On the quantum mechanical solutions with minimal length uncertainty

In this paper, we study two generalized uncertainty principles (GUPs) including [X, P] = i~(1+P2j) and [X, P] = i~(1+P2+k2P4) which imply minimal measurable lengths. Using two momentum representations, for the former GUP, we find eigenvalues and eigenfunctions of the free particle and the harmonic oscillator in terms of generalized trigonometric functions. Also, for the latter GUP, we obtain quantum mechanical solutions of a particle in a box and harmonic oscillator. Finally we investigate the statistical properties of the harmonic oscillator including partition function, internal energy, and heat capacity in the context of the first GUP.

Quantum gravity; generalized uncertainty principle; minimal length.