Knowledge Bank

We have shown that one and two-parameter analytical mapping functions such as $r(y;r¯,\alpha )=r¯[1+1\alpha \tan (\pi y/2)]$ and $r(y;r¯)=r¯[1+y1-y]$ transform the conventional radial Schr{" o}dinger equation into equivalent alternate forms \vspace{-2mm} $$ \frac{d^2\phi(y)}{dy^2}~=~ \left[\frac{\pi^2}{4}+\left(\frac{2\mu} {\hbar^2} \right) g^2(y) ,[E - U(r(y))]\right]\phi(y) \hspace{7mm} {\rm and} \hspace{7mm} \frac{d^2\phi(y)}{dy^2}~=~\left(\frac{2\mu}{\hbar^2}\right) g^2(y)\left[E - U(r(y)) \right]\phi(y) \vspace{-2mm} $$ % respectively, in which $g(y)=dr(y)/dy$. \textbf{78}, 052510 (2008).}~ Such transformed equations are defined on the finite domain $y\in [-1,1]$, 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 $as$ can be expressed in terms of the logarithmic derivative of the wave function $\varphi (y)$ at the right-hand boundary point: \vspace{-2mm} $$ a_s~=~\bar{r}\left[\frac{2} {\pi\alpha}~\frac{1} {\phi(y)}~\frac{d\phi(y)} {dy}+1\right]{y=1} \hspace{7mm} {\rm and} \hspace{7mm} a_s~=~\bar{r}\left[,2~\frac{1}{\phi(y)}~\frac{d\phi(y)}{dy}~-1\right]{y=1} $$ % The required logarithmic derivative of $\varphi (y)$ 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 $r\to \infty$. This zero-energy wavefunction may also be combined with wavefunctions for ordinary bound states generated in the same manner to calculate photoassociation absorption matrix elements using any appropriately modified Franck-Condon computer program. \smallskip VVM is grateful to INTAS grant 06-1000014-5964 for support.