Start from the 1" order Born approximation result given in Eq. (6.72 or 6.3.3), and showing all work - even integral eva

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Start From The 1 Order Born Approximation Result Given In Eq 6 72 Or 6 3 3 And Showing All Work Even Integral Eva 1 (30.38 KiB) Viewed 55 times

Based on Graduate Quantum Mechanics, Sakurai, see below:

Start From The 1 Order Born Approximation Result Given In Eq 6 72 Or 6 3 3 And Showing All Work Even Integral Eva 2 (17.46 KiB) Viewed 55 times

Start From The 1 Order Born Approximation Result Given In Eq 6 72 Or 6 3 3 And Showing All Work Even Integral Eva 3 (13.37 KiB) Viewed 55 times

Start From The 1 Order Born Approximation Result Given In Eq 6 72 Or 6 3 3 And Showing All Work Even Integral Eva 4 (14.27 KiB) Viewed 55 times

Start From The 1 Order Born Approximation Result Given In Eq 6 72 Or 6 3 3 And Showing All Work Even Integral Eva 5 (22.31 KiB) Viewed 55 times

Start from the 1" order Born approximation result given in Eq. (6.72 or 6.3.3), and showing all work - even integral evaluations, etc: a) Derive, in all details, the expression for scattering amplitude for a general spherically symmetric potential V(r'). This means to derive Eq. (6.74 or 6.3.5). b) Apply this result to find the scattering amplitude for a Yukawa potential in all of its details (Eq. 6.81 or 6.3.12). c) Reduce this result to the Coulomb scattering for an incoming particle of mass m and charge Ze, on a heavy target of charge Zle.
Taking the first term in the expansion, i.e. T= V or, equivalently, IV (+) = |k), is called the first-order Born approximation. In this case, the scattering amplitude is denoted by f("), where m f(1)(k',k)=- 270 | dx'e'l-K)x"[x') (6.72)
We can perform the angular integration in (6.72) explicitly to obtain 1 2m 1 f(!)(O)= V(r) (eign – e-ir) dr 2 ha iq 2m 1 rV(r) sin qr dr. 9 == Som vir)(eier ( S. - th? (6.74)
So, in the first Born approximation, the differential cross section for scattering by a Yukawa potential is given by do 2mV 1 (6.81) [2k+(1 - cos 6) +4212 dQ uh?
It is amusing to observe here that as u → 0, the Yukawa potential is reduced to the Coulomb potential, provided the ratio Volu is fixed, for example, to be ZZ'e?, in the limiting process. We see that the first Born differential cross section obtained in this manner becomes do (2m) (ZZ'ea)2 1 (6.82) dΩ 16k4 sinº(0/2)