Quantum Scattering

Experimentalists often shoot particles (that are effectively plane waves) at material and observe what comes out a relatively far distance away. Two common approaches for modeling such a scenario are the Born approximation and partial wave analysis. Below, we consider these techniques in the case of a stationary target with spherical symmetry.

Scattering is one of the key experimental techniques we have for probing the nature of matter. In most cases, we take the incident projectile to be a plane wave and the output wave is generally approximated in some way, as a means to infer material structure from scattering data. We consider the scattering center to be some localized potential, so we have the following (time dependent) Schrodinger equation.

$$ i\hbar \frac{d}{dt}\psi = \left[-\frac{\hbar^2}{2m}\nabla^2 +V(r)\right]\psi $$

We now consider a scenario in which we have an effectively stationary wave function in time. This may apply to scenarios where we run our apparatus for an extended period of time, such that we have a superposition of incident plane waves and scattered waves. This gives us the relevant time independent Schrodinger equation defined below.

$$ E\psi = \left[-\frac{\hbar^2}{2m}\nabla^2 +V(r)\right]\psi $$

We require that our solution for $\psi$ includes an incident plane wave, which we will generally choose to align with the z axis for simplicity. And now, we further suppose that the outgoing wave, the result of the scattering, is a spherical wave, with an amplitude potentially dependent on the solid angle (relative to the scattering center).

$$ \psi(r,\phi,\theta) \sim Ae^{ikz}+Bf(\phi,\theta)\frac{e^{ikr}}{r} $$

Note that this assumption only really works for observations made at a large radial distance from the scattering center. With this assumption in mind, working with this form allows us to make some headway.

Cross Sections

There are two types of relevant cross sections in scattering experiments with one target: the total cross section, that is, the area of the target as seen from the direction of the incident particle; and the differential cross section, which encodes how likely a scattered particle will be found in a certain range of solid angle.

The scattering amplitude of our model wave function is related to the differential cross section via the below relation.

$$ \vert f(\phi,\theta)\vert^2=\frac{d\sigma}{d\Omega} $$

Born Approximation

The Born approximation is a truncation of terms in the expansion of the scattering amplitude. Effectively, it's an application of time-dependent perturbation theory to a central potential. It tends to work better for high energy scattering.

The first order correction to the scattering amplitude in the Born series is given below as a function of incident wave vector $\vec{k}$ and scattered wave vector $\vec{k}'$.

$$ \begin{align} f(\vec{k},\vec{k}')_B^{(1)}=-\frac{m}{2\pi \hbar^2}\int_V d\vec(x)' e^{i(\vec{k}-\vec{k}')\cdot \vec{x}'}V(\vec{x}') \end{align} $$

Partial Wave Analysis

Our approach to partial wave here is to expand both the incident and scattered waves in terms of spherical harmonics, and then analyze coefficients in terms of the quantum numbers $\ell$ and $m$.

If we take the incident quantum projectile to be a plane wave aligned with the z-axis, we may expand it's wave function in terms of spherical harmonics as below.

$$ e^{ikz}=e^{ikr\cos\theta}=\sum_{\ell =0}^{\infty}(2\ell+1)i^{\ell} j_{\ell}(kr)P_{\ell}(\cos\theta ) $$

$$ f(\phi,\theta)=\sum_{\ell =0}^{\infty}\sum_{m=-\ell}^{\ell}f_{\ell}^mY^m_{\ell}(\phi,\theta) $$

$$ \begin{align} \psi(r,\theta,\phi)&=e^{ikr\cos\theta}+f(\theta,\phi) \frac{e^{ikr}}{r}\\ &=\sum_{\ell =0}^{\infty}\Big[(2\ell+1)i^{\ell} j_{\ell}(kr)P_{\ell}(\cos\theta ) + f_{\ell} j_{\ell}(kr)P_{\ell}(\cos\theta )\Big] \end{align} $$

Resources

Further resources include chapter 6 of Sakurai and Napolitano's Modern Quantum Mechanics (Third Edition), and Chapter 11 of David Griffith's Introduction to Quantum Mechanics (Second Edition). See also this thesis by Adam Lupu-Sax; and this set of lecture slides by Ben Simons which are accompanied by these notes.