تقرير
Rapid Accurate Calculation of the s-Wave Scattering Length
| العنوان: | Rapid Accurate Calculation of the s-Wave Scattering Length |
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| المؤلفون: | Meshkov, Vladimir V., Stolyarov, Andrey V., Roy, Robert J. Le |
| المصدر: | J. Chem. Phys. 135, 154108 (2011) |
| سنة النشر: | 2011 |
| المجموعة: | Condensed Matter Physics (Other) |
| مصطلحات موضوعية: | Physics - Atomic Physics, Condensed Matter - Quantum Gases, Physics - Computational Physics |
| الوصف: | Transformation of the conventional radial Schr\"odinger equation defined on the interval $\,r\in[0,\infty)$ into an equivalent form defined on the finite domain $\,y(r)\in [a,b]\,$ allows the s-wave scattering length $a_s$ to be exactly expressed in terms of a logarithmic derivative of the transformed wave function $\phi(y)$ at the outer boundary point $y=b$, which corresponds to $r=\infty$. In particular, for an arbitrary interaction potential that dies off as fast as $1/r^n$ for $n\geq 4$, the modified wave function $\phi(y)$ obtained by using the two-parameter mapping function $r(y;\bar{r},\beta) = \bar{r}[1+\frac{1}{\beta}\tan(\pi y/2)]$ has no singularities, and $$a_s=\bar{r}[1+\frac{2}{\pi\beta}\frac{1}{\phi(1)}\frac{d\phi(1)}{dy}].$$ For a well bound potential with equilibrium distance $r_e$, the optimal mapping parameters are $\,\bar{r}\approx r_e\,$ and $\,\beta\approx \frac{n}{2}-1$. An outward integration procedure based on Johnson's log-derivative algorithm [B.R.\ Johnson, J.\ Comp.\ Phys., \textbf{13}, 445 (1973)] combined with a Richardson extrapolation procedure is shown to readily yield high precision $a_s$-values both for model Lennard-Jones ($2n,n$) potentials and for realistic published potentials for the Xe--e$^-$, Cs$_2(a\,^3\Sigma_u^+$) and $^{3,4}$He$_2(X\,^1\Sigma_g^+)$ systems. Use of this same transformed Schr{\"o}dinger equation was previously shown [V.V. Meshkov et al., Phys.\ Rev.\ A, {\bf 78}, 052510 (2008)] to ensure the efficient calculation of all bound levels supported by a potential, including those lying extremely close to dissociation. Comment: 12 pages, 9 figures, to appear in J. Chem. Phys |
| نوع الوثيقة: | Working Paper |
| DOI: | 10.1063/1.3649946 |
| URL الوصول: | http://arxiv.org/abs/1109.6522 |
| رقم الأكسشن: | edsarx.1109.6522 |
| قاعدة البيانات: | arXiv |
| DOI: | 10.1063/1.3649946 |
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