Adaptive Analytic Mapping Procedures for Simple and Accurate Calculation of Scattering Lengths and Photoassociation Absorption Intensities

Details Adaptive Analytic Mapping Procedures for Simple and Accurate Calculation of Scattering Lengths and Photoassociation Absorption Intensities Adaptive Analytic Mapping Procedures for Simple and Accurate Calculation of Scattering Lengths and Photoassociation Absorption Intensities

Jun 2009

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=2009mss..confemi08l&link_type=abstract

"International Symposium On Molecular Spectroscopy, 64th Meeting, Held 22-26 June, 2009 at Ohio State University. http://molspec

Physics

Electronic

Scientific paper

We have shown that one and two-parameter analytical mapping functions such as
r(y;bar{r}, α)=bar{r}[1 + {1}/{α} tan(π y/2)] and r(y;bar{r})=bar{r} [ {1+ y}/{1-y} ] transform the conventional radial Schrödinger equation into equivalent alternate forms
{d^2φ(y)}/{dy^2} = [{π^2}/{4}+({2μ}/ {hbar^2} ) g^2(y) [E - U(r(y))]]φ(y) and {d^2φ(y)}/{dy^2} = ({2μ}/{hbar^2}) g^2(y)[E - U(r(y)) ]φ(y)
respectively, in which g(y)=dr(y)/dy . Such transformed equations are defined on the finite domain
yin [-1,1], and they may be solved routinely using standard numerical methods at all energies up to and including the potential asymptote. At the energy of the potential asymptote, the s-wave scattering length a_s can be expressed in terms of the logarithmic derivative of the wave function φ(y) at the right-hand boundary point:
a_s = bar{r}[{2}/ {πα} {1}/ {φ(y)} {dφ(y)}/ {dy}+1]_{y=1} and a_s = bar{r}[ 2 {1}/{φ(y)} {dφ(y)}/{dy} -1]_{y=1}
The required logarithmic derivative of φ(y) can be obtained efficiently by direct outward integration of the differential equation all the way to the end point y=1, which corresponds to the limit rto infty. This zero-energy wavefunction may also be combined with wavefunctions for ordinary bound states generated in the same manner^a to calculate photoassociation absorption matrix elements using any appropriately modified Franck-Condon computer program.
VVM is grateful to INTAS grant 06-1000014-5964 for support.
V.V. Meshkov, A.V. Stolyarov, and R.J. Le Roy, Phys. Rev. A 78, 052510 (2008)

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