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 ↓


Optical theorem


The optical theorem is a fundamental principle in quantum scattering theory that relates the forward scattering amplitude of a wave to the total cross section of the scattering process. It forms a bridge between theory and measurable physical quantities, providing important information about how particles interact. This theorem is based in optics more than a century ago and its roots extend to major discoveries in quantum mechanics. In this detailed discussion, we will dive into the main concepts, derivation, and applications of the optical theorem to provide a comprehensive understanding.

Basic concepts

Scattering theory is essential in quantum physics because it allows us to study the interaction of particles. The interaction is typically characterized by how a wave is deflected. In quantum mechanics, a wave function describes the probability amplitude of finding a particle at a given point. When a wave encounters a possible state, it can be influenced or changed, producing a scattering event.

Understanding scattering

To represent the scattered wave function, we use an asymptotic expression. Suppose a plane wave approaches a target potential, then it can be represented as follows:

Ψ_in(r) = e ik·r

Here, k is the wave vector, and r is the position vector. The wave function scattered away from the target looks like a spherical wave, given as:

Ψ_sc(r) = f(k') e ikr /r

The quantity f(k') is the scattering amplitude, and it provides important information regarding scattering intensity and angles.

Total scattering cross-section

The scattering cross-section is an important parameter in understanding interactions. For a target, the total scattering cross-section, σ_total, indicates the apparent area that can scatter the incoming wave. It is determined via an integral over all possible scattering directions:

σ_total = ∫ |f(k')|² dΩ

In this integral, dΩ denotes a differential solid angle. The relation between the cross-section and the scattering amplitude is obvious and important in scattering processes.

Derivation of the optical theorem

The optical theorem can be derived by examining the interference between incoming plane waves and outgoing spherical waves. Consider a wave function composed of both the incident plane wave and the scattered spherical wave:

Ψ_total(r) = Ψ_in(r) + Ψ_sc(r)

By applying the principle of conservation of probability, in particular for the infinite detector region, the following results are obtained:

Im[Ψ* ∇² Ψ] = 0

Using Green's theorem and manipulating this relation gives the optical theorem:

σ_total = (4π/k) Im[f(0)]

Thus, the imaginary part of the forward scattering amplitude, f(0), is directly related to the total cross-section, providing a simple but profound connection between observable measurements.

Visual example

Consider the following representation of a scattering event where a plane wave interacts with a target, producing a scattered wave. The diagram below shows the paths and amplitudes of the waves when analyzed graphically.

Incoming plane wave Scattered wave Scattered wave

In this illustration, the blue line represents the incoming plane wave, while the red lines represent how the waves scatter at different angles after interacting with the target (gray circle). This coherent summation of these scattered waves causes interference, which is crucial to the phenomenon described by the optical theorem.

Applications of optical theorem

The optical theorem is used in a variety of physics fields, providing insight into atomic, nuclear, and particle physics. Some practical applications include:

  • Nuclear physics: This theorem helps in determining the reaction rates in nuclear reactors and provides constraints on the nuclear cross section.
  • Particle physics: In high energy physics, it provides important constraints on scattering processes involving subatomic particles.
  • Medical physics: This helps optimize radiation therapy because it explains how different tissues scatter radiation.

Lesson example: Analysis of particle interactions

Imagine a particle beam falling on a hydrogen nucleus. This interaction can result in elastic scattering, where the particles are scattered but not absorbed. In this case, the optical theorem can play an essential role in predicting the scattering cross section for elastic events in particle detectors.

Elastic cross-section σ_el = (4π/k) Im[f_el(0)]

Here, f_el(0) is the forward elastic scattering amplitude.

Using the experimental data, the forward amplitude is calculated, resulting in the total elastic cross-section. This result is important when attempting to determine particle properties in collider experiments.

Mathematical insights into the optical theorem

The optical theorem is rooted in complex mathematical formulas. To understand these intricacies it is necessary to be familiar with boundary value problems, Green's functions, and asymptotic wave analysis.

g(r) = (exp(ikr) / r) [Sphereical Wavfunction]

where exp(ikr)/r is the spherical wave function solution and serves as the standard form used in scattering theory.

Conclusion

The optical theorem unleashes the power of fundamental concepts in quantum mechanics, linking observable phenomena with theoretical predictions. Combining theoretical calculations with forward scattering properties, it provides an essential tool for physicists exploring the microscopic world. The optical theorem is central in research and applications, from explaining nuclear reactions to probing interactions within particle accelerators.


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Optical theorem

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