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262
Chapter 4. Particle scattering
Figure 4.6: Momentum transfer and scattering angle.
The quantity q is related to scattering angle ϑ by
q = 2k sin (ϑ/2),
(4.100)
as is evident from Figure 4.6. Substituting, we find (4.99, 4.100)
mZze
2 2
1
σ(ϑ) =
2 .
(4.101)
h
2
2πf 0 ¯
4k 2 sin
2 (ϑ/2) + α 2
In the limit α → 0, we obtain (4.101)
Zze
2
2
σ(ϑ) =
,
(4.102)
16πf 0 H sin
2 (ϑ/2)
where H is the total energy eigenvalue of the incident particle. This
is equal to the kinetic energy of the particle outside of the scattering field. This is identical with the result for Rutherford scattering
(4.44, 4.54) by an unscreened nucleus, derived previously from classical mechanics. This remarkable equivalence between the classical
and quantum mechanical results only holds true for the scattering
potential energy inversely proportional to the separation r.
We now proceed to integrate the differential cross section σ over
�
262
Chapter 4. Particle scattering
Figure 4.6: Momentum transfer and scattering angle.
The quantity q is related to scattering angle ϑ by
q = 2k sin (ϑ/2),
(4.100)
as is evident from Figure 4.6. Substituting, we find (4.99, 4.100)
mZze
2 2
1
σ(ϑ) =
2 .
(4.101)
h
2
2πf 0 ¯
4k 2 sin
2 (ϑ/2) + α 2
In the limit α → 0, we obtain (4.101)
Zze
2
2
σ(ϑ) =
,
(4.102)
16πf 0 H sin
2 (ϑ/2)
where H is the total energy eigenvalue of the incident particle. This
is equal to the kinetic energy of the particle outside of the scattering field. This is identical with the result for Rutherford scattering
(4.44, 4.54) by an unscreened nucleus, derived previously from classical mechanics. This remarkable equivalence between the classical
and quantum mechanical results only holds true for the scattering
potential energy inversely proportional to the separation r.
We now proceed to integrate the differential cross section σ over