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
Top images from around the web for Scattering systems overview 3.5 Multiple Slit Diffraction (Diffraction Gratings) – Douglas College Physics 1207 View original
Is this image relevant?
electromagnetism - In scattering, how does a particle 'know' which direction it is being ... View original
Is this image relevant?
3.5 Multiple Slit Diffraction (Diffraction Gratings) – Douglas College Physics 1207 View original
Is this image relevant?
1 of 3
Top images from around the web for Scattering systems overview 3.5 Multiple Slit Diffraction (Diffraction Gratings) – Douglas College Physics 1207 View original
Is this image relevant?
electromagnetism - In scattering, how does a particle 'know' which direction it is being ... View original
Is this image relevant?
3.5 Multiple Slit Diffraction (Diffraction Gratings) – Douglas College Physics 1207 View original
Is this image relevant?
1 of 3
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 ψ = E ψ H\psi = E\psi H ψ = E ψ
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 ψ ( r ) ∼ e i k r / r \psi(r) \sim e^{ikr}/r ψ ( r ) ∼ e ik r / 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 σ t o t = ∫ d σ d Ω d Ω \sigma_{tot} = \int \frac{d\sigma}{d\Omega} d\Omega σ t o t = ∫ d Ω d σ d Ω
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 d σ d Ω = ∣ f ( θ , ϕ ) ∣ 2 \frac{d\sigma}{d\Omega} = |f(\theta, \phi)|^2 d Ω d σ = ∣ f ( θ , ϕ ) ∣ 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 σ t o t = 4 π k Im f ( 0 ) \sigma_{tot} = \frac{4\pi}{k} \text{Im}f(0) σ t o t = k 4 π 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 π i T S = 1 - 2\pi i T S = 1 − 2 πi T
Poles of S-matrix correspond to bound states and resonances
Unitarity and symmetry
Unitarity condition S † S = S S † = 1 S^\dagger S = SS^\dagger = 1 S † S = S S † = 1 ensures conservation of probability
Time-reversal symmetry implies S i j = S j i S_{ij} = S_{ji} 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 ( θ ) = ∑ l ( 2 l + 1 ) f l P l ( cos θ ) f(\theta) = \sum_l (2l+1)f_l P_l(\cos\theta) f ( θ ) = ∑ l ( 2 l + 1 ) f l P l ( cos θ )
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 ) ∼ sin ( k r − l π / 2 + δ l ) R_l(r) \sim \sin(kr - l\pi/2 + \delta_l) R l ( r ) ∼ sin ( k r − l π /2 + δ l )
Related to scattering amplitude by f l = 1 2 i k ( e 2 i δ l − 1 ) f_l = \frac{1}{2ik}(e^{2i\delta_l} - 1) f l = 2 ik 1 ( e 2 i δ 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 δ l ( 0 ) − δ l ( ∞ ) = n l π \delta_l(0) - \delta_l(\infty) = n_l\pi δ l ( 0 ) − δ l ( ∞ ) = n l π , 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 ( k , k ′ ) = − m 2 π ℏ 2 ∫ V ( r ) e i ( k − k ′ ) ⋅ r d 3 r 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 f ( k , k ′ ) = − 2 π ℏ 2 m ∫ V ( r ) e i ( k − k ′ ) ⋅ r d 3 r
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 ) + ⋯ f = f^{(1)} + f^{(2)} + f^{(3)} + \cdots f = f ( 1 ) + f ( 2 ) + f ( 3 ) + ⋯
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 ( ∣ r ∣ ) V(r) = V(|\mathbf{r}|) V ( r ) = V ( ∣ 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 1/r 1/ r potential
Scattering amplitude given by f ( θ ) = − η 2 k sin 2 ( θ / 2 ) e − i η ln ( sin 2 ( θ / 2 ) ) f(\theta) = -\frac{\eta}{2k\sin^2(\theta/2)}e^{-i\eta\ln(\sin^2(\theta/2))} f ( θ ) = − 2 k s i n 2 ( θ /2 ) η e − i η l n ( s i n 2 ( θ /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 e − μ r r V(r) = -g\frac{e^{-\mu r}}{r} V ( r ) = − g r e − μ r
Models screened Coulomb interactions in plasmas and nuclear forces
Characterized by screening length 1 / μ 1/\mu 1/ μ
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
Describes energy dependence of cross-section near a resonance
Given by σ ( E ) = σ b g + π k 2 Γ 2 ( E − E R ) 2 + Γ 2 / 4 \sigma(E) = \sigma_{bg} + \frac{\pi}{k^2}\frac{\Gamma^2}{(E-E_R)^2 + \Gamma^2/4} σ ( E ) = σ b g + k 2 π ( E − E R ) 2 + Γ 2 /4 Γ 2
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 Γ / 2 E = E_R - i\Gamma/2 E = E R − i Γ/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 Γ = ℏ / τ \Gamma = \hbar/\tau Γ = ℏ/ τ
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 Δ E Δ t ∼ ℏ \Delta E \Delta t \sim \hbar Δ E Δ t ∼ ℏ
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
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 ∑ k S i k S j k ∗ = δ i j \sum_k S_{ik}S_{jk}^* = \delta_{ij} ∑ k S ik S jk ∗ = δ 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 ∣ ψ ⟩ = ∣ ϕ ⟩ + G 0 V ∣ ψ ⟩ |\psi\rangle = |\phi\rangle + G_0V|\psi\rangle ∣ ψ ⟩ = ∣ ϕ ⟩ + G 0 V ∣ ψ ⟩ 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)
Defines transition operator T via V ∣ ψ ⟩ = T ∣ ϕ ⟩ V|\psi\rangle = T|\phi\rangle V ∣ ψ ⟩ = T ∣ ϕ ⟩
T-matrix satisfies T = V + V G 0 T T = V + VG_0T T = V + V G 0 T
Scattering amplitude related to T-matrix elements by f = − m 2 π ℏ 2 ⟨ k ′ ∣ T ∣ k ⟩ f = -\frac{m}{2\pi\hbar^2}\langle\mathbf{k'}|T|\mathbf{k}\rangle f = − 2 π ℏ 2 m ⟨ k ′ ∣ T ∣ k ⟩
Useful for developing systematic approximations and understanding multiple scattering
Resolvent operator
Green's function of the full Hamiltonian G ( z ) = ( z − H ) − 1 G(z) = (z-H)^{-1} G ( z ) = ( z − H ) − 1
Related to free particle Green's function by G = G 0 + G 0 V G G = G_0 + G_0VG G = G 0 + G 0 V G
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
Relates S-matrix elements to correlation functions of quantum fields
Expresses scattering amplitudes in terms of asymptotic in and out states
Given by ⟨ f ∣ S ∣ i ⟩ = ∫ ∏ j d 4 x j e i p j ⋅ x j ⟨ 0 ∣ T { ϕ ( x 1 ) ⋯ ϕ ( x n ) } ∣ 0 ⟩ \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 ⟨ f ∣ S ∣ i ⟩ = ∫ ∏ j d 4 x j e i p j ⋅ x j ⟨ 0∣ T { ϕ ( x 1 ) ⋯ ϕ ( x n )} ∣0 ⟩
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 − → μ + μ − e^+e^- \rightarrow \mu^+\mu^- e + e − → μ + μ − related to e − μ − → e − μ − e^-\mu^- \rightarrow e^-\mu^- e − μ − → e − μ − 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