Perturbation theory (quantum mechanics)

The method begins by writing the Hamiltonian as H = H₀ + λH′, where H₀ is exactly solvable and λ is a small parameter.¹ In the time-independent case, the energy eigenvalues E_n and eigenstates |ψ_n⟩ are expanded in powers of λ. The first-order correction to the energy is the expectation value of the perturbation in the unperturbed state: E_n⁽¹⁾ = ⟨ψ_n⁽⁰⁾| H′ |ψ_n⁽⁰⁾⟩. The first-order correction to the state involves a sum over all other states, weighted by matrix elements of H′ divided by energy denominators. For time-dependent perturbations, the Dyson series gives the evolution operator, and transition probabilities are computed via Fermi's golden rule.

Perturbation theory is essential for describing the Stark effect (electric field splitting of atomic levels) and the Zeeman effect (magnetic field splitting).¹ It also explains the fine structure of hydrogen, where relativistic corrections and spin–orbit coupling are treated as perturbations.² In molecular physics, the anharmonic oscillator uses perturbation theory to correct the harmonic oscillator approximation, yielding vibrational overtones. In quantum field theory, perturbative expansions in the coupling constant underlie the calculation of scattering amplitudes and Feynman diagrams.

The series is often asymptotic, not convergent, and may diverge for all non-zero λ, though the first few terms give excellent approximations.³ Degeneracy requires special treatment: degenerate perturbation theory diagonalizes the perturbation within the degenerate subspace.¹ Higher-order corrections involve sums over intermediate states that can become cumbersome, and the method fails when the perturbation is strong or when level crossings occur.² In such cases, non-perturbative methods like the variational method or numerical diagonalization are used.

Often called Rayleigh–Schrödinger perturbation theory, it was developed independently by Lord Rayleigh (for mechanical vibrations) and Erwin Schrödinger (for quantum mechanics) in 1926.⁴ The time-dependent variant was formalized by Paul Dirac, leading to the Dyson series and the interaction picture. A lesser-known nuance is that the series is typically an asymptotic series: for the anharmonic oscillator, the perturbation expansion diverges, yet the first few terms match experiment remarkably well.³ The method is also used in quantum chemistry (Møller–Plesset perturbation theory) and in lattice gauge theory as a weak-coupling expansion.

Hamiltonian

The operator that represents the total energy of a quantum system.

Eigenstate

A quantum state corresponding to a definite energy eigenvalue.

Perturbation

A small additional term added to the exactly solvable Hamiltonian.

Degeneracy

The situation where multiple eigenstates share the same energy eigenvalue.