A class of GUP solutions in deformed quantum mechanics

Various candidates of quantum gravity such as string theory, loop quantum gravity and black hole physics all predict the existence of a minimum observable length which modifies the Heisenberg uncertainty principle to so-called Generalized Uncertainty Principle (GUP). This approach results in the modification of the commutation relations and changes all Hamiltonians in quantum mechanics. In this paper, we present a class of physically acceptable solutions for a general commutation relation without directly solving the corresponding generalized Schrodinger equations. These solutions satisfy the boundary conditions and exhibit the effect of the deformed algebra on the energy spectrum. We show that, this procedure prevents us from doing equivalent but lengthy calculations.Comment: 9 pages, 1 figur

A Higher Order GUP with Minimal Length Uncertainty and Maximal Momentum II: Applications

In a recent paper, we presented a nonperturbative higher order generalized uncertainty principle (GUP) that is consistent with various proposals of quantum gravity such as string theory, loop quantum gravity, doubly special relativity, and predicts both a minimal length uncertainty and a maximal observable momentum. In this Letter, we find exact maximally localized states and present a formally self-adjoint and naturally perturbative representation of this modified algebra. Then we extend this GUP to D dimensions that will be shown it is noncommutative and find invariant density of states. We show that the presence of the maximal momentum results in upper bounds on the energy spectrum of the free particle and the particle in box. Moreover, this form of GUP modifies blackbody radiation spectrum at high frequencies and predicts a finite cosmological constant. Although it does not solve the cosmological constant problem, it gives a better estimation with respect to the presence of just the minimal length.Comment: 15 pages, 3 figures, second part of arXiv:1110.2999, to appear in Physics Letters

Exact Ultra Cold Neutrons' Energy Spectrum in Gravitational Quantum Mechanics

We find exact energy eigenvalues and eigenfunctions of the quantum bouncer in the presence of the minimal length uncertainty and the maximal momentum. This form of Generalized (Gravitational) Uncertainty Principle (GUP) agrees with various theories of quantum gravity and predicts a minimal length uncertainty proportional to $\hbar\sqrt{\beta}$ and a maximal momentum proportional to $1/\sqrt{\beta}$, where $\beta$ is the deformation parameter. We also find the semiclassical energy spectrum and discuss the effects of this GUP on the transition rate of the ultra cold neutrons in gravitational spectrometers. Then, based on the Nesvizhevsky's famous experiment, we obtain an upper bound on the dimensionless GUP parameter.Comment: 11 pages, 1 figure, to appear in European Physical Journal

On the boundary conditions in deformed quantum mechanics with minimal length uncertainty

We find the coordinate space wave functions, maximal localization states, and quasiposition wave functions in a GUP framework that implies a minimal length uncertainty using a formally self-adjoint representation. We show that how the boundary conditions in quasiposition space can be exactly determined from the boundary conditions in coordinate space.Comment: 9 pages, to appear in Advances in High Energy Physic