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

arXiv:1011.3690Physical Review ADec 6, 201018 citations

On an Exactly Solvable Two-Body Problem in Two-Dimensional Quantum Mechanics

Roman Ya. Kezerashvili, Jianning Luo, Claudio R. Malvino

Few-Body SystemsOct 3, 20239 citations

Universal simulation of Hamiltonian dynamics for quantum systems with finite-dimensional state spaces

Michael A. Nielsen, Michael J. Bremner, Jennifer L. Dodd +2 more

What interactions are sufficient to simulate arbitrary quantum dynamics in a composite quantum system? Dodd et al. [Phys. Rev. A 65, 040301(R) (2002)] provided a partial solution to this problem in the form of an efficient algorithm to simulate any desired two-body Hamiltonian evolution using any fixed two-body entangling N-qubit Hamiltonian, and local unitaries. We extend this result to the case where the component systems are qudits, that is, have D dimensions. As a consequence we explain how universal quantum computation can be performed with any fixed two-body entangling N-qudit Hamiltonian, and local unitaries.What interactions are sufficient to simulate arbitrary quantum dynamics in a composite quantum system? Dodd et al. [Phys. Rev. A 65, 040301(R) (2002)] provided a partial solution to this problem in the form of an efficient algorithm to simulate any desired two-body Hamiltonian evolution using any fixed two-body entangling N-qubit Hamiltonian, and local unitaries. We extend this result to the case where the component systems are qudits, that is, have D dimensions. As a consequence we explain how universal quantum computation can be performed with any fixed two-body entangling N-qudit Hamiltonian, and local unitaries.

arXiv:quant-ph/0109064Physical Review AAug 30, 2002127 citations

Quantum Mechanics of Many-Body Systems

Luca Salasnich

UNITEXT for physicsJan 1, 20226 citations

The Finite Quantum Many-Body Problem

Ricardo A. Broglia

World Scientific series in 20th century physicsMay 21, 20190 citations

The quantum

N

-body problem is studied in the context of nonrelativistic quantum mechanics with a one-dimensional deformed Heisenberg algebra of the form $[\mathrm{x\ifmmode \hat{}\else \^{}\fi{}},\mathrm{p\ifmmode \hat{}\else \^{}\fi{}}]=i(1+\ensuremath{\beta}{\mathrm{p\ifmmode \hat{}\else \^{}\fi{}}}^{2})$, leading to the existence of a minimal observable length $\sqrt{\ensuremath{\beta}}$. For a generic pairwise interaction potential, analytical formulas are obtained that allow estimation of 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 $\ensuremath{\beta}$-dependent term grows faster with increasing

N

than the $\ensuremath{\beta}$-independent term. Then, it is argued that such a behavior should also be observed with generic potentials and for

D

-dimensional systems. Consequently, quantum

N

-body bound states might be interesting places to look at nontrivial manifestations of a minimal length, since the more particles that are present, the more the system deviates from standard quantum-mechanical predictions.