Abstract
We consider a Generalized Uncertainty Principle (GUP) framework which predicts a maximal uncertainty in momentum and minimal uncertainties both in position and momentum. We apply supersymmetric quantum mechanics method and the shape invariance condition to obtain the exact harmonic oscillator eigenvalues in this GUP context. We find the supersymmetric partner Hamiltonians and show that the harmonic oscillator belongs to a hierarchy of Hamiltonians with a shift in momentum representation and different masses and frequencies. We also study the effect of a uniform electric field on the harmonic oscillator energy spectrum in this setup.
Highlights
The assumption of the continuity of spacetime manifold may be broken at high energy limit
We obtain the supersymmetric partner Hamiltonian of the harmonic oscillator and show that the harmonic oscillator belongs to a hierarchy of Hamiltonians with a shift in momentum representation and with different masses and frequencies
In this Letter, we have considered a Generalized Uncertainty Principle (GUP) framework that admits maximal momentum uncertainty and nonzero minimal position and momentum uncertainties
Summary
The assumption of the continuity of spacetime manifold may be broken at high energy limit. In ordinary quantum mechanics we can make ∆x arbitrarily small by letting ∆p grow correspondingly, but in the GUP framework there exists a nonzero and minimal position uncertainty. In this Letter, we apply SUSYQM method and the notion of the shape invariance on the eigenvalue problem of the harmonic oscillator in a GUP framework that predicts maximal uncertainty in momentum and minimal uncertainties in both position and momentum. In this case x and p satisfy the following modified commutation relation [x, p] = ih(1 + ρx2 + 2α2p2 − αp),. We study the effect of a uniform electric field on the harmonic oscillator energy spectrum in this setup
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