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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