Two trapped particles interacting by a finite-range two-body potential in two spatial dimensions

We examine the problem of two particles confined in an isotropic harmonic trap, which interact via a finite-range Gaussian-shaped potential in two spatial dimensions. We derive an approximative transcendental equation for the energy and study the resulting spectrum as a function of the interparticle interaction strength. Both the attractive and repulsive systems are analyzed. We study the impact of the potential's range on the ground-state energy. We also explicitly verify by a variational treatment that in the zero-range limit the positive $\delta$ potential in two dimensions only reproduces the noninteracting results, if the Hilbert space in not truncated, and demonstrate that an extremely large Hilbert space is required to approach the ground state when one is to tackle the limit of zero-range interaction numerically. Finally, we establish and discuss the connection between our finite-range treatment and regularized zero-range results from the literature. The present results indicate that a finite-range interparticle potential is numerically amenable for treating the statics and the nonequilibrium dynamics of interacting many-particle systems (bosons) in two dimensions.

Figure 1

Ground-state energy (in the center of mass frame) of two particles in a harmonic trap interacting via a (nonregularized) $\delta$ potential in 2D as a function of the size of the Hilbert space. The results are obtained by numerically solving Eq. (11) with ${\lambda }_0=1$, $l=1$ and different $N$. Notice the logarithmic scale on the $x$ axis. All quantities are dimensionless.

Figure 2

Energies of the first three states (in the center-of-mass frame) of two trapped particles in 2D interacting via a Gaussian-shaped potential versus the interaction strength ${\lambda }_0$. The dashed-dotted lines show the energies of the respective noninteracting system. Here $l=1$ and $s/l=0.1$. All quantities are dimensionless.

Figure 3

Energy of the ground state (in the center-of-mass frame) of two trapped particles in 2D interacting via a Gaussian-shaped potential versus the width of the interaction, $s$, for different choices of ${\lambda }_0/\hslash \omega$. See legend inside the graph. Notice the logarithmic scale on the $x$ axis. All quantities are dimensionless.

Figure 4

Plot of the LHS (thick line) and the RHS (dashed line) of Eq. (19) versus $E/\hslash \omega$ for $l=1$ and $s/l=0.1$. For $E/\hslash \omega >-1$ the two curves are practically indistinguishable. The vertical dashed-dotted lines show the poles of both sides of Eq. (19). Notice the logarithmic scale on the $x$ axis. See text for discussion. All quantities are dimensionless.

Figure 5

Comparison between the ground-state energy $E_0$ versus ${\lambda }_0$ obtained from Eq. (17) (lines) and that computed by direct diagonalization (symbols). The widths from top to bottom are $s=0.5$ (squares), $0.4$ (triangles), $0.2$ (crosses), $0.1$ (circles), and $l=1$. See text for more details. All quantities are dimensionless