Finite and infinite cross sections

[The_Duck]

The cross section for scattering by a Coulomb potential 1/r is the same for both classical and quantum mechanics, and the total cross section is infinite. I understand this classically as saying that no matter how large an impact parameter an incoming particle has, it will still be deflected at least a little bit by the potential, so the cross sectional area in which incoming particles are scattered at some angle > 0 is infinite.

I have seen the Born approximation of the quantum mechanical scattering cross section for a Yukawa potential e^(-mr)/r. Here the total cross section is finite. In the classical case, though, I feel like the same argument as was used for the Coulomb potential applies: although the potential falls off quickly, there is always some force at arbitrarily large distances from the origin, so all incoming particles should be deflected at least a little bit, no matter how large their impact parameters. So I expect that in classical mechanics, the total cross section for the Yukawa potential is infinite.

I'm somewhat uncomfortable with this cross section being infinite in classical mechanics, but finite in quantum mechanics. Is there a conflict here?

 

[tom.stoer]

No, for the Yukawa potential the classical total cross section is finite, too. The total cross section can is only in rare cases related to the "size" of the target.

 

[The_Duck]

I don't see how this can be :( In the classical case, consider the function b(\theta) which gives the impact parameter as a function of the scattering angle. For the Yukawa potential in particular and for a broad class of potentials I expect that this function has b(pi) = 0 (particles shot directly at the target bounce back) and b(theta->0) = infinity (you have to use arbitrarily large impact parameters to get arbitrarily small deflections), and b(theta) monotonically decreasing from theta=0 to theta=pi. Then write

\sigma = \int \frac{d\sigma}{d \Omega} d \Omega

My classical mechanics book tells me I can rewrite the differential scattering cross section:

= \int \left ( \frac{b}{\sin \theta} \left | \frac{d b}{d \theta} \right | \right ) (\sin \theta d \theta d \phi)

= 2 \pi \int_0^\pi b \left | \frac{db}{d \theta} \right | d \theta

If b satisfies the conditions I expect it to then this is

= -2 \pi \int_0^\pi b \frac{db}{d \theta} d \theta

= -2 \pi \int_\infty^0 db

= \infty

Where do I go wrong?

 

[tom.stoer]

Good question. You should really check if all the conditions are met.

If you do the calculation in Born approx. (which should be equivalent to a classical approx.) then you see that in the Yukawa case the cross sections are modified due to the "range" a in exp(-r/a). The theta-diveregence in the denominator (for the Rutherford cross section) is softened as you get something like 1/(sin² + 1/a²)². Therefore the theta-integration is finite for the Yukawa potential.

 

[The_Duck]

I understand the Born approximation calculation of the QM cross section of the Yukawa and Coulomb potentials. In what way is it equivalent to something classical?

 

[tom.stoer]

The Born approx. corresponds to tree graphs w/o quantum loops. You can see this for the Coulomb potential where the classical calculation based on trajectories and the qm Born approx. are identical.