Contents
Introduction Time-Independent Degenerate Perturbation Time-Dependent Fermi's Golden Rule Applications Misconceptions FAQ Sources Introduction
Most quantum mechanical problems cannot be solved exactly. Perturbation theory provides systematic approximations when the Hamiltonian can be split into a soluble part Ĥ₀ and a small perturbation Ĥ': Ĥ = Ĥ₀ + λĤ'. Expanding the solutions in powers of λ gives corrections to energies, wave functions, and transition probabilities. The technique is the workhorse for atomic physics, nuclear physics, condensed matter, and quantum field theory.
Time-Independent Perturbation Theory
For a stationary Hamiltonian Ĥ = Ĥ₀ + λĤ' with known eigenstates |n⁰⟩ of Ĥ₀ with energies En ⁰:
First-Order Energy
En ⁽¹⁾ = ⟨n⁰|Ĥ'|n⁰⟩
First-Order Wave Function
|n⁽¹⁾⟩ = Σm≠n [⟨m⁰|Ĥ'|n⁰⟩ / (En ⁰ − Em ⁰)] |m⁰⟩
Second-Order Energy
En ⁽²⁾ = Σm≠n |⟨m⁰|Ĥ'|n⁰⟩|² / (En ⁰ − Em ⁰)
The series in λ generally converges for small enough λ. For large λ, alternative methods are needed [1 ].
Degenerate Perturbation Theory
If |n⁰⟩ is degenerate (multiple states with same En ⁰), the first-order formulas have vanishing denominators. The fix: diagonalize the perturbation matrix within the degenerate subspace. The first-order corrections to energies are eigenvalues of this matrix; the corresponding eigenvectors are the proper zeroth-order states.
Example: Stark Effect
An electric field perturbs hydrogen. For n = 2 (4-fold degenerate), the perturbation lifts degeneracy linearly in the field — the linear Stark effect, characteristic of hydrogen. For non-degenerate states, the leading effect is quadratic.
Time-Dependent Perturbation Theory
For a time-dependent perturbation Ĥ'(t), the probability of finding the system in state |m⟩ at time t, given it started in |n⟩:
Pn→m (t) = |(1/iℏ) ∫₀t ⟨m|Ĥ'(t')|n⟩ exp(iωmn t') dt'|²
where ωmn = (Em − En )/ℏ. For a harmonic perturbation, transitions are most efficient at resonance (driving frequency = transition frequency).
Fermi's Golden Rule
For transitions from a discrete state into a continuum of states with density ρ(Ef ), the transition rate per unit time:
Γi→f = (2π/ℏ) |⟨f|Ĥ'|i⟩|² ρ(Ef )
Named "Fermi's golden rule" by Fermi, who used it extensively. It's the workhorse for calculating decay rates, scattering cross sections, ionization probabilities, and any process involving transitions to a continuum [2 ].
Applications
Atomic physics: Stark, Zeeman effects; line broadening; spontaneous emission rates.Nuclear physics: Beta decay rates; nuclear transitions.Condensed matter: Electron-phonon scattering; electrical resistivity from impurities.Particle physics: Decay rates, scattering amplitudes (Feynman diagrams are perturbation expansion).Chemistry: Molecular spectra, reaction rates. Common Misconceptions
"Perturbation theory always converges"
It often gives asymptotic series that diverge eventually. For QED, the perturbation series is asymptotic, with the asymptotic remainder bounded for many practical purposes.
"Higher-order corrections always improve accuracy"
For asymptotic series, only up to a certain order. Beyond optimal order, terms grow.
"Perturbation theory replaces exact solutions"
No — it gives approximate solutions when exact ones are unavailable. Where exact solutions exist, they are preferable.
"Fermi's golden rule applies to any transition"
It applies when the final states form a continuum and the perturbation is weak. Not for discrete-to-discrete transitions or strong driving.
FAQ
What's the small parameter?
Whatever makes the perturbation small compared to Ĥ₀. For Stark effect, the electric field strength relative to atomic field scales. For QED, the fine structure constant α ≈ 1/137.
When does perturbation theory fail?
Strong coupling regimes (lattice QCD, BCS pairing, etc.) require non-perturbative methods. Resonances and near-degeneracies require degenerate perturbation theory.
What's the link to Feynman diagrams?
Feynman diagrams are a pictorial representation of perturbation-theory terms. Each diagram corresponds to a term in the perturbation expansion of the S-matrix.
How is perturbation theory related to numerical methods?
For small perturbations, analytic expansion is faster than numerics. For large perturbations, direct numerical diagonalization is often easier and more accurate.
Sources
Sakurai, J. J., Napolitano, J. (2017). Modern Quantum Mechanics , 2nd ed.
Fermi, E. (1950). Nuclear Physics . University of Chicago Press.
Griffiths, D. J. (2018). Introduction to Quantum Mechanics , 3rd ed.
Cohen-Tannoudji, C., Diu, B., Laloë, F. (1977). Quantum Mechanics . Wiley.