Notes on Scattering Theory Lecture

Overview

• Covered approximate methods in quantum mechanics, transitioning to scattering theory.
• Scattering theory has classical roots, involving reduction of two-body problems to one-body problems in the center-of-mass (CM) frame.
• Cross-section is computed in the CM frame and then converted to the lab frame for experimental connections.

Classical Scattering Theory Recap

Two-Body Problem in CM Frame

• Two particles with masses M1 and M2 and initial velocities u1 and u2.
• Post-collision, particles scatter with final velocities.
• In 1D: 1 momentum conservation equation and 1 energy conservation equation.
• In 3D: 3 momentum conservation equations and 1 energy conservation equation.
• Number of unknowns is 6 (3 for each particle's initial velocities).

Scattering Geometry in CM Frame

• Total momentum is zero.
• Velocities represented as u1c, u2c, v1c, v2c in the CM frame.
• The scattering angle is denoted as Theta.

Quantum Mechanical Scattering

Collision Experiment Setup

• Incident beam represents particle 1 (mass 1).
• Target is particle 2, which is struck by the incoming particle.
• Detectors are positioned at various angles (Theta1, Theta2, etc.).

Scattering Cross-Section Definition

• Differential scattering cross-section: [ \sigma(\Theta, \Phi) = \frac{\text{Number of particles scattered in solid angle}}{\text{Incident intensity}} ]
• Total cross-section obtained by integrating over all angles: [ \Sigma = \int \sigma(\Theta, \Phi) d\Omega ]
• For spherically symmetric potentials, [ \Sigma = 2\pi \int_{0}^{\pi} \sin(\Theta) d\Theta \sigma(\Theta) ]

Quantum Scattering Properties

• Focus on elastic scattering (energy conserved) vs. inelastic scattering (energy not conserved).
• The scattering potential is translationally invariant; depends only on relative coordinates.
• Effective mass: [ \mu = \frac{M1 \cdot M2}{M1 + M2} ]

Wave Function Behavior

• Before collision (t -> -∞): behaves as a free wave.
• After collision (t -> +∞): consists of both free and scattered waves.
• Scattering amplitude (FK(Theta)): important for analyzing scattering.

Green's Function in Scattering Theory

• Green's function relates to the wave equation and helps solve scattering problems.
• Defined as a solution to [ (\nabla^2 + k^2) G(R, R') = \delta(R - R') ]
• The solution of the scattering problem can be expressed using Green's function.

Born Approximation

• Simplification for weak potentials, ignoring higher-order terms in potential.
• Scattering amplitude (FK(Theta)) related to the Fourier transform of the potential.
• Differential scattering cross-section: [ \sigma_B(\Theta) = |F_K(\Theta)|^2 ]

Conclusion

• Scattering theory connects quantum mechanical principles to experimental observations.
• Born approximation provides a fundamental approach to compute scattering cross-sections from potentials.
• Recap of approximate methods: perturbation theory, variational principle, WKB approximation, and time-dependent perturbation.

Key Takeaways

• Scattering theory analyzes the interaction between particles and provides methods to compute scattering cross-sections using quantum mechanics and Green's functions.
• Understanding the fundamentals of scattering is crucial for interpreting experimental results in quantum mechanics.