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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 [tex]\delta_l[/tex] of the partial waves in the partial wave expansion of the scattered wave.

One way (the semi-classical) to calculate it is through

[tex] \delta_l = \int^{r}_{r_0} k(r')dr' - kr [/tex],

where [tex]r_0[/tex] is the "border" of the classically allowed area for the particle(s) and [tex]k(r)[/tex] is the "local" wave number (right choice of word ).

The other way is through the 1st order Born approximation:

[tex]-\frac{tan ~\delta_l}{k} = <u^0_l | U | u^0_l> [/tex]

Now if I remember right the second method was presented as the for sure prefered 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 [tex]u_l[/tex] of a free particle, that is [tex]u_l^0[/tex]. It must be very depending on the kind of scattering potential we're dealing with, or?