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 (6)

• Hydrogen-atom spectrum under a minimal-length hypothesis
• Minimal Length Uncertainty Relation and gravitational quantum well
• Statistical physics in deformed spaces with minimal length
• Hydrogen atom in momentum space with a minimal length
• Remarks on the Noncommutative Gravitational Quantum Well
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