۱- Introduction

۲- Quantum mechanics on (anti)-de Sitter spaces

۳- Free particle

۴- Harmonic oscillator

۵- Pseudoharmonic oscillator

۶- Conclusion

References

Abstract

All commutation relations are modified in (anti)-de Sitter background and the Heisenberg uncertainty principle is changed to the so-called extended uncertainty principle (EUP). In this scenario, the commutators between position and momentum operators are functions of the position space variables, instead of a constant and the coordinate representation of the momentum operators for this model becomes coordinate dependent. In the AdS space, a lower bound on momentum uncertainty arises, which is not present in the dS space. In this paper, we present an exact solution of the D-dimensional free particle, the harmonic oscillator and pseudoharmonic oscillator in AdS and dS spaces. The eigenfunctions are determined for both cases and the energy eigenvalues are obtained.

Introduction

The idea of noncommutative spacetime is quite old. It has been incorporated in quantum fields by Snyder [1, 2] in order to regularize the divergences in quantum field theories. Snyder model admits a fundamental length scale and it is invariant under the Lorentz group action. However, this idea was largely forgotten due to the remarkable success of the renormalization theory in quantum electrodynamics. After the work of Kontsevich [3], Snyder model has attracted great attention. Modern candidates for a theory of quantum gravity and string theory suggests that the structure of spacetime may be noncommutative at scales close to the Planck length.