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On Some Quantum Correction to the Coulomb Potential in Generalized Uncertainty Principle Approach

M. Baradaran, L. M. Nieto, S. Zarrinkamar

Abstract

Taking into account the importance of the unified theory of quantum mechanics and gravity, and the existence of a minimal length of the order of the Planck scale, we consider a modified Schrödinger equation resulting from a generalized uncertainty principle, which finds applications from the realm of quantum information to large-scale physics, with a quantum mechanically corrected gravitational interaction proposed very recently. As the resulting equation cannot be solved by common exact approaches, we propose a Bethe ansatz approach, which will be applied and whose results we will discuss, commenting on the analogy of the present study with some other interesting physical problems.

On Some Quantum Correction to the Coulomb Potential in Generalized Uncertainty Principle Approach

Abstract

Taking into account the importance of the unified theory of quantum mechanics and gravity, and the existence of a minimal length of the order of the Planck scale, we consider a modified Schrödinger equation resulting from a generalized uncertainty principle, which finds applications from the realm of quantum information to large-scale physics, with a quantum mechanically corrected gravitational interaction proposed very recently. As the resulting equation cannot be solved by common exact approaches, we propose a Bethe ansatz approach, which will be applied and whose results we will discuss, commenting on the analogy of the present study with some other interesting physical problems.
Paper Structure (7 sections, 54 equations, 4 figures)

This paper contains 7 sections, 54 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The modified Schrödinger equation with a GUP and a quantum correction to the Coulomb interaction
  3. Schrödinger equation for a Coulomb--4 potential
  4. The Lie-algebraic approach
  5. The ordinary case as a double-confluent Heun equation
  6. Solutions of the Coulomb--4 model with a GUP
  7. Conclusions

Figures (4)

  • Figure 1: Allowed values of the potential parameters \ref{['PotConsOrdn0']}, corresponding to the ground state \ref{['QESenerExpn0']}, for $\alpha_1=-1/{10}$.
  • Figure 2: In the drawing on the left, three values of the negative energy of the ground state and their corresponding probability densities $|\psi_0(x)|^2$, differentiating each case by the color of the lines. Using the same criterion, the corresponding Coulomb--4 potentials \ref{['potV']} are represented in the drawing on the right. The values of the potential parameters are the following: $\alpha_1=-1/{10}$ in all cases and $(\alpha_2=-0.0776,\alpha_3=-0.0097,\alpha_4=0.0053)$ for the red curves, $(\alpha_2=-0.0603,\alpha_3=-0.0070,\alpha_4=0.0037)$ for the orange curves and $(\alpha_2=-0.0527,\alpha_3=-0.0102,\alpha_4=0.0053)$ for the blue curves, all of them determined from the potential constraint \ref{['PotConsOrdn0']}, cf. FIG. \ref{['figparametersground']}.
  • Figure 3: Allowed potential parameters in \ref{['PotConsOrdn1']}, corresponding to the first excited state of the ordinary case, for $\alpha_1=-\frac{1}{5}$.
  • Figure 4: On the left, three values of the negative energy of the first excited state and their corresponding probability densities $|\psi_1(x)|^2$, differentiating each case by the color of the lines. Using the same criterion, the corresponding Coulomb--4 potentials \ref{['potV']} are represented in the drawing on the right. The values of the potential parameters are the following: $\alpha_1=-\frac{1}{5}$ in all cases and $(\alpha_2=-0.0301,\alpha_3=-0.0002,\alpha_4=0.0029)$ for the red curves, $(\alpha_2=-0.0141,\alpha_3=-0.0003,\alpha_4=0.0569)$ for the orange curves and $(\alpha_2=-0.0880,\alpha_3=-0.0075,\alpha_4=0.0438)$ for the blue curves, all of them determined from the potential constraint \ref{['PotConsOrdn1']}, cf. FIG. \ref{['fig4restrictionsexcited']}.