Scattering Cross-section Problems in Shankar 19.3.2 and 19.3.3?

[Old Guy]

Homework Statement

Problems are Shankar 19.3.2 and 19.3.3 with spherically symmetric potentials V(r)=-V$$_{0}(r_{0}-r)\theta$$ and V(r)=V$$_{0}exp(-r_{2}/r^{0}_{2}^{})$$

Homework Equations

$$f\left( \theta \right) = - \frac{{2\mu }}{{\hbar ^2 }}\int\limits_{r_0 }^r {\frac{{\sin qr'}}{q}V\left( {r'} \right)r'dr'}$$
and
$$\frac{{d\sigma }}{{d\Omega }} = \left| {f\left( \theta \right)} \right|^2$$

The Attempt at a Solution

Don't I just substitute the potentials for V(r) and integrate? The example in Shankar seemed to do that successfully for the Yukawa potential. What am I missing?

 

[malawi_glenn]

Have you tried it?

 

[Old Guy]

Yes, I did try. Shankar gives the answer to the first one:
$$\frac{d\sigma}{d\Omega}= 4r_0^2 \left( {\frac{{\mu V_0 r_0^2 }}{{\hbar ^2 }}} \right)^2 \frac{{\left( {\sin qr_0 - qr_0 \cos qr_0 } \right)^2 }}{{\left( {qr_0 } \right)^6 }}$$


Questions that arise from this:
What happened to $$\theta$$?
What are the appropriate limits of integration - r$$_{0}$$ to $$\infty$$?

 

[huuha]

why to study the structure of nuclear scattering is often used as Compton scattering without using the other?

 

[Jolb]

I was just working through Shankar 19.3.3 and it's seriously a tough problem (if you don't use mathematica.) I thought it might be nice to put my hints on here.

For finding ∂σ/∂Ω, I used the following tricks (in the order listed):

1. Get rid of the r in the integrand by saying r sin(qr) = ∂/∂q[cos(qr)]

2. Write the cos as the sum of exponentials

3. Combine the two exponential integrals

4. Complete the square for the exponential integral

5. Change variables to turn it into a gaussian integral

And then to find σ

6. Use shankar's hint to change the integral over q into an integral over cosθ
I think it officially qualifies as a "tricky" problem.

13.3.2 just requires you to evaluate a limit using L'Hopital's rule--it's not nearly as challenging.