Analytical Solutions of the Schrödinger Equation for Two Confined Particles with the van der Waals Interaction

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发表日期：

2024年11月3日

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Ruijie Du

摘要

We derive exact analytical solutions to the Schrödinger equation featuring a dual-scale potential, namely, a blend of a van der Waals (vdW) potential and an isotropic harmonic potential. The asymptotic behaviors of these solutions as $r\rightarrow 0$ $r \to 0$ and $r\rightarrow \infty $ $r \to \infty$ are also elucidated. These results are obtained through the approach we recently developed [arXiv: 2207.09377]. Using our results, we further calculate the s-wave and p-wave energy spectrums of two particles confined in an isotropic harmonic trap, with vdW inter-particle interaction. We compare our exact results and the ones given by the zero-range pseudopotential (ZRP) approaches, with either energy-dependent or energy-independent s-wave scattering length $a_s$ $a_{s}$ or p-wave scattering volume $V_p$ $V_{p}$ . It is shown that the results of ZRP approaches with energy-dependent $a_s$ $a_{s}$ or $V_p$ $V_{p}$ consist well with our exact ones, when the length scale $\beta _6$ $β_{6}$ of the vdW potential equals to or less than the length scale $a_h$ $a_{h}$ of the confinement potential. Furthermore, when $\beta _6\gg a_h$ $β_{6} ≫ a_{h}$ (e.g., $\beta _6=10a_h$ $β_{6} = 10 a_{h}$ ) all the ZRP approaches fail. Our results are helpful for the research of confined ultracold atoms or molecules with strong vdW interactions.