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 ↓


S-matrix theory


S-matrix theory, or scattering matrix theory, is a central concept in quantum scattering theory dealing with the interactions of particles. This concept, which gained significant importance in the mid-20th century, considers the interactions of particles without relying on field-theoretical details. Instead, it focuses on the inputs and outcomes of scattering events, allowing physicists to calculate observable quantities such as cross-sections and transition rates.

Introduction to scattering theory

The process of scattering occurs when particles collide and interact, thereby altering their original paths and energies. These are fundamental phenomena studied in quantum mechanics, which are crucial to understanding everything from subatomic particles to the interactions of atomic nuclei.

In these processes, we generally consider two types of situations:

1. Initial State: The incoming particles before interaction. 2. Final State: The outgoing particles after interaction.

The main purpose of scattering theory is to connect these states, primarily by using mathematical tools that help predict the outcomes of particle interactions. The S-matrix provides an elegant solution to this by acting as a bridge between these initial and final states.

Definition of the S-matrix

The S-matrix or scattering matrix is a unitary matrix that encodes all possible outcomes of a given scattering process. Its components describe how an initial state transforms into various possible final states.

Mathematically, the S-matrix is represented as:

S_{fi} = langle f | S | i rangle

Here, |i> and |f> are the initial and final states, respectively. The S-matrix element S_{fi} gives the probability amplitude for a system prepared in the state |i> that can be observed in the state |f> after the interaction.

Unity and conservation laws

A key property of the S-matrix is its unity, which ensures the conservation of probability. This means that the total probability of all possible final states, given an initial state, is equal to one, which is a reflection of conservation laws (such as conservation of energy and momentum) in quantum mechanics.

The unity condition of the S-matrix can be expressed as:

S^dagger S = SS^dagger = I

where S^dagger is the Hermitian conjugate of the S-matrix and I is the identity matrix.

Visual representation

Simple 2-to-2 scattering process

To see how the S-matrix works, consider a simple 2-to-2 particle scattering process. In the initial state two particles intersect each other, interact through some potential, and depart as another set of particles.

ABCD

In this view, particles A and B collide, and particles C and D arise from the interaction. The S-matrix will allow us to calculate the probability amplitudes for the various possible configurations of C and D after the collision.

Text example: Simple elastic scattering

Consider an initial state |i> consisting of two particles. If |f> denotes the state where both particles scatter without changing identity or internal state (elastic scattering), then the S-matrix elements will be calculated to determine the probabilities and probability amplitudes of this outcome.

Let |i> = |A, B> and |f> = |C, D> where C = A and D = B (identical scattering). The probability amplitude is given as S_{fi} = langle C, D | S | A, B rangle.

Analytical framework of S-matrix theory

Setting up the S-matrix includes all possible overlaps of states within energy conservation constraints. It serves as a cornerstone for building theories where interaction mechanisms may be too complex to model explicitly.

The points to consider in S-matrix theory are as follows:

- Asymptotic States: Particles free before and after the collision. - Invariant Amplitude: Considering Lorentz invariance helps simplify descriptions. - Complex Plane Analysis: Analytic properties like poles correspond to bound states or resonances.

S-matrix in quantum field theory

In quantum field theory (QFT), the S-matrix plays an important role by extending previous applications from non-relativistic cases to relativistic situations, and also touches upon elementary particle physics where interactions are abundant.

The transition amplitude in QFT is calculated using Feynman diagrams, each of which corresponds to an S-matrix element:

- Each line in a Feynman diagram represents a particle's propagator. - Vertices represent points where interactions (forces) occur.

Illustration of a simple Feynman diagram

e⁻γe⁻

In this diagram, an electron ((e^-)) emits a photon ((gamma)), showing a fundamental QED interaction described by an S-matrix element.

Applications of S-matrix theory

S-matrix theory is used extensively in high-energy physics to understand particle collisions in accelerators, such as those carried out at CERN's Large Hadron Collider (LHC). By theorizing the S-matrix for given processes, researchers can make predictions on outcome probabilities and verify them against experimental data.

It is also important in studying resonances and particle stability, and links observations to fundamental forces and intrinsic properties of particles.

Benefits and challenges

The benefits of using S-matrix theory include:

- Unification of scattering processes without detailed dynamics. - Simplifies complex interactions into manageable calculations. - Directly relates to observable physical quantities.

However, challenges exist:

- Requires assumptions like asymptotic conditions and stability. - Some calculations can become mathematically intensive. - Less detailed insights into underlying interaction dynamics.

Despite these challenges, the S-matrix remains an elegant and effective tool in the physicist’s arsenal.

Conclusion

S-matrix theory provides a powerful framework within quantum mechanics to handle the complexities of particle interactions without having to delve deeply into the elusive details of each underlying process. Its application in both quantum and quantum field theories allows for the interpretation and prediction of phenomena at different scale levels in theoretical physics, providing direct links to practical results.

Future developments may enhance computational techniques for wider applications, and help bridge the gap between theoretical formulation and experimental validation.


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S-matrix theory

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