Partial wave expansion

$\includegraphics[width=0.75\columnwidth, clip]{C:/webpage/figs/partwave.eps}$

As in classical mechanics the two-body problem is conveniently described in the center-of-mass system of the particles where it is reduced to the equivalent problem of a single particle moving in a potential. The wave function solving the Schrödinger equation is at large distances (beyond the range of the interaction potential) represented by an incoming plane wave along the

axis and an outgoing spherical wave

$\begin{displaymath} \psi \sim e^{i k z}+f(\theta)e^{i kr}/r,\end{displaymath}$

(2)

where

is the magnitude of the relative wave-vector of the colliding particles. $f(\theta)$ is the energy dependent complex scattering amplitude which depends on the details of the interaction potential. In partial wave analysis the scattering amplitude is expanded as

$\begin{displaymath} f(\theta)=(1/2i k)\sum_{l=0}^{\infty}(2l+1)(e^{2\i\eta_l}-1)P_l(\cos \theta), \end{displaymath}$

(3)

where

is the Legendre polynomial of order

and $\eta_l$ are the partial wave phase shifts. The differential cross section is the squared modulus of the scattering amplitude

$\begin{displaymath} \frac{d\sigma}{d\Omega} =\vert f(\theta)\vert^2, \end{displaymath}$

(4)

and has an angular pattern which depends crucially on the quantum mechanical interference between the partial wave states as dictated by the phase shifts. The total elastic cross section can be expressed as

$\begin{displaymath} \sigma=\int\frac{d\sigma}{d\Omega}d\Omega =2\pi\int_{0}^\pi\vert f(\theta)\vert^2\sin\theta d\theta=\sum_l \sigma_l, \end{displaymath}$

(5)

where

$\begin{displaymath} \sigma_l=4\pi(2l+l)\sin^2\eta_l/k^2 \end{displaymath}$

(6)

are the partial cross sections.

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nk 2004-11-02