5.6. Summary

In this chapter, we have analysed scattering of particles by a potential localised in a finite region around some point, which we take as the origin. The starting point of the quantum mechanical analysis is the wave function far from the scattering centre, which reads: $$\psi({\bf r}) = e^{{\rm i}{\bf k}_{\text{i}} \cdot {\bf r}} + f(\vartheta,\varphi) \frac{e^{{\rm i}kr}}{r}.$$ The first term represents the incoming wave (the wave vector ${\bf k}_{\text{i}}$ is usually taken along the $z$-axis) and the second term represents the scattered wave whose amplitude can be measured as a function of $\vartheta$ and $\varphi$ at a detector -- this amplitude is given as the differential cross section: $$\frac{d\sigma}{d\Omega} = \left| f(\vartheta,\varphi)\right|^2.$$ The shape of the function $f(\vartheta,\varphi)$ is determined by the interaction potential $V({\bf r})$, which is often taken to be spherically symmetric: $V({\bf r}) = V(r)$. The total cross section is the integral of the differential cross section: $$\sigma_{\text{tot}} = \int \frac{d\sigma}{d\Omega} \sin\vartheta d\vartheta d\varphi.$$

The calculation of $f(\vartheta,\varphi)$ from the scattering potential $V$ can be performed quite easily provided the interaction potential is weak. In that case, we can use the exact Green's function solution: $$\psi({\bf r}) = \phi({\bf r}) + \int G_0({\bf r}, {\bf r}') V({\bf r}') \psi({\bf r}') \; d^3 r'$$ and its first Born approximation: $$\psi({\bf r}) = \phi({\bf r}) + \int G_0({\bf r}, {\bf r}') V({\bf r}') \phi({\bf r}') \; d^3 r'.$$ From this last equation, the form of $f(\vartheta,\varphi)$ can readily be derived: $$f(\vartheta, \varphi) = -\frac{m}{2\pi \hslash^2} \int V({\bf r}') e^{{\rm i}({\bf k}_{\text{i}}-{\bf k}_{\text{f}})\cdot {\bf r}'} \; d^3 r'.$$ We usually set ${\bf q}= {\bf k}_{\text{i}}-{\bf k}_{\text{f}}$. For a spherically symmetric potential, the integral depends only on the length $q$ of this vector. This length is given as $$q = 2 k \sin(\vartheta/2),$$ where $k$ is the length of the wave vector of the in- and outgoing waves, and $\vartheta$ is the scattering angle.

The Born approximation yields the exact result for scattering off a Coulomb potential (the Rutherford formula): $$\frac{d\sigma}{d\Omega} = \left[ \frac{q_1 q_2}{16 \pi \epsilon_0 E \sin^2 (\vartheta/2)}\right]^2.$$

We have derived the optical theorem as a consequence of the conservation of matter: $$\sigma_{\text{tot}} = {\text{Im}} \frac{4\pi f(\vartheta=0)}{k}.$$

We then have approached the quantum scattering problem from a very different angle: Partial Wave Analysis. In this formalism, we expans the incoming wave in terms of different $l$, and for each of these, we separately calculate the scattering waves and their contribution to the scattering cross sections. The central quantity which contains the information concerning the scattering of a wave with definite $l$, is the phase shift, $\delta_l$. This can be found from solving the radial Schrödinger equation with the scattering potential $V(r)$, and then identifying the radial part opf the solution for large $r$ as a shifted spherical Bessel function: $$R_l (r {\text{large}}) \rightarrow j_l (kr+\delta_l).$$ From these $\delta_l$ we obtain: $$\frac{d\sigma}{d\Omega} = \frac{1}{k^2} \left| \sum_{l=0}^\infty (2l+1) e^{i\delta_l} \sin(\delta_l) P_l (\cos \theta)\right|^2.$$ For the total cross section we find, using the orthonormality relations of the Legendre polynomials: $$\sigma_{\text{tot}} = 2 \pi \int d\theta \sin \theta \frac{d\sigma}{d\Omega}(\theta) = \frac{4\pi}{k^2} \sum_{l=0}^\infty (2l+1) \sin^2 \delta_l.$$