Minimal length and bouncing-particle spectrum

Abstract

In this paper we study the effects of the Generalized Uncertainty Principle (GUP) on the spectrum of a particle that is bouncing vertically and elastically on a smooth reflecting floor in the Earth's gravitational field (a quantum bouncer). We calculate energy levels and corresponding wave functions of this system in terms of the GUP parameter. We compare the outcomes of our study with the results obtained from elementary quantum mechanics. A potential application of the present study is discussed finally.

Singular inverse square potential in coordinate space with a minimal length

“…In most of these works, the Schrödinger equation has been solved in momentum space as the coordinate representation leads to a fourth order differential equation, whose resolution is generally not possible. This explains the use of the perturbation techniques in the study of certain problems in coordinate space [27,29,34,36,37]. However, there are some disagreements between the results of the coordinate and momentum representations, as that observed in the case of the hydrogen atom [35].…”

“…[25][26][27][28]. Particularly, special attention was given to the fundamental non-relativistic quantum systems, such as the harmonic oscillator [15,29,30], the hydrogen atom in one [31,32] and three [29,[33][34][35] dimensions, the singular inverse square potential [6,13] and the gravitational quantum well [36,37]. In most of these works, the Schrödinger equation has been solved in momentum space as the coordinate representation leads to a fourth order differential equation, whose resolution is generally not possible.…”

A note on coherent states with quantum gravity effects

“…But the story is changed by considering the modified uncertainty principle due to the quantum gravity which is now called the Generalized Uncertainty Principle (GUP). In fact, due to the non-zero effective cutoff, it is not possible to set (∆x) min = 0 whenever the effects of quantum gravity become important meaning that we should replace HUP by GUP [12][13][14][15][16][17][18][19][20][21][22][23][24][25][26].…”

Minimal Length and the Quantum Bouncer: A Nonperturbative Study

“…The introduction of this idea in the form of the Generalized Uncertainty Principle (GUP) has attracted much attention in recent years and many papers have been appeared in the literature to address the effects of GUP on various quantum mechanical systems [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]. In doubly special relativity theories, we consider the Planck energy as an additional invariant beside the velocity of light [18][19][20][21][22].…”