the [itex]l[/itex]-quantum number of the incoming wave. You often have a plane wave coming in to the target. The plane wave you write as a linear combination of spherical waves, and you call this "Partial Wave expansion". http://farside.ph.utexas.edu/teaching/qm/lectures/node70.html The [itex]l[/itex] that you see in eq 957 is then the "[itex]l[/itex] - QM number". And [itex]l[/itex] is denoted by, 0 = s, 1 = p, 2 = d, etc, same as in atomic physics notation. Semiclassicaly, you can see the [itex]l[/itex] as the classical angular momenta of the incoming particle with respect to the centre of the scattering potential. And also the [itex]l[/itex] is quantisized, so only some values of [itex]l[/itex] are allowed. Now since the sum goes to infinity in eq 957, we cut of where we expect no partial waves to contribute. And that is often assigned by [tex] l_{max} \approx R\cdot k [/tex] Where R is the range of the potential and k is the momenta of the incoming particle (wave number). Now the cross section is proportional to the scattering amplitude modulus square, i.e the modulus square of eq. 965 times a constant with alot of pi's hbar's etc. So the s-cross section, you only have [itex]l[/itex] = 0 in you sum, and p-cross section only [itex]l[/itex] = 1. etc. I hope you got the idea =)
Hello, To munch the QM into an analogy: Another way to look at is - how does one particle look to another. If you assume the target particle to be ball-like, then in its basic form (ground state), you'll get the classic 3D cross-section - this is how it will appear to the incoming particle and such it will be scattered from the target particle. But if the particle is excited to a higher state, then it will no longer appear as a ball but something else entirely. And vice versa. Smoochie