| Rayleigh–Schrödinger perturbation theory | |
|---|---|
| Name | Rayleigh–Schrödinger perturbation theory |
| Field | Quantum mechanics |
| Introduced | 1920s |
| Notable people | Lord Rayleigh; Erwin Schrödinger |
Rayleigh–Schrödinger perturbation theory
Rayleigh–Schrödinger perturbation theory is a formalism used in quantum mechanics to approximate eigenvalues and eigenstates of a Hamiltonian that differs slightly from an exactly solvable operator. It provides systematic expansions in a small parameter to compute energy corrections and state vectors, and underpins many analytical and computational methods across atomic, molecular and condensed matter physics.
Rayleigh–Schrödinger perturbation theory (often abbreviated RSPT) addresses problems where the full Hamiltonian H can be written as H0 + λV, with H0 solvable and V a perturbing operator. The method yields power-series corrections in the coupling parameter λ for eigenenergies and eigenstates of H. RSPT is central to the practical treatment of interactions in systems studied in atomic physics, molecular physics, solid-state physics, and quantum chemistry. Classic applications include the analysis of fine structure in the hydrogen atom, perturbative treatments in quantum field theory (as an inspiration for diagrammatic expansions), and corrections used in Hartree–Fock and post-Hartree–Fock methods.
The standard formulation begins by assuming a nondegenerate eigenpair (E^(0), |ψ^(0)>) of the unperturbed operator H0 satisfying H0|ψ^(0)> = E^(0)|ψ^(0)>. One seeks expansions E = Σ_n λ^n E^(n) and |ψ> = Σ_n λ^n |ψ^(n)>. Substitution into the time-independent Schrödinger equation and collection of orders in λ yields recursive relations. The first-order energy is E^(1) = ⟨ψ^(0)|V|ψ^(0)⟩ and higher orders involve sums over intermediate states using resolvent operators (1/(E^(0) − H0)) projected off the reference state. The approach uses linear algebraic tools such as projection operators, matrix elements, and completeness relations from the spectral theorem for self-adjoint operators.
For nondegenerate levels, RSPT provides explicit formulae: second-order energy corrections are sums E^(2) = Σ_{m≠0} |⟨ψ_m^(0)|V|ψ_0^(0)⟩|^2/(E_0^(0) − E_m^(0)). Corrections to the state vector are obtained using denominators (E_0^(0) − E_m^(0)), which can lead to large contributions near level crossings. Practical implementations appear in perturbative corrections in quantum chemistry such as Møller–Plesset perturbation theory (MP2 and higher) and in many-body perturbation theory (MBPT) approaches used in nuclear physics and condensed matter physics. The nondegenerate scheme assumes a well-separated spectrum and relies on the orthonormality of unperturbed eigenfunctions, often represented in a chosen basis (e.g., Slater determinant basis).
When H0 has degenerate eigenvalues, naive application of nondegenerate formulae fails because denominators vanish. Degenerate perturbation theory first diagonalizes the perturbation V within the degenerate subspace to find the proper zeroth-order basis that splits the degeneracy. This procedure leads to secular equations and the need to solve a finite-dimensional matrix eigenproblem. Degenerate RSPT is essential for treating phenomena like Zeeman effect splitting in atoms, crystal-field splitting in transition metal complexes, and level mixing in systems studied at CERN and other experimental facilities where symmetry-induced degeneracies occur.
RSPT is fundamentally a time-independent expansion for stationary states, but it connects to time-dependent perturbation theory through adiabatic switching and the interaction picture. Time-dependent perturbation theory, developed by Paul Dirac and others, addresses transitions induced by time-varying perturbations and yields transition probabilities via Fermi’s golden rule. Under adiabatic modulation of λ(t), the instantaneous eigenvalues from RSPT approximate quasistatic energies; conversely, time-dependent methods motivate resummation of secular terms encountered in naive time-dependent expansions. Diagrammatic techniques (e.g., Feynman diagram analogues in many-body theory) relate both formulations by organizing contributions in powers of the coupling.
Series produced by Rayleigh–Schrödinger expansions are often asymptotic rather than convergent. Classic examples demonstrate factorial growth of coefficients, requiring techniques such as Borel resummation, Padé approximants, or analytic continuation to extract physical results. Studies by George A. Baker Jr. and others established use of Padé methods in quantum chemistry; Borel summation figures in rigorous treatments of divergent perturbation series in quantum field theory and anharmonic oscillators. Practical many-body implementations use partial summations (e.g., coupled cluster theory) to capture infinite classes of diagrams and improve convergence properties.
RSPT underlies numerous concrete calculations: energy level shifts in the hydrogen atom due to external electric fields (Stark effect), perturbative corrections to molecular binding energies in quantum chemistry methods (Møller–Plesset perturbation theory), and electron correlation corrections in condensed matter computed via GW and MBPT schemes. It also forms the basis for pedagogical discussions of degenerate perturbations in the fine structure and for analytical estimates in model systems like the harmonic oscillator with anharmonic terms. High-precision spectroscopy and comparisons with experiments at institutions such as NIST use perturbative results combined with nonperturbative corrections to achieve agreement with measured transition frequencies.