronskians has been used to solve the second-order quasiposition eigenvalue equation. It is observed for all of these quantum systems that the introduction of a nonzero minimal length uncertainty induces a positive shift in the energy levels. It is shown that the calculation of energy levels in systems satisfying the minimal length uncertainty relation is not limited to a small number of problems like particle in a box and the harmonic oscillator but can be extended to a wider class of problems involving potentials such as the Pöschl–Teller and Gaussian wells.

The classical limit of minimal length uncertainty relation: revisit with the Hamilton-Jacobi method

The existence of a minimum measurable length could deform not only the standard quantum mechanics but also classical physics. The effects of the minimal length on classical orbits of particles in a gravitation field have been investigated before, using the deformed Poisson bracket or Schwarzschild metric. In this paper, we first use the Hamilton-Jacobi method to derive the deformed equations of motion in the context of Newtonian mechanics and general relativity. We then employ them to study the precession of planetary orbits, deflection of light, and time delay in radar propagation. We also set limits on the deformation parameter by comparing our results with the observational measurements. Finally, comparison with results from previous papers is given at the end of this paper.

Position and momentum uncertainties of a particle in a V-shaped potential under the minimal length uncertainty relation

We calculate the uncertainties in the position and momentum of a particle in the 1D potential [Formula: see text], [Formula: see text], when the position and momentum operators obey the deformed commutation relation [Formula: see text], [Formula: see text]. As in the harmonic oscillator case, which was investigated in a previous publication, the Hamiltonian [Formula: see text] admits discrete positive energy eigenstates for both positive and negative mass. The uncertainties for the positive mass states behave as [Formula: see text] as in the [Formula: see text] limit. For the negative mass states, however, in contrast to the harmonic oscillator case where we had [Formula: see text], both [Formula: see text] and [Formula: see text] diverge. We argue that the existence of the negative mass states and the divergence of their uncertainties can be understood by taking the classical limit of the theory. Comparison of our results is made with previous work by Benczik.

Massive fermions interacting via a harmonic oscillator in the presence of a minimal length uncertainty relation

We derive the relativistic energy spectrum for the modified Dirac equation by adding a harmonic oscillator potential where the coordinates and momenta are assumed to obey the commutation relation [Formula: see text]. In the nonrelativistic (NR) limit, our results are in agreement with the ones obtained previously. Furthermore, the extension to the construction of creation and annihilation operators for the harmonic oscillators with minimal length uncertainty relation is presented. Finally, we show that the commutation relation of the [Formula: see text] algebra is satisfied by the operators [Formula: see text] and [Formula: see text].

Entangled squeezed states in noncommutative spaces with minimal length uncertainty relations

We provide an explicit construction of entangled states in a noncommutative space with nonclassical states, particularly with the squeezed states. Noncommutative systems are found to be more entangled than the usual quantum mechanical systems. The noncommutative parameter provides an additional degree of freedom in the construction by which one can raise the entanglement of the noncommutative systems to fairly higher values beyond the usual systems. Despite of having classical-like behaviour, coherent states in noncommutative space produce little amount of entanglement and therefore they possess slight nonclassicality as well, which are not true for the coherent states of ordinary harmonic oscillator.

Hermitian versus non-Hermitian representations for minimal length uncertainty relations

We investigate four different types of representations of deformed canonical variables leading to generalized versions of Heisenberg’s uncertainty relations resulting from noncommutative spacetime structures. We demonstrate explicitly how the representations are related to each other and study three characteristically different solvable models on these spaces, the harmonic oscillator, the manifestly non-Hermitian Swanson model and an intrinsically noncommutative model with Pöschl–Teller type potential. We provide an analytical expression for the metric in terms of quantities specific to the generic solution procedure and show that when it is appropriately implemented expectation values are independent of the particular representation. A recently proposed inequivalent representation resulting from Jordan twists is shown to lead to unphysical models. We suggest an anti--symmetric modification to overcome this shortcoming.

Thermostatistics with minimal length uncertainty relation

The existence of minimal length is suggested in any quantum theory of gravity such as string theory, double special relativity and black hole physics. One way to impose minimal length is by deforming Heisenberg algebra in a phase space which is called the generalized uncertainty principle (GUP). In this paper, we develop statistical mechanics in the GUP framework. Our method is quite general and does not need to fix the generalized coordinates and momenta. We define a general transformation in phase space which transforms the usual Heisenberg algebra to a deformed one. In this method, quantum gravity effects only act on the structure of phase space and we relate these effects to the density of states. We find an interesting phenomenon in Maxwell–Boltzmann statistics which has no classical analogy. We show that there is an upper bound for the number of excited particles in the limit of high temperature which implies condensation. Also we study modification of Bose–Einstein condensation and the completely degenerate gas.

Squeezed coherent states for noncommutative spaces with minimal length uncertainty relations

We provide an explicit construction for Gazeau-Klauder coherent states related to non-Hermitian Hamiltonians with discrete bounded below and nondegenerate eigenspectrum. The underlying spacetime structure is taken to be of a noncommutative type with associated uncertainty relations implying minimal lengths. The uncertainty relations for the constructed states are shown to be saturated in a Hermitian as well as a non-Hermitian setting for a perturbed harmonic oscillator. The computed value of the Mandel parameter dictates that the coherent wave packets are assembled according to sub-Poissonian statistics. Fractional revival times, indicating the superposition of classical-like sub-wave packets, are clearly identified.

New approach to nonperturbative quantum mechanics with minimal length uncertainty

The existence of a minimal measurable length is a common feature of various approaches to quantum gravity such as string theory, loop quantum gravity and black-hole physics. In this scenario, all commutation relations are modified and the Heisenberg uncertainty principle is changed to the so-called Generalized (Gravitational) Uncertainty Principle (GUP). Here, we present a one-dimensional nonperturbative approach to quantum mechanics with minimal length uncertainty relation which implies X=x to all orders and P=p+(1/3)\beta p^3 to first order of GUP parameter \beta, where X and P are the generalized position and momentum operators and [x,p]=i\hbar. We show that this formalism is an equivalent representation of the seminal proposal by Kempf, Mangano, and Mann and predicts the same physics. However, this proposal reveals many significant aspects of the generalized uncertainty principle in a simple and comprehensive form and the existence of a maximal canonical momentum is manifest through this representation. The problems of the free particle and the harmonic oscillator are exactly solved in this GUP framework and the effects of GUP on the thermodynamics of these systems are also presented. Although X, P, and the Hamiltonian of the harmonic oscillator all are formally self-adjoint, the careful study of the domains of these operators shows that only the momentum operator remains self-adjoint in the presence of the minimal length uncertainty. We finally discuss the difficulties with the definition of potentials with infinitely sharp boundaries.

Position and momentum uncertainties of the normal and inverted harmonic oscillators under the minimal length uncertainty relation

We analyze the position and momentum uncertainties of the energy eigenstates of the harmonic oscillator in the context of a deformed quantum mechanics, namely, that in which the commutator between the position and momentum operators is given by [x,p]=i\hbar(1+\beta p^2). This deformed commutation relation leads to the minimal length uncertainty relation \Delta x > (\hbar/2)(1/\Delta p +\beta\Delta p), which implies that \Delta x ~ 1/\Delta p at small \Delta p while \Delta x ~ \Delta p at large \Delta p. We find that the uncertainties of the energy eigenstates of the normal harmonic oscillator (m>0), derived in Ref. [1], only populate the \Delta x ~ 1/\Delta p branch. The other branch, \Delta x ~ \Delta p, is found to be populated by the energy eigenstates of the `inverted' harmonic oscillator (m<0). The Hilbert space in the 'inverted' case admits an infinite ladder of positive energy eigenstates provided that \Delta x_{min} = \hbar\sqrt{\beta} > \sqrt{2} [\hbar^2/k|m|]^{1/4}. Correspondence with the classical limit is also discussed.

On the Minimal Length Uncertainty Relation and the Foundations of String Theory

We review our work on the minimal length uncertainty relation as suggested by perturbative string theory. We discuss simple phenomenological implications of the minimal length uncertainty relation and then argue that the combination of the principles of quantum theory and general relativity allow for a dynamical energy-momentum space. We discuss the implication of this for the problem of vacuum energy and the foundations of nonperturbative string theory.

Deformed time–energy uncertainty in string theory

The space-time extension of strings is studied in terms of the ordinary time–energy uncertainty relation. Based on the minimal length uncertainty relation, a deformed time–energy uncertainty relation is obtained.

Minimal length uncertainty relations and new shape invariant models

This paper identifies a new class of shape invariant models. These models are based on extensions of conventional quantum mechanics that satisfy a string-motivated minimal length uncertainty relation. An important feature of our construction is the pairing of operators that are not adjoints of each other. The results in this paper thus show the broader applicability of shape invariance to exactly solvable systems.