7.5 Scattering theory

Scattering theory is a crucial part of spectral theory, exploring how waves and particles interact with targets. It's used in quantum mechanics, acoustics, and electromagnetic theory to predict collision outcomes and understand material properties through particle interactions.

The theory involves analyzing incoming and outgoing waves, cross-sections, and the scattering matrix. It uses techniques like partial wave analysis, Born approximation, and Green's function methods to solve complex scattering problems in various fields of physics.

Fundamentals of scattering theory

• Scattering theory forms a crucial part of spectral theory, providing insights into the interaction of waves or particles with targets
• Applies to various fields including quantum mechanics, acoustics, and electromagnetic theory
• Helps predict outcomes of collision experiments and understand material properties through particle interactions

Scattering systems overview

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• Consists of an incident wave or particle, a target, and the resulting scattered waves
• Characterized by the scattering potential which represents the interaction between the incident particle and the target
• Scattering amplitude describes the probability of scattering in different directions
• Applications include particle physics experiments (Large Hadron Collider) and medical imaging techniques (X-ray scattering)

Time-dependent vs time-independent scattering

• Time-dependent scattering involves explicitly time-varying potentials or wave packets
• Time-independent scattering assumes steady-state conditions with time-harmonic waves
• Stationary scattering theory uses time-independent Schrödinger equation $H\psi = E\psi$
• Time-dependent approach necessary for studying non-stationary processes (ultrafast laser pulses)

Incoming and outgoing waves

• Incoming waves represent the initial state of the system before scattering occurs
• Outgoing waves describe the final state after the scattering interaction
• Asymptotic behavior of wavefunctions characterized by $\psi(r) \sim e^{ikr}/r$ for large distances r
• Boundary conditions crucial for determining unique solutions (Sommerfeld radiation condition)

Scattering cross-section

• Cross-section measures the effective area for scattering interactions
• Fundamental quantity in spectral theory for quantifying scattering probabilities
• Relates theoretical predictions to experimental measurements of scattering events

Total cross-section

• Represents the total probability of scattering in all directions
• Calculated by integrating the differential cross-section over all solid angles
• Expressed mathematically as $\sigma_{tot} = \int \frac{d\sigma}{d\Omega} d\Omega$
• Used to determine mean free path of particles in a medium (neutron transport in nuclear reactors)

Differential cross-section

• Describes the angular distribution of scattered particles
• Defined as the ratio of scattered flux to incident flux per unit solid angle
• Mathematically expressed as $\frac{d\sigma}{d\Omega} = |f(\theta, \phi)|^2$
• Measured in experiments to probe structure of matter (Rutherford scattering)

Optical theorem

• Relates the total cross-section to the forward scattering amplitude
• States that $\sigma_{tot} = \frac{4\pi}{k} \text{Im}f(0)$, where k wave number and f(0) forward scattering amplitude
• Consequence of unitarity in quantum mechanics
• Useful for checking consistency of scattering calculations and experimental data

Scattering matrix

• S-matrix connects incoming and outgoing states in scattering processes
• Central concept in spectral theory for describing quantum mechanical scattering
• Provides a complete description of the scattering process in terms of probability amplitudes

S-matrix properties

• Unitary matrix ensuring conservation of probability
• Elements S_{ij} represent transition amplitudes between initial and final states
• Related to the T-matrix by $S = 1 - 2\pi i T$
• Poles of S-matrix correspond to bound states and resonances

Unitarity and symmetry

• Unitarity condition $S^\dagger S = SS^\dagger = 1$ ensures conservation of probability
• Time-reversal symmetry implies $S_{ij} = S_{ji}$ for systems without magnetic fields
• Parity conservation leads to additional constraints on S-matrix elements
• Crossing symmetry relates scattering amplitudes in different channels (s-channel and t-channel)

Analytic structure

• S-matrix elements are analytic functions of energy in the complex plane
• Branch cuts correspond to thresholds for inelastic processes
• Poles on the real axis represent bound states
• Poles in the lower half-plane indicate resonances with finite lifetimes

Partial wave analysis

• Decomposes scattering amplitudes into contributions from different angular momenta
• Simplifies the scattering problem by exploiting rotational symmetry
• Essential technique in spectral theory for analyzing complex scattering processes

Angular momentum decomposition

• Expands scattering amplitude in terms of Legendre polynomials $f(\theta) = \sum_l (2l+1)f_l P_l(\cos\theta)$
• Each term in the expansion corresponds to a specific angular momentum state
• Allows separation of radial and angular parts of the scattering problem
• Particularly useful for central potentials (spherically symmetric interactions)

Phase shifts

• Characterize the effect of scattering potential on each partial wave
• Defined by the asymptotic behavior of radial wavefunctions $R_l(r) \sim \sin(kr - l\pi/2 + \delta_l)$
• Related to scattering amplitude by $f_l = \frac{1}{2ik}(e^{2i\delta_l} - 1)$
• Measure of the strength of scattering in each angular momentum channel

Levinson's theorem

• Relates the number of bound states to the behavior of phase shifts at zero energy
• States that $\delta_l(0) - \delta_l(\infty) = n_l\pi$, where n_l number of bound states with angular momentum l
• Provides a way to count bound states from scattering data
• Generalizations exist for multichannel scattering and relativistic systems

Born approximation

• Perturbative approach to scattering problems in quantum mechanics
• Assumes weak interaction between incident particle and scattering potential
• Provides analytical solutions for many scattering problems in spectral theory

First Born approximation

• Treats scattering potential as a small perturbation to free particle motion
• Scattering amplitude given by $f(\mathbf{k}, \mathbf{k'}) = -\frac{m}{2\pi\hbar^2}\int V(\mathbf{r})e^{i(\mathbf{k}-\mathbf{k'})\cdot\mathbf{r}}d^3r$
• Valid for high-energy scattering or weak potentials
• Widely used in atomic and nuclear physics (electron scattering from atoms)

Higher-order Born series

• Systematic expansion of scattering amplitude in powers of interaction potential
• Each term in the series represents multiple scattering events
• Expressed as $f = f^{(1)} + f^{(2)} + f^{(3)} + \cdots$
• Convergence of the series depends on the strength of the potential

Validity and limitations

• Breaks down for strong potentials or low-energy scattering
• Fails to describe bound states and resonances accurately
• Does not satisfy unitarity condition exactly
• Useful for quick estimates and qualitative understanding of scattering processes

Potential scattering

• Studies scattering from specific forms of interaction potentials
• Fundamental to understanding atomic, molecular, and nuclear interactions
• Provides insights into the relationship between potential shape and scattering observables

Central potentials

• Depend only on the distance between scatterer and target $V(r) = V(|\mathbf{r}|)$
• Allow separation of radial and angular parts of the Schrödinger equation
• Lead to conservation of angular momentum in scattering process
• Examples include gravitational and electrostatic potentials

Coulomb scattering

• Describes scattering from long-range $1/r$ potential
• Scattering amplitude given by $f(\theta) = -\frac{\eta}{2k\sin^2(\theta/2)}e^{-i\eta\ln(\sin^2(\theta/2))}$
• Exhibits divergence in forward scattering direction (θ = 0)
• Relevant for charged particle interactions (alpha particle scattering from nuclei)

Yukawa potential

• Short-range potential of the form $V(r) = -g\frac{e^{-\mu r}}{r}$
• Models screened Coulomb interactions in plasmas and nuclear forces
• Characterized by screening length $1/\mu$
• Reduces to Coulomb potential for μ → 0 and to contact interaction for μ → ∞

Resonances in scattering

• Occur when incident particle energy matches quasi-bound state of the system
• Manifest as sharp peaks in scattering cross-sections
• Important for understanding atomic and nuclear structure in spectral theory

Breit-Wigner formula

• Describes energy dependence of cross-section near a resonance
• Given by $\sigma(E) = \sigma_{bg} + \frac{\pi}{k^2}\frac{\Gamma^2}{(E-E_R)^2 + \Gamma^2/4}$
• Parameters include resonance energy E_R, width Γ, and background cross-section σ_bg
• Widely used in particle physics to characterize unstable particles

Complex energy poles

• Resonances correspond to poles of S-matrix in complex energy plane
• Located at $E = E_R - i\Gamma/2$ where E_R resonance energy and Γ width
• Real part gives resonance position, imaginary part related to lifetime
• Analytic continuation of S-matrix reveals these poles

Resonance width and lifetime

• Width Γ inversely proportional to resonance lifetime τ via $\Gamma = \hbar/\tau$
• Narrow resonances (small Γ) correspond to long-lived states
• Broad resonances (large Γ) indicate short-lived, unstable states
• Heisenberg uncertainty principle relates energy uncertainty to lifetime $\Delta E \Delta t \sim \hbar$

Multichannel scattering

• Involves multiple possible final states or reaction pathways
• Essential for describing complex reactions in atomic, molecular, and nuclear physics
• Extends single-channel formalism to account for coupled scattering processes

Coupled-channel formalism

• Describes interactions between different scattering channels
• Schrödinger equation becomes a set of coupled differential equations
• S-matrix becomes a matrix with elements S_{ij} for transitions between channels i and j
• Unitarity condition generalizes to $\sum_k S_{ik}S_{jk}^* = \delta_{ij}$

Inelastic scattering

• Involves energy transfer between internal degrees of freedom and relative motion
• Examples include rotational and vibrational excitations in molecular collisions
• Characterized by thresholds in scattering cross-sections
• Requires consideration of internal structure of colliding particles

Rearrangement collisions

• Involves change in identity of colliding particles (A + BC → AB + C)
• Requires careful treatment of asymptotic states and boundary conditions
• Important in chemical reactions and nuclear physics (transfer reactions)
• Described by transition operators connecting different arrangement channels

Green's function methods

• Powerful technique for solving scattering problems in spectral theory
• Relates scattering amplitudes to solutions of inhomogeneous differential equations
• Provides a formal framework for developing approximation schemes

Lippmann-Schwinger equation

• Integral equation formulation of the scattering problem
• Given by $|\psi\rangle = |\phi\rangle + G_0V|\psi\rangle$ where G_0 free particle Green's function
• Equivalent to Schrödinger equation with appropriate boundary conditions
• Serves as starting point for many approximation methods (Born series)

T-matrix formulation

• Defines transition operator T via $V|\psi\rangle = T|\phi\rangle$
• T-matrix satisfies $T = V + VG_0T$
• Scattering amplitude related to T-matrix elements by $f = -\frac{m}{2\pi\hbar^2}\langle\mathbf{k'}|T|\mathbf{k}\rangle$
• Useful for developing systematic approximations and understanding multiple scattering

Resolvent operator

• Green's function of the full Hamiltonian $G(z) = (z-H)^{-1}$
• Related to free particle Green's function by $G = G_0 + G_0VG$
• Poles of resolvent correspond to bound states and resonances
• Spectral representation provides connection to eigenstates of the system

Inverse scattering problem

• Aims to reconstruct interaction potential from scattering data
• Important in various fields including geophysics, medical imaging, and quantum mechanics
• Challenging due to non-uniqueness and ill-posedness of the problem

Reconstruction of potentials

• Uses scattering data (phase shifts, cross-sections) to infer underlying potential
• Methods include variable phase approach and quantum inversion techniques
• Requires careful analysis of available data and appropriate regularization
• Applications include determining nuclear force from nucleon-nucleon scattering data

Uniqueness and ambiguity

• Different potentials can produce identical scattering data (phase-equivalent potentials)
• Ambiguities arise from limited angular range or energy range of measurements
• Transformation methods (Gel'fand-Levitan, Marchenko) can generate families of equivalent potentials
• Additional physical constraints often needed to select physically relevant solutions

Gel'fand-Levitan method

• Technique for reconstructing one-dimensional potentials from spectral data
• Uses Jost function or S-matrix as input to derive integral equation for transformation kernel
• Potential obtained from solution of integral equation
• Generalizations exist for three-dimensional and multichannel problems

Scattering in quantum field theory

• Extends scattering theory to relativistic quantum systems
• Deals with creation and annihilation of particles in high-energy collisions
• Fundamental to particle physics and understanding of fundamental interactions

LSZ reduction formula

• Relates S-matrix elements to correlation functions of quantum fields
• Expresses scattering amplitudes in terms of asymptotic in and out states
• Given by $\langle f|S|i\rangle = \int \prod_j d^4x_j e^{ip_j\cdot x_j} \langle 0|T\{\phi(x_1)\cdots\phi(x_n)\}|0\rangle$
• Provides connection between field theory and scattering observables

Feynman diagrams in scattering

• Graphical representation of terms in perturbative expansion of S-matrix
• Each diagram corresponds to a specific scattering process
• Rules for translating diagrams into mathematical expressions
• Powerful tool for calculating scattering amplitudes in quantum electrodynamics and other field theories

Crossing symmetry

• Relates scattering amplitudes in different channels (s, t, u-channels)
• Allows prediction of new processes from known ones by analytic continuation
• Example $e^+e^- \rightarrow \mu^+\mu^-$ related to $e^-\mu^- \rightarrow e^-\mu^-$ by crossing
• Consequence of relativistic invariance and analytic properties of S-matrix

Computational methods

• Numerical techniques for solving scattering problems in spectral theory
• Essential for handling complex potentials and multichannel systems
• Complement analytical methods and provide solutions where exact results are not available

Partial wave summation

• Numerical implementation of partial wave expansion
• Truncates infinite sum to finite number of terms based on convergence criteria
• Accuracy improves with inclusion of higher angular momentum contributions
• Efficient for spherically symmetric potentials and low to moderate energies

Numerical integration techniques

• Solve radial Schrödinger equation using methods like Runge-Kutta or Numerov algorithm
• Adaptive step size methods improve efficiency and accuracy
• Boundary conditions implemented through appropriate choice of integration limits
• Challenges include handling singularities and matching asymptotic solutions

Monte Carlo methods for scattering

• Simulate scattering processes using probabilistic techniques
• Particularly useful for complex geometries and multiple scattering problems
• Examples include Monte Carlo N-Particle (MCNP) code for neutron transport
• Can handle both classical and quantum mechanical scattering processes