Scattering theory - phase shift - best approx.

[jonas_nilsson]

Hi all,

we're looking at scattering theory in the QM course right now, and I've got a question concerning the approximative ways of calculating the phase shift \delta_l of the partial waves in the partial wave expansion of the scattered wave.

One way (the semi-classical) to calculate it is through
\delta_l = \int^{r}_{r_0} k(r')dr' - kr,
where r_0 is the "border" of the classically allowed area for the particle(s) and k(r) is the "local" wave number (right choice of word ).

The other way is through the 1st order Born approximation:
-\frac{tan ~\delta_l}{k} = <u^0_l | U | u^0_l>

Now if I remember right the second method was presented as the for sure preferred one. My question is: how's that?. How can we be sure that this is the best way. The clue might be that the first is (semi-) classical, but on the other hand it seems quite rough to just use the u_l of a free particle, that is u_l^0. It must be very depending on the kind of scattering potential we're dealing with, or?

 

[DrClaude]

The criterion for the validity of the Born approximation is
$$
\frac{m | V_0 | a^2}{\hbar^2} \ll 1
$$
where $m$ is the mass of the particle, $V_0$ and $a$ the height and range of the potential, respectively. Therefore, if this condition is met, then the Born approximation is a good approximation, and the result will be better than using a semi-classical approximation. One can also go further and use the second Born approximation to refine the result.