A higher order GUP with minimal length uncertainty and maximal momentum

Abstract

We present a higher order generalized (gravitational) uncertainty principle (GUP) in the form [X, P ] = ih/(1 − βP 2 ). This form of GUP 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. We show that the presence of the maximal momentum results in an upper bound on the energy spectrum of the momentum eigenstates and the harmonic oscillator.

One-dimensional hydrogen atom with minimal length uncertainty and maximal momentum

“…In the framework of this generalized uncertainty principle, the absolutely smallest uncertainty in position is given by [31] (∆X) min = 3 √ 3 4h β.…”

“…However, the eigenfunctions obeying quantization condition (48) do not satisfy (31) and (47) and the Hermicity condition (20) therefore fails. Comparison between the two quantization conditions (29) and (48) shows that…”

“…The problems of the free particle, particle in box, harmonic oscillator, maximally localized states, black-body radiation, and cosmological constant have been studied in this framework [31,32]. Here, we solve the problem of the one-dimensional hydrogen atom in this deformed quantum mechanics and find the exact energy eigenvalues and eigenfunctions in the momentum space.…”

Generalized relativistic wave equations with intrinsic maximum momentum

“…While most of the studied MCR's incorporate a minimum position uncertainty and usually leads to the concept of minimal length [6,8,9] which generally known as the main prediction by generalized uncertainty principle (GUP), there are others that exhibit a maximum momentum cutoff [13][14][15][16][17][18][19][20] as in Doubly Special Relativity (DSR). We have investigated large classes of deformed quantum mechanics of the latter type and used it to study several quantum mechanical systems such as deformed harmonic oscillator (DHO) [20], a particle in potential well/barrier and their bound/scattering states [21] as well as generalized coherent states with maximum momentum [22].…”

“…Intrinsic maximum momentum arises when f (P ) has a singularity [19,20] or a zero at some P = P 0 [13][14][15][16][17][18]20]. One effect of maximum momentum is that the spectrum of bound states terminates at finite energy even for potentials like the harmonic oscillator [20]; this is in contrast to MCR's which exhibit instead a minimum position uncertainty [6,8,9].…”