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Physical Review A

Effective-range theory for quantum reflection amplitudes

Florian Arnecke, Harald Friedrich, Javier Madroñero

Published Dec 4, 2006• 37 citations
Publisher

Abstract

We formulate an effective-range theory for the near-threshold behavior of the amplitudes describing quantum reflection in attractive potential tails by adapting the effective-range theory of ordinary elastic scattering to the case of incoming boundary conditions at small distances. For homogeneous attractive potentials proportional to $\ensuremath{-}1∕{r}^{\ensuremath{\alpha}}$ with $\ensuremath{\alpha}>5$, the effective range turns out to be a real multiple (with known coefficient) of the complex scattering length which defines the leading, linear momentum dependence of phase and modulus of the quantum reflection amplitude. Analytical expressions are also given for the leading and next-to-leading terms in the near-threshold behavior of the quantum reflection amplitudes for homogeneous attractive potentials proportional to $\ensuremath{-}1∕{r}^{3}$, $\ensuremath{-}1∕{r}^{4}$, and $\ensuremath{-}1∕{r}^{5}$.

DOI
10.1103/physreva.74.062702

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We formulate an effective-range theory for the near-threshold behavior of the amplitudes describing quantum reflection in attractive potential tails by adapting the effective-range theory of ordinary elastic scattering to the case of incoming boundary conditions at small distances. For homogeneous attractive potentials proportional to \ensuremath−1∕r\ensuremathα\ensuremath{-}1∕{r}^{\ensuremath{\alpha}}\ensuremath−1∕r\ensuremathα with \ensuremathα>5\ensuremath{\alpha}>5\ensuremathα>5, the effective range turns out to be a real multiple (with known coefficient) of the complex scattering length which defines the leading, linear momentum dependence of phase and modulus of the quantum reflection amplitude. Analytical expressions are also given for the leading and next-to-leading terms in the near-threshold behavior of the quantum reflection amplitudes for homogeneous attractive potentials proportional to \ensuremath−1∕r3\ensuremath{-}1∕{r}^{3}\ensuremath−1∕r3, \ensuremath−1∕r4\ensuremath{-}1∕{r}^{4}\ensuremath−1∕r4, and \ensuremath−1∕r5\ensuremath{-}1∕{r}^{5}\ensuremath−1∕r5.
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We formulate the quantization rule ${n}_{\mathrm{th}}\ensuremath{-}n=F({E}_{n})$ for bound-state energies En{E}_{n}En​ in a deep potential well with a long-range attractive tail for an extended range of near-threshold energies. This makes it possible to accurately read off the scattering length aaa from the energy of a high-lying (not necessarily the highest-lying) bound state via $a=\overline{a}+b∕\mathrm{tan}[\ensuremath{\pi}F({E}_{n})]$. For a potential tail proportional to $\ensuremath{-}1∕{r}^{\ensuremath{\alpha}}$, $\ensuremath{\alpha}>3$, the constants a‾\overline{a}a and bbb are known and an explicit and universally valid expression for the function F(E)F(E)F(E) is given analytically.
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For potentials with attractive tails, as occur in typical atomic interactions, we present a simple formula for the sss-wave phase shift ${\ensuremath{\delta}}_{0}$. It exposes a universal dependence of ${\ensuremath{\delta}}_{0}(E)$ on the potential tail and the influence of effects specific to a given potential, which enter via the scattering length aaa, or equivalently, the noninteger part ${\ensuremath{\Delta}}_{\mathrm{th}}$ of the threshold quantum number nth{n}_{\mathrm{th}}nth​. The formula accurately reproduces ${\ensuremath{\delta}}_{0}(E)$ from threshold up to the semiclassical regime, far beyond the validity of the effective-range expansion. We derive the tail functions occurring in the formula for ${\ensuremath{\delta}}_{0}(E)$ and demonstrate the validity of the formula for attractive potential tails proportional to 1/r61/{r}^{6}1/r6 or to 1/r41/{r}^{4}1/r4, and also for a mixed potential tail consisting of a 1/r41/{r}^{4}1/r4 term together with a non-negligible 1/r61/{r}^{6}1/r6 contribution.