Position and momentum uncertainties of a particle in a V-shaped potential under the minimal length uncertainty relation
arXiv:1402.7191 · doi:10.1142/S0217751X15502061
Abstract
We calculate the uncertainties in the position and momentum of a particle in the 1D potential V(x)=F|x|, F>0, when the position and momentum operators obey the deformed commutation relation [x,p]=i\hbar(1+βp^2), β>0. As in the harmonic oscillator case, which was investigated in a previous publication, the Hamiltonian H_1 = p^2/2m + F|x| admits discrete positive energy eigenstates for both positive and negative mass. The uncertainties for the positive mass states behave as Δx ~ 1/Δp as in the β=0 limit. For the negative mass states, however, in contrast to the harmonic oscillator case where we had Δx ~ Δp, both Δx and Δp diverge. We argue that the existence of the negative mass states and the divergence of their uncertainties can be understood by taking the classical limit of the theory. Comparison of our results is made with previous work by Benczik.
25 pages revtex4-1, 36 pdf figures, updated introduction
References in corpus (34)
- Universality of Quantum Gravity Corrections
- Discreteness of Space from the Generalized Uncertainty Principle
- Quantum-corrected black hole thermodynamics to all orders in the Planck length
- The effects of minimal length and maximal momentum on the transition rate of ultra cold neutrons in gravitational field
- Hydrogen-atom spectrum under a minimal-length hypothesis
- Minimal Length Uncertainty Relation and gravitational quantum well
- Interpretation of Quantum Field Theories with a Minimal Length Scale
- A Higher Order GUP with Minimal Length Uncertainty and Maximal Momentum II: Applications
- Running Coupling with Minimal Length
- Statistical physics in deformed spaces with minimal length
- The Big-Bang Singularity in the framework of a Generalized Uncertainty Principle
- Thermodynamics of a black hole based on a generalized uncertainty principle
- Minimal Length and Bouncing Particle Spectrum
- Corrections to the ns-levels of hydrogen atom in deformed space with minimal length
- Lorentz-covariant deformed algebra with minimal length
- Quantum Dynamics of the Taub Universe in a Generalized Uncertainty Principle framework
- On the modification of Hamiltonians' spectrum in gravitational quantum mechanics
- Dilaton Cosmology, Noncommutativity and Generalized Uncertainty Principle
- Scattering problem in deformed space with minimal length
- Generalized uncertainty principle in Bianchi type I quantum cosmology
- Time-dependent q-deformed coherent states for generalized uncertainty relations
- Squeezed coherent states for noncommutative spaces with minimal length uncertainty relations
- PT-symmetric noncommutative spaces with minimal volume uncertainty relations
- Generalized uncertainty principle and Hořava-Lifshitz gravity
- Quantum Gravity Corrections and Entropy at the Planck time
- A Note on Quantum Field Theories with a Minimal Length Scale
- Reply to "Comment on 'Universality of Quantum Gravity Corrections' "
- One-dimensional hydrogen atom with minimal length uncertainty and maximal momentum
- Cosmology with minimal length uncertainty relations
- A note on the one-dimensional hydrogen atom with minimal length uncertainty
- Quantum Gravity, Dynamical Energy-Momentum Space and Vacuum Energy
- Lagrangian Formulation of a Magnetostatic Field in the Presence of a Minimal Length Scale Based on the Kempf Algebra
- Minimal areas from q-deformed oscillator algebras
- WKB approximation in deformed space with minimal length and minimal momentum