# Determining a Scattering Cross Section (Quantum Mechanics)

• CDL
In summary, scattering of a particle of mass ##m## on the potential occurs when the potential is localized in a region defined by ##b##. The cross section is given by ##\sigma## and is a function of the energy of the particle.
CDL

## Homework Statement

Consider scattering of a particle of mass ##m## on the potential
$$U(r) = \begin{cases} 0, & r \geq b\\ W, & r < b \\ \end{cases}$$
Where ##W## is some arbitrary chosen constant, and the radius ##b## is considered a small parameter. Find the cross section ##\sigma## in the limit ##b \to 0## (to avoid confusion, find the first non-zero term in the expansion of ##\sigma## over ##b##). Prove in particular, that the limit does not depend on the energy ##E## of the particle.

## Homework Equations

I am thinking of using partial wave analysis for this problem, since we have a "localised" potential. In particular, using

$$\sigma = \frac{4 \pi}{k^2} \sum_{l} (2l+1) \text{sin}^2 \delta_l$$
One thing that I am struggling to get my head around is how to find the phase shifts, ##\delta_l##. After some searching around, I found the following formulas (not that I really understand them), $$\text{tan}\delta_l = \frac{k a j'_l(kb) - \beta_{l +} j_l(kb)}{kay'_{l}(ka) - \beta_{l+} y_l(ka)}$$ Where ##j## and ##y## denote the spherical Bessel functions. Also, $$\beta_{l+} = \frac{1}{\mathcal{R}} \frac{d\mathcal{R}}{dr} \bigg\rvert_{r = b}$$ with $$\mathcal{R} = e^{i \delta_l} \left(\text{cos}\delta_l \ j_l(kr) -\text{sin}\delta_l \ y_l(kr)\right)$$

## The Attempt at a Solution

After using the equations in the above section, and 2 pages of working, I didn't get far. Those equation don't seem to use any information about the form of potential. I wanted to find the partial wave phase shifts until I hit the first non-zero one, and then sub it into the expansion for ##\sigma##. I'm not sure how this would correspond to the first term in the expansion of ##\sigma## over ##b## though.

My whole understanding of this topic is kind of shaky, and reading Griffiths and the lecture notes doesn't seem to be getting me very far. Any suggestions on how to get a good grasp of this stuff?

Well, I am still researching this but I thought I would get back to you with the one consideration I have found so far. That is, if ##b<<1/k## where ##k## is the incident wave vector then s wave scattering ##(l=0)## dominates. But then, maybe you already knew that.

## 1. What is a scattering cross section?

A scattering cross section is a measure of how likely it is for particles to scatter off of each other or off of a target. It is a fundamental quantity in quantum mechanics that helps us understand the interactions between particles.

## 2. How is a scattering cross section determined?

A scattering cross section is determined by using mathematical equations and experimental data. The equations take into account the properties of the particles and the target, as well as the energy and angle of the scattering. The experimental data is used to validate the calculations and refine the results.

## 3. What are the units of a scattering cross section?

The units of a scattering cross section depend on the system being studied. In quantum mechanics, it is typically measured in units of area, such as square meters or barns (10^-28 square meters).

## 4. How does the scattering cross section relate to other properties of particles?

The scattering cross section is related to other properties of particles, such as their size and charge. It can also provide information about the strength of their interactions and the nature of their internal structure.

## 5. What are the practical applications of determining a scattering cross section?

Determining a scattering cross section has many practical applications in fields such as nuclear physics, astrophysics, and materials science. It can help us understand the behavior of particles in different environments and can be used to design experiments and technologies that rely on particle interactions.

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