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) =…