Partial Wave Analysis
Partial-wave analysis decomposes a scattering problem into independent angular-momentum channels, each described by a single phase shift; for short-range potentials at low energy only a few channels matter, making the method especially powerful.
Definition
Partial-wave analysis is the method of expanding a scattering wavefunction into components of definite orbital angular momentum, the effect of the potential on each being summarized by a phase shift that determines the scattering amplitude and cross section.
Scope
The topic covers the expansion of the incident plane wave and the full scattering state in spherical waves of definite angular momentum, the phase shift that each channel acquires from the potential, the scattering amplitude and cross section expressed as sums over partial waves, the dominance of low angular momenta at low energy, resonances as rapid phase-shift variation, and the relation of the zero-energy phase shift to the scattering length.
Core questions
- How is a scattering state decomposed into angular-momentum channels?
- What is a phase shift and how does it encode the effect of the potential?
- Why do only a few partial waves contribute at low energy?
- How do resonances and the scattering length appear in partial-wave language?
Key concepts
- partial-wave expansion
- phase shift
- centrifugal barrier
- s-wave scattering
- scattering length
- resonance
Key theories
- Phase shifts
- A short-range potential leaves each partial wave asymptotically a free spherical wave shifted in phase, and the full scattering amplitude is a sum over angular momenta weighted by these phase shifts, so measuring or computing them fully characterizes the scattering.
- Low-energy dominance and resonances
- A centrifugal barrier suppresses high angular momenta at low energy, so often only the lowest few partial waves contribute; a phase shift passing rapidly through ninety degrees signals a resonance, and the zero-energy limit defines the scattering length governing ultracold collisions.
Clinical relevance
Partial-wave analysis is the standard language for low-energy nuclear and atomic collisions: nucleon-nucleon and electron-atom scattering are reported as phase shifts, and the s-wave scattering length it defines controls the interactions and stability of ultracold atomic gases and Bose-Einstein condensates.
History
The partial-wave expansion grew out of the classical theory of wave scattering by Rayleigh; Faxen and Holtsmark applied it to quantum electron-atom scattering in the 1920s, and Wigner and others developed the theory of resonances and threshold behavior that underpins nuclear reaction analysis.
Key figures
- John Strutt, Lord Rayleigh
- Hans Faxen
- John Holtsmark
- Eugene Wigner
Related topics
Seminal works
- taylor2006
- newton2002
Frequently asked questions
- Why does only s-wave scattering matter at very low energy?
- The centrifugal barrier keeps higher angular-momentum waves away from the short-range potential when the energy is small, so their phase shifts are negligible and the spherically symmetric s-wave channel dominates the cross section.
- What does a phase shift tell you?
- It measures how much the potential advances or retards a given partial wave relative to free propagation; the sign indicates attraction or repulsion, its energy dependence reveals resonances, and the full set of phase shifts reconstructs the scattering amplitude.