Minimal Length Quantum Mechanics Research Articles

Towards quantum mechanics on the curved cotangent bundle

The minimal-length paradigm is a cornerstone of quantum gravity phenomenology. Recently, it has been demonstrated that minimal-length quantum mechanics can alternatively be described as an undeformed theory set on a nontrivial momentum space. However, there is no fully consistent formulation of these theories beyond Cartesian coordinates in flat space and, in particular, no position representation. This paper is intended to take the first steps in bridging this gap. We find a natural position representation of the position and momentum operators on general curved cotangent bundles. In an expansion akin to Riemann normal coordinates with curvature in both position and momentum space, we apply the formalism perturbatively to the isotropic harmonic oscillator and the hydrogenic atom. Due to the symmetry of the harmonic oscillator under exchange of positions and momenta, we show that it is impossible to distinguish position- from momentum-space curvature with oscillators alone. Thus, we obtain an instantiation of Born reciprocity on the curved cotangent bundle, i.e. in precisely the way Born originally envisioned. It manifests itself as a symmetry mixing UV and IR physics, reminiscent of T-duality in string theory.

Position in Minimal Length Quantum Mechanics

Several approaches to quantum gravity imply the presence of a minimal measurable length at high energies. This is in tension with the Heisenberg Uncertainty Principle. Such a contrast is then considered in phenomenological approaches to quantum gravity by introducing a minimal length in quantum mechanics via the Generalized Uncertainty Principle. Several features of the standard theory are affected by such a modification. For example, position eigenstates are no longer included in models of quantum mechanics with a minimal length. Furthermore, while the momentum-space description can still be realized in a relatively straightforward way, the (quasi-)position representation acquires numerous issues. Here, we will review such issues, clarifying aspects regarding models with a minimal length. Finally, we will consider the effects of such models on simple quantum mechanical systems.

Q-Deformed superstatistic thermodynamics in the presence of minimal length quantum mechanics

In this paper, we study the thermodynamic properties of a quantum oscillator in the presence of the minimal length scale in terms of the q-deformed superstatistics of statistical mechanics. We evaluated the partition function from the Boltzmann factor and obtained other thermodynamic properties such as internal energy, Helmholtz free energy, entropy, and specific heat capacity. We have also shown graphically the effects of the minimal length and the q-statistical properties on the thermodynamic properties of the system.

Generalized relativistic harmonic oscillator in minimal length quantum mechanics

We solve the generalized relativistic harmonic oscillator in 1  +  1 dimensions in the presence of a minimal length. Using the momentum space representation, we explore all the possible signs of the potentials and discuss their bound-state solutions for fermions and antifermions. Furthermore, we also find an isolated solution from the Sturm–Liouville scheme. All cases already analyzed in the literature are obtained as particular cases.

Dirac equation in minimal length quantum mechanics with energy- dependent harmonic potential

In the rest of the article we want to introduce modified Dirac equation with energy depended harmonic potential(EDP) with the framework of minimal length quantum mechanics.The purpose of this paper is to provide a general equation of the basic ideas and results of quantum gravity phenomenology in the nuclear physics. The energy of defined system has been reported.

The spin–orbit interaction in minimal length quantum mechanics; the case of a (2+1)-dimensional Dirac oscillator

We solve the (2+1)-dimensional Dirac oscillator in the presence of spin–orbit interaction within the framework of minimal length quantum mechanics. To report an exact analytical solution, we transform the problem into the momentum space and polar coordinates and thereby obtain the eigenfunctions and eigenenergies of the system.

Scattering states of Cusp potential in minimal length Dirac equation

In this article, we investigate the scattering state of spin-half particle within the framework of minimal length quantum mechanics for Cusp potential in terms of Whittaker functions. We calculate the reflection and transmission coefficients for spin and pseudospin symmetry limits using the conditions of equation of continuity of the wave functions.

Minimal Length Quantum Mechanics of Dirac Particles in Noncommutative Space

We study the two-dimensional harmonic oscillator in commutative and noncommutative space within the framework of minimal length quantum mechanics for spin-1/2 particles. The energy spectra and the eigenfunction are obtained in both cases. Special cases are also deduced.

Cusp Interaction in Minimal Length Quantum Mechanics

The modified Schrodinger equation with a minimal length is considered under a Cusp potential which includes the exponential interaction. Next, exact analytical solutions of the problem are reported and thereby the scattering states as well as the corresponding transmission and reflection coefficients are reported.

Potential well and step potential within the framework of minimal length quantum mechanics

The potential well and the step potential are studied under the modified Schrodinger equation due to minimal length in an exact analytical manner. The scattering problems as well as the reflection and transmission parameters are reported.

SCATTERING STATES OF HULTHÉN INTERACTION IN MINIMAL LENGTH QUANTUM MECHANICS

We first revisit the nonrelativistic minimal length quantum mechanics and reveal an interesting symmetry of the problem. In fact, we will show that the cumbersome problem can be cast into the ordinary Schrödinger equation with a new effective potential. Next, as a typical example, we show the minimal length Schrödinger equation in the presence of a nonminimal Hulthén vector interaction. The transmission and reflection coefficients are reported as well.

A simple efficient methodology for Dirac equation in minimal length quantum mechanics

We solve the modified Dirac equation by adding a harmonic oscillator potential and implementing the Nikiforov–Uvarov technique. The closed forms of solutions are reported in a quite simple and systematic manner.

Scattering states of Woods–Saxon interaction in minimal length quantum mechanics

We propose a very simple approach to deal with the problems of the modified Schrödinger equation due to minimal length and thereby solve the minimal length Schrödinger equation in the presence of a non-minimal Woods–Saxon interaction. The transmission and reflection coefficients are reported as well.

Lifetime Broadening of a Nuclear Magnetic Resonance Peak under Minimal Length Uncertainty Analysis

It is expected that some low energy relics of high energy phenomena may be verified by nowadays experiments, and some phenomenological low energy effects may be explored in solvable models at a level of quantum mechanics in a minimal length scale [1–9]. In literature, the minimal length quantum mechanics have been extensively studied [10, 1, 2]. In this letter we study the possibility of testing minimal length scale via nuclear magnetic resonance spectroscopy taking into account the solution of broadening of a magnetic resonance peak by minimal length uncertainty. One manifestation of the uncertainty principle is the effect of decreasing the lifetime of a nucleus in a given environment on the nuclear magnetic resonance spectrum of the same nucleus or one to which it is coupled. These kinetic effects may have experimental significance by giving rise to spectra whose shift-state values or line widths may be used as a measure, or to spectra which are simplified so that the resonances are more suitable for quantitative measurements. The shifting and merging of spectra by kinetic process may also be used to shift an increasing resonance away from a resonance of interest if the interfering species can be made to undergo rapid exchange. The magnitude of the effect can be estimated from uncertainty principle, ∆E∆t ≥ ~. Since ∆E = ~∆ν and ∆t may be identified with τ , the average lifetime of the excited state in the various exchanging states, the relationship becomes τ∆ν ≈ 1/2π. Thus the magnitude of the rate constant that will give an observable effect will vary for different types of spectrometry, depending on the frequency difference involved. The widest application of this principle has been in nuclear magnetic resonance spectrometry, especially proton NMR. When a system