Few-body physics — the quantum / classical dynamics of systems of 2-5 particles, distinguished from both many-body physics (statistical-mechanical limit) and few-level problems (internal structure abstracted). Foundational mathematical…
few-body-physics
Pair count: C(n,2) = n·(n-1)/2
In an n-particle system, the number of distinct unordered pairs is the binomial coefficient C(n,2) = n·(n-1)/2 — the leading…
Efimov geometric scaling: E_{n+1}/E_n = exp(-2π/s_0)
Efimov (1970) discovered an infinite geometric tower of three-body bound states in the unitary limit (two-body scattering length a → ±∞,…
Jacobi 3-body reduced mass: μ_{12,3} = (m1+m2)·m3/(m1+m2+m3)
Jacobi coordinates decompose an n-body center-of-mass-frame into (n-1) independent coordinates. For three particles: ρ = r_2 - r_1…
Pair count at n=3: C(3,2) = 3
Sympy-exact witness of the smallest non-trivial three-body pair count. Setup: C(n,2) = n·(n-1)/2 as a symbolic one-parameter expression. …
Efimov ratio anchor: E_{n+1}/E_n(s_0=1) = exp(-2π)
Sympy-exact witness of the Efimov geometric-scaling canonical anchor at s_0 = 1. Setup: E_{n+1}/E_n = exp(-2π/s_0). Identity: at the s_0…
Jacobi equal-mass: μ_{12,3}/m (m_i=m) = 2/3
Sympy-exact witness of the identical-mass Jacobi 3-body reduced-mass ratio. Setup: μ_{12,3} = (m_1+m_2)·m_3/(m_1+m_2+m_3) as a…
Faddeev-Yakubovsky 3-body decomposition; Cayley n^{n-2} spanning trees
Faddeev-Yakubovsky 3-body decomposition framework (Faddeev 1961 Sov Phys JETP 12, 1014; Yakubovsky 1967 Sov J Nucl Phys 5, 937). Setup: the…
Wigner-Eckart / Schur-function decomposition for SU(2) coupling
Wigner-Eckart / Schur-function framework for SU(2) angular-momentum coupling (Wigner 1927 Z Phys 43, 624; Racah 1942 Phys Rev 62, 438).…
Slater determinant antisymmetry; Pauli exclusion via dual-space dual-vector wedge
Slater determinant antisymmetrisation framework (Slater 1929 Phys Rev 34, 1293). Setup: a system of N identical fermions has antisymmetric…
Theorem: 3^{3-2} - 3 = 0 (Cayley K_3 spanning-trees count)
Theorem (Cayley-K3 canonical): K_3 (complete graph on 3 vertices, the triangle) has exactly 3 spanning trees - the 3 paths obtained by…
Theorem: s_2(1, 1) - 3 = 0 (Schur s_2 evaluated at unit point)
Theorem (Schur-s_2 canonical): the Schur function s_lambda for lambda = (2) (single-row partition) in 2 variables is s_(2)(x, y) = x^2 + xy…
Theorem: det([[1, 1], [1, 1]]) = 0 (Slater-determinant Pauli vanishing)
Theorem (Slater-Pauli canonical): when two single-particle orbitals coincide (psi_1 = psi_2), the Slater determinant reduces to the…
Efimov trimer (Efimov 1970)
V Efimov 1970: 3-body bound states with Borromean topology + log-periodic geometric scaling factor 22.7; observed in Cs cold atoms (Kraemer…
Faddeev equations (1960)
L Faddeev 1960: 3-body Schrodinger reformulated as coupled integral equations on dyad-channels; basis of nuclear + atomic 3-body…
Feshbach resonance (1958)
H Feshbach 1958 nuclear-physics; transposed to BEC by Tiesinga 1993; magnetically-tunable scattering-length a(B) -> control unitary regime…
Halo nuclei (Tanihata 1985)
I Tanihata 1985 11Li / 6He / 11Be: low binding-energy + extended neutron-halo wavefunction; Borromean 3-body structure; benchmark for…
Universality (large scattering length)
Braaten-Hammer 2006 universality regime |a| >> r_0: cross-sections scale as a^2; inset-rich Efimov physics; cold-atoms + nuclear-halo +…
Hyperspherical harmonics
Smith 1960 + Macek 1968 hyperspherical adiabatic: 3-body wavefunction in (rho, Omega_5) coordinates; basis of dipositronium + Efimov-3-body…
Efimov (1970)
V Efimov 1970 trimer-states scaling-symmetry; modern modern foundational text + 2006 Innsbruck Cs-trimer experimental-confirmation.
First Efimov observation (Kraemer 2006)
T Kraemer 2006 + Innsbruck Cs-trimer experimental-Efimov; modern modern foundational + 2024 multiple-Feshbach-tuned-system Efimov…
Borromean (Borromean rings 1450s+)
Borromean-symbol classical + Hammer 1996 11Li nuclear-borromean; modern modern foundational + halo-nuclei + 6He / 9Be 2024 ab-initio.
Faddeev (1960)
L Faddeev 1960 three-body equations; modern modern foundational text + Yakubovsky 4-body + 2024 chiral-EFT few-body-nuclear.
Halo nucleus (Tanihata 1985)
I Tanihata 1985 11Li radii-anomaly halo-nucleus; modern modern foundational text + 2024 modern reaction-cross-section + 2-neutron halo…
Universality (Braaten-Hammer 2006)
E Braaten-H-W Hammer 2006 universality-review; modern modern foundational text + scaling + 2024 nuclear-cold-atom benchmark cross-system.