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Scattering theory - phase shift - best approx.

  1. Jun 15, 2005 #1
    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 [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 :confused:).

    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?
  2. jcsd
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