In this paper we calculate the energy levels of two trapped atoms interacting through a spherical two-body potential $V(r)$ under external confinement quantum mechanically. We then replace $V(r)$ by either a regularized energy-independent or an energy-dependent \ensuremath{\delta}-function potential. A comparison shows that the use of an energy-dependent pseudopotential improves significantly upon the use of an energy-independent pseudopotential. The accuracy of the pseudopotential approximation is approximately determined by the ratio ${\ensuremath{\beta}}_{6}{/a}_{\mathrm{ho}},$ where ${\ensuremath{\beta}}_{6}$ and ${a}_{\mathrm{ho}}$ denote the characteristic length scale of the two-body interaction potential $V(r)$ and of the trapping potential, respectively.In this paper we calculate the energy levels of two trapped atoms interacting through a spherical two-body potential $V(r)$ under external confinement quantum mechanically. We then replace $V(r)$ by either a regularized energy-independent or an energy-dependent \ensuremath{\delta}-function potential. A comparison shows that the use of an energy-dependent pseudopotential improves significantly upon the use of an energy-independent pseudopotential. The accuracy of the pseudopotential approximation is approximately determined by the ratio ${\ensuremath{\beta}}_{6}{/a}_{\mathrm{ho}},$ where ${\ensuremath{\beta}}_{6}$ and ${a}_{\mathrm{ho}}$ denote the characteristic length scale of the two-body interaction potential $V(r)$ and of the trapping potential, respectively.

Physical Review AApr 1, 2002132 citations

Energy-dependent ℓ -wave confinement-induced resonances

Benjamin Heß, Panagiotis Giannakeas, Peter Schmelcher

The universal aspects of two-body collisions in the presence of a harmonic confinement are investigated for both bosons and fermions. The main focus of this study are the confinement-induced resonances (CIRs) which are attributed to different angular momentum states $\ensuremath{\ell}$, and we explicitly show that in alkaline collisions only four universal $\ensuremath{\ell}$-wave CIRs emerge given that the interatomic potential is deep enough. Going beyond the single mode regime the energy dependence of $\ensuremath{\ell}$-wave CIRs is studied. In particular we show that all the $\ensuremath{\ell}$-wave CIRs may emerge even when the underlying two-body potential cannot support any bound state. We observe that the intricate dependence on the energy yields resonant features where the colliding system within the confining potential experiences an effective free-space scattering. Our analysis is done within the framework of the generalized $K$ -matrix theory, and the relevant analytical calculations are in very good agreement with the corresponding ab initio numerical scattering simulations.The universal aspects of two-body collisions in the presence of a harmonic confinement are investigated for both bosons and fermions. The main focus of this study are the confinement-induced resonances (CIRs) which are attributed to different angular momentum states $\ensuremath{\ell}$, and we explicitly show that in alkaline collisions only four universal $\ensuremath{\ell}$-wave CIRs emerge given that the interatomic potential is deep enough. Going beyond the single mode regime the energy dependence of $\ensuremath{\ell}$-wave CIRs is studied. In particular we show that all the $\ensuremath{\ell}$-wave CIRs may emerge even when the underlying two-body potential cannot support any bound state. We observe that the intricate dependence on the energy yields resonant features where the colliding system within the confining potential experiences an effective free-space scattering. Our analysis is done within the framework of the generalized $K$-matrix theory, and the relevant analytical calculations are in very good agreement with the corresponding ab initio numerical scattering simulations.

arXiv:1402.6685Physical Review AMay 29, 201415 citations

The Many-Body Problem in Quantum Mechanics

N. H. March, William H. L. Young, S. Sampanthar +1 more

American Journal of PhysicsJan 1, 1969463 citations

The many-body problem in quantum mechanics

Advances in MathematicsApr 1, 197619 citations

The helical quantum two-body problem and its wave packet dynamics

Peter Schmelcher

We explore the helical quantum two-body problem i.e. two repulsively Coulomb interacting particles confined to move along a helix. The effective potential possesses a tunable number of potential wells superimposed on the repulsive Coulomb interaction that can be varied by changing the ratio of the pitch and radius of the helix. The anharmonicity of these wells depends crucially on this ratio and on the order of the well which can be seen also by analyzing the individual wells energy eigenvalue spacing. Our main focus is the investigation of the quantum dynamics of differently prepared wave packets that scatter from the multi-well potential landscape. We show that there exists a rich pattern forming transient evolution which depends also on the number of bound states of the individual wells. We demonstrate how the multiple wells leave their fingerprints in the dynamics leading, among others, to oscillatory structures on different spatial scales, the formation of beats and pulsed emission from single well localized wave packets due to their intrawell dynamics.We explore the helical quantum two-body problem i.e. two repulsively Coulomb interacting particles confined to move along a helix. The effective potential possesses a tunable number of potential wells superimposed on the repulsive Coulomb interaction that can be varied by changing the ratio of the pitch and radius of the helix. The anharmonicity of these wells depends crucially on this ratio and on the order of the well which can be seen also by analyzing the individual wells energy eigenvalue spacing. Our main focus is the investigation of the quantum dynamics of differently prepared wave packets that scatter from the multi-well potential landscape. We show that there exists a rich pattern forming transient evolution which depends also on the number of bound states of the individual wells. We demonstrate how the multiple wells leave their fingerprints in the dynamics leading, among others, to oscillatory structures on different spatial scales, the formation of beats and pulsed emission from single well localized wave packets due to their intrawell dynamics.

quant-pharXiv:2603.09093APS Open ScienceMar 11, 20260 citations

In this paper we calculate the energy levels of two trapped atoms interacting through a spherical two-body potential

V(r)

under external confinement quantum mechanically. We then replace

V(r)

by either a regularized energy-independent or an energy-dependent \ensuremath{\delta}-function potential. A comparison shows that the use of an energy-dependent pseudopotential improves significantly upon the use of an energy-independent pseudopotential. The accuracy of the pseudopotential approximation is approximately determined by the ratio ${\ensuremath{\beta}}_{6}{/a}_{\mathrm{ho}},$ where ${\ensuremath{\beta}}_{6}$ and

{a}_{\mathrm{ho}}

denote the characteristic length scale of the two-body interaction potential

V(r)

and of the trapping potential, respectively.

K

-matrix theory, and the relevant analytical calculations are in very good agreement with the corresponding ab initio numerical scattering simulations.