A class of solutions under GUP for a harmonic oscillator and a particle in a box

Resumo

The existence of a minimal observable length is the common prediction of different theories of quantum gravity such as string theory, loop quantum gravity and black hole physics. The ordinary Heisenberg uncertainty principle (HUP) is not compatible with this prediction, so the HUP was generalized to the Generalized Uncertainty Principle (GUP) in literature. The consequences of this generalization are the generalized commutation relations between operators and so the generalized Hamiltonian for systems. In this paper, we represent the solutions of a harmonic oscillator and a particle in a box in the sense of a generalized commutation relation. We show that using of the Hellmann-Feynman (HF) theorem leads to the approximate energy spectrum of a harmonic oscillator and the exact energy spectrum of a particle in a box. In these cases, we do not solve the corresponding generalized Schro¨ dinger equation directly, but the HF theorem exhibit the effect of GUP on the energy spectrum. Also we obtain the uncertainties in both position and momentum for a harmonic oscillator and derive the GUP.