Regularization of the singular inverse square potential in quantum mechanics with a minimal length

Abstract

We study the problem of the attractive inverse square potential in quantum mechanics with a generalized uncertainty relation. Using the momentum representation, we show that this potential is regular in this framework. We solve analytically the s-wave bound states equation in terms of Heun's functions. We discuss in detail the bound states spectrum for a specific form of the generalized uncertainty relation. The minimal length may be interpreted as characterizing the dimension of the system.

On the quantum dynamics of a point particle in conical space

“…This pathological potential has deserved some attention recently (see for example [9,10] and references therein) although it has been addressed already in 1950 by K. M. Case [11]. The main problem with this potential is that the energy levels are unbounded from below making the bound states unstable.…”

Hydrogen atom in momentum space with a minimal length

“…[4][5][6][7][8][9][10][11][12][13]. The idea of modifying the standard Heisenberg uncertainty relation in such a way that it includes a minimal length has first been proposed in the context of quantum gravity and string theory [14,15].…”

Pseudoharmonic oscillator in quantum mechanics with a minimal length

“…However, it has been argued that, in relativistic or nonrelativistic quantum mechanics, this elementary length incorporated in the GUP may be viewed as an intrinsic scale characterizing the system under study [8,18]. Consequently, the formalism based on these deformed commutation relations may provide a new model for an effective description of complex systems such as quasiparticles, nuclei, and molecules [8].…”

“…Only a few problems have been solved exactly in the formalism of quantum mechanics with a minimal length, such as the Schrödinger equation for the harmonic oscillator [13] and for the singular inverse square potential [18,19]. In the hydrogen atom problem, for instance, the effect of the minimal length is assumed to be too small and studied perturbatively in coordinate space [12,[14][15][16].…”

“…In recent years, various topics were studied in this modified version of quantum mechanics, amongst other, we cite, the Schrödinger equation for, the harmonic oscillator potential [7,12,13], the hydrogen atom problem [12,[14][15][16][17], the singular inverse square potential [18,19], the problem of a particle in a gravitational quantum well [20,21]. In relativistic equations, the minimal length was introduced in the Dirac equation, with a constant magnetic field [22,23], with vector and scalar linear potentials [24], for the hydrogen atom potential [25], and the Dirac oscillator [26,27].…”