Quantum mechanics

Quantum scattering

Scattering relates an incident wave to an angular distribution of outgoing amplitudes; the Schrödinger equation and Born approximation predict how an interaction potential shapes it.

In elastic scattering, a particle with definite asymptotic energy encounters a potential $V(r)$ and is detected in an outgoing direction. The basic observable is the differential cross section $d\sigma/d\Omega$ , not a classical trajectory for each particle.

\psi(\mathbf r)\underset{r\to\infty}{\sim}e^{ikz}+f(\theta)\frac{e^{ikr}}{r},\qquad\frac{d\sigma}{d\Omega}=|f(\theta)|^2

Definition: Wave packets and cross sections

A wave packet has a finite momentum spread and models a beam more realistically than an ideal plane wave. Its center approaches the scattering region; deflected amplitude forms a far-field probability pattern. A cross section has units of area and encodes scattering probability per incident flux and solid angle.

Control the packet and Yukawa/Coulomb potential to inspect probability density, interference, and scattering directions.

The Born approximation

If the potential is weak compared with the kinetic energy, the incident wave can replace the exact scattering wave in the Lippmann–Schwinger integral. First Born amplitude is proportional to the Fourier transform of $V$ at momentum transfer $\mathbf q=\mathbf k_f-\mathbf k_i$ . It can fail at low energy, for strong potentials, or near resonances.

Example: Example: Coulomb scattering

An unscreened Coulomb potential is long-ranged and its Rutherford cross section is strongly forward-peaked. A Yukawa potential $V(r)\propto e^{-r/a}/r$ adds screening length $a$ , making the interaction effectively finite-ranged; increasing $a$ restores more small-angle scattering.

Solution

The factor $e^{-r/a}$ suppresses the interaction at distances much larger than $a$ ; the limit $a\to\infty$ recovers the unscreened Coulomb potential.

For elastic scattering by a central potential, the amplitude has a partial-wave expansion $f(\theta)=\sum_l(2l+1)f_lP_l(\cos\theta)$ . Each channel carries a phase shift $\delta_l$ ; for a real potential, $f_l=e^{i\delta_l}\sin\delta_l/k$ . The total cross section sums channel contributions, expressing probability conservation through unitarity of the S-matrix. At low energy, large-angular-momentum channels are usually suppressed by the centrifugal barrier, so the s-wave can dominate. Energy-dependent cross sections reveal scattering lengths and resonant structure.

Quick check

In scattering, the differential cross section is proportional to what?

What distinguishes Yukawa from unscreened Coulomb?

References

Eugen Merzbacher (1998). Quantum Mechanics