The quantum N-body problem with a minimal length

Abstract

The quantum $N$-body problem is studied in the context of nonrelativistic quantum mechanics with a one-dimensional deformed Heisenberg algebra of the form $[\hat{x},\hat{p}]=i(1+\beta {\hat{p}}^2)$, leading to the existence of a minimal observable length $\sqrt{\beta }$. For a generic pairwise interaction potential, analytical formulas are obtained that allow to estimate the ground-state energy of the $N$-body system by finding the ground-state energy of a corresponding two-body problem. It is first shown that, in the harmonic oscillator case, the $\beta$-dependent term grows faster with $N$ than the $\beta$-independent one. Then, it is argued that such a behavior should be observed also with generic potentials and for $D$-dimensional systems. In consequence, quantum $N$-body bound states might be interesting places to look at nontrivial manifestations of a minimal length since, the more particles are present, the more the system deviates from standard quantum mechanical predictions.

To appear in PRA

References in corpus (26)

• String Theory and Noncommutative Geometry
• Hilbert Space Representation of the Minimal Length Uncertainty Relation
• Gravity coupled with matter and foundation of non-commutative geometry
• The algebraic structure of the generalized uncertainty principle
• Nonpointlike Particles in Harmonic Oscillators
• Minimal Length Uncertainty Relation and Hydrogen Atom
• Uncertainty Relation in Quantum Mechanics with Quantum Group Symmetry
• Exact Solution of the Harmonic Oscillator in Arbitrary Dimensions with Minimal Length Uncertainty Relations
• Maximal Localisation in the Presence of Minimal Uncertainties in Positions and Momenta
• Dirac oscillator with nonzero minimal uncertainty in position
• Hydrogen-atom spectrum under a minimal-length hypothesis
• Minimal Length Uncertainty Relation and gravitational quantum well
• Composite system in deformed space with minimal length
• Lorentz-covariant deformed algebra with minimal length and application to the 1+1-dimensional Dirac oscillator
• Statistical physics in deformed spaces with minimal length
• Harmonic oscillator with minimal length uncertainty relations and ladder operators
• Modification of Heisenberg uncertainty relations in non-commutative Snyder space-time geometry
• Hydrogen atom in momentum space with a minimal length
• On the Space-Time Uncertainty Relations of Liouville Strings and D Branes
• Remarks on the Noncommutative Gravitational Quantum Well
• More on a SUSYQM approach to the harmonic oscillator with nonzero minimal uncertainties in position and/or momentum
• Quantum gravity effects on statistics and compact star configurations
• The quantum N-body problem and the auxiliary field method
• Auxiliary fields as a tool for computing analytical solutions of the Schrödinger equation
• Eigenstates with the auxiliary field method
• Quantum-gravitational corrections to the hydrogen atom and harmonic oscillator

Cited by in corpus (8)

• Deformed Heisenberg algebra with minimal length and equivalence principle
• Electroweak Theory with a Minimal Length
• Deformed Heisenberg algebra and minimal length
• Approximate solutions for N-body Hamiltonians with identical particles in D dimensions
• Relation of deformed nonlinear algebras with linear ones
• Exactly solvable dynamical models with a minimal length uncertainty
• Two- and three-body calculations within the dominantly orbital state method
• Theory of a many-boson system with deformed Heisenberg algebra