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.