💫Intro to Quantum Mechanics II Unit 8 – Scattering Theory & Partial Wave Analysis

Key Concepts and Foundations

• Scattering theory describes the interaction between particles or waves and a target
• Assumes particles or waves are far from the target, allowing for asymptotic approximations
• Incident particles or waves are represented by plane waves with well-defined momentum and energy
• Scattered particles or waves are represented by spherical waves emanating from the target
• Scattering amplitude $f(\theta, \phi)$ encodes the probability amplitude of scattering in a given direction
  • Depends on the scattering angle $\theta$ and azimuthal angle $\phi$
• Differential cross section $\frac{d\sigma}{d\Omega} = |f(\theta, \phi)|^2$ measures the probability of scattering into a solid angle $d\Omega$
• Total cross section $\sigma = \int \frac{d\sigma}{d\Omega} d\Omega$ represents the total probability of scattering

Scattering Theory Basics

• Scattering process is described by the Schrödinger equation with a scattering potential $V(\mathbf{r})$
• Asymptotic behavior of the wavefunction is a superposition of incident and scattered waves
• Incident wave is a plane wave $e^{i\mathbf{k} \cdot \mathbf{r}}$ with wave vector $\mathbf{k}$
• Scattered wave is a spherical wave $\frac{e^{ikr}}{r} f(\theta, \phi)$ with scattering amplitude $f(\theta, \phi)$
• Born approximation assumes weak scattering potential and expresses $f(\theta, \phi)$ in terms of $V(\mathbf{r})$
  • First-order Born approximation: $f(\theta, \phi) \propto \int e^{-i\mathbf{q} \cdot \mathbf{r}} V(\mathbf{r}) d^3r$
  • Higher-order Born approximations involve iterative solutions of the Schrödinger equation
• Partial wave analysis expands the scattering amplitude in terms of angular momentum eigenstates

Partial Wave Analysis

• Partial wave analysis decomposes the scattering problem into angular momentum channels
• Incident and scattered waves are expanded in terms of spherical harmonics $Y_l^m(\theta, \phi)$
• Radial wavefunction $R_l(r)$ satisfies the radial Schrödinger equation for each angular momentum $l$
• Scattering amplitude is expressed as a sum over partial waves: $f(\theta) = \sum_{l=0}^{\infty} (2l+1) f_l P_l(\cos \theta)$
  • $f_l$ is the partial wave scattering amplitude for angular momentum $l$
  • $P_l(\cos \theta)$ are Legendre polynomials
• Phase shift $\delta_l$ characterizes the effect of the scattering potential on each partial wave
  • Related to the asymptotic behavior of the radial wavefunction: $R_l(r) \sim \sin(kr - l\pi/2 + \delta_l)$
• Scattering cross section is expressed in terms of phase shifts: $\sigma = \frac{4\pi}{k^2} \sum_{l=0}^{\infty} (2l+1) \sin^2 \delta_l$

Cross Sections and Phase Shifts

• Differential cross section $\frac{d\sigma}{d\Omega}$ measures the probability of scattering into a solid angle $d\Omega$
  • Expressed in terms of the scattering amplitude: $\frac{d\sigma}{d\Omega} = |f(\theta, \phi)|^2$
• Total cross section $\sigma$ represents the total probability of scattering
  • Obtained by integrating the differential cross section over all solid angles: $\sigma = \int \frac{d\sigma}{d\Omega} d\Omega$
• Partial wave expansion of the cross section: $\sigma = \frac{4\pi}{k^2} \sum_{l=0}^{\infty} (2l+1) \sin^2 \delta_l$
  • Contributions from each angular momentum channel are determined by the phase shifts $\delta_l$
• Phase shifts $\delta_l$ encode the effect of the scattering potential on each partial wave
  • Related to the asymptotic behavior of the radial wavefunction: $R_l(r) \sim \sin(kr - l\pi/2 + \delta_l)$
  • Can be calculated from the radial Schrödinger equation or approximated using Born approximation
• Resonances occur when a phase shift passes through $\pi/2$, leading to enhanced scattering
  • Correspond to the formation of quasi-bound states or virtual states in the scattering potential

Optical Theorem and Its Applications

• Optical theorem relates the total cross section to the forward scattering amplitude
  • $\sigma = \frac{4\pi}{k} \text{Im} f(\theta=0)$
  • Provides a powerful tool to calculate total cross sections from the imaginary part of the forward scattering amplitude
• Useful in various scattering processes, including particle physics and nuclear physics
• Enables the determination of total cross sections from experimental measurements of differential cross sections
• Connects the scattering amplitude to the absorption of the incident wave by the target
• Generalizations of the optical theorem exist for inelastic scattering and multi-channel scattering
• Plays a crucial role in the study of scattering resonances and bound states
• Provides constraints on the scattering amplitude based on unitarity and analyticity

Practical Examples and Problem Solving

• Scattering of particles by a square well potential
  • Analytically solvable example that demonstrates the concepts of partial wave analysis and phase shifts
• Scattering of electrons by atoms (electron-atom scattering)
  • Relevant for understanding atomic structure and electron transport in materials
• Scattering of neutrons by nuclei (neutron-nucleus scattering)
  • Important for studying nuclear structure and interactions
• Scattering of light by small particles (Rayleigh scattering)
  • Explains the blue color of the sky and the reddening of the sun at sunset
• Scattering of X-rays by crystals (X-ray diffraction)
  • Fundamental technique for determining the atomic structure of materials
• Solving scattering problems involves setting up the Schrödinger equation, applying boundary conditions, and calculating observables such as cross sections and phase shifts

Advanced Topics and Extensions

• Multichannel scattering theory deals with scattering processes involving multiple open channels
  • Includes inelastic scattering, where the internal states of the particles can change
• Relativistic scattering theory extends the formalism to relativistic quantum mechanics
  • Necessary for describing high-energy scattering processes in particle physics
• Eikonal approximation provides a semiclassical description of scattering at high energies
  • Useful for studying scattering in strong fields or at small angles
• Inverse scattering theory aims to reconstruct the scattering potential from the observed scattering data
  • Has applications in imaging and remote sensing
• Scattering in the presence of external fields (electric, magnetic) modifies the scattering cross sections
• Non-perturbative methods, such as the R-matrix theory, are used for scattering involving strong interactions
• Effective field theories provide a systematic framework for describing low-energy scattering processes

Connections to Other Areas of Physics

• Scattering theory is a fundamental tool in various branches of physics
• Quantum field theory: Scattering amplitudes are calculated using Feynman diagrams and perturbation theory
• Particle physics: Scattering experiments probe the fundamental interactions and properties of elementary particles
  • Examples include deep inelastic scattering, electron-positron annihilation, and hadron collisions
• Nuclear physics: Scattering experiments reveal the structure and interactions of nuclei
  • Examples include elastic and inelastic scattering of electrons, protons, and neutrons by nuclei
• Atomic and molecular physics: Scattering of electrons and photons by atoms and molecules provides information about their structure and dynamics
• Condensed matter physics: Scattering techniques (neutron, X-ray) are used to study the structure and excitations of materials
• Optics and acoustics: Scattering of light and sound waves is described by similar principles
• Scattering theory also has applications in astrophysics, geophysics, and medical imaging