Solutions of the Schr\"{o}dinger equation for anisotropic dipole-dipole interaction plus isotropic van der Waals interaction

• URL: http://arxiv.org/abs/2207.09377v3
• Date: Thu, 21 Jul 2022 07:06:43 GMT
• Title: Solutions of the Schr\"{o}dinger equation for anisotropic dipole-dipole interaction plus isotropic van der Waals interaction
• Authors: Ruijie Du, Ran Qi, Peng Zhang
• Abstract summary: We derive the solutions for the Schr"odinger equation for an anisotropic dipole-dipole interaction plus an isotropic attractive vdW potential. Results can be used in the research of collisions and chemical reactions between ultra-cold polar molecules in a static electric field.
• Score: 3.7530059578901147
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: By generalizing Bo Gao's approach [Phys. Rev. A 58, 1728 (1998)] for solving the Schr\"{o}dinger equation for an isotropic van der Waals (vdW) potential to the systems with a multi-scale anisotropic long-range interaction, we derive the solutions for the Schr\"{o}dinger equation for an anisotropic dipole-dipole interaction plus an isotropic attractive vdW potential, i.e., ${C_d(1-3\cos^2\theta)}/{r^3}-{C_6}/{r^6}$, which is projected to the subspace with angular momentum $l\leq l_{\rm cut}$, with $l_{\rm cut}$ being an arbitrary angular-momentum cutoff. Here $\theta$ is the polar angle of the coordinate $\boldsymbol{r}$ and $r=|\boldsymbol{r}|$. The asymptotic behaviors of these solutions for $r\rightarrow 0$ and $r\rightarrow \infty$ are obtained. These results can be used in the research of collisions and chemical reactions between ultra-cold polar molecules in a static electric field. Our approach to derive the solutions can be applied to the systems with a general long-range potential $\sum_{\lambda= 2}^{\lambda_{\rm max}} {V_\lambda(\theta,\varphi)}/{r^\lambda}$, with $\varphi$ being the azimuthal angle of $\boldsymbol{r}$, and thus can be used in various problems on molecule-molecule interaction.

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