What is Scattering Theory?

Explanation¹

What is Scattering Theory?

Scattering theory studies the effects that an inhomogeneous medium has on incoming particles or waves. It explains why the sky is blue and Rutherford’s alpha particle scattering experiment, and is also applied in medical imaging technologies such as tomography.

Problems Addressed

Scattering theory can broadly be divided into quantum scattering theory and classical scattering theory. A fundamental problem in classical scattering theory involves “boundaries with impenetrable obstacles” and “media that can be penetrated and have compact support” causing scattering of time-harmonic acoustic/electromagnetic waves.

In scattering theory, the total field $u$ is divided into the incident field $u^{i}$ and the scattered field $u^{s}$.

$$ u(x,t) = u^{i}(x,t) + u^{s}(x,t) $$

To understand, let’s consider a simple example.

In the above situation, $u^{i}, u^{s}$ are as follows.

$$ u^{i}(x) = \begin{cases} a & x \le x_{0} \\ 0 & x \gt x_{0} \end{cases} \quad \text{and} \quad u^{s}(x) = \begin{cases} 0 & x \le x_{0} \\ cx+d & x \gt x_{0} \end{cases} $$

Then, the total field $u$ is as follows.

$$ u(x) = u^{i}(x) + u^{s}(x) = \begin{cases} a & x \le x_{0} \\ cx+d & x \gt x_{0} \end{cases} $$

What is Scattering Theory?

Explanation1