Graduate
  1. Classical mechanics  1.1. Advanced Kinematics  1.1.1. Generalized coordinate systems  1.1.2. Parametric equations of motion  1.1.3. Non-inertial frames and fictitious forces  1.1.4. Covariant formulation of momentum  1.1.5. Action-angle variable  1.2. Lagrangian and Hamiltonian mechanics  1.2.1. Principle of minimum action  1.2.2. Euler–Lagrange equations  1.2.3. Noether's theorem and conservation laws  1.2.4. Normalized Speed  1.2.5. Canonical conversion  1.2.6. Poisson bracket and Hamilton–Jacobi theory  1.2.7. Hamiltonian chaos and integrability  1.3. Rigid body mobility  1.3.1. Euler's equations of motion  1.3.2. Principal moments of inertia  1.3.3. Gyroscopic motion and precession  1.3.4. Inertia tensor  1.3.5. Stability of rotational speed  1.4. Nonlinear dynamics and chaos  1.4.1. Phase space and stability analysis  1.4.2. Poincaré sections and bifurcation theory  1.4.3. Strange Attractors and Fractals  1.4.4. Lyapunov exponent  1.4.5. KAM theorem and quasi-periodic motion
  2. Electromagnetism  2.1. Advanced Electrodynamics  2.1.1. Green's function and potential theory  2.1.2. Multipole expansion  2.1.3. Image charging method  2.1.4. Boundary conditions and uniqueness theorem  2.2. Electromagnetic wave propagation  2.2.1. Plane waves in dielectrics and conductors  2.2.2. Waveguides and cavity resonators  2.2.3. Radiation pressure and optical tweezers  2.2.4. Polarization and Jones calculus  2.3. Relativistic electrodynamics  2.3.1. Covariant formulation of electrodynamics  2.3.2. Lienard–Wiechert potentials  2.3.3. Synchrotron radiation and bremsstrahlung  2.3.4. Electromagnetic field tensor and Lorentz transformations  2.4. Plasma physics  2.4.1. Magnetohydrodynamics  2.4.2. Debye shielding and plasma oscillations  2.4.3. Alfvén waves and tokamak confinement  2.4.4. Plasma instabilities and turbulence
  3. Statistical mechanics and thermodynamics  3.1. Advanced Thermodynamics  3.1.1. Legendre transforms and thermodynamic potentials  3.1.2. Fluctuation–dissipation theorem  3.1.3. Onsager interpersonal relations  3.1.4. Critical events and phase transitions  3.2. Quantum statistical mechanics  3.2.1. Density Matrix and Ensemble Theory  3.2.2. Bose–Einstein condensates  3.2.3. Quantum phase transition  3.2.4. Fermi–Dirac and Bose–Einstein statistics  3.2.5. Partition functions and grand canonical ensembles
  4. Quantum mechanics  4.1. Advanced wave mechanics  4.1.1. WKB approximation  4.1.2. Path integral formulation  4.1.3. Quantum harmonic oscillator and coherent states  4.2. Angular momentum and spin  4.2.1. Spherical Harmonics  4.2.2. Clebsch–Gordan coefficient  4.2.3. Wigner–Eckart theorem  4.2.4. Spin–orbit coupling  4.3. Quantum scattering theory  4.3.1. Born approximation  4.3.2. Partial wave analysis  4.3.3. Optical theorem  4.3.4. S-matrix theory  4.4. Quantum field theory  4.4.1. Second Quantization  4.4.2. Feynman diagrams and propagators  4.4.3. Renormalization theory  4.4.4. Gauge theory and Yang–Mills fields  4.4.5. Spontaneous symmetry breaking and the Higgs mechanism
  5. General relativity and cosmology  5.1. Tensor calculus and differential geometry  5.1.1. Einstein field equations  5.1.2. Schwarzschild and Kerr metrics  5.1.3. Geodesics and Christoffel symbols  5.2. Cosmology and the Universe  5.2.1. The FLRW metric and cosmic inflation  5.2.2. Dark energy and structure formation  5.2.3. Cosmic microwave background radiation
  6. Condensed matter physics  6.1. Band structure and transport theory  6.1.1. Block theorem and the Kronig–Penney model  6.1.2. Fermi surface topology  6.1.3. Quantum Hall Effect  6.2. Superconductivity  6.2.1. BCS theory and Cooper pairs  6.2.2. Meissner effect and flux quantization  6.2.3. The Josephson Effect and SQUIDs  6.3. Topological phases of matter  6.3.1. Topological Insulators  6.3.2. Majorana fermions in topological phases of matter
  7. Nuclear and Particle Physics  7.1. Quantum Chromodynamics (QCD)  7.1.1. Quark–gluon plasma  7.1.2. Asymptotic freedom  7.1.3. Confinement and hadronization  7.2. Beyond the Standard Model  7.2.1. Grand Unified Theories  7.2.2. Supersymmetry and extra dimensions  7.2.3. Neutrino oscillations

Graduate → Quantum mechanics ↓


Quantum scattering theory


Quantum scattering theory is a fundamental part of quantum mechanics that deals with the interaction of particles. It is mainly used to understand and describe how particles such as electrons, protons or even atoms and molecules scatter after interacting with each other or with a potential. This theory is essential in fields such as atomic, nuclear and condensed matter physics.

Basic concepts

To understand quantum scattering theory, it is important to first understand basic quantum theories such as the wave function and the Schrödinger equation. The wave function describes the quantum state of a system and is usually represented by the symbol ψ (psi). The Schrödinger equation governs the evolution of these wave functions:

iℏ (∂ψ/∂t) = Hψ

Where:

  • i is the imaginary unit.
  • ℏ (h-bar) is the reduced Planck constant.
  • H is the Hamiltonian operator representing the total energy of the system.

Scattering process

In a scattering event, an incoming wave, such as a beam of particles, falls on a target or potential. The particles interact with this target, resulting in scattered waves traveling in different directions. An essential aspect of this process is determining how likely the particles are to scatter in a particular direction. This probability is described by the scattering amplitude, denoted as f(θ, φ), where θ and φ are angles indicating the direction of scattering.

Wavefunction in scattering

In quantum mechanics, we typically use two types of wave functions to describe scattering:

  • Incoming plane wave: represents incoming particles and is usually written as: 
    ψin = ei(𝐤·𝐫 - ωt)
  • Scattered wave: Describes the particles after scattering. It is often spherically symmetric and is given as follows: 
    ψsc = f(θ, φ) (eikr /r)
Here, k is the wave number, which is related to the energy of the particles, and r is the distance from the scattering center.

Partial wave analysis

One method of simplifying scattering problems is partial wave analysis. In this method, the wave function is expanded into a series of spherical harmonics. This becomes particularly useful because each term of the series (called a partial wave) can be analyzed independently. The total wave function can be expressed as:

ψ(r, θ, φ) = Σ (2l + 1) il eiδl Pl (cos θ) (eikr /r)

Where the conditions are:

  • Pl are Legendre polynomials.
  • δl is the phase shift, which contains information about the potential.
  • Σ represents the sum of the different angular momentum states.

Optical theorem

The optical theorem is a remarkable and very useful result in scattering theory that relates the total cross-section to the forward scattering amplitude. It is given as:

σtotal = (4π/k) Im[f(0)]

Here, σtotal is the cross-section, Im indicates the imaginary part, and f(0) is the forward scattering amplitude.

Lippmann–Schwinger equation

Another central equation in quantum scattering theory is the Lippmann-Schwinger equation. This is an integral equation that provides a way to calculate the scattered wave function given the interaction potential. It is written as:

ψ+ = φ + (1/E - H0 + iε) Vψ

In this equation:

  • ψ+ is the outgoing wave function.
  • φ is the incident wave function.
  • V is the interaction potential.
  • H0 is the free Hamiltonian.
  • ε is a very small positive number used to ensure convergence of the integral.

Visual representation

To better understand scattering, let's consider a visual representation. Here, we depict an incoming plane wave that scatters from a spherical potential, resulting in a spherical outgoing wave:

The incoming wave Scattered waves

Examples and calculations

To understand how quantum scattering theory is applied, let's look at some simple examples:

1. Scattering from a hard sphere

Consider a scenario where particles are scattered by a rigid ball. This interaction can be modeled as a potential that is zero everywhere except at the surface of the sphere. In such cases, the phase shift for each partial wave is given by matching the boundary conditions on the surface.

2. Yukawa potential

The Yukawa potential is another classic example, commonly found in nuclear physics. It is given as:

V(r) = -g (e-αr /r)

Here, g is the strength of the interaction, and α sets the threshold. Calculating the scattering amplitude with this potential involves solving the Lippmann–Schwinger equation numerically.

Born approximation

For weak potentials, the Born approximation provides a way to calculate the scattering amplitudes. The first-order Born approximation for the scattering amplitude is:

f(θ, φ) ≈ -(2π²m/ħ²k) ∫V(r')ei(kk')·r' d³r'

This approximation is valid when the potential is weak or the wavelength of the incoming particle is much larger than the range of the potential.

Conclusion

Quantum scattering theory is a versatile and powerful framework that provides deep insights into the nature of quantum interactions. It forms the basis for many practical applications in various fields of physics. Whether it is the beautiful formulation of wave functions, partial wave analysis, or numerical solutions of integral equations, the theory is an indispensable part of the physicist's toolkit.


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Quantum scattering theory

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