Systematic series-expansion methods for eigenvalues and eigenstates of Hamiltonians of the form H = H_0 + λV, where H_0 has a known spectrum and λ is a small formal parameter. Rayleigh 1894 secular-equation origin, Schrödinger 1926 formal…
perturbation-theory
Rayleigh-Schrödinger time-independent perturbation theory
Rayleigh 1894 / Schrödinger 1926 expansion for eigenvalues of H = H_0 + λV under the assumption that λV is a small perturbation of the…
Dyson 1949 time-ordered interaction-picture propagator series
Dyson 1949 (Phys. Rev. 75:486) expressed the time-evolution operator in the interaction picture as an infinite time-ordered exponential. …
Brillouin-Wigner self-consistent perturbation expansion
Brillouin 1933 / Wigner 1935 variant of Rayleigh-Schrödinger PT in which the exact eigenvalue E_n appears in the energy denominators…
RS 1st-order energy shift for λx² perturbation of harmonic oscillator
For H_0 = (p̂² + ω²x̂²)/2 with spectrum E_n^(0) = ω(n+1/2) (ℏ=m=1 units) and perturbation V = λ·x̂², the first-order RS energy shift is…
RS 2nd-order energies for 2-level system with off-diagonal perturbation
Textbook exact check of the RS 2nd-order formula. Take H_0 = diag(0, Δ) and V = v·σ_x = v·[[0,1],[1,0]] (purely off-diagonal, real v > 0).…
1st-order ground-state shift for λx⁴ perturbation of harmonic oscillator
Textbook Bender-Wu problem: H_0 = (p̂² + ω²x̂²)/2 plus V = λ·x̂⁴. Evaluate ⟨0|x̂⁴|0⟩ using x̂ = (â+â†)/√(2mω): x̂⁴ = (â+â†)⁴ / (4m²ω²) =…
Pade-Borel resummation: convert divergent perturbative series into convergent rational approximants
Pade-Borel resummation framework. Setup: a formal power series f(z) = sum_{n>=0} a_n z^n with coefficients a_n growing factorially (|a_n| ~…
WKB semiclassical expansion: psi(x) ~ A(x)*exp(i*S(x)/hbar) with S expanded in powers of hbar
Wentzel-Kramers-Brillouin (WKB) semiclassical expansion. Setup: stationary 1D Schrodinger equation -hbar^2/(2m) psi'' + V(x) psi = E psi.…
Van Vleck / Lowdin quasi-degenerate perturbation: block-diagonalise H via Schur-lemma decomposition
Van Vleck / Lowdin quasi-degenerate perturbation theory. Setup: Hamiltonian H = H_0 + V where H_0 has a low-energy subspace P (dim d)…
Theorem: Pade[1/1](e^x) = (2+x)/(2-x) agrees with Taylor series through O(x^2)
Theorem (Pade-resummation exactness, diagonal case): the [1/1] Pade approximant of exp(x), defined as the unique rational function…
Theorem: Bohr-Sommerfeld quantization gives exact E_n = hbar*omega*(n + 1/2) for harmonic oscillator
Theorem (WKB Bohr-Sommerfeld exactness on harmonic oscillator): for the classical action of a 1D harmonic oscillator V(q) = (1/2)…
Theorem: 2x2 Van Vleck block has trace E1+E2, det E1*E2 - V^2, second-order shift V^2/(E1-E2)
Theorem (Van Vleck 2x2 quasi-degenerate canonical form): for the real-symmetric Hamiltonian H = [[E_1, V], [V, E_2]] the linear-algebra…
Rayleigh-Schrodinger PT (1926)
Rayleigh-Schrodinger 1926 standard nondegenerate + degenerate perturbation theory; Brillouin-Wigner alternative; basis of all-quantum…
Regularization (t Hooft 1972)
G t Hooft 1972 dimensional-regularization (Nobel 1999); minimal-subtraction MS-bar scheme; modern UV-divergence handling in QFT.
RG (Wilson 1971)
K Wilson 1971 (Nobel 1982) RG block-spin + epsilon-expansion; basis of universality + critical-phenomena exponents.
Borel summation (1899)
E Borel 1899 Borel-summation of divergent series; modern resurgence theory Ecalle 1981; basis of asymptotic-perturbative-physics treatments.
Instanton tunneling
BPST 1975 + Coleman 1977 instantons; non-perturbative tunneling-amplitude; modern resurgence + Lefschetz-thimble approaches.
Convergence radius
Dyson 1952 argued QED perturbation-series is asymptotic + zero-convergence; modern resurgence Marino 2014 sums divergent + non-perturbative.
Rayleigh-Schrödinger (1926)
Rayleigh 1894 + Schrödinger 1926 nondegenerate-perturbation; modern modern foundational text + 2024 ML-driven Padé-perturbation…
Brillouin-Wigner (1933)
L Brillouin 1932 + E Wigner 1935 BW-perturbation; modern modern foundational text + 2024 multireference Brueckner / CCSD-T benchmark BW.
Time-dependent (Dirac 1927)
P Dirac 1927 (Nobel 1933) time-dependent-perturbation; modern modern foundational text + Fermi-golden-rule + 2024 ML-driven…
Borel summability (Bender-Wu 1969)
C Bender-T-T Wu 1969 anharmonic-oscillator divergent-PT; modern modern foundational text + 2024 resurgence + trans-series Écalle-modern.
WKB (Wentzel-Kramers-Brillouin-Jeffreys 1923-1926)
G Wentzel-H Kramers-L Brillouin 1926 + Jeffreys 1923 WKB; modern modern foundational text + connection-formula + 2024 quasinormal-modes BH.
Instanton (Feynman 1965)
R Feynman 1965 (Nobel 1965) + Polyakov 1976 instanton; modern modern foundational text + 2024 trans-series + Lefschetz-thimble Monte-Carlo.