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 ↓


Partial wave analysis


Partial wave analysis is a technique used in quantum mechanics, particularly in the study of quantum scattering theory. It is a detailed method for solving quantum mechanics scattering problems and is useful in understanding how particles interact when they collide or scatter from each other. This technique is quite general and can be applied to a wide variety of scattering scenarios.

To understand partial wave analysis, we must first recall some basic concepts in quantum mechanics. The basic equation describing quantum mechanical systems is the Schrödinger equation:

        iħ ∂Ψ/∂t = HΨ
    iħ ∂Ψ/∂t = HΨ

Here, Ψ is the wave function of the involved particles, ħ is the Planck constant, and H is the Hamiltonian, an operator corresponding to the total energy of the system.

Basics of scattering theory

In quantum scattering theory, we are often interested in how a particle, such as an electron, interacts with a target, such as an atom or another particle. When a particle approaches a target, it can be treated as a plane wave:

        Ψ_incoming = A e^(ikz)
    Ψ_incoming = A e^(ikz)

where A is the amplitude, k is the wave vector, and z denotes the direction of incidence. After the interaction, the wave will scatter, and part of it will flow back out as a spherical wave:

        Ψ_scattered = f(θ, φ) e^(ikr)/r
    Ψ_scattered = f(θ, φ) e^(ikr)/r

Here, f(θ, φ) is the scattering amplitude, which describes how the target affects the wave, and (r, θ, φ) are spherical coordinates.

What is partial wave analysis?

Partial wave analysis simplifies the process of solving the scattering problem by decomposing the scattered wave into a series of spherical waves, each of which is characterized by an integer angular momentum quantum number, l. The idea is to express the wave function as a sum of these partial waves:

        Ψ_scattered = ∑ (2l + 1) i^l a_l P_l(cos θ) e^(iσ_l) e^(ikr)/r
    Ψ_scattered = ∑ (2l + 1) i^l a_l P_l(cos θ) e^(iσ_l) e^(ikr)/r

In this expression, a_l represents the scattering amplitudes for each partial wave, P_l are the Legendre polynomials, and σ_l is the phase shift experienced by the target on the incoming wave. This can be visualized using a series of concentric spherical waves representing different l values.

One of the main advantages of partial wave analysis is that it neatly separates the contributions to the scattering according to the angular momentum, which is a conserved quantity in many physical situations. This means that only certain values of l will make significant contributions at a given energy.

Connecting with physical intuition

To understand the idea of partial wave analysis, consider the famous double-slit experiment. When electrons are fired through a double slit, an interference pattern emerges, highlighting the wave nature of the particles. Similarly, when particles scatter, the amplitude of the scattered waves depends on the interference of different partial waves.

For example, think about how waves on a pond spread out from the point where a stone is dropped. The waves spread out in concentric circles, just like our circular waves. By breaking down the scattered wave into its component parts (partial waves), we can better understand the contribution of each to the final pattern.

More about phase change

When a wave scatters off a target, the potential affects the phase of the outgoing wave. This effect is captured by the phase shift σ_l associated with each partial wave. Physically, the phase shift represents the delay imposed on each partial wave due to the interaction potential.

The total cross section, which measures the probability of scattering at any angle, can be expressed as the phase shift for each partial wave:

        σ_total = (4π/k^2) ∑ (2l + 1) sin^2(σ_l)
    σ_total = (4π/k^2) ∑ (2l + 1) sin^2(σ_l)

The expression shows that the cross section is composed of contributions at each l. For example, if each σ_l is zero, it means that no scattering occurs, and the wave does not change its phase due to the interaction with the target.

An example: scattering by a hard sphere

A classic problem in partial wave analysis is scattering by a rigid sphere, which approximates a few simple interactions. Imagine that an incoming wave collides with a perfectly rigid, impenetrable sphere. This setup serves as an excellent starting point for understanding the effect of a target on an incident quantum wave.

For a rigid sphere with radius a, the boundary conditions on the surface of the sphere dictate that the wave function must vanish. In this case, the phase change can be determined analytically. The partial waves for this scenario are spherically symmetric:

        σ_l = arctan(j_l(ka)/n_l(ka))
    σ_l = arctan(j_l(ka)/n_l(ka))

Here j_l and n_l are spherical Bessel functions representing solutions of the radial Schrödinger equation for spherical potentials.

Visualization of partial waves

To understand the mechanics of partial waves, let us consider a simple visual representation. Imagine a series of concentric circles representing the wave fronts of partial waves. The central circle, which is smaller in radius, represents low l (for example, l=0) and the outer circles with larger radii represent higher l values. The interference of these partial waves contributes to the overall scattering amplitude in a defined way.

Each circle can be thought of as a single partial wave in action. The interaction between these wavefronts, as they scatter off the target and form an interference pattern, is what partial wave analysis seeks to describe and quantify.

Conclusion

Partial wave analysis is a powerful and practical method used in quantum scattering to break down and study the interactions of waves with the target. By evaluating the phase shift of each partial wave, we gain an understanding of the scattering efficiency and cross section. Furthermore, partial wave analysis helps to separate the contributions of different angular momentum states.

As we delve deeper into quantum mechanics, fractional wave analysis serves as a useful bridge from classical wave ideas to the quantum realm. Its application is wide-ranging and spans a variety of problems in physics, providing insights into atomic collisions, nuclear physics, and much more.


Graduate → 4.3.2


U
username
0%
completed in Graduate

← Prev (4.3.1)
Born approximation
Next (4.3.3) →
Optical theorem

Comments 



Partial wave analysis

https://www.physicsflow.com/g/4.3.2?pfal=en