| perturbation theory | |
|---|---|
| Name | Perturbation theory |
| Field | Quantum physics |
| Introduced | 1920s |
| Related | Quantum mechanics, Quantum field theory, Perturbation (physics) |
perturbation theory
Perturbation theory is an approximation method used to find approximate solutions to problems that cannot be solved exactly, by starting from a solvable Hamiltonian and adding a small additional term. In Quantum mechanics and Quantum field theory, it permits calculation of energy shifts, transition amplitudes, and scattering cross sections when exact diagonalization or nonperturbative methods are infeasible. It underpins many computational techniques in theoretical and experimental contexts, from atomic spectroscopy to predictions in the Standard Model.
Perturbation theory treats a physical system described by a Hamiltonian H = H0 + λV, where H0 is solvable and V is a perturbation multiplied by a bookkeeping parameter λ. Series expansions in powers of λ produce corrections to eigenvalues and eigenstates; truncation yields practical approximations. The method links to linear algebraic perturbation of operators, Rayleigh–Schrödinger methods, and variational approaches such as Brillouin–Wigner perturbation theory. Key figures in the development include Pascual Jordan, Paul Dirac, and John von Neumann, and the approach is foundational for calculations performed at institutions like CERN and Los Alamos National Laboratory.
Time-independent perturbation theory addresses stationary states of H0 perturbed by a static V. The canonical formalism, Rayleigh–Schrödinger perturbation theory, yields corrections E^(n) to energies and |ψ^(n)⟩ to eigenstates by iterative formulas involving resolvents (Green's functions). For nondegenerate states the first-order energy shift is ⟨ψ0|V|ψ0⟩, while second-order shifts sum over virtual intermediate states. This section connects to techniques in atomic physics developed by Arnold Sommerfeld and refined in textbooks by Dirac and Landau and Lifshitz. Computational packages such as Gaussian and methods in quantum chemistry employ these expansions for molecular energies and response properties.
Time-dependent perturbation theory treats V(t) and yields transition amplitudes via the time-evolution operator or Dyson series. The lowest-order result leads to Fermi's golden rule for transition rates between initial and final states coupled to a continuum: rate ∝ |⟨f|V|i⟩|^2 times density of final states. Applications include electromagnetic interactions treated by semiclassical perturbations and time-dependent perturbations in NMR and Atomic clocks. The formalism extends to interaction-picture methods introduced by Paul Dirac and to time-ordered exponentials used in nonrelativistic and relativistic contexts.
When H0 has degenerate eigenvalues, naive expansions break down; degenerate perturbation theory diagonalizes V within the degenerate subspace. Symmetry plays a central role: perturbations that respect or break symmetries of H0 determine selection rules and splitting patterns (e.g., fine structure, Zeeman and Stark effects). Group-theoretical methods from Eugène Wigner and Hermann Weyl are commonly used to classify degeneracies and irreducible representations. In solid-state physics, crystal-field splitting and band-structure perturbations use symmetry-adapted basis sets; in molecular spectroscopy, Hund's rules and point-group symmetry guide level classification.
In Quantum field theory, perturbation theory is formulated as an expansion in a coupling constant (e.g., the fine-structure constant α). Perturbative amplitudes are organized by Feynman diagrams, introduced by Richard Feynman, which represent contributions to S-matrix elements. Renormalization procedures developed by Julian Schwinger, Sin-Itiro Tomonaga, and Freeman Dyson remove ultraviolet divergences order-by-order and introduce running couplings via the Renormalization group. Perturbation theory underlies precision predictions in Quantum electrodynamics and aspects of the Standard Model tested at accelerators like Fermilab and CERN.
Many perturbative series in quantum physics are asymptotic rather than convergent. Techniques such as Borel resummation, Padé approximants, and renormalon analysis seek to extract finite results from divergent series. Nonperturbative phenomena—instantons, tunneling, and confinement in Quantum chromodynamics—require methods beyond perturbation theory or combined approaches (e.g., semiclassical expansions, lattice gauge theory at CERN's lattice collaborations). Theoretical work by Gerard 't Hooft and Michael Berry elucidates large-order behavior and connections to complex saddles.
Perturbation theory is applied across domains: in atomic and molecular spectroscopy it predicts fine and hyperfine structure, Lamb shifts, and Stark/Zeeman splittings measured in precision experiments at institutions such as NIST. In scattering theory, Born series expansions give cross sections and are used in electron–atom scattering and low-energy nuclear scattering models. Many-body perturbation theory (MBPT), including Hartree–Fock plus perturbative corrections and diagrammatic techniques like the GW approximation and Random phase approximation, underpins electronic structure methods in condensed matter and quantum chemistry. Extensions include coupling to computational platforms (e.g., Quantum ESPRESSO) and integration with nonperturbative techniques such as Density matrix renormalization group and Quantum Monte Carlo.